# Copyright (C) [2024] [Mattia Bennati]
# DATASET 1 GlobClus_prop
#### ASTRO DATA: 20 different measures for 147 globular star clusters in the Milky Way Galaxy
# https://astrostatistics.psu.edu/MSMA/datasets/index.html
# Properties include Galactic location, integrated stellar luminosity, metallicity, ellipticity,
# central surface brightness, color, and seven measures of dynamical state
# (core and tidal radius, concentration, central star density and relaxation time,
# central velocity dispersion and escape velocity).
# https://search.r-project.org/CRAN/refmans/astrodatR/html/GlobClus_prop.html
# data description: Appendix C.7 of Feigelson & Babu (2012)
# with statistical analysis in Chapter 8.
# Importing packages
#install.packages("igraph")
#install.packages("RcppEigen")
#install.packages("RcppArmadillo")
#install.packages("gRbase")
#install.packages("gRain")
#install.packages("gRim")
library(gRbase)
library(gRain)
library(gRim)
library(bnlearn)
##
## Attaching package: 'bnlearn'
## The following objects are masked from 'package:gRbase':
##
## ancestors, children, parents
# Clears the current environment
rm(list=ls(all=TRUE))
# Sets the current working directory
setwd("~/Desktop/university/AI/FOSM/project/GlobusClus_prop-Analysis/")
# Installing the "astrodatR" dataset
# install.packages("astrodatR")
# Importing the dataset
require(astrodatR)
## Loading required package: astrodatR
# Parsing the dataset data
data(GlobClus_prop)
# Saves the data into the dat variable
dat <- GlobClus_prop
# Prints the first 5 rows of the dataset
# print(head(dat, n=5))
# CHOSEN OBJECTIVE:
# Study of the dynamics of globular clusters
# Identifying which parameters influence the dispersion of the
# central velocity of a globular star cluster and how they affect it
# PHASE 1:
# IDENTIFICATION OF VARIABLES
# Name: Common name
# Gal.long: Galactic longitude (degrees)
# Gal.lat: Galactic latitude (degrees)
# R.sol: Distance from the Sun (kiloparsecs, kpc)
# R.GC: Distance from the Galactic Center (kpc)
# Metal: Log metallicity with respect to solar metallicity
# Mv: Absolute magnitude
# r.core: Core radius (parsecs, pc)
# r.tidal: Tidal radius (pc)
# Conc: Core concentration parameter
# log.t: Log central relaxation timescale (years)
# log.rho: Log central density (solar masses per cubic parsec)
# S0: Central velocity dispersion (kilometers per second)
# V.esc: Central escape velocity (km/s)
# VHB: Level of the horizontal branch (magnitude)
# E.B-V: Color excess (magnitude)
# B-V: Color index (magnitude)
# Ellipt: Ellipticity
# V.t: Integrated V magnitude (magnitude)
# CSBt: Central surface brightness (magnitude per square arcsecond)
# PHASE 2:
# CLEANING THE DATASET FROM NAN VALUES (WITH INCOMPLETE DATA) AND EXCLUDING
# THE NAME COLUMN AS IT IS SUPERFLUOUS FOR THE STUDY
dat <- na.omit(dat[,-1])
# First 5 records of the dataset
head(dat, n=5)
## Gal.long Gal.lat R.sol R.GC Metal Mv r.core r.tidal Conc log.t log.rho
## 1 305.90 -44.89 4.6 8.1 -0.8 -9.6 0.5 60.3 2.1 7.9 5.0
## 2 151.15 -89.38 8.2 12.1 -1.4 -6.6 4.0 37.0 0.9 9.0 2.0
## 3 301.53 -46.25 8.7 9.9 -1.4 -8.4 0.5 25.9 1.7 7.8 4.8
## 4 270.54 -52.13 16.1 18.3 -1.2 -7.8 1.9 37.1 1.3 8.6 3.2
## 7 258.36 -48.47 116.4 117.9 -1.7 -4.5 9.5 81.2 0.9 9.3 0.0
## S0 V.esc VHB E.B.V B.V Ellipt V.t CSBt
## 1 13.2 56.8 14.1 0 1.0 3.8 3 5.6
## 2 2.8 10.0 15.3 0 0.9 8.1 1 11.0
## 3 10.3 41.8 15.4 0 0.9 6.4 6 6.1
## 4 5.5 20.7 16.6 0 0.9 8.2 8 8.7
## 7 0.7 2.4 20.9 0 0.8 15.8 4 15.0
# PHASE 3:
# DEFINITION OF VARIABLES
y = dat$S0
x1 = dat$Gal.long
x2 = dat$Gal.lat
x3 = dat$R.sol
x4 = dat$R.GC
x5 = dat$Metal
x6 = dat$Mv
x7 = dat$r.core
x8 = dat$r.tidal
x9 = dat$Conc
x10 = dat$log.t
x11 = dat$log.rho
x12 = dat$V.esc
x13 = dat$VHB
x14 = dat$E.B.V
x15 = dat$B.V
x16 = dat$Ellipt
x17 = dat$V.t
x18 = dat$CSBt
# PHASE 4:
# ANALYSIS OF SIGNIFICANT VARIABLES:
# Definition of the complete model:
mq <- lm(y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13 + x14 + x15 + x16 + x17 + x18)
summary(mq)
##
## Call:
## lm(formula = y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 +
## x10 + x11 + x12 + x13 + x14 + x15 + x16 + x17 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.35240 -0.08275 -0.01367 0.06688 0.70644
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.4932971 4.0122361 0.123 0.90241
## x1 -0.0001159 0.0001097 -1.056 0.29370
## x2 0.0008409 0.0006320 1.331 0.18654
## x3 0.0058477 0.0058386 1.002 0.31913
## x4 -0.0027320 0.0046645 -0.586 0.55949
## x5 -0.1743371 0.1230087 -1.417 0.15971
## x6 0.4107678 0.4499448 0.913 0.36362
## x7 -0.0277227 0.0175895 -1.576 0.11836
## x8 -0.0001507 0.0010495 -0.144 0.88616
## x9 -0.0349403 0.2545558 -0.137 0.89112
## x10 1.1866765 0.3453261 3.436 0.00088 ***
## x11 -0.3031051 0.3517006 -0.862 0.39098
## x12 0.2364541 0.0032508 72.736 < 2e-16 ***
## x13 -0.1351995 0.4372948 -0.309 0.75787
## x14 1.4366685 0.4871607 2.949 0.00402 **
## x15 0.5159395 0.4768871 1.082 0.28207
## x16 0.1026885 0.4341540 0.237 0.81354
## x17 -0.0029176 0.0080131 -0.364 0.71659
## x18 -0.5989269 0.1782506 -3.360 0.00113 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1661 on 94 degrees of freedom
## Multiple R-squared: 0.998, Adjusted R-squared: 0.9976
## F-statistic: 2604 on 18 and 94 DF, p-value: < 2.2e-16
# Definition of the null model (intercept only):
mq0 <- lm(y ~ 1)
summary(mq0)
##
## Call:
## lm(formula = y ~ 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.5283 -2.3283 -0.6283 1.9717 12.8717
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.2283 0.3201 19.46 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.402 on 112 degrees of freedom
# 1st - P-VALUE STRATEGY WITH "BACKWARD" METHOD:
# Starting from the complete model, iteratively eliminate the variable
# with the highest p-value (Pr(>|t|)) until there are no non-significant variables
mq1 <- update(mq, .~. -x9)
summary(mq1)
##
## Call:
## lm(formula = y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x10 +
## x11 + x12 + x13 + x14 + x15 + x16 + x17 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.35253 -0.08095 -0.01351 0.06982 0.70550
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.2358663 3.5285428 0.067 0.946845
## x1 -0.0001171 0.0001088 -1.077 0.284318
## x2 0.0008386 0.0006285 1.334 0.185311
## x3 0.0058135 0.0058030 1.002 0.318987
## x4 -0.0026871 0.0046290 -0.580 0.562954
## x5 -0.1765506 0.1213157 -1.455 0.148883
## x6 0.4220482 0.4400854 0.959 0.339986
## x7 -0.0276866 0.0174965 -1.582 0.116880
## x8 -0.0001718 0.0010327 -0.166 0.868230
## x10 1.2283975 0.1630443 7.534 2.85e-11 ***
## x11 -0.2940487 0.3436676 -0.856 0.394361
## x12 0.2363726 0.0031795 74.343 < 2e-16 ***
## x13 -0.1401001 0.4335784 -0.323 0.747311
## x14 1.4483989 0.4771228 3.036 0.003097 **
## x15 0.5259681 0.4688171 1.122 0.264731
## x16 0.1074227 0.4305412 0.250 0.803508
## x17 -0.0030333 0.0079274 -0.383 0.702847
## x18 -0.6052511 0.1713015 -3.533 0.000636 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1653 on 95 degrees of freedom
## Multiple R-squared: 0.998, Adjusted R-squared: 0.9976
## F-statistic: 2786 on 17 and 95 DF, p-value: < 2.2e-16
mq1 <- update(mq1, .~. -x8)
summary(mq1)
##
## Call:
## lm(formula = y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x10 + x11 +
## x12 + x13 + x14 + x15 + x16 + x17 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.35442 -0.08096 -0.01251 0.07026 0.70668
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.3351778 3.4600193 0.097 0.923030
## x1 -0.0001178 0.0001082 -1.089 0.278953
## x2 0.0008111 0.0006034 1.344 0.182008
## x3 0.0058245 0.0057732 1.009 0.315563
## x4 -0.0028468 0.0045053 -0.632 0.528968
## x5 -0.1782262 0.1202830 -1.482 0.141689
## x6 0.4362329 0.4295541 1.016 0.312397
## x7 -0.0278998 0.0173609 -1.607 0.111328
## x10 1.2320327 0.1607530 7.664 1.45e-11 ***
## x11 -0.3033374 0.3373798 -0.899 0.370850
## x12 0.2365381 0.0030045 78.728 < 2e-16 ***
## x13 -0.1332178 0.4294089 -0.310 0.757055
## x14 1.4653611 0.4637341 3.160 0.002111 **
## x15 0.5335312 0.4642385 1.149 0.253304
## x16 0.1006465 0.4264339 0.236 0.813921
## x17 -0.0030075 0.0078856 -0.381 0.703762
## x18 -0.6130698 0.1638913 -3.741 0.000312 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1644 on 96 degrees of freedom
## Multiple R-squared: 0.998, Adjusted R-squared: 0.9977
## F-statistic: 2991 on 16 and 96 DF, p-value: < 2.2e-16
mq1 <- update(mq1, .~. -x16)
summary(mq1)
##
## Call:
## lm(formula = y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x10 + x11 +
## x12 + x13 + x14 + x15 + x17 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.35641 -0.08289 -0.01025 0.06667 0.70961
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.1996146 3.3953637 0.059 0.953240
## x1 -0.0001167 0.0001075 -1.085 0.280620
## x2 0.0008166 0.0006000 1.361 0.176647
## x3 0.0057490 0.0057362 1.002 0.318725
## x4 -0.0027745 0.0044729 -0.620 0.536525
## x5 -0.1730569 0.1176951 -1.470 0.144694
## x6 0.5369299 0.0496387 10.817 < 2e-16 ***
## x7 -0.0276555 0.0172454 -1.604 0.112045
## x10 1.2355184 0.1592921 7.756 8.81e-12 ***
## x11 -0.2949216 0.3338532 -0.883 0.379211
## x12 0.2365294 0.0029896 79.117 < 2e-16 ***
## x13 -0.0320368 0.0245645 -1.304 0.195255
## x14 1.4713379 0.4607828 3.193 0.001898 **
## x15 0.5154334 0.4556279 1.131 0.260735
## x17 -0.0041297 0.0062600 -0.660 0.511005
## x18 -0.6094081 0.1623592 -3.753 0.000297 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1636 on 97 degrees of freedom
## Multiple R-squared: 0.998, Adjusted R-squared: 0.9977
## F-statistic: 3221 on 15 and 97 DF, p-value: < 2.2e-16
mq1 <- update(mq1, .~. -x4)
summary(mq1)
##
## Call:
## lm(formula = y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x11 + x12 +
## x13 + x14 + x15 + x17 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.35257 -0.08184 -0.00729 0.06178 0.71043
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.1741975 3.3844423 0.051 0.959056
## x1 -0.0001181 0.0001072 -1.102 0.273374
## x2 0.0008270 0.0005979 1.383 0.169705
## x3 0.0024359 0.0020852 1.168 0.245566
## x5 -0.1684015 0.1170863 -1.438 0.153544
## x6 0.5395782 0.0492993 10.945 < 2e-16 ***
## x7 -0.0277383 0.0171907 -1.614 0.109837
## x10 1.2366363 0.1587811 7.788 7.16e-12 ***
## x11 -0.3001491 0.3326975 -0.902 0.369180
## x12 0.2364865 0.0029794 79.373 < 2e-16 ***
## x13 -0.0243882 0.0211786 -1.152 0.252308
## x14 1.4698092 0.4593275 3.200 0.001853 **
## x15 0.5130562 0.4541794 1.130 0.261388
## x17 -0.0035160 0.0061618 -0.571 0.569575
## x18 -0.6148681 0.1616107 -3.805 0.000247 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1631 on 98 degrees of freedom
## Multiple R-squared: 0.998, Adjusted R-squared: 0.9977
## F-statistic: 3473 on 14 and 98 DF, p-value: < 2.2e-16
mq1 <- update(mq1, .~. -x17)
summary(mq1)
##
## Call:
## lm(formula = y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x11 + x12 +
## x13 + x14 + x15 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.36375 -0.07854 -0.00903 0.06905 0.72228
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.1749390 3.3728946 0.052 0.958740
## x1 -0.0001227 0.0001065 -1.152 0.252102
## x2 0.0007614 0.0005847 1.302 0.195842
## x3 0.0022720 0.0020583 1.104 0.272345
## x5 -0.1743596 0.1162218 -1.500 0.136737
## x6 0.5400045 0.0491254 10.992 < 2e-16 ***
## x7 -0.0263506 0.0169597 -1.554 0.123443
## x10 1.2313887 0.1579737 7.795 6.58e-12 ***
## x11 -0.3062268 0.3313924 -0.924 0.357701
## x12 0.2366298 0.0029587 79.978 < 2e-16 ***
## x13 -0.0224821 0.0208421 -1.079 0.283350
## x14 1.4453521 0.4557630 3.171 0.002021 **
## x15 0.5381038 0.4505107 1.194 0.235163
## x18 -0.6163417 0.1610387 -3.827 0.000227 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1626 on 99 degrees of freedom
## Multiple R-squared: 0.998, Adjusted R-squared: 0.9977
## F-statistic: 3766 on 13 and 99 DF, p-value: < 2.2e-16
mq1 <- update(mq1, .~. -x11)
summary(mq1)
##
## Call:
## lm(formula = y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x12 + x13 +
## x14 + x15 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.40040 -0.07510 -0.00540 0.06222 0.72767
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.8889521 0.6181588 -4.673 9.26e-06 ***
## x1 -0.0001284 0.0001063 -1.209 0.22970
## x2 0.0007332 0.0005835 1.257 0.21180
## x3 0.0026776 0.0020095 1.332 0.18573
## x5 -0.1262483 0.1038314 -1.216 0.22689
## x6 0.5336481 0.0486059 10.979 < 2e-16 ***
## x7 -0.0277277 0.0168818 -1.642 0.10363
## x10 1.3579486 0.0786667 17.262 < 2e-16 ***
## x12 0.2361010 0.0029007 81.395 < 2e-16 ***
## x13 -0.0254384 0.0205801 -1.236 0.21933
## x14 1.2120506 0.3791654 3.197 0.00186 **
## x15 0.3320253 0.3911531 0.849 0.39800
## x18 -0.4735346 0.0452432 -10.466 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1624 on 100 degrees of freedom
## Multiple R-squared: 0.998, Adjusted R-squared: 0.9977
## F-statistic: 4086 on 12 and 100 DF, p-value: < 2.2e-16
mq1 <- update(mq1, .~. -x15)
summary(mq1)
##
## Call:
## lm(formula = y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x12 + x13 +
## x14 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.40103 -0.07663 -0.00220 0.06881 0.73009
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.4772228 0.3826742 -6.473 3.5e-09 ***
## x1 -0.0001360 0.0001057 -1.287 0.201
## x2 0.0007073 0.0005819 1.216 0.227
## x3 0.0026031 0.0020048 1.298 0.197
## x5 -0.0425866 0.0326172 -1.306 0.195
## x6 0.5250776 0.0474798 11.059 < 2e-16 ***
## x7 -0.0277405 0.0168584 -1.646 0.103
## x10 1.3418453 0.0762393 17.600 < 2e-16 ***
## x12 0.2362477 0.0028915 81.704 < 2e-16 ***
## x13 -0.0268126 0.0204879 -1.309 0.194
## x14 1.5176539 0.1187795 12.777 < 2e-16 ***
## x18 -0.4647963 0.0439954 -10.565 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1622 on 101 degrees of freedom
## Multiple R-squared: 0.9979, Adjusted R-squared: 0.9977
## F-statistic: 4469 on 11 and 101 DF, p-value: < 2.2e-16
mq1 <- update(mq1, .~. -x2)
summary(mq1)
##
## Call:
## lm(formula = y ~ x1 + x3 + x5 + x6 + x7 + x10 + x12 + x13 + x14 +
## x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.39884 -0.07960 0.00595 0.06595 0.73189
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.5026243 0.3829965 -6.534 2.55e-09 ***
## x1 -0.0001293 0.0001058 -1.222 0.225
## x3 0.0023564 0.0019991 1.179 0.241
## x5 -0.0461214 0.0325632 -1.416 0.160
## x6 0.5140892 0.0467202 11.004 < 2e-16 ***
## x7 -0.0248856 0.0167330 -1.487 0.140
## x10 1.3349987 0.0762086 17.518 < 2e-16 ***
## x12 0.2354596 0.0028245 83.364 < 2e-16 ***
## x13 -0.0261963 0.0205295 -1.276 0.205
## x14 1.5251507 0.1188966 12.828 < 2e-16 ***
## x18 -0.4649954 0.0440980 -10.545 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1626 on 102 degrees of freedom
## Multiple R-squared: 0.9979, Adjusted R-squared: 0.9977
## F-statistic: 4893 on 10 and 102 DF, p-value: < 2.2e-16
mq1 <- update(mq1, .~. -x3)
summary(mq1)
##
## Call:
## lm(formula = y ~ x1 + x5 + x6 + x7 + x10 + x12 + x13 + x14 +
## x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.40344 -0.07574 0.00478 0.06350 0.73465
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.6547450 0.3612783 -7.348 4.91e-11 ***
## x1 -0.0001312 0.0001060 -1.238 0.219
## x5 -0.0478541 0.0325915 -1.468 0.145
## x6 0.5211999 0.0464166 11.229 < 2e-16 ***
## x7 -0.0159458 0.0149436 -1.067 0.288
## x10 1.3297908 0.0762241 17.446 < 2e-16 ***
## x12 0.2358231 0.0028129 83.836 < 2e-16 ***
## x13 -0.0085629 0.0140858 -0.608 0.545
## x14 1.4782901 0.1122645 13.168 < 2e-16 ***
## x18 -0.4677794 0.0441178 -10.603 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1629 on 103 degrees of freedom
## Multiple R-squared: 0.9979, Adjusted R-squared: 0.9977
## F-statistic: 5416 on 9 and 103 DF, p-value: < 2.2e-16
mq1 <- update(mq1, .~. -x13)
summary(mq1)
##
## Call:
## lm(formula = y ~ x1 + x5 + x6 + x7 + x10 + x12 + x14 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.39788 -0.07868 0.00587 0.06454 0.75390
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.7721244 0.3044254 -9.106 6.66e-15 ***
## x1 -0.0001289 0.0001056 -1.221 0.225
## x5 -0.0490928 0.0324290 -1.514 0.133
## x6 0.5310377 0.0433721 12.244 < 2e-16 ***
## x7 -0.0179257 0.0145401 -1.233 0.220
## x10 1.3495416 0.0687440 19.631 < 2e-16 ***
## x12 0.2354506 0.0027370 86.025 < 2e-16 ***
## x14 1.4877723 0.1108382 13.423 < 2e-16 ***
## x18 -0.4791671 0.0398222 -12.033 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1624 on 104 degrees of freedom
## Multiple R-squared: 0.9979, Adjusted R-squared: 0.9977
## F-statistic: 6131 on 8 and 104 DF, p-value: < 2.2e-16
mq1 <- update(mq1, .~. -x1)
summary(mq1)
##
## Call:
## lm(formula = y ~ x5 + x6 + x7 + x10 + x12 + x14 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.42602 -0.07545 0.00083 0.06638 0.74943
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.743152 0.304207 -9.017 9.77e-15 ***
## x5 -0.046819 0.032451 -1.443 0.152
## x6 0.534608 0.043374 12.325 < 2e-16 ***
## x7 -0.018145 0.014573 -1.245 0.216
## x10 1.353216 0.068838 19.658 < 2e-16 ***
## x12 0.235025 0.002721 86.373 < 2e-16 ***
## x14 1.498104 0.110772 13.524 < 2e-16 ***
## x18 -0.483881 0.039727 -12.180 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1628 on 105 degrees of freedom
## Multiple R-squared: 0.9979, Adjusted R-squared: 0.9977
## F-statistic: 6974 on 7 and 105 DF, p-value: < 2.2e-16
mq1 <- update(mq1, .~. -x7)
summary(mq1)
##
## Call:
## lm(formula = y ~ x5 + x6 + x10 + x12 + x14 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.42766 -0.08150 0.01102 0.06488 0.74583
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.462762 0.205057 -12.010 <2e-16 ***
## x5 -0.040194 0.032095 -1.252 0.213
## x6 0.550194 0.041636 13.214 <2e-16 ***
## x10 1.361043 0.068728 19.803 <2e-16 ***
## x12 0.234268 0.002659 88.098 <2e-16 ***
## x14 1.566623 0.096386 16.254 <2e-16 ***
## x18 -0.511718 0.032924 -15.542 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1632 on 106 degrees of freedom
## Multiple R-squared: 0.9978, Adjusted R-squared: 0.9977
## F-statistic: 8094 on 6 and 106 DF, p-value: < 2.2e-16
mq1 <- update(mq1, .~. -x5)
summary(mq1)
##
## Call:
## lm(formula = y ~ x6 + x10 + x12 + x14 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.44914 -0.07025 0.01326 0.06725 0.75342
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.404720 0.200281 -12.01 <2e-16 ***
## x6 0.536888 0.040365 13.30 <2e-16 ***
## x10 1.346157 0.067872 19.83 <2e-16 ***
## x12 0.233623 0.002616 89.32 <2e-16 ***
## x14 1.541982 0.094606 16.30 <2e-16 ***
## x18 -0.506816 0.032777 -15.46 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1636 on 107 degrees of freedom
## Multiple R-squared: 0.9978, Adjusted R-squared: 0.9977
## F-statistic: 9661 on 5 and 107 DF, p-value: < 2.2e-16
# IDENTIFIED MODEL:
# lm(formula = y ~ x6 + x10 + x12 + x14 + x18)
# Summary information of the model:
summary(mq1)
##
## Call:
## lm(formula = y ~ x6 + x10 + x12 + x14 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.44914 -0.07025 0.01326 0.06725 0.75342
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.404720 0.200281 -12.01 <2e-16 ***
## x6 0.536888 0.040365 13.30 <2e-16 ***
## x10 1.346157 0.067872 19.83 <2e-16 ***
## x12 0.233623 0.002616 89.32 <2e-16 ***
## x14 1.541982 0.094606 16.30 <2e-16 ***
## x18 -0.506816 0.032777 -15.46 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1636 on 107 degrees of freedom
## Multiple R-squared: 0.9978, Adjusted R-squared: 0.9977
## F-statistic: 9661 on 5 and 107 DF, p-value: < 2.2e-16
# 2nd - P-VALUE STRATEGY WITH "FORWARD" METHOD:
# starting from the null model (intercept only), iteratively add
# a variable with the lowest p-value
mq2 <- update(mq0, .~. +x1)
summary(mq2)
##
## Call:
## lm(formula = y ~ x1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.7523 -2.1354 -0.5117 2.0944 12.4285
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.803021 0.481501 12.052 <2e-16 ***
## x1 0.002513 0.002129 1.181 0.24
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.396 on 111 degrees of freedom
## Multiple R-squared: 0.0124, Adjusted R-squared: 0.003504
## F-statistic: 1.394 on 1 and 111 DF, p-value: 0.2403
mq2 <- update(mq0, .~. +x2)
summary(mq2)
##
## Call:
## lm(formula = y ~ x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.3923 -2.3107 -0.5819 1.7492 12.8901
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.228898 0.321407 19.380 <2e-16 ***
## x2 0.002819 0.011232 0.251 0.802
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.417 on 111 degrees of freedom
## Multiple R-squared: 0.0005672, Adjusted R-squared: -0.008437
## F-statistic: 0.063 on 1 and 111 DF, p-value: 0.8023
mq2 <- update(mq0, .~. +x3)
summary(mq2)
##
## Call:
## lm(formula = y ~ x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.9255 -2.3452 -0.6054 1.6366 12.8528
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.98328 0.39625 17.623 < 2e-16 ***
## x3 -0.05453 0.01793 -3.042 0.00294 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.283 on 111 degrees of freedom
## Multiple R-squared: 0.07693, Adjusted R-squared: 0.06862
## F-statistic: 9.251 on 1 and 111 DF, p-value: 0.002937
mq2 <- update(mq0, .~. +x4)
summary(mq2)
##
## Call:
## lm(formula = y ~ x4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.9886 -2.3528 -0.6407 1.7819 12.5366
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.87286 0.36236 18.967 < 2e-16 ***
## x4 -0.05526 0.01657 -3.335 0.00116 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.258 on 111 degrees of freedom
## Multiple R-squared: 0.09106, Adjusted R-squared: 0.08287
## F-statistic: 11.12 on 1 and 111 DF, p-value: 0.001162
mq2 <- update(mq0, .~. +x5)
summary(mq2)
##
## Call:
## lm(formula = y ~ x5)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.6772 -2.3410 -0.7176 1.9016 12.3845
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.0730 0.9287 7.616 9.36e-12 ***
## x5 0.5958 0.6150 0.969 0.335
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.403 on 111 degrees of freedom
## Multiple R-squared: 0.008386, Adjusted R-squared: -0.0005474
## F-statistic: 0.9387 on 1 and 111 DF, p-value: 0.3347
mq2 <- update(mq0, .~. +x6)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.1414 -1.2443 -0.1092 0.9834 7.3929
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -10.2611 1.1210 -9.154 3.17e-15 ***
## x6 -2.2190 0.1488 -14.915 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.972 on 111 degrees of freedom
## Multiple R-squared: 0.6671, Adjusted R-squared: 0.6641
## F-statistic: 222.5 on 1 and 111 DF, p-value: < 2.2e-16
mq2 <- update(mq0, .~. +x7)
summary(mq2)
##
## Call:
## lm(formula = y ~ x7)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.1296 -2.0943 -0.8296 1.4996 12.0996
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.3881 0.3630 20.352 < 2e-16 ***
## x7 -0.6462 0.1233 -5.243 7.63e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.06 on 111 degrees of freedom
## Multiple R-squared: 0.1985, Adjusted R-squared: 0.1913
## F-statistic: 27.49 on 1 and 111 DF, p-value: 7.625e-07
mq2 <- update(mq0, .~. +x8)
summary(mq2)
##
## Call:
## lm(formula = y ~ x8)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.5330 -2.3352 -0.6233 1.9406 12.8074
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.527942 0.495343 13.179 <2e-16 ***
## x8 -0.007220 0.009099 -0.793 0.429
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.408 on 111 degrees of freedom
## Multiple R-squared: 0.00564, Adjusted R-squared: -0.003318
## F-statistic: 0.6296 on 1 and 111 DF, p-value: 0.4292
mq2 <- update(mq0, .~. +x9)
summary(mq2)
##
## Call:
## lm(formula = y ~ x9)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.3943 -1.8993 -0.7773 1.2430 11.9040
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.3619 1.1252 0.322 0.748
## x9 3.7967 0.7043 5.391 3.99e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.042 on 111 degrees of freedom
## Multiple R-squared: 0.2075, Adjusted R-squared: 0.2003
## F-statistic: 29.06 on 1 and 111 DF, p-value: 3.992e-07
mq2 <- update(mq0, .~. +x10)
summary(mq2)
##
## Call:
## lm(formula = y ~ x10)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.4198 -2.4288 -0.8747 1.6892 12.8794
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 15.0509 3.2848 4.582 1.21e-05 ***
## x10 -1.0902 0.4041 -2.698 0.00806 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.311 on 111 degrees of freedom
## Multiple R-squared: 0.06154, Adjusted R-squared: 0.05309
## F-statistic: 7.279 on 1 and 111 DF, p-value: 0.008064
mq2 <- update(mq0, .~. +x11)
summary(mq2)
##
## Call:
## lm(formula = y ~ x11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.1329 -1.5530 -0.2982 1.1070 9.8971
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.6017 0.7075 -0.85 0.397
## x11 1.8499 0.1811 10.21 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.454 on 111 degrees of freedom
## Multiple R-squared: 0.4844, Adjusted R-squared: 0.4798
## F-statistic: 104.3 on 1 and 111 DF, p-value: < 2.2e-16
mq2 <- update(mq0, .~. +x12)
summary(mq2)
##
## Call:
## lm(formula = y ~ x12)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.36563 -0.16324 0.00488 0.17539 1.40265
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.30958 0.07152 4.329 3.3e-05 ***
## x12 0.23609 0.00248 95.210 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3759 on 111 degrees of freedom
## Multiple R-squared: 0.9879, Adjusted R-squared: 0.9878
## F-statistic: 9065 on 1 and 111 DF, p-value: < 2.2e-16
mq2 <- update(mq0, .~. +x13)
summary(mq2)
##
## Call:
## lm(formula = y ~ x13)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.3583 -2.3759 -0.6308 1.8702 12.9087
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.32559 3.27505 2.237 0.0273 *
## x13 -0.06595 0.19589 -0.337 0.7370
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.416 on 111 degrees of freedom
## Multiple R-squared: 0.00102, Adjusted R-squared: -0.00798
## F-statistic: 0.1133 on 1 and 111 DF, p-value: 0.737
mq2 <- update(mq0, .~. +x14)
summary(mq2)
##
## Call:
## lm(formula = y ~ x14)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.0138 -2.4336 -0.7138 1.9457 12.9259
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.6336 0.4071 13.839 <2e-16 ***
## x14 1.8018 0.7845 2.297 0.0235 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.339 on 111 degrees of freedom
## Multiple R-squared: 0.04537, Adjusted R-squared: 0.03677
## F-statistic: 5.275 on 1 and 111 DF, p-value: 0.02351
mq2 <- update(mq0, .~. +x15)
summary(mq2)
##
## Call:
## lm(formula = y ~ x15)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.1711 -2.3065 -0.6711 2.1466 12.6524
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.3826 0.8704 5.035 1.86e-06 ***
## x15 1.5884 0.6986 2.274 0.0249 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.341 on 111 degrees of freedom
## Multiple R-squared: 0.0445, Adjusted R-squared: 0.0359
## F-statistic: 5.17 on 1 and 111 DF, p-value: 0.0249
mq2 <- update(mq0, .~. +x16)
summary(mq2)
##
## Call:
## lm(formula = y ~ x16)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.0060 -2.0173 -0.4343 0.9868 11.6864
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.9745 1.1424 11.358 < 2e-16 ***
## x16 -0.7887 0.1295 -6.089 1.67e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.959 on 111 degrees of freedom
## Multiple R-squared: 0.2504, Adjusted R-squared: 0.2436
## F-statistic: 37.08 on 1 and 111 DF, p-value: 1.666e-08
mq2 <- update(mq0, .~. +x17)
summary(mq2)
##
## Call:
## lm(formula = y ~ x17)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.3912 -2.1912 -0.7825 1.6349 13.2001
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.3261 0.6327 8.418 1.51e-13 ***
## x17 0.1913 0.1160 1.649 0.102
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.376 on 111 degrees of freedom
## Multiple R-squared: 0.0239, Adjusted R-squared: 0.01511
## F-statistic: 2.718 on 1 and 111 DF, p-value: 0.102
mq2 <- update(mq0, .~. +x18)
summary(mq2)
##
## Call:
## lm(formula = y ~ x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.6436 -1.3141 -0.4282 0.5847 10.8293
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 16.6882 1.0316 16.18 <2e-16 ***
## x18 -1.1431 0.1099 -10.40 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.432 on 111 degrees of freedom
## Multiple R-squared: 0.4935, Adjusted R-squared: 0.4889
## F-statistic: 108.1 on 1 and 111 DF, p-value: < 2.2e-16
# Add x6 to the final model (minimum p-value)
mq2f <- update(mq0, .~. +x6)
# Continue iteratively:
mq2 <- update(mq2f, .~. +x1)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.1501 -1.2848 -0.0582 1.0189 7.4326
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.025e+01 1.127e+00 -9.100 4.51e-15 ***
## x6 -2.225e+00 1.512e-01 -14.715 < 2e-16 ***
## x1 -3.033e-04 1.256e-03 -0.242 0.81
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.98 on 110 degrees of freedom
## Multiple R-squared: 0.6673, Adjusted R-squared: 0.6613
## F-statistic: 110.3 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x2)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.8824 -1.1688 -0.1992 1.0145 7.1102
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -10.79519 1.11515 -9.681 <2e-16 ***
## x6 -2.29044 0.14804 -15.472 <2e-16 ***
## x2 -0.01627 0.00645 -2.522 0.0131 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.926 on 110 degrees of freedom
## Multiple R-squared: 0.6853, Adjusted R-squared: 0.6796
## F-statistic: 119.8 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x3)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.6609 -1.2479 -0.2373 0.9161 7.5433
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.35925 1.13152 -8.271 3.39e-13 ***
## x6 -2.15385 0.14600 -14.753 < 2e-16 ***
## x3 -0.03016 0.01056 -2.855 0.00514 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.911 on 110 degrees of freedom
## Multiple R-squared: 0.6901, Adjusted R-squared: 0.6845
## F-statistic: 122.5 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x4)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.332 -1.189 -0.284 1.074 7.403
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.331641 1.122644 -8.312 2.74e-13 ***
## x6 -2.141164 0.145685 -14.697 < 2e-16 ***
## x4 -0.030093 0.009821 -3.064 0.00275 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.901 on 110 degrees of freedom
## Multiple R-squared: 0.6933, Adjusted R-squared: 0.6877
## F-statistic: 124.3 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x5)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x5)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.5828 -1.4524 -0.1586 1.1689 6.3493
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.0022 1.1394 -7.901 2.28e-12 ***
## x6 -2.2660 0.1432 -15.820 < 2e-16 ***
## x5 1.1343 0.3430 3.307 0.00128 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.889 on 110 degrees of freedom
## Multiple R-squared: 0.6972, Adjusted R-squared: 0.6917
## F-statistic: 126.7 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x7)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x7)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.1930 -1.1683 -0.4440 0.9228 7.2713
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -8.31248 1.02430 -8.115 7.58e-13 ***
## x6 -2.06048 0.13176 -15.639 < 2e-16 ***
## x7 -0.42936 0.07035 -6.103 1.59e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.712 on 110 degrees of freedom
## Multiple R-squared: 0.7513, Adjusted R-squared: 0.7468
## F-statistic: 166.2 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x8)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x8)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.6280 -1.1807 -0.3168 0.9252 6.3427
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -11.058044 0.911817 -12.127 < 2e-16 ***
## x6 -2.519571 0.126352 -19.941 < 2e-16 ***
## x8 -0.034613 0.004471 -7.742 5.14e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.594 on 110 degrees of freedom
## Multiple R-squared: 0.7845, Adjusted R-squared: 0.7806
## F-statistic: 200.3 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x9)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x9)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.5490 -1.2200 -0.1636 0.8601 7.4614
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -11.1769 1.1230 -9.952 < 2e-16 ***
## x6 -2.0424 0.1549 -13.184 < 2e-16 ***
## x9 1.4421 0.4753 3.034 0.00301 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.903 on 110 degrees of freedom
## Multiple R-squared: 0.6928, Adjusted R-squared: 0.6873
## F-statistic: 124.1 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x10)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x10)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.7396 -1.1031 -0.1862 0.5767 7.2223
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.5478 1.8207 0.301 0.764
## x6 -2.2921 0.1251 -18.318 < 2e-16 ***
## x10 -1.4027 0.2024 -6.931 2.99e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.652 on 110 degrees of freedom
## Multiple R-squared: 0.7683, Adjusted R-squared: 0.7641
## F-statistic: 182.4 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x11)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.1229 -0.9135 -0.3397 0.4354 6.8664
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -10.5850 0.8653 -12.233 < 2e-16 ***
## x6 -1.7167 0.1283 -13.383 < 2e-16 ***
## x11 1.0987 0.1255 8.755 2.75e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.521 on 110 degrees of freedom
## Multiple R-squared: 0.8038, Adjusted R-squared: 0.8003
## F-statistic: 225.4 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x12)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.43240 -0.14140 -0.00139 0.18146 1.28741
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.271747 0.277411 -0.980 0.3294
## x6 -0.102734 0.047422 -2.166 0.0324 *
## x12 0.228826 0.004146 55.191 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3698 on 110 degrees of freedom
## Multiple R-squared: 0.9884, Adjusted R-squared: 0.9882
## F-statistic: 4686 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x13)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x13)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.3930 -1.1510 -0.0407 1.0131 7.3431
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -11.33333 2.27424 -4.983 2.34e-06 ***
## x6 -2.22514 0.14968 -14.866 < 2e-16 ***
## x13 0.06171 0.11376 0.542 0.589
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.978 on 110 degrees of freedom
## Multiple R-squared: 0.668, Adjusted R-squared: 0.662
## F-statistic: 110.7 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x14)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x14)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.5003 -1.0947 -0.2265 1.1109 7.3546
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -11.3108 1.0221 -11.066 < 2e-16 ***
## x6 -2.2616 0.1334 -16.958 < 2e-16 ***
## x14 2.2218 0.4152 5.351 4.82e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.764 on 110 degrees of freedom
## Multiple R-squared: 0.7359, Adjusted R-squared: 0.7311
## F-statistic: 153.2 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x15)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x15)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.3039 -1.0562 -0.2362 1.0021 6.9502
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.1632 1.1069 -11.892 < 2e-16 ***
## x6 -2.2808 0.1313 -17.365 < 2e-16 ***
## x15 2.1028 0.3640 5.777 7.18e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.735 on 110 degrees of freedom
## Multiple R-squared: 0.7446, Adjusted R-squared: 0.74
## F-statistic: 160.4 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x16)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x16)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.3912 -1.1506 -0.0378 1.0161 7.3403
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -11.27216 2.19822 -5.128 1.27e-06 ***
## x6 -2.28561 0.19431 -11.763 < 2e-16 ***
## x16 0.06034 0.11273 0.535 0.594
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.978 on 110 degrees of freedom
## Multiple R-squared: 0.668, Adjusted R-squared: 0.662
## F-statistic: 110.7 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x17)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x17)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.0562 -1.3128 -0.1708 0.9670 7.4564
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -10.30992 1.13395 -9.092 4.69e-15 ***
## x6 -2.21016 0.15146 -14.592 < 2e-16 ***
## x17 0.02429 0.06898 0.352 0.725
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.98 on 110 degrees of freedom
## Multiple R-squared: 0.6675, Adjusted R-squared: 0.6615
## F-statistic: 110.4 on 2 and 110 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x18)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.1281 -1.1594 -0.3613 0.8767 7.0361
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.0223 2.1589 -0.937 0.351
## x6 -1.6973 0.1828 -9.287 1.68e-15 ***
## x18 -0.4767 0.1095 -4.355 3.00e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.829 on 110 degrees of freedom
## Multiple R-squared: 0.7161, Adjusted R-squared: 0.7109
## F-statistic: 138.7 on 2 and 110 DF, p-value: < 2.2e-16
# Add x12 (minimum p-value)
mq2f <- update(mq2f, .~. +x12)
# New iteration
mq2 <- update(mq2f, .~. +x1)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.44345 -0.14391 -0.00267 0.17740 1.29428
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.2701619 0.2785682 -0.970 0.3343
## x6 -0.1044433 0.0478729 -2.182 0.0313 *
## x12 0.2288017 0.0041634 54.955 <2e-16 ***
## x1 -0.0000808 0.0002355 -0.343 0.7322
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3713 on 109 degrees of freedom
## Multiple R-squared: 0.9884, Adjusted R-squared: 0.9881
## F-statistic: 3099 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x2)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.43551 -0.14245 -0.00234 0.17972 1.28751
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.777e-01 2.917e-01 -0.952 0.3433
## x6 -1.038e-01 4.995e-02 -2.078 0.0401 *
## x12 2.288e-01 4.287e-03 53.362 <2e-16 ***
## x2 -8.835e-05 1.281e-03 -0.069 0.9451
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3715 on 109 degrees of freedom
## Multiple R-squared: 0.9884, Adjusted R-squared: 0.9881
## F-statistic: 3095 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x3)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.43402 -0.14516 0.00206 0.17391 1.27943
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.248924 0.277041 -0.899 0.3709
## x6 -0.109427 0.047539 -2.302 0.0232 *
## x12 0.227457 0.004262 53.373 <2e-16 ***
## x3 -0.002762 0.002101 -1.315 0.1914
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3686 on 109 degrees of freedom
## Multiple R-squared: 0.9886, Adjusted R-squared: 0.9883
## F-statistic: 3145 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x4)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.42562 -0.14535 -0.00505 0.17647 1.29249
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.258846 0.277266 -0.934 0.3526
## x6 -0.108922 0.047668 -2.285 0.0242 *
## x12 0.227526 0.004295 52.978 <2e-16 ***
## x4 -0.002255 0.001979 -1.140 0.2569
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3693 on 109 degrees of freedom
## Multiple R-squared: 0.9885, Adjusted R-squared: 0.9882
## F-statistic: 3133 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x5)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x5)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.36746 -0.15277 0.01266 0.17534 1.29135
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.23306 0.27674 -0.842 0.4016
## x6 -0.12448 0.04914 -2.533 0.0127 *
## x12 0.22696 0.00429 52.905 <2e-16 ***
## x5 0.10826 0.06948 1.558 0.1221
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3674 on 109 degrees of freedom
## Multiple R-squared: 0.9887, Adjusted R-squared: 0.9883
## F-statistic: 3165 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x7)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x7)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.40595 -0.17790 0.02701 0.19071 1.19800
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.218483 0.277406 -0.788 0.433
## x6 -0.074407 0.050263 -1.480 0.142
## x12 0.233038 0.004878 47.778 <2e-16 ***
## x7 0.028776 0.017876 1.610 0.110
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3671 on 109 degrees of freedom
## Multiple R-squared: 0.9887, Adjusted R-squared: 0.9884
## F-statistic: 3170 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x8)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x8)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.42751 -0.15002 0.01715 0.16617 1.26693
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.812583 0.295672 -2.748 0.007014 **
## x6 -0.234637 0.056012 -4.189 5.71e-05 ***
## x12 0.218838 0.004668 46.877 < 2e-16 ***
## x8 -0.004553 0.001168 -3.897 0.000168 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.348 on 109 degrees of freedom
## Multiple R-squared: 0.9898, Adjusted R-squared: 0.9895
## F-statistic: 3532 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x9)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.77194 -0.09375 0.05089 0.11431 0.68422
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.023302 0.179079 5.714 9.7e-08 ***
## x6 -0.055204 0.027120 -2.036 0.0442 *
## x12 0.245654 0.002602 94.420 < 2e-16 ***
## x9 -0.882591 0.057964 -15.227 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2101 on 109 degrees of freedom
## Multiple R-squared: 0.9963, Adjusted R-squared: 0.9962
## F-statistic: 9757 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x10)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x10)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.03501 -0.15868 0.02099 0.16530 0.95279
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.097817 0.338754 -6.193 1.07e-08 ***
## x6 0.110020 0.048723 2.258 0.0259 *
## x12 0.249825 0.004464 55.962 < 2e-16 ***
## x10 0.355942 0.048765 7.299 4.98e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3044 on 109 degrees of freedom
## Multiple R-squared: 0.9922, Adjusted R-squared: 0.992
## F-statistic: 4626 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x11)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.29883 -0.19587 -0.00159 0.16828 1.09479
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.427092 0.316294 1.350 0.17972
## x6 -0.036368 0.047730 -0.762 0.44774
## x12 0.243773 0.005461 44.640 < 2e-16 ***
## x11 -0.157200 0.040198 -3.911 0.00016 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3479 on 109 degrees of freedom
## Multiple R-squared: 0.9898, Adjusted R-squared: 0.9895
## F-statistic: 3535 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x13)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x13)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.44496 -0.13628 -0.00455 0.15979 1.24610
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.135475 0.473051 0.286 0.775
## x6 -0.097556 0.047644 -2.048 0.043 *
## x12 0.229143 0.004154 55.157 <2e-16 ***
## x13 -0.022640 0.021309 -1.062 0.290
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3696 on 109 degrees of freedom
## Multiple R-squared: 0.9885, Adjusted R-squared: 0.9882
## F-statistic: 3128 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x14)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x14)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.40654 -0.13205 -0.00347 0.20254 1.30772
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.465315 0.306648 -1.517 0.1321
## x6 -0.132436 0.051445 -2.574 0.0114 *
## x12 0.225904 0.004592 49.196 <2e-16 ***
## x14 0.139689 0.096385 1.449 0.1501
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3679 on 109 degrees of freedom
## Multiple R-squared: 0.9886, Adjusted R-squared: 0.9883
## F-statistic: 3156 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x15)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x15)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.38951 -0.13162 0.00449 0.20165 1.30790
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.627728 0.348736 -1.800 0.07462 .
## x6 -0.140150 0.052155 -2.687 0.00833 **
## x12 0.225239 0.004645 48.490 < 2e-16 ***
## x15 0.144471 0.086923 1.662 0.09937 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3668 on 109 degrees of freedom
## Multiple R-squared: 0.9887, Adjusted R-squared: 0.9884
## F-statistic: 3175 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x16)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x16)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.44560 -0.13734 -0.00427 0.15915 1.24832
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.114204 0.459691 0.248 0.804
## x6 -0.075335 0.054075 -1.393 0.166
## x12 0.229136 0.004155 55.153 <2e-16 ***
## x16 -0.022227 0.021115 -1.053 0.295
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3696 on 109 degrees of freedom
## Multiple R-squared: 0.9885, Adjusted R-squared: 0.9882
## F-statistic: 3127 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x17)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x17)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.42687 -0.13918 -0.00359 0.17431 1.29364
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.275793 0.280470 -0.983 0.3276
## x6 -0.102290 0.047763 -2.142 0.0345 *
## x12 0.228809 0.004167 54.911 <2e-16 ***
## x17 0.001647 0.012949 0.127 0.8990
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3714 on 109 degrees of freedom
## Multiple R-squared: 0.9884, Adjusted R-squared: 0.9881
## F-statistic: 3096 on 3 and 109 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x18)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.31685 -0.17950 0.01691 0.18996 1.12690
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.431054 0.414322 -3.454 0.000788 ***
## x6 -0.130060 0.045632 -2.850 0.005226 **
## x12 0.235933 0.004397 53.664 < 2e-16 ***
## x18 0.085031 0.023465 3.624 0.000443 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3509 on 109 degrees of freedom
## Multiple R-squared: 0.9896, Adjusted R-squared: 0.9894
## F-statistic: 3473 on 3 and 109 DF, p-value: < 2.2e-16
# Add x9 (minimum p-value)
mq2f <- update(mq2f, .~. +x9)
# New iteration
mq2 <- update(mq2f, .~. +x1)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.79696 -0.09560 0.03489 0.12107 0.69838
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.0368556 0.1778955 5.828 5.88e-08 ***
## x6 -0.0594654 0.0270364 -2.199 0.030 *
## x12 0.2457082 0.0025820 95.164 < 2e-16 ***
## x9 -0.8889203 0.0576475 -15.420 < 2e-16 ***
## x1 -0.0002175 0.0001325 -1.641 0.104
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2084 on 108 degrees of freedom
## Multiple R-squared: 0.9964, Adjusted R-squared: 0.9962
## F-statistic: 7432 on 4 and 108 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x2)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.7437 -0.1100 0.0497 0.1197 0.6785
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.0965677 0.1873341 5.854 5.24e-08 ***
## x6 -0.0438592 0.0284291 -1.543 0.126
## x12 0.2465274 0.0026806 91.966 < 2e-16 ***
## x9 -0.8894987 0.0580341 -15.327 < 2e-16 ***
## x2 0.0009362 0.0007251 1.291 0.199
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2094 on 108 degrees of freedom
## Multiple R-squared: 0.9963, Adjusted R-squared: 0.9962
## F-statistic: 7363 on 4 and 108 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x3)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.76286 -0.08805 0.04025 0.10289 0.66162
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.080406 0.170348 6.342 5.4e-09 ***
## x6 -0.064495 0.025814 -2.498 0.013981 *
## x12 0.243878 0.002512 97.101 < 2e-16 ***
## x9 -0.897993 0.055067 -16.307 < 2e-16 ***
## x3 -0.004176 0.001138 -3.671 0.000378 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.199 on 108 degrees of freedom
## Multiple R-squared: 0.9967, Adjusted R-squared: 0.9966
## F-statistic: 8158 on 4 and 108 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x4)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.74643 -0.06977 0.04584 0.10312 0.68030
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.072640 0.170469 6.292 6.84e-09 ***
## x6 -0.064847 0.025871 -2.507 0.013681 *
## x12 0.243772 0.002523 96.620 < 2e-16 ***
## x9 -0.901099 0.055237 -16.313 < 2e-16 ***
## x4 -0.003877 0.001073 -3.615 0.000459 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1993 on 108 degrees of freedom
## Multiple R-squared: 0.9967, Adjusted R-squared: 0.9966
## F-statistic: 8131 on 4 and 108 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x5)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x5)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.78254 -0.09466 0.04556 0.11652 0.67636
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.027103 0.179533 5.721 9.57e-08 ***
## x6 -0.048641 0.028608 -1.700 0.092 .
## x12 0.246360 0.002779 88.656 < 2e-16 ***
## x9 -0.892494 0.059630 -14.967 < 2e-16 ***
## x5 -0.030024 0.040868 -0.735 0.464
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2105 on 108 degrees of freedom
## Multiple R-squared: 0.9963, Adjusted R-squared: 0.9962
## F-statistic: 7287 on 4 and 108 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x7)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x7)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.71667 -0.06769 0.01021 0.08369 0.75606
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.129043 0.162027 6.968 2.65e-10 ***
## x6 -0.100759 0.025861 -3.896 0.00017 ***
## x12 0.240430 0.002541 94.628 < 2e-16 ***
## x9 -1.022701 0.058545 -17.469 < 2e-16 ***
## x7 -0.053941 0.010331 -5.221 8.69e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1886 on 108 degrees of freedom
## Multiple R-squared: 0.997, Adjusted R-squared: 0.9969
## F-statistic: 9087 on 4 and 108 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x8)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x8)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.79824 -0.08617 0.03007 0.09396 0.69833
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.6312470 0.1906641 3.311 0.00127 **
## x6 -0.1389107 0.0320662 -4.332 3.32e-05 ***
## x12 0.2387281 0.0029224 81.690 < 2e-16 ***
## x9 -0.8434107 0.0547228 -15.412 < 2e-16 ***
## x8 -0.0028163 0.0006657 -4.231 4.90e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1955 on 108 degrees of freedom
## Multiple R-squared: 0.9968, Adjusted R-squared: 0.9967
## F-statistic: 8456 on 4 and 108 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x10)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x10)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.77841 -0.05428 0.03065 0.07882 0.76133
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.059885 0.451208 6.782 6.58e-10 ***
## x6 -0.200526 0.038873 -5.158 1.14e-06 ***
## x12 0.236936 0.002976 79.614 < 2e-16 ***
## x9 -1.291647 0.099633 -12.964 < 2e-16 ***
## x10 -0.279980 0.057837 -4.841 4.33e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1913 on 108 degrees of freedom
## Multiple R-squared: 0.997, Adjusted R-squared: 0.9968
## F-statistic: 8829 on 4 and 108 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x11)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.71731 -0.04655 0.02823 0.06939 0.77004
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.685088 0.169941 4.031 0.000104 ***
## x6 -0.109118 0.025914 -4.211 5.29e-05 ***
## x12 0.235280 0.002967 79.310 < 2e-16 ***
## x9 -1.137521 0.068872 -16.517 < 2e-16 ***
## x11 0.160224 0.028842 5.555 2.01e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1861 on 108 degrees of freedom
## Multiple R-squared: 0.9971, Adjusted R-squared: 0.997
## F-statistic: 9330 on 4 and 108 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x13)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x13)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.78334 -0.08513 0.04633 0.11684 0.61500
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.652433 0.277100 5.963 3.17e-08 ***
## x6 -0.046850 0.026393 -1.775 0.0787 .
## x12 0.246324 0.002528 97.457 < 2e-16 ***
## x9 -0.892679 0.056183 -15.889 < 2e-16 ***
## x13 -0.034154 0.011741 -2.909 0.0044 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2032 on 108 degrees of freedom
## Multiple R-squared: 0.9966, Adjusted R-squared: 0.9964
## F-statistic: 7821 on 4 and 108 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x14)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x14)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.76344 -0.07728 0.05012 0.11976 0.69875
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.915995 0.197025 4.649 9.48e-06 ***
## x6 -0.070613 0.029567 -2.388 0.0187 *
## x12 0.244053 0.002876 84.848 < 2e-16 ***
## x9 -0.876416 0.057988 -15.114 < 2e-16 ***
## x14 0.070900 0.055051 1.288 0.2005
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2094 on 108 degrees of freedom
## Multiple R-squared: 0.9963, Adjusted R-squared: 0.9962
## F-statistic: 7362 on 4 and 108 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x15)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x15)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.76464 -0.08432 0.05347 0.11741 0.69456
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.912112 0.224969 4.054 9.51e-05 ***
## x6 -0.066230 0.030316 -2.185 0.0311 *
## x12 0.244505 0.002959 82.621 < 2e-16 ***
## x9 -0.876004 0.058606 -14.947 < 2e-16 ***
## x15 0.041203 0.050326 0.819 0.4147
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2104 on 108 degrees of freedom
## Multiple R-squared: 0.9963, Adjusted R-squared: 0.9962
## F-statistic: 7296 on 4 and 108 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x16)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x16)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.78425 -0.08625 0.04472 0.11937 0.61667
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.633549 0.269983 6.051 2.12e-08 ***
## x6 -0.012424 0.029968 -0.415 0.67929
## x12 0.246336 0.002525 97.545 < 2e-16 ***
## x9 -0.893327 0.056143 -15.912 < 2e-16 ***
## x16 -0.034237 0.011624 -2.945 0.00395 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.203 on 108 degrees of freedom
## Multiple R-squared: 0.9966, Adjusted R-squared: 0.9964
## F-statistic: 7835 on 4 and 108 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x17)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x17)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.71382 -0.08869 0.03234 0.10133 0.73003
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.006786 0.176907 5.691 1.1e-07 ***
## x6 -0.050617 0.026861 -1.884 0.0622 .
## x12 0.245752 0.002568 95.705 < 2e-16 ***
## x9 -0.895510 0.057566 -15.556 < 2e-16 ***
## x17 0.014438 0.007273 1.985 0.0497 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2073 on 108 degrees of freedom
## Multiple R-squared: 0.9964, Adjusted R-squared: 0.9963
## F-statistic: 7516 on 4 and 108 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x18)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.74519 -0.08646 0.02632 0.10232 0.69880
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.05906 0.32835 6.271 7.57e-09 ***
## x6 -0.02799 0.02672 -1.048 0.297190
## x12 0.24338 0.00254 95.833 < 2e-16 ***
## x9 -1.02789 0.06759 -15.208 < 2e-16 ***
## x18 -0.06033 0.01638 -3.684 0.000361 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1989 on 108 degrees of freedom
## Multiple R-squared: 0.9967, Adjusted R-squared: 0.9966
## F-statistic: 8165 on 4 and 108 DF, p-value: < 2.2e-16
# Add x11 (minimum p-value)
mq2f <- update(mq2f, .~. +x11)
# New iteration
mq2 <- update(mq2f, .~. +x1)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.72938 -0.05485 0.03316 0.06899 0.77366
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.001e-01 1.714e-01 4.084 8.56e-05 ***
## x6 -1.094e-01 2.597e-02 -4.214 5.25e-05 ***
## x12 2.356e-01 3.000e-03 78.531 < 2e-16 ***
## x9 -1.133e+00 6.924e-02 -16.366 < 2e-16 ***
## x11 1.558e-01 2.947e-02 5.287 6.63e-07 ***
## x1 -9.186e-05 1.209e-04 -0.760 0.449
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1865 on 107 degrees of freedom
## Multiple R-squared: 0.9971, Adjusted R-squared: 0.997
## F-statistic: 7435 on 5 and 107 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x2)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.68934 -0.05608 0.02936 0.07388 0.76462
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.7444611 0.1783356 4.174 6.10e-05 ***
## x6 -0.0999040 0.0272390 -3.668 0.000383 ***
## x12 0.2360704 0.0030516 77.359 < 2e-16 ***
## x9 -1.1393966 0.0688340 -16.553 < 2e-16 ***
## x11 0.1581443 0.0288809 5.476 2.91e-07 ***
## x2 0.0007026 0.0006452 1.089 0.278647
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.186 on 107 degrees of freedom
## Multiple R-squared: 0.9971, Adjusted R-squared: 0.997
## F-statistic: 7477 on 5 and 107 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x3)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.71951 -0.04905 0.02961 0.07630 0.76152
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.7142786 0.1850077 3.861 0.000194 ***
## x6 -0.1069910 0.0265318 -4.033 0.000104 ***
## x12 0.2356908 0.0031438 74.969 < 2e-16 ***
## x9 -1.1235461 0.0771648 -14.560 < 2e-16 ***
## x11 0.1501109 0.0381216 3.938 0.000147 ***
## x3 -0.0005735 0.0014064 -0.408 0.684219
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1868 on 107 degrees of freedom
## Multiple R-squared: 0.9971, Adjusted R-squared: 0.997
## F-statistic: 7406 on 5 and 107 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x4)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.71717 -0.04900 0.02883 0.07507 0.76444
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.7116525 0.1836588 3.875 0.000184 ***
## x6 -0.1071992 0.0264752 -4.049 9.75e-05 ***
## x12 0.2356456 0.0031218 75.483 < 2e-16 ***
## x9 -1.1248655 0.0763534 -14.732 < 2e-16 ***
## x11 0.1507311 0.0377980 3.988 0.000122 ***
## x4 -0.0005128 0.0013125 -0.391 0.696773
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1868 on 107 degrees of freedom
## Multiple R-squared: 0.9971, Adjusted R-squared: 0.997
## F-statistic: 7405 on 5 and 107 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x5)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x5)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.73521 -0.05168 0.01791 0.08078 0.75836
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.680300 0.168873 4.028 0.000105 ***
## x6 -0.098740 0.026601 -3.712 0.000328 ***
## x12 0.236236 0.003011 78.454 < 2e-16 ***
## x9 -1.165011 0.070683 -16.482 < 2e-16 ***
## x11 0.165860 0.028886 5.742 8.86e-08 ***
## x5 -0.056152 0.036186 -1.552 0.123673
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1849 on 107 degrees of freedom
## Multiple R-squared: 0.9972, Adjusted R-squared: 0.997
## F-statistic: 7562 on 5 and 107 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x7)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x7)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.70985 -0.05390 0.02139 0.06674 0.77462
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.852503 0.196842 4.331 3.36e-05 ***
## x6 -0.112195 0.025778 -4.352 3.09e-05 ***
## x12 0.236377 0.003018 78.333 < 2e-16 ***
## x9 -1.116164 0.069549 -16.049 < 2e-16 ***
## x11 0.104804 0.044155 2.374 0.0194 *
## x7 -0.025726 0.015609 -1.648 0.1023
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1847 on 107 degrees of freedom
## Multiple R-squared: 0.9972, Adjusted R-squared: 0.9971
## F-statistic: 7583 on 5 and 107 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x8)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x8)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.73249 -0.05179 0.02669 0.06731 0.76115
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.6292997 0.1817498 3.462 0.000771 ***
## x6 -0.1237363 0.0308829 -4.007 0.000114 ***
## x12 0.2349454 0.0029945 78.458 < 2e-16 ***
## x9 -1.0892258 0.0884195 -12.319 < 2e-16 ***
## x11 0.1365173 0.0396491 3.443 0.000822 ***
## x8 -0.0007602 0.0008714 -0.872 0.384912
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1863 on 107 degrees of freedom
## Multiple R-squared: 0.9971, Adjusted R-squared: 0.997
## F-statistic: 7447 on 5 and 107 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x10)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.51527 -0.06311 0.01123 0.07509 0.76495
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.806484 2.941230 -3.334 0.001176 **
## x6 0.329184 0.125133 2.631 0.009779 **
## x12 0.233187 0.002877 81.041 < 2e-16 ***
## x9 -0.314002 0.239618 -1.310 0.192856
## x11 0.797302 0.180422 4.419 2.38e-05 ***
## x10 1.257455 0.351987 3.572 0.000532 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1767 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8279 on 5 and 107 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x13)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x13)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.72812 -0.06004 0.01888 0.08413 0.72645
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.041637 0.281701 3.698 0.000345 ***
## x6 -0.099951 0.026381 -3.789 0.000251 ***
## x12 0.236558 0.003055 77.431 < 2e-16 ***
## x9 -1.119892 0.069302 -16.160 < 2e-16 ***
## x11 0.145856 0.030052 4.854 4.15e-06 ***
## x13 -0.017710 0.011203 -1.581 0.116868
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1848 on 107 degrees of freedom
## Multiple R-squared: 0.9972, Adjusted R-squared: 0.997
## F-statistic: 7568 on 5 and 107 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x14)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x14)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.71920 -0.06078 0.02538 0.07557 0.76608
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.742412 0.177787 4.176 6.07e-05 ***
## x6 -0.101395 0.026847 -3.777 0.000261 ***
## x12 0.235629 0.002981 79.034 < 2e-16 ***
## x9 -1.166892 0.073921 -15.786 < 2e-16 ***
## x11 0.175448 0.032037 5.476 2.90e-07 ***
## x14 -0.059109 0.054340 -1.088 0.279144
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.186 on 107 degrees of freedom
## Multiple R-squared: 0.9971, Adjusted R-squared: 0.997
## F-statistic: 7477 on 5 and 107 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x15)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x15)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.72386 -0.05873 0.01717 0.08363 0.76191
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.853168 0.197782 4.314 3.59e-05 ***
## x6 -0.095265 0.027091 -3.516 0.000643 ***
## x12 0.236066 0.002984 79.122 < 2e-16 ***
## x9 -1.185455 0.074431 -15.927 < 2e-16 ***
## x11 0.182353 0.031692 5.754 8.39e-08 ***
## x15 -0.079594 0.048918 -1.627 0.106664
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1847 on 107 degrees of freedom
## Multiple R-squared: 0.9972, Adjusted R-squared: 0.9971
## F-statistic: 7578 on 5 and 107 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x16)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x16)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.72874 -0.05916 0.01862 0.08536 0.72697
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.033964 0.275174 3.757 0.00028 ***
## x6 -0.081883 0.030819 -2.657 0.00909 **
## x12 0.236589 0.003056 77.417 < 2e-16 ***
## x9 -1.119672 0.069273 -16.163 < 2e-16 ***
## x11 0.145492 0.030070 4.838 4.42e-06 ***
## x16 -0.017828 0.011109 -1.605 0.11147
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1848 on 107 degrees of freedom
## Multiple R-squared: 0.9972, Adjusted R-squared: 0.9971
## F-statistic: 7573 on 5 and 107 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x17)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x17)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.69063 -0.05249 0.01616 0.07319 0.78930
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.690837 0.169899 4.066 9.15e-05 ***
## x6 -0.104607 0.026231 -3.988 0.000122 ***
## x12 0.235757 0.002997 78.656 < 2e-16 ***
## x9 -1.133422 0.068925 -16.444 < 2e-16 ***
## x11 0.153605 0.029469 5.213 9.14e-07 ***
## x17 0.007188 0.006672 1.077 0.283764
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.186 on 107 degrees of freedom
## Multiple R-squared: 0.9971, Adjusted R-squared: 0.997
## F-statistic: 7475 on 5 and 107 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x18)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.71353 -0.05722 0.01883 0.07820 0.76401
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.185553 0.368657 3.216 0.00172 **
## x6 -0.089470 0.028790 -3.108 0.00242 **
## x12 0.235786 0.002967 79.469 < 2e-16 ***
## x9 -1.164082 0.070625 -16.482 < 2e-16 ***
## x11 0.137097 0.032420 4.229 4.97e-05 ***
## x18 -0.026308 0.017226 -1.527 0.12965
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.185 on 107 degrees of freedom
## Multiple R-squared: 0.9972, Adjusted R-squared: 0.997
## F-statistic: 7556 on 5 and 107 DF, p-value: < 2.2e-16
# Add x10 (lowest p-value)
mq2f <- update(mq2f, .~. +x10)
# Final iteration
mq2 <- update(mq2f, .~. +x1)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10 + x1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.49169 -0.06137 0.01462 0.06378 0.76990
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.008e+01 2.948e+00 -3.419 0.000892 ***
## x6 3.410e-01 1.254e-01 2.719 0.007654 **
## x12 2.336e-01 2.893e-03 80.731 < 2e-16 ***
## x9 -2.849e-01 2.407e-01 -1.183 0.239326
## x11 8.090e-01 1.805e-01 4.482 1.88e-05 ***
## x10 1.293e+00 3.529e-01 3.663 0.000392 ***
## x1 -1.292e-04 1.149e-04 -1.124 0.263378
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1765 on 106 degrees of freedom
## Multiple R-squared: 0.9975, Adjusted R-squared: 0.9973
## F-statistic: 6917 on 6 and 106 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x2)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10 + x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.51283 -0.07031 0.00972 0.07710 0.76097
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.5565782 2.9592538 -3.229 0.001652 **
## x6 0.3275380 0.1253027 2.614 0.010251 *
## x12 0.2338226 0.0029747 78.605 < 2e-16 ***
## x9 -0.3315242 0.2407826 -1.377 0.171456
## x11 0.7832747 0.1813844 4.318 3.55e-05 ***
## x10 1.2328540 0.3535880 3.487 0.000713 ***
## x2 0.0005283 0.0006160 0.858 0.393036
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.177 on 106 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 6882 on 6 and 106 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x3)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10 + x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.51590 -0.06244 0.01079 0.07741 0.76314
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.7744978 2.9756395 -3.285 0.001384 **
## x6 0.3285635 0.1259006 2.610 0.010372 *
## x12 0.2332803 0.0030648 76.116 < 2e-16 ***
## x9 -0.3130354 0.2409683 -1.299 0.196738
## x11 0.7935769 0.1857972 4.271 4.25e-05 ***
## x10 1.2543693 0.3552397 3.531 0.000614 ***
## x3 -0.0001226 0.0013426 -0.091 0.927405
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1776 on 106 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 6835 on 6 and 106 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x4)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10 + x4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.51534 -0.06308 0.01113 0.07529 0.76483
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.803e+00 2.982e+00 -3.287 0.001373 **
## x6 3.291e-01 1.261e-01 2.610 0.010350 *
## x12 2.332e-01 3.047e-03 76.537 < 2e-16 ***
## x9 -3.140e-01 2.408e-01 -1.304 0.195097
## x11 7.969e-01 1.865e-01 4.274 4.21e-05 ***
## x10 1.257e+00 3.559e-01 3.532 0.000612 ***
## x4 -1.113e-05 1.255e-03 -0.009 0.992943
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1776 on 106 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 6835 on 6 and 106 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x5)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10 + x5)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.49098 -0.05739 -0.00121 0.07721 0.75347
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.776319 2.919701 -3.348 0.001126 **
## x6 0.337936 0.124334 2.718 0.007674 **
## x12 0.234135 0.002916 80.284 < 2e-16 ***
## x9 -0.343781 0.238578 -1.441 0.152543
## x11 0.800729 0.179110 4.471 1.96e-05 ***
## x10 1.253275 0.349413 3.587 0.000508 ***
## x5 -0.055237 0.034334 -1.609 0.110631
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1754 on 106 degrees of freedom
## Multiple R-squared: 0.9975, Adjusted R-squared: 0.9973
## F-statistic: 7002 on 6 and 106 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x7)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10 + x7)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.51737 -0.06645 0.00246 0.06932 0.76914
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.415048 2.933487 -3.210 0.001760 **
## x6 0.316321 0.124615 2.538 0.012590 *
## x12 0.234213 0.002935 79.789 < 2e-16 ***
## x9 -0.313989 0.238090 -1.319 0.190084
## x11 0.733211 0.184028 3.984 0.000124 ***
## x10 1.228419 0.350249 3.507 0.000665 ***
## x7 -0.022922 0.014867 -1.542 0.126086
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1756 on 106 degrees of freedom
## Multiple R-squared: 0.9975, Adjusted R-squared: 0.9973
## F-statistic: 6989 on 6 and 106 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x8)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10 + x8)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.50911 -0.06348 0.00993 0.07093 0.75717
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.7812150 2.9462008 -3.320 0.001235 **
## x6 0.3132250 0.1268878 2.469 0.015166 *
## x12 0.2329080 0.0029028 80.236 < 2e-16 ***
## x9 -0.2773696 0.2442634 -1.136 0.258712
## x11 0.7719437 0.1834281 4.208 5.4e-05 ***
## x10 1.2485469 0.3527343 3.540 0.000596 ***
## x8 -0.0006685 0.0008283 -0.807 0.421449
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.177 on 106 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 6877 on 6 and 106 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x13)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10 + x13)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.53524 -0.06405 0.00911 0.07310 0.73104
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.129835 2.978665 -3.065 0.002760 **
## x6 0.319740 0.124961 2.559 0.011917 *
## x12 0.234266 0.002988 78.405 < 2e-16 ***
## x9 -0.331428 0.239258 -1.385 0.168889
## x11 0.761912 0.181944 4.188 5.84e-05 ***
## x10 1.209788 0.352838 3.429 0.000865 ***
## x13 -0.013855 0.010738 -1.290 0.199767
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1762 on 106 degrees of freedom
## Multiple R-squared: 0.9975, Adjusted R-squared: 0.9973
## F-statistic: 6943 on 6 and 106 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x14)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10 + x14)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.52785 -0.06158 0.00979 0.07439 0.76238
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.526784 2.969195 -3.209 0.001765 **
## x6 0.324352 0.125530 2.584 0.011131 *
## x12 0.233472 0.002906 80.328 < 2e-16 ***
## x9 -0.352803 0.245316 -1.438 0.153335
## x11 0.792988 0.180854 4.385 2.75e-05 ***
## x10 1.228585 0.354650 3.464 0.000769 ***
## x14 -0.040036 0.052037 -0.769 0.443385
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1771 on 106 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 6873 on 6 and 106 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x15)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10 + x15)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.52826 -0.06506 0.00912 0.07208 0.75869
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.268550 2.957104 -3.134 0.002228 **
## x6 0.323284 0.124735 2.592 0.010894 *
## x12 0.233892 0.002914 80.272 < 2e-16 ***
## x9 -0.383809 0.244259 -1.571 0.119089
## x11 0.790310 0.179812 4.395 2.64e-05 ***
## x10 1.208975 0.352489 3.430 0.000862 ***
## x15 -0.063192 0.046874 -1.348 0.180489
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1761 on 106 degrees of freedom
## Multiple R-squared: 0.9975, Adjusted R-squared: 0.9973
## F-statistic: 6952 on 6 and 106 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x16)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10 + x16)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.53589 -0.06302 0.00988 0.07363 0.73164
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.11891 2.97891 -3.061 0.002794 **
## x6 0.33299 0.12476 2.669 0.008804 **
## x12 0.23429 0.00299 78.350 < 2e-16 ***
## x9 -0.33278 0.23928 -1.391 0.167212
## x11 0.76057 0.18204 4.178 6.06e-05 ***
## x10 1.20758 0.35293 3.422 0.000886 ***
## x16 -0.01387 0.01065 -1.302 0.195760
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1762 on 106 degrees of freedom
## Multiple R-squared: 0.9975, Adjusted R-squared: 0.9973
## F-statistic: 6944 on 6 and 106 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x17)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10 + x17)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.52291 -0.05896 0.01057 0.07494 0.77187
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.557347 3.021370 -3.163 0.002036 **
## x6 0.320453 0.127614 2.511 0.013544 *
## x12 0.233405 0.002942 79.323 < 2e-16 ***
## x9 -0.331952 0.244942 -1.355 0.178228
## x11 0.779961 0.186527 4.181 5.98e-05 ***
## x10 1.227838 0.361470 3.397 0.000961 ***
## x17 0.002537 0.006511 0.390 0.697543
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1774 on 106 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 6845 on 6 and 106 DF, p-value: < 2.2e-16
mq2 <- update(mq2f, .~. +x18)
summary(mq2)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.53104 -0.06386 0.01086 0.06894 0.76057
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.026514 3.005359 -3.003 0.003330 **
## x6 0.327339 0.124877 2.621 0.010047 *
## x12 0.233651 0.002897 80.656 < 2e-16 ***
## x9 -0.365582 0.242902 -1.505 0.135282
## x11 0.755445 0.183352 4.120 7.53e-05 ***
## x10 1.209436 0.353488 3.421 0.000886 ***
## x18 -0.019940 0.016529 -1.206 0.230353
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1764 on 106 degrees of freedom
## Multiple R-squared: 0.9975, Adjusted R-squared: 0.9973
## F-statistic: 6929 on 6 and 106 DF, p-value: < 2.2e-16
# No further significant variables present.
# IDENTIFIED MODEL:
# lm(formula = y ~ x6 + x12 + x9 + x11 + x10)
# Summary information:
summary(mq2f)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.51527 -0.06311 0.01123 0.07509 0.76495
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.806484 2.941230 -3.334 0.001176 **
## x6 0.329184 0.125133 2.631 0.009779 **
## x12 0.233187 0.002877 81.041 < 2e-16 ***
## x9 -0.314002 0.239618 -1.310 0.192856
## x11 0.797302 0.180422 4.419 2.38e-05 ***
## x10 1.257455 0.351987 3.572 0.000532 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1767 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8279 on 5 and 107 DF, p-value: < 2.2e-16
# 3rd Approach: P-VALUE METHOD "MIXED"
# • The procedure starts with the forward method considering the null model and
# iteratively adds the variable with a significant effect that has the smallest p-value.
# • If after adding significant effect variables, others in the
# model become non-significant, they are removed.
# • The procedure ends when there are no more variables to add or remove
# based on the p-value.
# Setting the starting model based on the previous "FORWARD" strategy
# (evaluating if adding a variable makes another one non-significant)
mq3f <- update(mq0, .~. +x6 + x12)
summary(mq3f)
##
## Call:
## lm(formula = y ~ x6 + x12)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.43240 -0.14140 -0.00139 0.18146 1.28741
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.271747 0.277411 -0.980 0.3294
## x6 -0.102734 0.047422 -2.166 0.0324 *
## x12 0.228826 0.004146 55.191 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3698 on 110 degrees of freedom
## Multiple R-squared: 0.9884, Adjusted R-squared: 0.9882
## F-statistic: 4686 on 2 and 110 DF, p-value: < 2.2e-16
# The significance of x6 is slightly reduced, but sufficient to keep
# both.
# Adding the third identified variable:
mq3f <- update(mq3f, .~. +x9)
summary(mq3f)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.77194 -0.09375 0.05089 0.11431 0.68422
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.023302 0.179079 5.714 9.7e-08 ***
## x6 -0.055204 0.027120 -2.036 0.0442 *
## x12 0.245654 0.002602 94.420 < 2e-16 ***
## x9 -0.882591 0.057964 -15.227 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2101 on 109 degrees of freedom
## Multiple R-squared: 0.9963, Adjusted R-squared: 0.9962
## F-statistic: 9757 on 3 and 109 DF, p-value: < 2.2e-16
# The significance is not altered.
# Adding the fourth identified significant variable
mq3f <- update(mq3f, .~. +x11)
summary(mq3f)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.71731 -0.04655 0.02823 0.06939 0.77004
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.685088 0.169941 4.031 0.000104 ***
## x6 -0.109118 0.025914 -4.211 5.29e-05 ***
## x12 0.235280 0.002967 79.310 < 2e-16 ***
## x9 -1.137521 0.068872 -16.517 < 2e-16 ***
## x11 0.160224 0.028842 5.555 2.01e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1861 on 108 degrees of freedom
## Multiple R-squared: 0.9971, Adjusted R-squared: 0.997
## F-statistic: 9330 on 4 and 108 DF, p-value: < 2.2e-16
# The significance of all variables has now increased
# Adding the fifth significant variable
mq3f <- update(mq3f, .~. +x10)
summary(mq3f)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.51527 -0.06311 0.01123 0.07509 0.76495
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.806484 2.941230 -3.334 0.001176 **
## x6 0.329184 0.125133 2.631 0.009779 **
## x12 0.233187 0.002877 81.041 < 2e-16 ***
## x9 -0.314002 0.239618 -1.310 0.192856
## x11 0.797302 0.180422 4.419 2.38e-05 ***
## x10 1.257455 0.351987 3.572 0.000532 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1767 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8279 on 5 and 107 DF, p-value: < 2.2e-16
# Now x9 becomes non-significant, removing it
mq3f <- update(mq3f, .~. -x9)
summary(mq3f)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.54633 -0.05866 0.00924 0.07217 0.76854
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.514137 0.806207 -16.76 <2e-16 ***
## x6 0.485352 0.038286 12.68 <2e-16 ***
## x12 0.232624 0.002854 81.50 <2e-16 ***
## x11 1.015592 0.069539 14.61 <2e-16 ***
## x10 1.701196 0.096391 17.65 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1773 on 108 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 1.028e+04 on 4 and 108 DF, p-value: < 2.2e-16
# Since there is a difference from the previous "FORWARD" procedure, now checking
# for the presence of other significant variables
mq3 <- update(mq3f, .~. +x1)
summary(mq3)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10 + x1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.51686 -0.07425 0.01495 0.05828 0.77368
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.343e+01 8.065e-01 -16.657 <2e-16 ***
## x6 4.824e-01 3.826e-02 12.609 <2e-16 ***
## x12 2.331e-01 2.872e-03 81.167 <2e-16 ***
## x11 1.006e+00 6.977e-02 14.418 <2e-16 ***
## x10 1.695e+00 9.628e-02 17.600 <2e-16 ***
## x1 -1.438e-04 1.144e-04 -1.257 0.212
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1768 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8269 on 5 and 107 DF, p-value: < 2.2e-16
mq3 <- update(mq3f, .~. +x2)
summary(mq3)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10 + x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.54571 -0.06362 -0.00261 0.07186 0.76528
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.348e+01 8.095e-01 -16.65 <2e-16 ***
## x6 4.915e-01 3.924e-02 12.52 <2e-16 ***
## x12 2.331e-01 2.946e-03 79.14 <2e-16 ***
## x11 1.014e+00 6.972e-02 14.54 <2e-16 ***
## x10 1.701e+00 9.659e-02 17.61 <2e-16 ***
## x2 4.563e-04 6.164e-04 0.74 0.461
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1777 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8190 on 5 and 107 DF, p-value: < 2.2e-16
mq3 <- update(mq3f, .~. +x3)
summary(mq3)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10 + x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.54720 -0.06082 0.00899 0.06976 0.76558
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.344e+01 9.396e-01 -14.308 <2e-16 ***
## x6 4.836e-01 4.032e-02 11.994 <2e-16 ***
## x12 2.328e-01 3.050e-03 76.320 <2e-16 ***
## x11 1.008e+00 8.490e-02 11.877 <2e-16 ***
## x10 1.694e+00 1.085e-01 15.619 <2e-16 ***
## x3 -1.992e-04 1.346e-03 -0.148 0.883
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1781 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8150 on 5 and 107 DF, p-value: < 2.2e-16
mq3 <- update(mq3f, .~. +x4)
summary(mq3)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10 + x4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.54659 -0.05924 0.00897 0.07173 0.76808
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.350e+01 9.299e-01 -14.516 <2e-16 ***
## x6 4.850e-01 4.019e-02 12.065 <2e-16 ***
## x12 2.327e-01 3.028e-03 76.824 <2e-16 ***
## x11 1.014e+00 8.419e-02 12.044 <2e-16 ***
## x10 1.700e+00 1.076e-01 15.794 <2e-16 ***
## x4 -4.242e-05 1.259e-03 -0.034 0.973
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1781 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8148 on 5 and 107 DF, p-value: < 2.2e-16
mq3 <- update(mq3f, .~. +x5)
summary(mq3)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10 + x5)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.52646 -0.07047 -0.00075 0.07733 0.75818
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.813256 0.826269 -16.718 <2e-16 ***
## x6 0.507277 0.040800 12.433 <2e-16 ***
## x12 0.233455 0.002892 80.713 <2e-16 ***
## x11 1.038044 0.070760 14.670 <2e-16 ***
## x10 1.736465 0.098710 17.592 <2e-16 ***
## x5 -0.051398 0.034402 -1.494 0.138
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1763 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8319 on 5 and 107 DF, p-value: < 2.2e-16
mq3 <- update(mq3f, .~. +x7)
summary(mq3)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10 + x7)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.54842 -0.07179 0.01009 0.06365 0.77274
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.122529 0.840731 -15.608 <2e-16 ***
## x6 0.472482 0.038958 12.128 <2e-16 ***
## x12 0.233649 0.002914 80.178 <2e-16 ***
## x11 0.951489 0.080720 11.787 <2e-16 ***
## x10 1.672139 0.097638 17.126 <2e-16 ***
## x7 -0.022923 0.014918 -1.537 0.127
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1762 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8328 on 5 and 107 DF, p-value: < 2.2e-16
mq3 <- update(mq3f, .~. +x8)
summary(mq3)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10 + x8)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.53398 -0.06481 0.01082 0.06448 0.75820
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.294e+01 9.803e-01 -13.196 < 2e-16 ***
## x6 4.422e-01 5.658e-02 7.816 4.02e-12 ***
## x12 2.324e-01 2.865e-03 81.090 < 2e-16 ***
## x11 9.515e-01 9.312e-02 10.217 < 2e-16 ***
## x10 1.625e+00 1.215e-01 13.374 < 2e-16 ***
## x8 -8.433e-04 8.150e-04 -1.035 0.303
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1773 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8230 on 5 and 107 DF, p-value: < 2.2e-16
mq3 <- update(mq3f, .~. +x13)
summary(mq3)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10 + x13)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.56670 -0.07297 0.00859 0.07856 0.73688
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.071787 0.883806 -14.790 <2e-16 ***
## x6 0.484622 0.038209 12.683 <2e-16 ***
## x12 0.233608 0.002962 78.856 <2e-16 ***
## x11 0.993726 0.071710 13.858 <2e-16 ***
## x10 1.679551 0.097838 17.167 <2e-16 ***
## x13 -0.013015 0.010767 -1.209 0.229
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1769 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8260 on 5 and 107 DF, p-value: < 2.2e-16
mq3 <- update(mq3f, .~. +x14)
summary(mq3)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10 + x14)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.55643 -0.05897 0.01077 0.07274 0.76724
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.624011 0.840632 -16.207 <2e-16 ***
## x6 0.494258 0.042641 11.591 <2e-16 ***
## x12 0.232756 0.002878 80.880 <2e-16 ***
## x11 1.029543 0.075560 13.626 <2e-16 ***
## x10 1.717181 0.102271 16.791 <2e-16 ***
## x14 -0.024651 0.051179 -0.482 0.631
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.178 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8166 on 5 and 107 DF, p-value: < 2.2e-16
mq3 <- update(mq3f, .~. +x15)
summary(mq3)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10 + x15)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.56130 -0.06589 0.00677 0.07269 0.76443
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.729715 0.832619 -16.490 <2e-16 ***
## x6 0.507050 0.043673 11.610 <2e-16 ***
## x12 0.233060 0.002885 80.789 <2e-16 ***
## x11 1.046865 0.075841 13.803 <2e-16 ***
## x10 1.738969 0.103085 16.869 <2e-16 ***
## x15 -0.047578 0.046122 -1.032 0.305
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1773 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8229 on 5 and 107 DF, p-value: < 2.2e-16
mq3 <- update(mq3f, .~. +x16)
summary(mq3)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10 + x16)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.56736 -0.07224 0.00831 0.07822 0.73758
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.078257 0.880799 -14.848 <2e-16 ***
## x6 0.497649 0.039520 12.592 <2e-16 ***
## x12 0.233623 0.002965 78.806 <2e-16 ***
## x11 0.993438 0.071742 13.847 <2e-16 ***
## x10 1.679357 0.097844 17.164 <2e-16 ***
## x16 -0.012978 0.010681 -1.215 0.227
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1769 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8261 on 5 and 107 DF, p-value: < 2.2e-16
mq3 <- update(mq3f, .~. +x17)
summary(mq3)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10 + x17)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.54959 -0.05685 0.00601 0.07171 0.77101
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.350e+01 8.153e-01 -16.559 <2e-16 ***
## x6 4.854e-01 3.846e-02 12.620 <2e-16 ***
## x12 2.327e-01 2.906e-03 80.079 <2e-16 ***
## x11 1.014e+00 7.093e-02 14.294 <2e-16 ***
## x10 1.700e+00 9.743e-02 17.446 <2e-16 ***
## x17 8.781e-04 6.420e-03 0.137 0.891
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1781 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8150 on 5 and 107 DF, p-value: < 2.2e-16
mq3 <- update(mq3f, .~. +x18)
summary(mq3)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.56262 -0.06196 0.00829 0.07138 0.76559
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.380743 0.818676 -16.344 <2e-16 ***
## x6 0.503931 0.043000 11.719 <2e-16 ***
## x12 0.232913 0.002872 81.100 <2e-16 ***
## x11 1.010910 0.069744 14.495 <2e-16 ***
## x10 1.720606 0.098571 17.455 <2e-16 ***
## x18 -0.015561 0.016367 -0.951 0.344
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1774 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8217 on 5 and 107 DF, p-value: < 2.2e-16
# NO FURTHER VARIABLES TO ADD/REMOVE.
# IDENTIFIED MODEL:
# lm(formula = y ~ x6 + x12 + x11 + x10)
# Summary information:
summary(mq3f)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.54633 -0.05866 0.00924 0.07217 0.76854
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.514137 0.806207 -16.76 <2e-16 ***
## x6 0.485352 0.038286 12.68 <2e-16 ***
## x12 0.232624 0.002854 81.50 <2e-16 ***
## x11 1.015592 0.069539 14.61 <2e-16 ***
## x10 1.701196 0.096391 17.65 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1773 on 108 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 1.028e+04 on 4 and 108 DF, p-value: < 2.2e-16
# 4th Approach: LIKELIHOOD FUNCTIONS COMPARISON (FORWARD WITH K=0)
# Starting from the null model
forw_lik <- step(mq0, scope=formula(mq), direction="forward", k=0)
## Start: AIC=275.71
## y ~ 1
##
## Df Sum of Sq RSS AIC
## + x12 1 1280.71 15.68 -223.16
## + x6 1 864.86 431.53 151.41
## + x18 1 639.73 656.66 198.86
## + x11 1 627.98 668.41 200.86
## + x16 1 324.59 971.80 243.15
## + x9 1 268.98 1027.41 249.44
## + x7 1 257.31 1039.08 250.71
## + x4 1 118.04 1178.34 264.93
## + x3 1 99.74 1196.65 266.67
## + x10 1 79.79 1216.60 268.54
## + x14 1 58.82 1237.57 270.47
## + x15 1 57.69 1238.69 270.57
## + x17 1 30.99 1265.40 272.98
## + x1 1 16.08 1280.31 274.30
## + x5 1 10.87 1285.52 274.76
## + x8 1 7.31 1289.08 275.07
## + x13 1 1.32 1295.07 275.60
## + x2 1 0.74 1295.65 275.65
## <none> 1296.39 275.71
##
## Step: AIC=-223.16
## y ~ x12
##
## Df Sum of Sq RSS AIC
## + x9 1 10.6896 4.9926 -352.50
## + x10 1 5.1070 10.5752 -267.68
## + x11 1 2.4221 13.2602 -242.12
## + x18 1 1.2584 14.4238 -232.61
## + x7 1 0.6956 14.9866 -228.29
## + x6 1 0.6417 15.0405 -227.88
## + x16 1 0.5279 15.1543 -227.03
## + x8 1 0.3560 15.3262 -225.75
## + x13 1 0.2233 15.4590 -224.78
## + x3 1 0.1568 15.5255 -224.29
## + x4 1 0.1068 15.5754 -223.93
## + x5 1 0.1033 15.5789 -223.91
## + x2 1 0.0468 15.6354 -223.50
## + x15 1 0.0417 15.6405 -223.46
## + x14 1 0.0289 15.6533 -223.37
## + x17 1 0.0112 15.6711 -223.24
## + x1 1 0.0019 15.6803 -223.17
## <none> 15.6822 -223.16
##
## Step: AIC=-352.5
## y ~ x12 + x9
##
## Df Sum of Sq RSS AIC
## + x18 1 0.67626 4.3163 -368.94
## + x11 1 0.63758 4.3550 -367.93
## + x7 1 0.61248 4.3801 -367.29
## + x16 1 0.53335 4.4593 -365.26
## + x3 1 0.46914 4.5235 -363.65
## + x4 1 0.45227 4.5403 -363.23
## + x13 1 0.40222 4.5904 -361.99
## + x17 1 0.19959 4.7930 -357.11
## + x6 1 0.18284 4.8098 -356.71
## + x2 1 0.15157 4.8410 -355.98
## + x8 1 0.14968 4.8429 -355.94
## + x1 1 0.08969 4.9029 -354.54
## + x5 1 0.07865 4.9140 -354.29
## + x10 1 0.06663 4.9260 -354.01
## + x14 1 0.00542 4.9872 -352.62
## + x15 1 0.00127 4.9913 -352.52
## <none> 4.9926 -352.50
##
## Step: AIC=-368.94
## y ~ x12 + x9 + x18
##
## Df Sum of Sq RSS AIC
## + x14 1 0.57258 3.7438 -385.02
## + x11 1 0.32484 3.9915 -377.78
## + x15 1 0.32043 3.9959 -377.66
## + x7 1 0.26037 4.0560 -375.97
## + x4 1 0.18348 4.1329 -373.85
## + x3 1 0.17923 4.1371 -373.73
## + x2 1 0.17023 4.1461 -373.49
## + x17 1 0.10459 4.2118 -371.71
## + x8 1 0.07737 4.2390 -370.99
## + x10 1 0.04404 4.2723 -370.10
## + x6 1 0.04342 4.2729 -370.08
## + x1 1 0.03796 4.2784 -369.94
## + x16 1 0.02984 4.2865 -369.73
## + x5 1 0.02532 4.2910 -369.61
## + x13 1 0.00257 4.3138 -369.01
## <none> 4.3163 -368.94
##
## Step: AIC=-385.02
## y ~ x12 + x9 + x18 + x14
##
## Df Sum of Sq RSS AIC
## + x11 1 0.46634 3.2774 -400.06
## + x10 1 0.37854 3.3652 -397.07
## + x6 1 0.21610 3.5277 -391.74
## + x2 1 0.15441 3.5894 -389.78
## + x15 1 0.13998 3.6038 -389.33
## + x5 1 0.12387 3.6199 -388.83
## + x16 1 0.10409 3.6397 -388.21
## + x8 1 0.07438 3.6694 -387.29
## + x17 1 0.04280 3.7010 -386.32
## + x13 1 0.00461 3.7392 -385.16
## + x1 1 0.00440 3.7394 -385.16
## + x7 1 0.00267 3.7411 -385.10
## + x3 1 0.00076 3.7430 -385.05
## + x4 1 0.00039 3.7434 -385.04
## <none> 3.7438 -385.02
##
## Step: AIC=-400.06
## y ~ x12 + x9 + x18 + x14 + x11
##
## Df Sum of Sq RSS AIC
## + x6 1 0.108660 3.1688 -403.87
## + x7 1 0.053948 3.2235 -401.93
## + x5 1 0.035170 3.2423 -401.28
## + x2 1 0.034323 3.2431 -401.25
## + x13 1 0.031203 3.2462 -401.14
## + x15 1 0.029466 3.2480 -401.08
## + x8 1 0.023633 3.2538 -400.87
## + x1 1 0.016823 3.2606 -400.64
## + x17 1 0.012552 3.2649 -400.49
## + x16 1 0.008206 3.2692 -400.34
## + x10 1 0.003379 3.2740 -400.17
## + x4 1 0.002516 3.2749 -400.14
## + x3 1 0.002240 3.2752 -400.13
## <none> 3.2774 -400.06
##
## Step: AIC=-403.87
## y ~ x12 + x9 + x18 + x14 + x11 + x6
##
## Df Sum of Sq RSS AIC
## + x10 1 0.32012 2.8486 -415.90
## + x2 1 0.05939 3.1094 -406.01
## + x5 1 0.03656 3.1322 -405.18
## + x7 1 0.03283 3.1359 -405.04
## + x16 1 0.02809 3.1407 -404.87
## + x13 1 0.02716 3.1416 -404.84
## + x15 1 0.01545 3.1533 -404.42
## + x1 1 0.01518 3.1536 -404.41
## + x17 1 0.00866 3.1601 -404.18
## + x4 1 0.00787 3.1609 -404.15
## + x3 1 0.00758 3.1612 -404.14
## + x8 1 0.00037 3.1684 -403.88
## <none> 3.1688 -403.87
##
## Step: AIC=-415.9
## y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10
##
## Df Sum of Sq RSS AIC
## + x5 1 0.039604 2.8090 -417.48
## + x2 1 0.034882 2.8138 -417.29
## + x1 1 0.032374 2.8163 -417.19
## + x7 1 0.029136 2.8195 -417.06
## + x13 1 0.020397 2.8283 -416.71
## + x16 1 0.020351 2.8283 -416.71
## + x15 1 0.016779 2.8319 -416.57
## + x3 1 0.001520 2.8471 -415.96
## + x4 1 0.000816 2.8478 -415.93
## + x17 1 0.000454 2.8482 -415.92
## + x8 1 0.000208 2.8484 -415.91
## <none> 2.8487 -415.90
##
## Step: AIC=-417.48
## y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10 + x5
##
## Df Sum of Sq RSS AIC
## + x15 1 0.048922 2.7601 -419.47
## + x7 1 0.041810 2.7672 -419.18
## + x1 1 0.036431 2.7726 -418.96
## + x2 1 0.027843 2.7812 -418.61
## + x13 1 0.017891 2.7912 -418.21
## + x16 1 0.017752 2.7913 -418.20
## + x4 1 0.002365 2.8067 -417.58
## + x3 1 0.002073 2.8070 -417.57
## + x8 1 0.001347 2.8077 -417.54
## + x17 1 0.000605 2.8084 -417.51
## <none> 2.8090 -417.48
##
## Step: AIC=-419.47
## y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10 + x5 + x15
##
## Df Sum of Sq RSS AIC
## + x7 1 0.037507 2.7226 -421.01
## + x2 1 0.032349 2.7278 -420.80
## + x1 1 0.030108 2.7300 -420.71
## + x13 1 0.011770 2.7483 -419.95
## + x16 1 0.011469 2.7487 -419.94
## + x4 1 0.001919 2.7582 -419.55
## + x3 1 0.001255 2.7589 -419.52
## + x8 1 0.000277 2.7598 -419.48
## + x17 1 0.000174 2.7599 -419.48
## <none> 2.7601 -419.47
##
## Step: AIC=-421.01
## y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10 + x5 + x15 + x7
##
## Df Sum of Sq RSS AIC
## + x2 1 0.039221 2.6834 -422.65
## + x1 1 0.029649 2.6930 -422.25
## + x13 1 0.004810 2.7178 -421.21
## + x16 1 0.004510 2.7181 -421.20
## + x3 1 0.003694 2.7189 -421.17
## + x8 1 0.003529 2.7191 -421.16
## + x4 1 0.002923 2.7197 -421.14
## + x17 1 0.000967 2.7216 -421.05
## <none> 2.7226 -421.01
##
## Step: AIC=-422.65
## y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10 + x5 + x15 + x7 + x2
##
## Df Sum of Sq RSS AIC
## + x1 1 0.032456 2.6509 -424.03
## + x3 1 0.007529 2.6759 -422.97
## + x4 1 0.006694 2.6767 -422.94
## + x17 1 0.004438 2.6790 -422.84
## + x13 1 0.002708 2.6807 -422.77
## + x16 1 0.002430 2.6810 -422.76
## + x8 1 0.001066 2.6823 -422.70
## <none> 2.6834 -422.65
##
## Step: AIC=-424.03
## y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10 + x5 + x15 + x7 + x2 +
## x1
##
## Df Sum of Sq RSS AIC
## + x3 1 0.0063232 2.6446 -424.30
## + x4 1 0.0059010 2.6450 -424.28
## + x13 1 0.0037863 2.6471 -424.19
## + x16 1 0.0034497 2.6475 -424.18
## + x17 1 0.0030676 2.6479 -424.16
## + x8 1 0.0011118 2.6498 -424.08
## <none> 2.6509 -424.03
##
## Step: AIC=-424.3
## y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10 + x5 + x15 + x7 + x2 +
## x1 + x3
##
## Df Sum of Sq RSS AIC
## + x13 1 0.0301187 2.6145 -425.59
## + x16 1 0.0284085 2.6162 -425.52
## + x17 1 0.0032204 2.6414 -424.44
## + x4 1 0.0000409 2.6446 -424.30
## + x8 1 0.0000401 2.6446 -424.30
## <none> 2.6446 -424.30
##
## Step: AIC=-425.59
## y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10 + x5 + x15 + x7 + x2 +
## x1 + x3 + x13
##
## Df Sum of Sq RSS AIC
## + x4 1 0.0078665 2.6066 -425.93
## + x17 1 0.0075750 2.6069 -425.92
## + x16 1 0.0058415 2.6086 -425.85
## + x8 1 0.0008129 2.6137 -425.63
## <none> 2.6145 -425.59
##
## Step: AIC=-425.93
## y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10 + x5 + x15 + x7 + x2 +
## x1 + x3 + x13 + x4
##
## Df Sum of Sq RSS AIC
## + x17 1 0.0102530 2.5964 -426.38
## + x16 1 0.0080577 2.5986 -426.28
## + x8 1 0.0000583 2.6066 -425.94
## <none> 2.6066 -425.93
##
## Step: AIC=-426.38
## y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10 + x5 + x15 + x7 + x2 +
## x1 + x3 + x13 + x4 + x17
##
## Df Sum of Sq RSS AIC
## + x16 1 0.00136872 2.5950 -426.44
## + x8 1 0.00039342 2.5960 -426.40
## <none> 2.5964 -426.38
##
## Step: AIC=-426.44
## y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10 + x5 + x15 + x7 + x2 +
## x1 + x3 + x13 + x4 + x17 + x16
##
## Df Sum of Sq RSS AIC
## + x8 1 0.00056879 2.5944 -426.46
## <none> 2.5950 -426.44
##
## Step: AIC=-426.46
## y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10 + x5 + x15 + x7 + x2 +
## x1 + x3 + x13 + x4 + x17 + x16 + x8
# ALWAYS CHOOSES THE COMPLETE MODEL:
# y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10 + x5 + x15 + x7 + x2 +
# x1 + x3 + x13 + x4 + x17 + x16 + x8
# Summary information
summary(forw_lik)
##
## Call:
## lm(formula = y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10 + x5 +
## x15 + x7 + x2 + x1 + x3 + x13 + x4 + x17 + x16 + x8)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.35240 -0.08275 -0.01367 0.06688 0.70644
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.4932971 4.0122361 0.123 0.90241
## x12 0.2364541 0.0032508 72.736 < 2e-16 ***
## x9 -0.0349403 0.2545558 -0.137 0.89112
## x18 -0.5989269 0.1782506 -3.360 0.00113 **
## x14 1.4366685 0.4871607 2.949 0.00402 **
## x11 -0.3031051 0.3517006 -0.862 0.39098
## x6 0.4107678 0.4499448 0.913 0.36362
## x10 1.1866765 0.3453261 3.436 0.00088 ***
## x5 -0.1743371 0.1230087 -1.417 0.15971
## x15 0.5159395 0.4768871 1.082 0.28207
## x7 -0.0277227 0.0175895 -1.576 0.11836
## x2 0.0008409 0.0006320 1.331 0.18654
## x1 -0.0001159 0.0001097 -1.056 0.29370
## x3 0.0058477 0.0058386 1.002 0.31913
## x13 -0.1351995 0.4372948 -0.309 0.75787
## x4 -0.0027320 0.0046645 -0.586 0.55949
## x17 -0.0029176 0.0080131 -0.364 0.71659
## x16 0.1026885 0.4341540 0.237 0.81354
## x8 -0.0001507 0.0010495 -0.144 0.88616
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1661 on 94 degrees of freedom
## Multiple R-squared: 0.998, Adjusted R-squared: 0.9976
## F-statistic: 2604 on 18 and 94 DF, p-value: < 2.2e-16
# 5th Approach: AIC PENALIZATION CRITERION (FORWARD WITH K=2)
# [Akaike Information Criterion:]
# The penalization term used is penAIC(d) = 2|d|, where given a model Fd(θ),
# the information index is AICd = 2|d| − ℓ(ˆθd; X).[estimator of theta in d]
# The best model within the explored class of models is the one with
# the minimum AICd value.
forw_aic <- step(mq0, scope=formula(mq), direction="forward", k=2)
## Start: AIC=277.71
## y ~ 1
##
## Df Sum of Sq RSS AIC
## + x12 1 1280.71 15.68 -219.16
## + x6 1 864.86 431.53 155.41
## + x18 1 639.73 656.66 202.86
## + x11 1 627.98 668.41 204.86
## + x16 1 324.59 971.80 247.15
## + x9 1 268.98 1027.41 253.44
## + x7 1 257.31 1039.08 254.71
## + x4 1 118.04 1178.34 268.93
## + x3 1 99.74 1196.65 270.67
## + x10 1 79.79 1216.60 272.54
## + x14 1 58.82 1237.57 274.47
## + x15 1 57.69 1238.69 274.57
## + x17 1 30.99 1265.40 276.98
## <none> 1296.39 277.71
## + x1 1 16.08 1280.31 278.30
## + x5 1 10.87 1285.52 278.76
## + x8 1 7.31 1289.08 279.07
## + x13 1 1.32 1295.07 279.60
## + x2 1 0.74 1295.65 279.65
##
## Step: AIC=-219.16
## y ~ x12
##
## Df Sum of Sq RSS AIC
## + x9 1 10.6896 4.9926 -346.50
## + x10 1 5.1070 10.5752 -261.68
## + x11 1 2.4221 13.2602 -236.12
## + x18 1 1.2584 14.4238 -226.61
## + x7 1 0.6956 14.9866 -222.29
## + x6 1 0.6417 15.0405 -221.88
## + x16 1 0.5279 15.1543 -221.03
## + x8 1 0.3560 15.3262 -219.75
## <none> 15.6822 -219.16
## + x13 1 0.2233 15.4590 -218.78
## + x3 1 0.1568 15.5255 -218.29
## + x4 1 0.1068 15.5754 -217.93
## + x5 1 0.1033 15.5789 -217.91
## + x2 1 0.0468 15.6354 -217.50
## + x15 1 0.0417 15.6405 -217.46
## + x14 1 0.0289 15.6533 -217.37
## + x17 1 0.0112 15.6711 -217.24
## + x1 1 0.0019 15.6803 -217.17
##
## Step: AIC=-346.5
## y ~ x12 + x9
##
## Df Sum of Sq RSS AIC
## + x18 1 0.67626 4.3163 -360.94
## + x11 1 0.63758 4.3550 -359.93
## + x7 1 0.61248 4.3801 -359.29
## + x16 1 0.53335 4.4593 -357.26
## + x3 1 0.46914 4.5235 -355.65
## + x4 1 0.45227 4.5403 -355.23
## + x13 1 0.40222 4.5904 -353.99
## + x17 1 0.19959 4.7930 -349.11
## + x6 1 0.18284 4.8098 -348.71
## + x2 1 0.15157 4.8410 -347.98
## + x8 1 0.14968 4.8429 -347.94
## + x1 1 0.08969 4.9029 -346.54
## <none> 4.9926 -346.50
## + x5 1 0.07865 4.9140 -346.29
## + x10 1 0.06663 4.9260 -346.01
## + x14 1 0.00542 4.9872 -344.62
## + x15 1 0.00127 4.9913 -344.52
##
## Step: AIC=-360.94
## y ~ x12 + x9 + x18
##
## Df Sum of Sq RSS AIC
## + x14 1 0.57258 3.7438 -375.02
## + x11 1 0.32484 3.9915 -367.78
## + x15 1 0.32043 3.9959 -367.66
## + x7 1 0.26037 4.0560 -365.97
## + x4 1 0.18348 4.1329 -363.85
## + x3 1 0.17923 4.1371 -363.73
## + x2 1 0.17023 4.1461 -363.49
## + x17 1 0.10459 4.2118 -361.71
## + x8 1 0.07737 4.2390 -360.99
## <none> 4.3163 -360.94
## + x10 1 0.04404 4.2723 -360.10
## + x6 1 0.04342 4.2729 -360.08
## + x1 1 0.03796 4.2784 -359.94
## + x16 1 0.02984 4.2865 -359.73
## + x5 1 0.02532 4.2910 -359.61
## + x13 1 0.00257 4.3138 -359.01
##
## Step: AIC=-375.02
## y ~ x12 + x9 + x18 + x14
##
## Df Sum of Sq RSS AIC
## + x11 1 0.46634 3.2774 -388.06
## + x10 1 0.37854 3.3652 -385.07
## + x6 1 0.21610 3.5277 -379.74
## + x2 1 0.15441 3.5894 -377.78
## + x15 1 0.13998 3.6038 -377.33
## + x5 1 0.12387 3.6199 -376.83
## + x16 1 0.10409 3.6397 -376.21
## + x8 1 0.07438 3.6694 -375.29
## <none> 3.7438 -375.02
## + x17 1 0.04280 3.7010 -374.32
## + x13 1 0.00461 3.7392 -373.16
## + x1 1 0.00440 3.7394 -373.16
## + x7 1 0.00267 3.7411 -373.10
## + x3 1 0.00076 3.7430 -373.05
## + x4 1 0.00039 3.7434 -373.04
##
## Step: AIC=-388.06
## y ~ x12 + x9 + x18 + x14 + x11
##
## Df Sum of Sq RSS AIC
## + x6 1 0.108660 3.1688 -389.87
## <none> 3.2774 -388.06
## + x7 1 0.053948 3.2235 -387.93
## + x5 1 0.035170 3.2423 -387.28
## + x2 1 0.034323 3.2431 -387.25
## + x13 1 0.031203 3.2462 -387.14
## + x15 1 0.029466 3.2480 -387.08
## + x8 1 0.023633 3.2538 -386.87
## + x1 1 0.016823 3.2606 -386.64
## + x17 1 0.012552 3.2649 -386.49
## + x16 1 0.008206 3.2692 -386.34
## + x10 1 0.003379 3.2740 -386.17
## + x4 1 0.002516 3.2749 -386.14
## + x3 1 0.002240 3.2752 -386.13
##
## Step: AIC=-389.87
## y ~ x12 + x9 + x18 + x14 + x11 + x6
##
## Df Sum of Sq RSS AIC
## + x10 1 0.32012 2.8486 -399.90
## + x2 1 0.05939 3.1094 -390.01
## <none> 3.1688 -389.87
## + x5 1 0.03656 3.1322 -389.18
## + x7 1 0.03283 3.1359 -389.04
## + x16 1 0.02809 3.1407 -388.87
## + x13 1 0.02716 3.1416 -388.84
## + x15 1 0.01545 3.1533 -388.42
## + x1 1 0.01518 3.1536 -388.41
## + x17 1 0.00866 3.1601 -388.18
## + x4 1 0.00787 3.1609 -388.15
## + x3 1 0.00758 3.1612 -388.14
## + x8 1 0.00037 3.1684 -387.88
##
## Step: AIC=-399.9
## y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10
##
## Df Sum of Sq RSS AIC
## <none> 2.8487 -399.90
## + x5 1 0.039604 2.8090 -399.48
## + x2 1 0.034882 2.8138 -399.29
## + x1 1 0.032374 2.8163 -399.19
## + x7 1 0.029136 2.8195 -399.06
## + x13 1 0.020397 2.8283 -398.71
## + x16 1 0.020351 2.8283 -398.71
## + x15 1 0.016779 2.8319 -398.57
## + x3 1 0.001520 2.8471 -397.96
## + x4 1 0.000816 2.8478 -397.93
## + x17 1 0.000454 2.8482 -397.92
## + x8 1 0.000208 2.8484 -397.91
# CHOOSES THE MODEL:
# y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10
# Summary information
summary(forw_aic)
##
## Call:
## lm(formula = y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.41987 -0.06391 0.01433 0.07466 0.75100
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.453316 3.649208 0.124 0.901376
## x12 0.234117 0.002708 86.455 < 2e-16 ***
## x9 -0.103748 0.235824 -0.440 0.660885
## x18 -0.575286 0.137484 -4.184 5.95e-05 ***
## x14 1.752429 0.431097 4.065 9.30e-05 ***
## x11 -0.221445 0.295084 -0.750 0.454664
## x6 0.487450 0.123099 3.960 0.000137 ***
## x10 1.135744 0.330636 3.435 0.000850 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1647 on 105 degrees of freedom
## Multiple R-squared: 0.9978, Adjusted R-squared: 0.9977
## F-statistic: 6811 on 7 and 105 DF, p-value: < 2.2e-16
# 6th Approach: BIC PENALIZATION CRITERION (FORWARD WITH k=log(length(y)))
# [Bayesian Information Criterion]
# Penalization term: penBIC(d) = |d| log n.
# The criterion suggests choosing the model Fd(θ; X) with
# minimum BICd = |d| log n − ℓ(ˆθd; X).
forw_bic <- step(mq0, scope=formula(mq), direction="forward", k=log(length(y)))
## Start: AIC=280.44
## y ~ 1
##
## Df Sum of Sq RSS AIC
## + x12 1 1280.71 15.68 -213.70
## + x6 1 864.86 431.53 160.87
## + x18 1 639.73 656.66 208.31
## + x11 1 627.98 668.41 210.31
## + x16 1 324.59 971.80 252.60
## + x9 1 268.98 1027.41 258.89
## + x7 1 257.31 1039.08 260.17
## + x4 1 118.04 1178.34 274.38
## + x3 1 99.74 1196.65 276.12
## + x10 1 79.79 1216.60 277.99
## + x14 1 58.82 1237.57 279.92
## + x15 1 57.69 1238.69 280.02
## <none> 1296.39 280.44
## + x17 1 30.99 1265.40 282.44
## + x1 1 16.08 1280.31 283.76
## + x5 1 10.87 1285.52 284.22
## + x8 1 7.31 1289.08 284.53
## + x13 1 1.32 1295.07 285.05
## + x2 1 0.74 1295.65 285.11
##
## Step: AIC=-213.7
## y ~ x12
##
## Df Sum of Sq RSS AIC
## + x9 1 10.6896 4.9926 -338.31
## + x10 1 5.1070 10.5752 -253.50
## + x11 1 2.4221 13.2602 -227.93
## + x18 1 1.2584 14.4238 -218.43
## + x7 1 0.6956 14.9866 -214.10
## <none> 15.6822 -213.70
## + x6 1 0.6417 15.0405 -213.70
## + x16 1 0.5279 15.1543 -212.85
## + x8 1 0.3560 15.3262 -211.57
## + x13 1 0.2233 15.4590 -210.60
## + x3 1 0.1568 15.5255 -210.11
## + x4 1 0.1068 15.5754 -209.75
## + x5 1 0.1033 15.5789 -209.72
## + x2 1 0.0468 15.6354 -209.31
## + x15 1 0.0417 15.6405 -209.28
## + x14 1 0.0289 15.6533 -209.19
## + x17 1 0.0112 15.6711 -209.06
## + x1 1 0.0019 15.6803 -208.99
##
## Step: AIC=-338.31
## y ~ x12 + x9
##
## Df Sum of Sq RSS AIC
## + x18 1 0.67626 4.3163 -350.03
## + x11 1 0.63758 4.3550 -349.02
## + x7 1 0.61248 4.3801 -348.38
## + x16 1 0.53335 4.4593 -346.35
## + x3 1 0.46914 4.5235 -344.74
## + x4 1 0.45227 4.5403 -344.32
## + x13 1 0.40222 4.5904 -343.08
## <none> 4.9926 -338.31
## + x17 1 0.19959 4.7930 -338.20
## + x6 1 0.18284 4.8098 -337.80
## + x2 1 0.15157 4.8410 -337.07
## + x8 1 0.14968 4.8429 -337.03
## + x1 1 0.08969 4.9029 -335.63
## + x5 1 0.07865 4.9140 -335.38
## + x10 1 0.06663 4.9260 -335.10
## + x14 1 0.00542 4.9872 -333.71
## + x15 1 0.00127 4.9913 -333.61
##
## Step: AIC=-350.03
## y ~ x12 + x9 + x18
##
## Df Sum of Sq RSS AIC
## + x14 1 0.57258 3.7438 -361.39
## + x11 1 0.32484 3.9915 -354.15
## + x15 1 0.32043 3.9959 -354.02
## + x7 1 0.26037 4.0560 -352.34
## + x4 1 0.18348 4.1329 -350.21
## + x3 1 0.17923 4.1371 -350.10
## <none> 4.3163 -350.03
## + x2 1 0.17023 4.1461 -349.85
## + x17 1 0.10459 4.2118 -348.08
## + x8 1 0.07737 4.2390 -347.35
## + x10 1 0.04404 4.2723 -346.46
## + x6 1 0.04342 4.2729 -346.45
## + x1 1 0.03796 4.2784 -346.30
## + x16 1 0.02984 4.2865 -346.09
## + x5 1 0.02532 4.2910 -345.97
## + x13 1 0.00257 4.3138 -345.37
##
## Step: AIC=-361.39
## y ~ x12 + x9 + x18 + x14
##
## Df Sum of Sq RSS AIC
## + x11 1 0.46634 3.2774 -371.69
## + x10 1 0.37854 3.3652 -368.71
## + x6 1 0.21610 3.5277 -363.38
## + x2 1 0.15441 3.5894 -361.42
## <none> 3.7438 -361.39
## + x15 1 0.13998 3.6038 -360.97
## + x5 1 0.12387 3.6199 -360.46
## + x16 1 0.10409 3.6397 -359.85
## + x8 1 0.07438 3.6694 -358.93
## + x17 1 0.04280 3.7010 -357.96
## + x13 1 0.00461 3.7392 -356.80
## + x1 1 0.00440 3.7394 -356.79
## + x7 1 0.00267 3.7411 -356.74
## + x3 1 0.00076 3.7430 -356.68
## + x4 1 0.00039 3.7434 -356.67
##
## Step: AIC=-371.69
## y ~ x12 + x9 + x18 + x14 + x11
##
## Df Sum of Sq RSS AIC
## <none> 3.2774 -371.69
## + x6 1 0.108660 3.1688 -370.78
## + x7 1 0.053948 3.2235 -368.84
## + x5 1 0.035170 3.2423 -368.18
## + x2 1 0.034323 3.2431 -368.16
## + x13 1 0.031203 3.2462 -368.05
## + x15 1 0.029466 3.2480 -367.99
## + x8 1 0.023633 3.2538 -367.78
## + x1 1 0.016823 3.2606 -367.55
## + x17 1 0.012552 3.2649 -367.40
## + x16 1 0.008206 3.2692 -367.25
## + x10 1 0.003379 3.2740 -367.08
## + x4 1 0.002516 3.2749 -367.05
## + x3 1 0.002240 3.2752 -367.04
# CHOOSES THE MODEL:
# y ~ x12 + x9 + x18 + x14 + x11
# Summary information
summary(forw_bic)
##
## Call:
## lm(formula = y ~ x12 + x9 + x18 + x14 + x11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.64191 -0.06187 0.02281 0.06056 0.74776
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.410137 0.925946 6.923 3.41e-10 ***
## x12 0.234033 0.002577 90.830 < 2e-16 ***
## x9 -0.974535 0.076614 -12.720 < 2e-16 ***
## x18 -0.362493 0.066296 -5.468 3.01e-07 ***
## x14 1.082834 0.224265 4.828 4.61e-06 ***
## x11 -0.428942 0.109931 -3.902 0.000167 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.175 on 107 degrees of freedom
## Multiple R-squared: 0.9975, Adjusted R-squared: 0.9974
## F-statistic: 8443 on 5 and 107 DF, p-value: < 2.2e-16
# 7th Approach: LIKELIHOOD FUNCTIONS COMPARISON (BOTH WITH K=0)
# STARTING FROM THE COMPLETE MODEL
both_lik <- step(mq, scope=formula(mq), direction="both", k=0)
## Start: AIC=-426.46
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 +
## x12 + x13 + x14 + x15 + x16 + x17 + x18
##
## Df Sum of Sq RSS AIC
## <none> 2.594 -426.46
## - x9 1 0.001 2.595 -426.44
## - x8 1 0.001 2.595 -426.44
## - x16 1 0.002 2.596 -426.40
## - x13 1 0.003 2.597 -426.35
## - x17 1 0.004 2.598 -426.30
## - x4 1 0.009 2.604 -426.05
## - x11 1 0.021 2.615 -425.57
## - x6 1 0.023 2.617 -425.47
## - x3 1 0.028 2.622 -425.26
## - x1 1 0.031 2.625 -425.13
## - x15 1 0.032 2.627 -425.07
## - x2 1 0.049 2.643 -424.36
## - x5 1 0.055 2.650 -424.07
## - x7 1 0.069 2.663 -423.52
## - x14 1 0.240 2.834 -416.46
## - x18 1 0.312 2.906 -413.65
## - x10 1 0.326 2.920 -413.09
## - x12 1 146.022 148.616 30.96
# ALWAYS CHOOSES THE COMPLETE MODEL:
# y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 +
# x12 + x13 + x14 + x15 + x16 + x17 + x18
# Summary information
summary(both_lik)
##
## Call:
## lm(formula = y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 +
## x10 + x11 + x12 + x13 + x14 + x15 + x16 + x17 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.35240 -0.08275 -0.01367 0.06688 0.70644
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.4932971 4.0122361 0.123 0.90241
## x1 -0.0001159 0.0001097 -1.056 0.29370
## x2 0.0008409 0.0006320 1.331 0.18654
## x3 0.0058477 0.0058386 1.002 0.31913
## x4 -0.0027320 0.0046645 -0.586 0.55949
## x5 -0.1743371 0.1230087 -1.417 0.15971
## x6 0.4107678 0.4499448 0.913 0.36362
## x7 -0.0277227 0.0175895 -1.576 0.11836
## x8 -0.0001507 0.0010495 -0.144 0.88616
## x9 -0.0349403 0.2545558 -0.137 0.89112
## x10 1.1866765 0.3453261 3.436 0.00088 ***
## x11 -0.3031051 0.3517006 -0.862 0.39098
## x12 0.2364541 0.0032508 72.736 < 2e-16 ***
## x13 -0.1351995 0.4372948 -0.309 0.75787
## x14 1.4366685 0.4871607 2.949 0.00402 **
## x15 0.5159395 0.4768871 1.082 0.28207
## x16 0.1026885 0.4341540 0.237 0.81354
## x17 -0.0029176 0.0080131 -0.364 0.71659
## x18 -0.5989269 0.1782506 -3.360 0.00113 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1661 on 94 degrees of freedom
## Multiple R-squared: 0.998, Adjusted R-squared: 0.9976
## F-statistic: 2604 on 18 and 94 DF, p-value: < 2.2e-16
# 8th Approach: AIC (BOTH WITH K=2)
both_aic <- step(mq, scope=formula(mq), direction="both", k=2)
## Start: AIC=-388.46
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 +
## x12 + x13 + x14 + x15 + x16 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x9 1 0.001 2.595 -390.44
## - x8 1 0.001 2.595 -390.44
## - x16 1 0.002 2.596 -390.40
## - x13 1 0.003 2.597 -390.35
## - x17 1 0.004 2.598 -390.30
## - x4 1 0.009 2.604 -390.05
## - x11 1 0.021 2.615 -389.57
## - x6 1 0.023 2.617 -389.47
## - x3 1 0.028 2.622 -389.26
## - x1 1 0.031 2.625 -389.13
## - x15 1 0.032 2.627 -389.07
## <none> 2.594 -388.46
## - x2 1 0.049 2.643 -388.36
## - x5 1 0.055 2.650 -388.07
## - x7 1 0.069 2.663 -387.52
## - x14 1 0.240 2.834 -380.46
## - x18 1 0.312 2.906 -377.65
## - x10 1 0.326 2.920 -377.09
## - x12 1 146.022 148.616 66.96
##
## Step: AIC=-390.44
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x10 + x11 + x12 +
## x13 + x14 + x15 + x16 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x8 1 0.001 2.596 -392.41
## - x16 1 0.002 2.597 -392.37
## - x13 1 0.003 2.598 -392.32
## - x17 1 0.004 2.599 -392.27
## - x4 1 0.009 2.604 -392.04
## - x11 1 0.020 2.615 -391.57
## - x6 1 0.025 2.620 -391.35
## - x3 1 0.027 2.622 -391.25
## - x1 1 0.032 2.627 -391.07
## - x15 1 0.034 2.629 -390.95
## <none> 2.595 -390.44
## - x2 1 0.049 2.644 -390.34
## - x5 1 0.058 2.653 -389.95
## - x7 1 0.068 2.663 -389.50
## + x9 1 0.001 2.594 -388.46
## - x14 1 0.252 2.847 -381.98
## - x18 1 0.341 2.936 -378.49
## - x10 1 1.551 4.145 -339.51
## - x12 1 150.967 153.562 68.66
##
## Step: AIC=-392.41
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x10 + x11 + x12 + x13 +
## x14 + x15 + x16 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x16 1 0.002 2.597 -394.34
## - x13 1 0.003 2.598 -394.30
## - x17 1 0.004 2.600 -394.24
## - x4 1 0.011 2.607 -393.94
## - x11 1 0.022 2.618 -393.46
## - x3 1 0.028 2.623 -393.22
## - x6 1 0.028 2.624 -393.20
## - x1 1 0.032 2.628 -393.02
## - x15 1 0.036 2.631 -392.86
## <none> 2.596 -392.41
## - x2 1 0.049 2.645 -392.30
## - x5 1 0.059 2.655 -391.85
## - x7 1 0.070 2.666 -391.41
## + x8 1 0.001 2.595 -390.44
## + x9 1 0.001 2.595 -390.44
## - x14 1 0.270 2.866 -383.23
## - x18 1 0.378 2.974 -379.03
## - x10 1 1.588 4.184 -340.46
## - x12 1 167.589 170.184 78.27
##
## Step: AIC=-394.34
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x10 + x11 + x12 + x13 +
## x14 + x15 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x4 1 0.010 2.608 -395.90
## - x17 1 0.012 2.609 -395.84
## - x11 1 0.021 2.618 -395.44
## - x3 1 0.027 2.624 -395.18
## - x1 1 0.032 2.629 -394.98
## - x15 1 0.034 2.631 -394.86
## - x13 1 0.046 2.643 -394.38
## <none> 2.597 -394.34
## - x2 1 0.050 2.647 -394.21
## - x5 1 0.058 2.655 -393.85
## - x7 1 0.069 2.666 -393.39
## + x16 1 0.002 2.596 -392.41
## + x9 1 0.001 2.596 -392.38
## + x8 1 0.001 2.597 -392.37
## - x14 1 0.273 2.870 -385.05
## - x18 1 0.377 2.974 -381.02
## - x10 1 1.611 4.208 -341.81
## - x6 1 3.133 5.730 -306.93
## - x12 1 167.601 170.198 76.28
##
## Step: AIC=-395.9
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x11 + x12 + x13 + x14 +
## x15 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x17 1 0.009 2.616 -397.52
## - x11 1 0.022 2.629 -396.96
## - x1 1 0.032 2.640 -396.51
## - x15 1 0.034 2.641 -396.43
## - x13 1 0.035 2.643 -396.38
## - x3 1 0.036 2.644 -396.33
## <none> 2.608 -395.90
## - x2 1 0.051 2.658 -395.71
## - x5 1 0.055 2.663 -395.54
## - x7 1 0.069 2.677 -394.93
## + x4 1 0.010 2.597 -394.34
## + x8 1 0.002 2.606 -393.98
## + x16 1 0.001 2.607 -393.94
## + x9 1 0.001 2.607 -393.92
## - x14 1 0.272 2.880 -386.67
## - x18 1 0.385 2.993 -382.33
## - x10 1 1.614 4.221 -343.45
## - x6 1 3.187 5.795 -307.66
## - x12 1 167.630 170.238 74.31
##
## Step: AIC=-397.52
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x11 + x12 + x13 + x14 +
## x15 + x18
##
## Df Sum of Sq RSS AIC
## - x11 1 0.023 2.639 -398.55
## - x13 1 0.031 2.647 -398.20
## - x3 1 0.032 2.648 -398.14
## - x1 1 0.035 2.651 -398.02
## - x15 1 0.038 2.654 -397.90
## - x2 1 0.045 2.661 -397.60
## <none> 2.616 -397.52
## - x5 1 0.059 2.676 -396.98
## - x7 1 0.064 2.680 -396.80
## + x17 1 0.009 2.608 -395.90
## + x4 1 0.007 2.609 -395.84
## + x16 1 0.007 2.609 -395.81
## + x9 1 0.002 2.614 -395.59
## + x8 1 0.001 2.615 -395.57
## - x14 1 0.266 2.882 -388.59
## - x18 1 0.387 3.003 -383.93
## - x10 1 1.606 4.222 -345.44
## - x6 1 3.193 5.809 -309.38
## - x12 1 169.034 171.650 73.24
##
## Step: AIC=-398.55
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x15 +
## x18
##
## Df Sum of Sq RSS AIC
## - x15 1 0.019 2.658 -399.74
## - x1 1 0.039 2.677 -398.91
## - x5 1 0.039 2.678 -398.89
## - x13 1 0.040 2.679 -398.84
## - x2 1 0.042 2.680 -398.78
## - x3 1 0.047 2.686 -398.56
## <none> 2.639 -398.55
## - x7 1 0.071 2.710 -397.54
## + x11 1 0.023 2.616 -397.52
## + x17 1 0.010 2.629 -396.96
## + x4 1 0.008 2.631 -396.89
## + x16 1 0.005 2.634 -396.77
## + x8 1 0.003 2.635 -396.69
## + x9 1 0.000 2.639 -396.56
## - x14 1 0.270 2.908 -389.56
## - x18 1 2.891 5.529 -316.96
## - x6 1 3.181 5.820 -311.18
## - x10 1 7.863 10.502 -244.47
## - x12 1 174.820 177.459 75.00
##
## Step: AIC=-399.74
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x2 1 0.039 2.697 -400.10
## - x1 1 0.044 2.701 -399.90
## - x3 1 0.044 2.702 -399.87
## - x5 1 0.045 2.703 -399.85
## - x13 1 0.045 2.703 -399.84
## <none> 2.658 -399.74
## - x7 1 0.071 2.729 -398.75
## + x15 1 0.019 2.639 -398.55
## + x17 1 0.012 2.646 -398.26
## + x4 1 0.007 2.651 -398.03
## + x16 1 0.004 2.653 -397.92
## + x11 1 0.004 2.654 -397.90
## + x8 1 0.004 2.654 -397.89
## + x9 1 0.003 2.655 -397.86
## - x18 1 2.937 5.595 -317.63
## - x6 1 3.218 5.876 -312.08
## - x14 1 4.296 6.954 -293.06
## - x10 1 8.152 10.809 -243.21
## - x12 1 175.662 178.320 73.55
##
## Step: AIC=-400.1
## y ~ x1 + x3 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x3 1 0.037 2.733 -400.57
## - x1 1 0.039 2.736 -400.46
## - x13 1 0.043 2.740 -400.31
## <none> 2.697 -400.10
## - x5 1 0.053 2.750 -399.90
## + x2 1 0.039 2.658 -399.74
## - x7 1 0.058 2.755 -399.67
## + x15 1 0.016 2.680 -398.78
## + x4 1 0.009 2.688 -398.48
## + x17 1 0.005 2.692 -398.30
## + x11 1 0.003 2.693 -398.24
## + x16 1 0.002 2.694 -398.20
## + x9 1 0.001 2.696 -398.14
## + x8 1 0.000 2.697 -398.10
## - x18 1 2.940 5.636 -318.79
## - x6 1 3.201 5.898 -313.67
## - x14 1 4.350 7.047 -293.55
## - x10 1 8.113 10.810 -245.21
## - x12 1 183.729 186.426 76.57
##
## Step: AIC=-400.57
## y ~ x1 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x13 1 0.010 2.743 -402.16
## - x7 1 0.030 2.764 -401.33
## - x1 1 0.041 2.774 -400.90
## <none> 2.733 -400.57
## - x5 1 0.057 2.791 -400.23
## + x3 1 0.037 2.697 -400.10
## + x2 1 0.031 2.702 -399.87
## + x4 1 0.021 2.712 -399.44
## + x15 1 0.014 2.719 -399.16
## + x11 1 0.009 2.724 -398.96
## + x8 1 0.004 2.729 -398.74
## + x17 1 0.002 2.731 -398.66
## + x16 1 0.001 2.732 -398.62
## + x9 1 0.000 2.733 -398.59
## - x18 1 2.983 5.717 -319.19
## - x6 1 3.346 6.079 -312.24
## - x14 1 4.601 7.335 -291.03
## - x10 1 8.077 10.810 -247.20
## - x12 1 186.520 189.254 76.27
##
## Step: AIC=-402.16
## y ~ x1 + x5 + x6 + x7 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x1 1 0.039 2.782 -402.56
## - x7 1 0.040 2.783 -402.52
## <none> 2.743 -402.16
## - x5 1 0.060 2.804 -401.70
## + x2 1 0.034 2.710 -401.56
## + x15 1 0.018 2.725 -400.91
## + x13 1 0.010 2.733 -400.57
## + x16 1 0.010 2.734 -400.56
## + x11 1 0.008 2.735 -400.49
## + x4 1 0.004 2.739 -400.32
## + x3 1 0.003 2.740 -400.31
## + x17 1 0.001 2.742 -400.22
## + x8 1 0.001 2.742 -400.21
## + x9 1 0.001 2.742 -400.21
## - x18 1 3.819 6.562 -305.61
## - x6 1 3.954 6.697 -303.30
## - x14 1 4.752 7.496 -290.58
## - x10 1 10.165 12.909 -229.15
## - x12 1 195.196 197.939 79.34
##
## Step: AIC=-402.56
## y ~ x5 + x6 + x7 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x7 1 0.041 2.824 -402.90
## <none> 2.782 -402.56
## - x5 1 0.055 2.838 -402.34
## + x1 1 0.039 2.743 -402.16
## + x2 1 0.030 2.753 -401.76
## + x15 1 0.022 2.760 -401.47
## + x13 1 0.008 2.774 -400.90
## + x11 1 0.008 2.774 -400.90
## + x16 1 0.008 2.774 -400.89
## + x4 1 0.005 2.778 -400.75
## + x3 1 0.004 2.778 -400.73
## + x17 1 0.003 2.779 -400.69
## + x9 1 0.003 2.780 -400.67
## + x8 1 0.001 2.782 -400.59
## - x18 1 3.931 6.714 -305.02
## - x6 1 4.026 6.808 -303.45
## - x14 1 4.847 7.629 -290.58
## - x10 1 10.240 13.023 -230.16
## - x12 1 197.697 200.479 78.79
##
## Step: AIC=-402.9
## y ~ x5 + x6 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x5 1 0.042 2.865 -403.24
## <none> 2.824 -402.90
## + x7 1 0.041 2.782 -402.56
## + x1 1 0.040 2.783 -402.52
## + x15 1 0.026 2.797 -401.96
## + x2 1 0.023 2.800 -401.84
## + x13 1 0.018 2.806 -401.62
## + x16 1 0.018 2.806 -401.61
## + x11 1 0.007 2.817 -401.18
## + x9 1 0.003 2.820 -401.04
## + x8 1 0.002 2.821 -401.00
## + x17 1 0.002 2.822 -400.96
## + x4 1 0.001 2.822 -400.95
## + x3 1 0.001 2.822 -400.95
## - x6 1 4.651 7.475 -294.89
## - x18 1 6.435 9.258 -270.71
## - x14 1 7.037 9.861 -263.59
## - x10 1 10.446 13.270 -230.03
## - x12 1 206.741 209.565 81.79
##
## Step: AIC=-403.24
## y ~ x6 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## <none> 2.865 -403.24
## + x5 1 0.042 2.824 -402.90
## + x1 1 0.035 2.830 -402.64
## + x2 1 0.030 2.835 -402.44
## + x7 1 0.028 2.838 -402.34
## + x15 1 0.021 2.845 -402.05
## + x13 1 0.019 2.846 -402.01
## + x16 1 0.019 2.846 -402.01
## + x11 1 0.011 2.854 -401.69
## + x9 1 0.001 2.864 -401.30
## + x17 1 0.001 2.864 -401.28
## + x8 1 0.001 2.865 -401.27
## + x3 1 0.000 2.865 -401.26
## + x4 1 0.000 2.865 -401.24
## - x6 1 4.738 7.603 -294.97
## - x18 1 6.403 9.268 -272.59
## - x14 1 7.114 9.979 -264.24
## - x10 1 10.534 13.400 -230.93
## - x12 1 213.635 216.500 83.47
# CHOOSES THE MODEL:
# y ~ x6 + x10 + x12 + x14 + x18
# Summary information
summary(both_aic)
##
## Call:
## lm(formula = y ~ x6 + x10 + x12 + x14 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.44914 -0.07025 0.01326 0.06725 0.75342
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.404720 0.200281 -12.01 <2e-16 ***
## x6 0.536888 0.040365 13.30 <2e-16 ***
## x10 1.346157 0.067872 19.83 <2e-16 ***
## x12 0.233623 0.002616 89.32 <2e-16 ***
## x14 1.541982 0.094606 16.30 <2e-16 ***
## x18 -0.506816 0.032777 -15.46 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1636 on 107 degrees of freedom
## Multiple R-squared: 0.9978, Adjusted R-squared: 0.9977
## F-statistic: 9661 on 5 and 107 DF, p-value: < 2.2e-16
# 9th Approach: BIC (BOTH WITH K=ln(length(y)))
both_bic <- step(mq, scope=formula(mq), direction="both", k=log(length(y)))
## Start: AIC=-336.64
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 +
## x12 + x13 + x14 + x15 + x16 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x9 1 0.001 2.595 -341.35
## - x8 1 0.001 2.595 -341.35
## - x16 1 0.002 2.596 -341.30
## - x13 1 0.003 2.597 -341.26
## - x17 1 0.004 2.598 -341.21
## - x4 1 0.009 2.604 -340.96
## - x11 1 0.021 2.615 -340.48
## - x6 1 0.023 2.617 -340.37
## - x3 1 0.028 2.622 -340.17
## - x1 1 0.031 2.625 -340.04
## - x15 1 0.032 2.627 -339.97
## - x2 1 0.049 2.643 -339.26
## - x5 1 0.055 2.650 -338.98
## - x7 1 0.069 2.663 -338.42
## <none> 2.594 -336.64
## - x14 1 0.240 2.834 -331.37
## - x18 1 0.312 2.906 -328.55
## - x10 1 0.326 2.920 -328.00
## - x12 1 146.022 148.616 116.05
##
## Step: AIC=-341.35
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x10 + x11 + x12 +
## x13 + x14 + x15 + x16 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x8 1 0.001 2.596 -346.04
## - x16 1 0.002 2.597 -346.00
## - x13 1 0.003 2.598 -345.95
## - x17 1 0.004 2.599 -345.90
## - x4 1 0.009 2.604 -345.68
## - x11 1 0.020 2.615 -345.21
## - x6 1 0.025 2.620 -344.99
## - x3 1 0.027 2.622 -344.89
## - x1 1 0.032 2.627 -344.71
## - x15 1 0.034 2.629 -344.59
## - x2 1 0.049 2.644 -343.98
## - x5 1 0.058 2.653 -343.58
## - x7 1 0.068 2.663 -343.14
## <none> 2.595 -341.35
## + x9 1 0.001 2.594 -336.64
## - x14 1 0.252 2.847 -335.61
## - x18 1 0.341 2.936 -332.12
## - x10 1 1.551 4.145 -293.14
## - x12 1 150.967 153.562 115.02
##
## Step: AIC=-346.04
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x10 + x11 + x12 + x13 +
## x14 + x15 + x16 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x16 1 0.002 2.597 -350.70
## - x13 1 0.003 2.598 -350.66
## - x17 1 0.004 2.600 -350.60
## - x4 1 0.011 2.607 -350.30
## - x11 1 0.022 2.618 -349.82
## - x3 1 0.028 2.623 -349.58
## - x6 1 0.028 2.624 -349.56
## - x1 1 0.032 2.628 -349.38
## - x15 1 0.036 2.631 -349.23
## - x2 1 0.049 2.645 -348.66
## - x5 1 0.059 2.655 -348.22
## - x7 1 0.070 2.666 -347.77
## <none> 2.596 -346.04
## + x8 1 0.001 2.595 -341.35
## + x9 1 0.001 2.595 -341.35
## - x14 1 0.270 2.866 -339.59
## - x18 1 0.378 2.974 -335.39
## - x10 1 1.588 4.184 -296.82
## - x12 1 167.589 170.184 121.91
##
## Step: AIC=-350.7
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x10 + x11 + x12 + x13 +
## x14 + x15 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x4 1 0.010 2.608 -354.98
## - x17 1 0.012 2.609 -354.93
## - x11 1 0.021 2.618 -354.53
## - x3 1 0.027 2.624 -354.27
## - x1 1 0.032 2.629 -354.07
## - x15 1 0.034 2.631 -353.95
## - x13 1 0.046 2.643 -353.47
## - x2 1 0.050 2.647 -353.29
## - x5 1 0.058 2.655 -352.94
## - x7 1 0.069 2.666 -352.48
## <none> 2.597 -350.70
## + x16 1 0.002 2.596 -346.04
## + x9 1 0.001 2.596 -346.01
## + x8 1 0.001 2.597 -346.00
## - x14 1 0.273 2.870 -344.14
## - x18 1 0.377 2.974 -340.11
## - x10 1 1.611 4.208 -300.90
## - x6 1 3.133 5.730 -266.02
## - x12 1 167.601 170.198 117.19
##
## Step: AIC=-354.98
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x11 + x12 + x13 + x14 +
## x15 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x17 1 0.009 2.616 -359.34
## - x11 1 0.022 2.629 -358.78
## - x1 1 0.032 2.640 -358.32
## - x15 1 0.034 2.641 -358.25
## - x13 1 0.035 2.643 -358.19
## - x3 1 0.036 2.644 -358.15
## - x2 1 0.051 2.658 -357.53
## - x5 1 0.055 2.663 -357.35
## - x7 1 0.069 2.677 -356.75
## <none> 2.608 -354.98
## + x4 1 0.010 2.597 -350.70
## + x8 1 0.002 2.606 -350.34
## + x16 1 0.001 2.607 -350.30
## + x9 1 0.001 2.607 -350.28
## - x14 1 0.272 2.880 -348.48
## - x18 1 0.385 2.993 -344.14
## - x10 1 1.614 4.221 -305.27
## - x6 1 3.187 5.795 -269.47
## - x12 1 167.630 170.238 112.49
##
## Step: AIC=-359.34
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x11 + x12 + x13 + x14 +
## x15 + x18
##
## Df Sum of Sq RSS AIC
## - x11 1 0.023 2.639 -363.09
## - x13 1 0.031 2.647 -362.74
## - x3 1 0.032 2.648 -362.68
## - x1 1 0.035 2.651 -362.56
## - x15 1 0.038 2.654 -362.45
## - x2 1 0.045 2.661 -362.15
## - x5 1 0.059 2.676 -361.52
## - x7 1 0.064 2.680 -361.34
## <none> 2.616 -359.34
## + x17 1 0.009 2.608 -354.98
## + x4 1 0.007 2.609 -354.93
## + x16 1 0.007 2.609 -354.90
## + x9 1 0.002 2.614 -354.68
## + x8 1 0.001 2.615 -354.66
## - x14 1 0.266 2.882 -353.13
## - x18 1 0.387 3.003 -348.47
## - x10 1 1.606 4.222 -309.99
## - x6 1 3.193 5.809 -273.92
## - x12 1 169.034 171.650 108.70
##
## Step: AIC=-363.09
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x15 +
## x18
##
## Df Sum of Sq RSS AIC
## - x15 1 0.019 2.658 -367.01
## - x1 1 0.039 2.677 -366.18
## - x5 1 0.039 2.678 -366.16
## - x13 1 0.040 2.679 -366.11
## - x2 1 0.042 2.680 -366.05
## - x3 1 0.047 2.686 -365.83
## - x7 1 0.071 2.710 -364.81
## <none> 2.639 -363.09
## + x11 1 0.023 2.616 -359.34
## + x17 1 0.010 2.629 -358.78
## + x4 1 0.008 2.631 -358.70
## + x16 1 0.005 2.634 -358.59
## + x8 1 0.003 2.635 -358.51
## + x9 1 0.000 2.639 -358.38
## - x14 1 0.270 2.908 -356.83
## - x18 1 2.891 5.529 -284.23
## - x6 1 3.181 5.820 -278.45
## - x10 1 7.863 10.502 -211.74
## - x12 1 174.820 177.459 107.73
##
## Step: AIC=-367.01
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x2 1 0.039 2.697 -370.10
## - x1 1 0.044 2.701 -369.90
## - x3 1 0.044 2.702 -369.87
## - x5 1 0.045 2.703 -369.85
## - x13 1 0.045 2.703 -369.84
## - x7 1 0.071 2.729 -368.75
## <none> 2.658 -367.01
## + x15 1 0.019 2.639 -363.09
## + x17 1 0.012 2.646 -362.80
## + x4 1 0.007 2.651 -362.58
## + x16 1 0.004 2.653 -362.47
## + x11 1 0.004 2.654 -362.45
## + x8 1 0.004 2.654 -362.43
## + x9 1 0.003 2.655 -362.41
## - x18 1 2.937 5.595 -287.63
## - x6 1 3.218 5.876 -282.08
## - x14 1 4.296 6.954 -263.06
## - x10 1 8.152 10.809 -213.21
## - x12 1 175.662 178.320 103.55
##
## Step: AIC=-370.1
## y ~ x1 + x3 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x3 1 0.037 2.733 -373.30
## - x1 1 0.039 2.736 -373.18
## - x13 1 0.043 2.740 -373.03
## - x5 1 0.053 2.750 -372.62
## - x7 1 0.058 2.755 -372.40
## <none> 2.697 -370.10
## + x2 1 0.039 2.658 -367.01
## + x15 1 0.016 2.680 -366.05
## + x4 1 0.009 2.688 -365.75
## + x17 1 0.005 2.692 -365.57
## + x11 1 0.003 2.693 -365.51
## + x16 1 0.002 2.694 -365.47
## + x9 1 0.001 2.696 -365.42
## + x8 1 0.000 2.697 -365.37
## - x18 1 2.940 5.636 -291.52
## - x6 1 3.201 5.898 -286.40
## - x14 1 4.350 7.047 -266.28
## - x10 1 8.113 10.810 -217.93
## - x12 1 183.729 186.426 103.85
##
## Step: AIC=-373.3
## y ~ x1 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x13 1 0.010 2.743 -377.62
## - x7 1 0.030 2.764 -376.78
## - x1 1 0.041 2.774 -376.35
## - x5 1 0.057 2.791 -375.68
## <none> 2.733 -373.30
## + x3 1 0.037 2.697 -370.10
## + x2 1 0.031 2.702 -369.87
## + x4 1 0.021 2.712 -369.44
## + x15 1 0.014 2.719 -369.16
## + x11 1 0.009 2.724 -368.96
## + x8 1 0.004 2.729 -368.74
## + x17 1 0.002 2.731 -368.66
## + x16 1 0.001 2.732 -368.62
## + x9 1 0.000 2.733 -368.58
## - x18 1 2.983 5.717 -294.64
## - x6 1 3.346 6.079 -287.69
## - x14 1 4.601 7.335 -266.48
## - x10 1 8.077 10.810 -222.65
## - x12 1 186.520 189.254 100.82
##
## Step: AIC=-377.62
## y ~ x1 + x5 + x6 + x7 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x1 1 0.039 2.782 -380.74
## - x7 1 0.040 2.783 -380.71
## - x5 1 0.060 2.804 -379.88
## <none> 2.743 -377.62
## + x2 1 0.034 2.710 -374.28
## + x15 1 0.018 2.725 -373.63
## + x13 1 0.010 2.733 -373.30
## + x16 1 0.010 2.734 -373.28
## + x11 1 0.008 2.735 -373.21
## + x4 1 0.004 2.739 -373.05
## + x3 1 0.003 2.740 -373.03
## + x17 1 0.001 2.742 -372.95
## + x8 1 0.001 2.742 -372.94
## + x9 1 0.001 2.742 -372.93
## - x18 1 3.819 6.562 -283.79
## - x6 1 3.954 6.697 -281.48
## - x14 1 4.752 7.496 -268.76
## - x10 1 10.165 12.909 -207.33
## - x12 1 195.196 197.939 101.16
##
## Step: AIC=-380.74
## y ~ x5 + x6 + x7 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x7 1 0.041 2.824 -383.81
## - x5 1 0.055 2.838 -383.25
## <none> 2.782 -380.74
## + x1 1 0.039 2.743 -377.62
## + x2 1 0.030 2.753 -377.22
## + x15 1 0.022 2.760 -376.92
## + x13 1 0.008 2.774 -376.35
## + x11 1 0.008 2.774 -376.35
## + x16 1 0.008 2.774 -376.34
## + x4 1 0.005 2.778 -376.20
## + x3 1 0.004 2.778 -376.19
## + x17 1 0.003 2.779 -376.14
## + x9 1 0.003 2.780 -376.13
## + x8 1 0.001 2.782 -376.04
## - x18 1 3.931 6.714 -285.93
## - x6 1 4.026 6.808 -284.35
## - x14 1 4.847 7.629 -271.49
## - x10 1 10.240 13.023 -211.06
## - x12 1 197.697 200.479 97.88
##
## Step: AIC=-383.81
## y ~ x5 + x6 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x5 1 0.042 2.865 -386.88
## <none> 2.824 -383.81
## + x7 1 0.041 2.782 -380.74
## + x1 1 0.040 2.783 -380.71
## + x15 1 0.026 2.797 -380.14
## + x2 1 0.023 2.800 -380.02
## + x13 1 0.018 2.806 -379.80
## + x16 1 0.018 2.806 -379.79
## + x11 1 0.007 2.817 -379.36
## + x9 1 0.003 2.820 -379.22
## + x8 1 0.002 2.821 -379.18
## + x17 1 0.002 2.822 -379.14
## + x4 1 0.001 2.822 -379.13
## + x3 1 0.001 2.822 -379.13
## - x6 1 4.651 7.475 -278.52
## - x18 1 6.435 9.258 -254.35
## - x14 1 7.037 9.861 -247.22
## - x10 1 10.446 13.270 -213.67
## - x12 1 206.741 209.565 98.16
##
## Step: AIC=-386.88
## y ~ x6 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## <none> 2.865 -386.88
## + x5 1 0.042 2.824 -383.81
## + x1 1 0.035 2.830 -383.55
## + x2 1 0.030 2.835 -383.35
## + x7 1 0.028 2.838 -383.25
## + x15 1 0.021 2.845 -382.96
## + x13 1 0.019 2.846 -382.92
## + x16 1 0.019 2.846 -382.92
## + x11 1 0.011 2.854 -382.60
## + x9 1 0.001 2.864 -382.21
## + x17 1 0.001 2.864 -382.19
## + x8 1 0.001 2.865 -382.17
## + x3 1 0.000 2.865 -382.17
## + x4 1 0.000 2.865 -382.15
## - x6 1 4.738 7.603 -281.33
## - x18 1 6.403 9.268 -258.96
## - x14 1 7.114 9.979 -250.60
## - x10 1 10.534 13.400 -217.30
## - x12 1 213.635 216.500 97.11
# CHOOSES THE MODEL:
# y ~ x6 + x10 + x12 + x14 + x18
# Summary information
summary(both_bic)
##
## Call:
## lm(formula = y ~ x6 + x10 + x12 + x14 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.44914 -0.07025 0.01326 0.06725 0.75342
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.404720 0.200281 -12.01 <2e-16 ***
## x6 0.536888 0.040365 13.30 <2e-16 ***
## x10 1.346157 0.067872 19.83 <2e-16 ***
## x12 0.233623 0.002616 89.32 <2e-16 ***
## x14 1.541982 0.094606 16.30 <2e-16 ***
## x18 -0.506816 0.032777 -15.46 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1636 on 107 degrees of freedom
## Multiple R-squared: 0.9978, Adjusted R-squared: 0.9977
## F-statistic: 9661 on 5 and 107 DF, p-value: < 2.2e-16
# 10th Approach: LIKELIHOOD FUNCTIONS COMPARISON (BACKWARD WITH K=0)
# STARTING FROM THE COMPLETE MODEL
back_lik <- step(mq, scope=formula(mq), direction="backward", k=0)
## Start: AIC=-426.46
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 +
## x12 + x13 + x14 + x15 + x16 + x17 + x18
##
## Df Sum of Sq RSS AIC
## <none> 2.594 -426.46
## - x9 1 0.001 2.595 -426.44
## - x8 1 0.001 2.595 -426.44
## - x16 1 0.002 2.596 -426.40
## - x13 1 0.003 2.597 -426.35
## - x17 1 0.004 2.598 -426.30
## - x4 1 0.009 2.604 -426.05
## - x11 1 0.021 2.615 -425.57
## - x6 1 0.023 2.617 -425.47
## - x3 1 0.028 2.622 -425.26
## - x1 1 0.031 2.625 -425.13
## - x15 1 0.032 2.627 -425.07
## - x2 1 0.049 2.643 -424.36
## - x5 1 0.055 2.650 -424.07
## - x7 1 0.069 2.663 -423.52
## - x14 1 0.240 2.834 -416.46
## - x18 1 0.312 2.906 -413.65
## - x10 1 0.326 2.920 -413.09
## - x12 1 146.022 148.616 30.96
# ALWAYS CHOOSES THE COMPLETE MODEL:
# y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 +
# x12 + x13 + x14 + x15 + x16 + x17 + x18
# Summary information
summary(back_lik)
##
## Call:
## lm(formula = y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 +
## x10 + x11 + x12 + x13 + x14 + x15 + x16 + x17 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.35240 -0.08275 -0.01367 0.06688 0.70644
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.4932971 4.0122361 0.123 0.90241
## x1 -0.0001159 0.0001097 -1.056 0.29370
## x2 0.0008409 0.0006320 1.331 0.18654
## x3 0.0058477 0.0058386 1.002 0.31913
## x4 -0.0027320 0.0046645 -0.586 0.55949
## x5 -0.1743371 0.1230087 -1.417 0.15971
## x6 0.4107678 0.4499448 0.913 0.36362
## x7 -0.0277227 0.0175895 -1.576 0.11836
## x8 -0.0001507 0.0010495 -0.144 0.88616
## x9 -0.0349403 0.2545558 -0.137 0.89112
## x10 1.1866765 0.3453261 3.436 0.00088 ***
## x11 -0.3031051 0.3517006 -0.862 0.39098
## x12 0.2364541 0.0032508 72.736 < 2e-16 ***
## x13 -0.1351995 0.4372948 -0.309 0.75787
## x14 1.4366685 0.4871607 2.949 0.00402 **
## x15 0.5159395 0.4768871 1.082 0.28207
## x16 0.1026885 0.4341540 0.237 0.81354
## x17 -0.0029176 0.0080131 -0.364 0.71659
## x18 -0.5989269 0.1782506 -3.360 0.00113 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1661 on 94 degrees of freedom
## Multiple R-squared: 0.998, Adjusted R-squared: 0.9976
## F-statistic: 2604 on 18 and 94 DF, p-value: < 2.2e-16
# 11th Approach: AIC (BOTH WITH K=2)
back_aic <- step(mq, scope=formula(mq), direction="backward", k=2)
## Start: AIC=-388.46
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 +
## x12 + x13 + x14 + x15 + x16 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x9 1 0.001 2.595 -390.44
## - x8 1 0.001 2.595 -390.44
## - x16 1 0.002 2.596 -390.40
## - x13 1 0.003 2.597 -390.35
## - x17 1 0.004 2.598 -390.30
## - x4 1 0.009 2.604 -390.05
## - x11 1 0.021 2.615 -389.57
## - x6 1 0.023 2.617 -389.47
## - x3 1 0.028 2.622 -389.26
## - x1 1 0.031 2.625 -389.13
## - x15 1 0.032 2.627 -389.07
## <none> 2.594 -388.46
## - x2 1 0.049 2.643 -388.36
## - x5 1 0.055 2.650 -388.07
## - x7 1 0.069 2.663 -387.52
## - x14 1 0.240 2.834 -380.46
## - x18 1 0.312 2.906 -377.65
## - x10 1 0.326 2.920 -377.09
## - x12 1 146.022 148.616 66.96
##
## Step: AIC=-390.44
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x10 + x11 + x12 +
## x13 + x14 + x15 + x16 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x8 1 0.001 2.596 -392.41
## - x16 1 0.002 2.597 -392.37
## - x13 1 0.003 2.598 -392.32
## - x17 1 0.004 2.599 -392.27
## - x4 1 0.009 2.604 -392.04
## - x11 1 0.020 2.615 -391.57
## - x6 1 0.025 2.620 -391.35
## - x3 1 0.027 2.622 -391.25
## - x1 1 0.032 2.627 -391.07
## - x15 1 0.034 2.629 -390.95
## <none> 2.595 -390.44
## - x2 1 0.049 2.644 -390.34
## - x5 1 0.058 2.653 -389.95
## - x7 1 0.068 2.663 -389.50
## - x14 1 0.252 2.847 -381.98
## - x18 1 0.341 2.936 -378.49
## - x10 1 1.551 4.145 -339.51
## - x12 1 150.967 153.562 68.66
##
## Step: AIC=-392.41
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x10 + x11 + x12 + x13 +
## x14 + x15 + x16 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x16 1 0.002 2.597 -394.34
## - x13 1 0.003 2.598 -394.30
## - x17 1 0.004 2.600 -394.24
## - x4 1 0.011 2.607 -393.94
## - x11 1 0.022 2.618 -393.46
## - x3 1 0.028 2.623 -393.22
## - x6 1 0.028 2.624 -393.20
## - x1 1 0.032 2.628 -393.02
## - x15 1 0.036 2.631 -392.86
## <none> 2.596 -392.41
## - x2 1 0.049 2.645 -392.30
## - x5 1 0.059 2.655 -391.85
## - x7 1 0.070 2.666 -391.41
## - x14 1 0.270 2.866 -383.23
## - x18 1 0.378 2.974 -379.03
## - x10 1 1.588 4.184 -340.46
## - x12 1 167.589 170.184 78.27
##
## Step: AIC=-394.34
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x10 + x11 + x12 + x13 +
## x14 + x15 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x4 1 0.010 2.608 -395.90
## - x17 1 0.012 2.609 -395.84
## - x11 1 0.021 2.618 -395.44
## - x3 1 0.027 2.624 -395.18
## - x1 1 0.032 2.629 -394.98
## - x15 1 0.034 2.631 -394.86
## - x13 1 0.046 2.643 -394.38
## <none> 2.597 -394.34
## - x2 1 0.050 2.647 -394.21
## - x5 1 0.058 2.655 -393.85
## - x7 1 0.069 2.666 -393.39
## - x14 1 0.273 2.870 -385.05
## - x18 1 0.377 2.974 -381.02
## - x10 1 1.611 4.208 -341.81
## - x6 1 3.133 5.730 -306.93
## - x12 1 167.601 170.198 76.28
##
## Step: AIC=-395.9
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x11 + x12 + x13 + x14 +
## x15 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x17 1 0.009 2.616 -397.52
## - x11 1 0.022 2.629 -396.96
## - x1 1 0.032 2.640 -396.51
## - x15 1 0.034 2.641 -396.43
## - x13 1 0.035 2.643 -396.38
## - x3 1 0.036 2.644 -396.33
## <none> 2.608 -395.90
## - x2 1 0.051 2.658 -395.71
## - x5 1 0.055 2.663 -395.54
## - x7 1 0.069 2.677 -394.93
## - x14 1 0.272 2.880 -386.67
## - x18 1 0.385 2.993 -382.33
## - x10 1 1.614 4.221 -343.45
## - x6 1 3.187 5.795 -307.66
## - x12 1 167.630 170.238 74.31
##
## Step: AIC=-397.52
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x11 + x12 + x13 + x14 +
## x15 + x18
##
## Df Sum of Sq RSS AIC
## - x11 1 0.023 2.639 -398.55
## - x13 1 0.031 2.647 -398.20
## - x3 1 0.032 2.648 -398.14
## - x1 1 0.035 2.651 -398.02
## - x15 1 0.038 2.654 -397.90
## - x2 1 0.045 2.661 -397.60
## <none> 2.616 -397.52
## - x5 1 0.059 2.676 -396.98
## - x7 1 0.064 2.680 -396.80
## - x14 1 0.266 2.882 -388.59
## - x18 1 0.387 3.003 -383.93
## - x10 1 1.606 4.222 -345.44
## - x6 1 3.193 5.809 -309.38
## - x12 1 169.034 171.650 73.24
##
## Step: AIC=-398.55
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x15 +
## x18
##
## Df Sum of Sq RSS AIC
## - x15 1 0.019 2.658 -399.74
## - x1 1 0.039 2.677 -398.91
## - x5 1 0.039 2.678 -398.89
## - x13 1 0.040 2.679 -398.84
## - x2 1 0.042 2.680 -398.78
## - x3 1 0.047 2.686 -398.56
## <none> 2.639 -398.55
## - x7 1 0.071 2.710 -397.54
## - x14 1 0.270 2.908 -389.56
## - x18 1 2.891 5.529 -316.96
## - x6 1 3.181 5.820 -311.18
## - x10 1 7.863 10.502 -244.47
## - x12 1 174.820 177.459 75.00
##
## Step: AIC=-399.74
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x2 1 0.039 2.697 -400.10
## - x1 1 0.044 2.701 -399.90
## - x3 1 0.044 2.702 -399.87
## - x5 1 0.045 2.703 -399.85
## - x13 1 0.045 2.703 -399.84
## <none> 2.658 -399.74
## - x7 1 0.071 2.729 -398.75
## - x18 1 2.937 5.595 -317.63
## - x6 1 3.218 5.876 -312.08
## - x14 1 4.296 6.954 -293.06
## - x10 1 8.152 10.809 -243.21
## - x12 1 175.662 178.320 73.55
##
## Step: AIC=-400.1
## y ~ x1 + x3 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x3 1 0.037 2.733 -400.57
## - x1 1 0.039 2.736 -400.46
## - x13 1 0.043 2.740 -400.31
## <none> 2.697 -400.10
## - x5 1 0.053 2.750 -399.90
## - x7 1 0.058 2.755 -399.67
## - x18 1 2.940 5.636 -318.79
## - x6 1 3.201 5.898 -313.67
## - x14 1 4.350 7.047 -293.55
## - x10 1 8.113 10.810 -245.21
## - x12 1 183.729 186.426 76.57
##
## Step: AIC=-400.57
## y ~ x1 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x13 1 0.010 2.743 -402.16
## - x7 1 0.030 2.764 -401.33
## - x1 1 0.041 2.774 -400.90
## <none> 2.733 -400.57
## - x5 1 0.057 2.791 -400.23
## - x18 1 2.983 5.717 -319.19
## - x6 1 3.346 6.079 -312.24
## - x14 1 4.601 7.335 -291.03
## - x10 1 8.077 10.810 -247.20
## - x12 1 186.520 189.254 76.27
##
## Step: AIC=-402.16
## y ~ x1 + x5 + x6 + x7 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x1 1 0.039 2.782 -402.56
## - x7 1 0.040 2.783 -402.52
## <none> 2.743 -402.16
## - x5 1 0.060 2.804 -401.70
## - x18 1 3.819 6.562 -305.61
## - x6 1 3.954 6.697 -303.30
## - x14 1 4.752 7.496 -290.58
## - x10 1 10.165 12.909 -229.15
## - x12 1 195.196 197.939 79.34
##
## Step: AIC=-402.56
## y ~ x5 + x6 + x7 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x7 1 0.041 2.824 -402.90
## <none> 2.782 -402.56
## - x5 1 0.055 2.838 -402.34
## - x18 1 3.931 6.714 -305.02
## - x6 1 4.026 6.808 -303.45
## - x14 1 4.847 7.629 -290.58
## - x10 1 10.240 13.023 -230.16
## - x12 1 197.697 200.479 78.79
##
## Step: AIC=-402.9
## y ~ x5 + x6 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x5 1 0.042 2.865 -403.24
## <none> 2.824 -402.90
## - x6 1 4.651 7.475 -294.89
## - x18 1 6.435 9.258 -270.71
## - x14 1 7.037 9.861 -263.59
## - x10 1 10.446 13.270 -230.03
## - x12 1 206.741 209.565 81.79
##
## Step: AIC=-403.24
## y ~ x6 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## <none> 2.865 -403.24
## - x6 1 4.738 7.603 -294.97
## - x18 1 6.403 9.268 -272.59
## - x14 1 7.114 9.979 -264.24
## - x10 1 10.534 13.400 -230.93
## - x12 1 213.635 216.500 83.47
# CHOOSES THE MODEL:
# y ~ x6 + x10 + x12 + x14 + x18
# Summary information
summary(back_aic)
##
## Call:
## lm(formula = y ~ x6 + x10 + x12 + x14 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.44914 -0.07025 0.01326 0.06725 0.75342
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.404720 0.200281 -12.01 <2e-16 ***
## x6 0.536888 0.040365 13.30 <2e-16 ***
## x10 1.346157 0.067872 19.83 <2e-16 ***
## x12 0.233623 0.002616 89.32 <2e-16 ***
## x14 1.541982 0.094606 16.30 <2e-16 ***
## x18 -0.506816 0.032777 -15.46 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1636 on 107 degrees of freedom
## Multiple R-squared: 0.9978, Adjusted R-squared: 0.9977
## F-statistic: 9661 on 5 and 107 DF, p-value: < 2.2e-16
# 12th Approach: BIC (BOTH WITH K=ln(length(y)))
back_bic <- step(mq, scope=formula(mq), direction="backward", k=log(length(y)))
## Start: AIC=-336.64
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 +
## x12 + x13 + x14 + x15 + x16 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x9 1 0.001 2.595 -341.35
## - x8 1 0.001 2.595 -341.35
## - x16 1 0.002 2.596 -341.30
## - x13 1 0.003 2.597 -341.26
## - x17 1 0.004 2.598 -341.21
## - x4 1 0.009 2.604 -340.96
## - x11 1 0.021 2.615 -340.48
## - x6 1 0.023 2.617 -340.37
## - x3 1 0.028 2.622 -340.17
## - x1 1 0.031 2.625 -340.04
## - x15 1 0.032 2.627 -339.97
## - x2 1 0.049 2.643 -339.26
## - x5 1 0.055 2.650 -338.98
## - x7 1 0.069 2.663 -338.42
## <none> 2.594 -336.64
## - x14 1 0.240 2.834 -331.37
## - x18 1 0.312 2.906 -328.55
## - x10 1 0.326 2.920 -328.00
## - x12 1 146.022 148.616 116.05
##
## Step: AIC=-341.35
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x10 + x11 + x12 +
## x13 + x14 + x15 + x16 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x8 1 0.001 2.596 -346.04
## - x16 1 0.002 2.597 -346.00
## - x13 1 0.003 2.598 -345.95
## - x17 1 0.004 2.599 -345.90
## - x4 1 0.009 2.604 -345.68
## - x11 1 0.020 2.615 -345.21
## - x6 1 0.025 2.620 -344.99
## - x3 1 0.027 2.622 -344.89
## - x1 1 0.032 2.627 -344.71
## - x15 1 0.034 2.629 -344.59
## - x2 1 0.049 2.644 -343.98
## - x5 1 0.058 2.653 -343.58
## - x7 1 0.068 2.663 -343.14
## <none> 2.595 -341.35
## - x14 1 0.252 2.847 -335.61
## - x18 1 0.341 2.936 -332.12
## - x10 1 1.551 4.145 -293.14
## - x12 1 150.967 153.562 115.02
##
## Step: AIC=-346.04
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x10 + x11 + x12 + x13 +
## x14 + x15 + x16 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x16 1 0.002 2.597 -350.70
## - x13 1 0.003 2.598 -350.66
## - x17 1 0.004 2.600 -350.60
## - x4 1 0.011 2.607 -350.30
## - x11 1 0.022 2.618 -349.82
## - x3 1 0.028 2.623 -349.58
## - x6 1 0.028 2.624 -349.56
## - x1 1 0.032 2.628 -349.38
## - x15 1 0.036 2.631 -349.23
## - x2 1 0.049 2.645 -348.66
## - x5 1 0.059 2.655 -348.22
## - x7 1 0.070 2.666 -347.77
## <none> 2.596 -346.04
## - x14 1 0.270 2.866 -339.59
## - x18 1 0.378 2.974 -335.39
## - x10 1 1.588 4.184 -296.82
## - x12 1 167.589 170.184 121.91
##
## Step: AIC=-350.7
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x10 + x11 + x12 + x13 +
## x14 + x15 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x4 1 0.010 2.608 -354.98
## - x17 1 0.012 2.609 -354.93
## - x11 1 0.021 2.618 -354.53
## - x3 1 0.027 2.624 -354.27
## - x1 1 0.032 2.629 -354.07
## - x15 1 0.034 2.631 -353.95
## - x13 1 0.046 2.643 -353.47
## - x2 1 0.050 2.647 -353.29
## - x5 1 0.058 2.655 -352.94
## - x7 1 0.069 2.666 -352.48
## <none> 2.597 -350.70
## - x14 1 0.273 2.870 -344.14
## - x18 1 0.377 2.974 -340.11
## - x10 1 1.611 4.208 -300.90
## - x6 1 3.133 5.730 -266.02
## - x12 1 167.601 170.198 117.19
##
## Step: AIC=-354.98
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x11 + x12 + x13 + x14 +
## x15 + x17 + x18
##
## Df Sum of Sq RSS AIC
## - x17 1 0.009 2.616 -359.34
## - x11 1 0.022 2.629 -358.78
## - x1 1 0.032 2.640 -358.32
## - x15 1 0.034 2.641 -358.25
## - x13 1 0.035 2.643 -358.19
## - x3 1 0.036 2.644 -358.15
## - x2 1 0.051 2.658 -357.53
## - x5 1 0.055 2.663 -357.35
## - x7 1 0.069 2.677 -356.75
## <none> 2.608 -354.98
## - x14 1 0.272 2.880 -348.48
## - x18 1 0.385 2.993 -344.14
## - x10 1 1.614 4.221 -305.27
## - x6 1 3.187 5.795 -269.47
## - x12 1 167.630 170.238 112.49
##
## Step: AIC=-359.34
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x11 + x12 + x13 + x14 +
## x15 + x18
##
## Df Sum of Sq RSS AIC
## - x11 1 0.023 2.639 -363.09
## - x13 1 0.031 2.647 -362.74
## - x3 1 0.032 2.648 -362.68
## - x1 1 0.035 2.651 -362.56
## - x15 1 0.038 2.654 -362.45
## - x2 1 0.045 2.661 -362.15
## - x5 1 0.059 2.676 -361.52
## - x7 1 0.064 2.680 -361.34
## <none> 2.616 -359.34
## - x14 1 0.266 2.882 -353.13
## - x18 1 0.387 3.003 -348.47
## - x10 1 1.606 4.222 -309.99
## - x6 1 3.193 5.809 -273.92
## - x12 1 169.034 171.650 108.70
##
## Step: AIC=-363.09
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x15 +
## x18
##
## Df Sum of Sq RSS AIC
## - x15 1 0.019 2.658 -367.01
## - x1 1 0.039 2.677 -366.18
## - x5 1 0.039 2.678 -366.16
## - x13 1 0.040 2.679 -366.11
## - x2 1 0.042 2.680 -366.05
## - x3 1 0.047 2.686 -365.83
## - x7 1 0.071 2.710 -364.81
## <none> 2.639 -363.09
## - x14 1 0.270 2.908 -356.83
## - x18 1 2.891 5.529 -284.23
## - x6 1 3.181 5.820 -278.45
## - x10 1 7.863 10.502 -211.74
## - x12 1 174.820 177.459 107.73
##
## Step: AIC=-367.01
## y ~ x1 + x2 + x3 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x2 1 0.039 2.697 -370.10
## - x1 1 0.044 2.701 -369.90
## - x3 1 0.044 2.702 -369.87
## - x5 1 0.045 2.703 -369.85
## - x13 1 0.045 2.703 -369.84
## - x7 1 0.071 2.729 -368.75
## <none> 2.658 -367.01
## - x18 1 2.937 5.595 -287.63
## - x6 1 3.218 5.876 -282.08
## - x14 1 4.296 6.954 -263.06
## - x10 1 8.152 10.809 -213.21
## - x12 1 175.662 178.320 103.55
##
## Step: AIC=-370.1
## y ~ x1 + x3 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x3 1 0.037 2.733 -373.30
## - x1 1 0.039 2.736 -373.18
## - x13 1 0.043 2.740 -373.03
## - x5 1 0.053 2.750 -372.62
## - x7 1 0.058 2.755 -372.40
## <none> 2.697 -370.10
## - x18 1 2.940 5.636 -291.52
## - x6 1 3.201 5.898 -286.40
## - x14 1 4.350 7.047 -266.28
## - x10 1 8.113 10.810 -217.93
## - x12 1 183.729 186.426 103.85
##
## Step: AIC=-373.3
## y ~ x1 + x5 + x6 + x7 + x10 + x12 + x13 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x13 1 0.010 2.743 -377.62
## - x7 1 0.030 2.764 -376.78
## - x1 1 0.041 2.774 -376.35
## - x5 1 0.057 2.791 -375.68
## <none> 2.733 -373.30
## - x18 1 2.983 5.717 -294.64
## - x6 1 3.346 6.079 -287.69
## - x14 1 4.601 7.335 -266.48
## - x10 1 8.077 10.810 -222.65
## - x12 1 186.520 189.254 100.82
##
## Step: AIC=-377.62
## y ~ x1 + x5 + x6 + x7 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x1 1 0.039 2.782 -380.74
## - x7 1 0.040 2.783 -380.71
## - x5 1 0.060 2.804 -379.88
## <none> 2.743 -377.62
## - x18 1 3.819 6.562 -283.79
## - x6 1 3.954 6.697 -281.48
## - x14 1 4.752 7.496 -268.76
## - x10 1 10.165 12.909 -207.33
## - x12 1 195.196 197.939 101.16
##
## Step: AIC=-380.74
## y ~ x5 + x6 + x7 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x7 1 0.041 2.824 -383.81
## - x5 1 0.055 2.838 -383.25
## <none> 2.782 -380.74
## - x18 1 3.931 6.714 -285.93
## - x6 1 4.026 6.808 -284.35
## - x14 1 4.847 7.629 -271.49
## - x10 1 10.240 13.023 -211.06
## - x12 1 197.697 200.479 97.88
##
## Step: AIC=-383.81
## y ~ x5 + x6 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## - x5 1 0.042 2.865 -386.88
## <none> 2.824 -383.81
## - x6 1 4.651 7.475 -278.52
## - x18 1 6.435 9.258 -254.35
## - x14 1 7.037 9.861 -247.22
## - x10 1 10.446 13.270 -213.67
## - x12 1 206.741 209.565 98.16
##
## Step: AIC=-386.88
## y ~ x6 + x10 + x12 + x14 + x18
##
## Df Sum of Sq RSS AIC
## <none> 2.865 -386.88
## - x6 1 4.738 7.603 -281.33
## - x18 1 6.403 9.268 -258.96
## - x14 1 7.114 9.979 -250.60
## - x10 1 10.534 13.400 -217.30
## - x12 1 213.635 216.500 97.11
# CHOOSES THE MODEL:
# y ~ x6 + x10 + x12 + x14 + x18
# Summary information
summary(back_bic)
##
## Call:
## lm(formula = y ~ x6 + x10 + x12 + x14 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.44914 -0.07025 0.01326 0.06725 0.75342
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.404720 0.200281 -12.01 <2e-16 ***
## x6 0.536888 0.040365 13.30 <2e-16 ***
## x10 1.346157 0.067872 19.83 <2e-16 ***
## x12 0.233623 0.002616 89.32 <2e-16 ***
## x14 1.541982 0.094606 16.30 <2e-16 ***
## x18 -0.506816 0.032777 -15.46 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1636 on 107 degrees of freedom
## Multiple R-squared: 0.9978, Adjusted R-squared: 0.9977
## F-statistic: 9661 on 5 and 107 DF, p-value: < 2.2e-16
# IDENTIFIED MODELS:
# BACKWARD (p-value): y ~ x6 + x10 + x12 + x14 + x18 [All extremely significant]
# FORWARD (p-value): y ~ x6 + x12 + x9 + x11 + x10 [All extremely significant except x9]
# MIXED (p-value): y ~ x6 + x12 + x11 + x10 [All extremely significant]
# FORWARD (LIKELIHOODS COMPARISON): y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10 + x5 + x15 + x7 + x2 + x1 + x3 + x13 + x4 + x17 + x16 + x8
# FORWARD (AIC): y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10
# FORWARD (BIC): y ~ x12 + x9 + x18 + x14 + x11
# BOTH (LIKELIHOODS COMPARISON): y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13 + x14 + x15 + x16 + x17 + x18
# BOTH (AIC): y ~ x6 + x10 + x12 + x14 + x18
# BOTH (BIC): y ~ x6 + x10 + x12 + x14 + x18
# BACKWARD (LIKELIHOODS COMPARISON): y ~ x6 + x10 + x12 + x14 + x18
# BACKWARD (AIC): y ~ x6 + x10 + x12 + x14 + x18
# BACKWARD (BIC): y ~ x6 + x10 + x12 + x14 + x18
# POTENTIALLY VALID MODELS:
# EXCLUDING COMPLETE AND DUPLICATE MODELS:
# A - 6 occurrences: y ~ x6 + x10 + x12 + x14 + x18
# B - 1 occurrence: y ~ x6 + x12 + x9 + x11 + x10
# C - 1 occurrence: y ~ x6 + x12 + x11 + x10
# D - 1 occurrence: y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10
# E - 1 occurrence: y ~ x12 + x9 + x18 + x14 + x11
# A
a <- lm(y ~ x6 + x10 + x12 + x14 + x18)
summary(a)
##
## Call:
## lm(formula = y ~ x6 + x10 + x12 + x14 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.44914 -0.07025 0.01326 0.06725 0.75342
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.404720 0.200281 -12.01 <2e-16 ***
## x6 0.536888 0.040365 13.30 <2e-16 ***
## x10 1.346157 0.067872 19.83 <2e-16 ***
## x12 0.233623 0.002616 89.32 <2e-16 ***
## x14 1.541982 0.094606 16.30 <2e-16 ***
## x18 -0.506816 0.032777 -15.46 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1636 on 107 degrees of freedom
## Multiple R-squared: 0.9978, Adjusted R-squared: 0.9977
## F-statistic: 9661 on 5 and 107 DF, p-value: < 2.2e-16
# B
b <- lm(y ~ x6 + x12 + x9 + x11 + x10)
summary(b)
##
## Call:
## lm(formula = y ~ x6 + x12 + x9 + x11 + x10)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.51527 -0.06311 0.01123 0.07509 0.76495
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.806484 2.941230 -3.334 0.001176 **
## x6 0.329184 0.125133 2.631 0.009779 **
## x12 0.233187 0.002877 81.041 < 2e-16 ***
## x9 -0.314002 0.239618 -1.310 0.192856
## x11 0.797302 0.180422 4.419 2.38e-05 ***
## x10 1.257455 0.351987 3.572 0.000532 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1767 on 107 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 8279 on 5 and 107 DF, p-value: < 2.2e-16
# C
c <- lm(y ~ x6 + x12 + x11 + x10)
summary(c)
##
## Call:
## lm(formula = y ~ x6 + x12 + x11 + x10)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.54633 -0.05866 0.00924 0.07217 0.76854
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.514137 0.806207 -16.76 <2e-16 ***
## x6 0.485352 0.038286 12.68 <2e-16 ***
## x12 0.232624 0.002854 81.50 <2e-16 ***
## x11 1.015592 0.069539 14.61 <2e-16 ***
## x10 1.701196 0.096391 17.65 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1773 on 108 degrees of freedom
## Multiple R-squared: 0.9974, Adjusted R-squared: 0.9973
## F-statistic: 1.028e+04 on 4 and 108 DF, p-value: < 2.2e-16
# D
d <- lm(y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10)
summary(d)
##
## Call:
## lm(formula = y ~ x12 + x9 + x18 + x14 + x11 + x6 + x10)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.41987 -0.06391 0.01433 0.07466 0.75100
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.453316 3.649208 0.124 0.901376
## x12 0.234117 0.002708 86.455 < 2e-16 ***
## x9 -0.103748 0.235824 -0.440 0.660885
## x18 -0.575286 0.137484 -4.184 5.95e-05 ***
## x14 1.752429 0.431097 4.065 9.30e-05 ***
## x11 -0.221445 0.295084 -0.750 0.454664
## x6 0.487450 0.123099 3.960 0.000137 ***
## x10 1.135744 0.330636 3.435 0.000850 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1647 on 105 degrees of freedom
## Multiple R-squared: 0.9978, Adjusted R-squared: 0.9977
## F-statistic: 6811 on 7 and 105 DF, p-value: < 2.2e-16
# E
e <- lm(y ~ x12 + x9 + x18 + x14 + x11)
summary(e)
##
## Call:
## lm(formula = y ~ x12 + x9 + x18 + x14 + x11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.64191 -0.06187 0.02281 0.06056 0.74776
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.410137 0.925946 6.923 3.41e-10 ***
## x12 0.234033 0.002577 90.830 < 2e-16 ***
## x9 -0.974535 0.076614 -12.720 < 2e-16 ***
## x18 -0.362493 0.066296 -5.468 3.01e-07 ***
## x14 1.082834 0.224265 4.828 4.61e-06 ***
## x11 -0.428942 0.109931 -3.902 0.000167 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.175 on 107 degrees of freedom
## Multiple R-squared: 0.9975, Adjusted R-squared: 0.9974
## F-statistic: 8443 on 5 and 107 DF, p-value: < 2.2e-16
# The most significant models are A and C (based on the p-values)
# CHOSEN FINAL MODEL:
# A with 6 occurrences: y ~ x6 + x10 + x12 + x14 + x18
summary(a)
##
## Call:
## lm(formula = y ~ x6 + x10 + x12 + x14 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.44914 -0.07025 0.01326 0.06725 0.75342
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.404720 0.200281 -12.01 <2e-16 ***
## x6 0.536888 0.040365 13.30 <2e-16 ***
## x10 1.346157 0.067872 19.83 <2e-16 ***
## x12 0.233623 0.002616 89.32 <2e-16 ***
## x14 1.541982 0.094606 16.30 <2e-16 ***
## x18 -0.506816 0.032777 -15.46 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1636 on 107 degrees of freedom
## Multiple R-squared: 0.9978, Adjusted R-squared: 0.9977
## F-statistic: 9661 on 5 and 107 DF, p-value: < 2.2e-16
# DEFINITION OF THE IDENTIFIED MODEL:
mq <- lm(y ~ x6 + x10 + x12 + x14 + x18)
print(summary(mq))
##
## Call:
## lm(formula = y ~ x6 + x10 + x12 + x14 + x18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.44914 -0.07025 0.01326 0.06725 0.75342
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.404720 0.200281 -12.01 <2e-16 ***
## x6 0.536888 0.040365 13.30 <2e-16 ***
## x10 1.346157 0.067872 19.83 <2e-16 ***
## x12 0.233623 0.002616 89.32 <2e-16 ***
## x14 1.541982 0.094606 16.30 <2e-16 ***
## x18 -0.506816 0.032777 -15.46 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1636 on 107 degrees of freedom
## Multiple R-squared: 0.9978, Adjusted R-squared: 0.9977
## F-statistic: 9661 on 5 and 107 DF, p-value: < 2.2e-16
# PHASE 5:
# PRELIMINARY ANALYSIS OF THE MODEL AND RESIDUALS:
# LIST OF SIGNIFICANT VARIABLES FOR CENTRAL VELOCITY DISPERSION (Km/s):
# Mv: Absolute magnitude
# log.t: Central relaxation time (years)
# V.esc: Central escape velocity (Km/s)
# E.B.V: Color excess (Magnitude)
# CSBt: Central surface brightness (Magnitude per square arcsecond)
# VARIABLE ANALYSIS:
# ALL VARIABLES ARE EXTREMELY SIGNIFICANT
# EFFECT OF VARIABLES ON CENTRAL VELOCITY DISPERSION:
# Mv: an increase in magnitude of one unit leads to an increase of approximately 0.536888 Km/s
# log.t: one year of central relaxation on a logarithmic scale leads to an increase of 1.346157 Km/s
# V.esc: an increase in escape velocity of 1 Km/s results in an increase of 0.233623 Km/s
# E.B.V: an increase of one unit in the magnitude of color excess results in an increase of 1.541982 Km/s
# CSBt: a one-unit increase in central surface brightness results in a decrease of 0.506816 Km/s
# ANALYSIS OF MODEL RESULTS
# STANDARD ERROR ESTIMATE: 0.1636 with 107 degrees of freedom [being relatively
# small, indicates good precision of the estimates.]
# DETERMINATION COEFFICIENT: 0.9977 [being very high, indicates that the
# model almost completely explains the variance of the dependent variable y.]
# F-TEST: EXTREMELY SIGNIFICANT [9661 out of 5 and 107 DF]
# Residuals are well distributed, with the median close to zero.
# A very high F-value and a very low p-value (as indicated in the
# results provided, p < 2.2e-16) reject the null hypothesis that the model with
# only the intercept term is equally good at explaining the variability
# in the dependent variable.]:
# IN THIS CASE, THE MODEL SHOWS AN EXCELLENT FIT!
# CREATING THE MODEL GRAPH:
plot(mq)
# The residuals appear to have an approximately normal distribution.
# The graph shows the presence of some extreme points
# that potentially interfere with the model (outliers).
# ANALYSIS OF POTENTIAL OUTLIERS:
boxplot(mq$residuals)
# GRAPH ANALYSIS:
# Residuals are centered: The median near zero indicates that the regression model is
# not biased either high or low.
# The concentration of data near the median and symmetry suggest
# that most of the data follows the identified model.
# However, there are potential outliers: Points outside the whiskers (at the extremes)
# Calculation of standardized residuals
std_res <- rstandard(mq)
# Identification of outliers
# The number "2" is a threshold conventionally chosen for the normal
# (or Gaussian) distribution. In a standardized normal distribution:
# About 95% of the data lies within 2 standard deviations from the mean.
outliers <- which(abs(std_res) > 2)
# Removing outliers from the dataset
clean_dat <- dat[-outliers, ]
# Verification of outlier removal
print(nrow(dat))
## [1] 113
print(nrow(clean_dat))
## [1] 108
# Updating the model without outliers
clean_mq <- lm(y ~ x6 + x10 + x12 + x14 + x18, data=clean_dat)
# Graph of residuals with and without outliers
par(mfrow=c(1,2))
plot(mq$fitted.values, mq$residuals, main="With Outliers")
abline(h=0, col="red")
plot(clean_mq$fitted.values, clean_mq$residuals, main="Without Outliers")
abline(h=0, col="blue")
# Resetting graph creation settings
par(mfrow=c(1,1))
# GRAPH ANALYSIS:
# Outliers do not seem to have a significant impact on the model, as the graphs
# are practically identical. This indicates that the model is quite
# robust and has a high tolerance for anomalous values.
# I will keep the model unchanged.
# DATA RESULTING FROM THE USE OF A GLM (Generalized Linear Model)
# DEFINITION OF THE IDENTIFIED MODEL:
mqg <- glm(y ~ x6 + x10 + x12 + x14 + x18)
print(summary(mqg))
##
## Call:
## glm(formula = y ~ x6 + x10 + x12 + x14 + x18)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.404720 0.200281 -12.01 <2e-16 ***
## x6 0.536888 0.040365 13.30 <2e-16 ***
## x10 1.346157 0.067872 19.83 <2e-16 ***
## x12 0.233623 0.002616 89.32 <2e-16 ***
## x14 1.541982 0.094606 16.30 <2e-16 ***
## x18 -0.506816 0.032777 -15.46 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.02677899)
##
## Null deviance: 1296.3894 on 112 degrees of freedom
## Residual deviance: 2.8654 on 107 degrees of freedom
## AIC: -80.561
##
## Number of Fisher Scoring iterations: 2
# THE IDENTIFIED DISTRIBUTION FAMILY IS GAUSSIAN
# WITH IDENTITY LINK FUNCTION
print(mqg$family)
##
## Family: gaussian
## Link function: identity
# The glm model, using the same formula as the linear model (y ~ x6 + x10 + x12 + x14 + x18),
# suggests similar results in terms of variable significance,
# as indicated by the coefficients and p-values. This indicates that all the selected
# variables are statistically significant in the model.
# The main difference lies in the use of glm, which can be adapted to
# model a wider range of residual distributions compared to lm.
# The model behaves similarly to lm for normally distributed data.
# The dispersion parameter and AIC value also indicate good model fit to the data.
# EVALUATION OF THE EFFECT OF INDIVIDUAL VARIABLES ON CENTRAL
# VELOCITY DISPERSION:
# ABSOLUTE MAGNITUDE
mqgx6 <- glm(y ~ x6)
summary(mqgx6)
##
## Call:
## glm(formula = y ~ x6)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -10.2611 1.1210 -9.154 3.17e-15 ***
## x6 -2.2190 0.1488 -14.915 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 3.887633)
##
## Null deviance: 1296.39 on 112 degrees of freedom
## Residual deviance: 431.53 on 111 degrees of freedom
## AIC: 478.09
##
## Number of Fisher Scoring iterations: 2
# The variable's high significance is confirmed.
# An increase in magnitude level leads to a reduction in velocity
# dispersion of 2.2190 Km/s
# CENTRAL RELAXATION TIME:
mqgx10 <- glm(y ~ x10)
summary(mqgx10)
##
## Call:
## glm(formula = y ~ x10)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 15.0509 3.2848 4.582 1.21e-05 ***
## x10 -1.0902 0.4041 -2.698 0.00806 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 10.96039)
##
## Null deviance: 1296.4 on 112 degrees of freedom
## Residual deviance: 1216.6 on 111 degrees of freedom
## AIC: 595.22
##
## Number of Fisher Scoring iterations: 2
# High significance is confirmed again, albeit slightly
# reduced compared to the model containing all variables.
# Each year of relaxation results in a velocity reduction of 1.0902 Km/s
# CENTRAL ESCAPE VELOCITY:
mqgx12 <- glm(y ~ x12)
summary(mqgx12)
##
## Call:
## glm(formula = y ~ x12)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.30958 0.07152 4.329 3.3e-05 ***
## x12 0.23609 0.00248 95.210 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.1412813)
##
## Null deviance: 1296.389 on 112 degrees of freedom
## Residual deviance: 15.682 on 111 degrees of freedom
## AIC: 103.52
##
## Number of Fisher Scoring iterations: 2
# High significance is confirmed.
# An increase in escape velocity of 1 Km/s corresponds to an increase in
# velocity dispersion of 0.23609 Km/s
# COLOR EXCESS MAGNITUDE
mqgx14 <- glm(y ~ x14)
summary(mqgx14)
##
## Call:
## glm(formula = y ~ x14)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.6336 0.4071 13.839 <2e-16 ***
## x14 1.8018 0.7845 2.297 0.0235 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 11.14931)
##
## Null deviance: 1296.4 on 112 degrees of freedom
## Residual deviance: 1237.6 on 111 degrees of freedom
## AIC: 597.15
##
## Number of Fisher Scoring iterations: 2
# Significant, but reduced compared to the model containing
# all variables.
# An increase of one unit in color excess magnitude results in an
# increase in velocity dispersion of 1.80018 Km/s
# CENTRAL SURFACE BRIGHTNESS:
mqgx18 <- glm(y ~ x18)
summary(mqgx18)
##
## Call:
## glm(formula = y ~ x18)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 16.6882 1.0316 16.18 <2e-16 ***
## x18 -1.1431 0.1099 -10.40 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 5.915889)
##
## Null deviance: 1296.39 on 112 degrees of freedom
## Residual deviance: 656.66 on 111 degrees of freedom
## AIC: 525.54
##
## Number of Fisher Scoring iterations: 2
# Highly significant.
# An increase in central surface brightness by one unit (magnitude per square arcsecond)
# leads to a reduction in velocity dispersion of 1.1231 Km/s
# PHASE 6:
# CORRELATION ANALYSIS AMONG THE MODEL VARIABLES:
# CREATING CORRELATION GRAPHS AMONG THE MODEL VARIABLES
plot(x6, x10)
plot(x6, x12)
plot(x6, x14)
plot(x6, x18)
plot(x10, x12)
plot(x10, x14)
plot(x10, x18)
plot(x12, x14)
plot(x12, x18)
plot(x14, x18)
# GRAPH ANALYSIS:
# POSSIBLE IDENTIFIED CORRELATIONS:
# x6-x12: x12 decreases as x6 increases, with a negative exponential trend
# x6-x18: Increase linearly
# x10-x18: Increase linearly
# x12-x18: x18 decreases as x12 increases, with a negative exponential trend
# EVALUATION OF MORE COMPLEX REGRESSION MODELS CONSIDERING
# POTENTIAL CORRELATIONS:
mqgx6x12 <- glm(y ~ x6*x12 + x10 + x14 + x18)
summary(mqgx6x12)
##
## Call:
## glm(formula = y ~ x6 * x12 + x10 + x14 + x18)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.763692 0.410018 -4.301 3.79e-05 ***
## x6 0.572460 0.044646 12.822 < 2e-16 ***
## x12 0.198314 0.019937 9.947 < 2e-16 ***
## x10 1.382440 0.070192 19.695 < 2e-16 ***
## x14 1.744244 0.146951 11.870 < 2e-16 ***
## x18 -0.561024 0.044428 -12.628 < 2e-16 ***
## x6:x12 -0.003210 0.001797 -1.786 0.0769 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.02624184)
##
## Null deviance: 1296.3894 on 112 degrees of freedom
## Residual deviance: 2.7816 on 106 degrees of freedom
## AIC: -81.911
##
## Number of Fisher Scoring iterations: 2
# The p-value for the interaction term is 0.0769, which is above the
# threshold of 0.05. This suggests that while the individual variables are significant,
# the additional effect of their interaction is not statistically
# significant.
# CORRELATION INDEX
cor(x6, x12) # Indicates a strong negative correlation between the two
## [1] -0.8085964
# Mv and CSBt
mqgx6x18 <- glm(y ~ x6*x18 + x10 + x12 + x14)
summary(mqgx6x18)
##
## Call:
## glm(formula = y ~ x6 * x18 + x10 + x12 + x14)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.480091 0.647729 -0.741 0.46022
## x6 0.845940 0.106623 7.934 2.33e-12 ***
## x18 -0.638334 0.052722 -12.108 < 2e-16 ***
## x10 1.279510 0.068698 18.625 < 2e-16 ***
## x12 0.246281 0.004782 51.498 < 2e-16 ***
## x14 1.244678 0.131926 9.435 1.05e-15 ***
## x6:x18 -0.028592 0.009187 -3.112 0.00239 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.02476848)
##
## Null deviance: 1296.3894 on 112 degrees of freedom
## Residual deviance: 2.6255 on 106 degrees of freedom
## AIC: -88.441
##
## Number of Fisher Scoring iterations: 2
# The effect of absolute magnitude on velocity dispersion varies based on
# different levels of central surface brightness and vice versa.
# The interaction between these two measures is significant, and an increase in it leads to a
# reduction in velocity dispersion of 0.028592 Km/s
# CORRELATION INDEX:
cor(x6, x18) # confirms a positive correlation
## [1] 0.6554871
# CONFIDENCE INTERVAL:
# With a test statistic (DEV) of 63.453 and a p-value very close to 0,
# a strong dependency between the two variables is highlighted
ci.dumpx6x18 <- ciTest(dat, set=c("Mv", "CSBt"))
print(ci.dumpx6x18)
## Testing Mv _|_ CSBt
## Statistic (DEV): 63.453 df: 1 p-value: 0.0000 method: CHISQ
# log.t and CSBt
mqgx10x18 <- glm(y ~ x10*x18 + x6 + x12 + x14)
summary(mqgx10x18)
##
## Call:
## glm(formula = y ~ x10 * x18 + x6 + x12 + x14)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.010253 0.794629 -6.305 6.75e-09 ***
## x10 1.608991 0.101250 15.891 < 2e-16 ***
## x18 -0.174181 0.103318 -1.686 0.09476 .
## x6 0.517715 0.038949 13.292 < 2e-16 ***
## x12 0.235078 0.002534 92.775 < 2e-16 ***
## x14 1.364585 0.104471 13.062 < 2e-16 ***
## x10:x18 -0.035867 0.010617 -3.378 0.00102 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.02440424)
##
## Null deviance: 1296.3894 on 112 degrees of freedom
## Residual deviance: 2.5868 on 106 degrees of freedom
## AIC: -90.115
##
## Number of Fisher Scoring iterations: 2
# With a p-value of 0.00102, there is a significant dependency between central relaxation time and
# central surface brightness in the context of velocity dispersion.
# An increase in the interaction leads to a reduction in velocity dispersion
# of 0.035867 Km/s
# CORRELATION INDEX:
cor(x10, x18) # confirms a positive correlation
## [1] 0.5149092
# CONFIDENCE INTERVAL:
# Shows a test statistic (DEV) of 34.811 with a p-value very close to 0,
# denoting a strong dependency between the two variables
ci.dumpx6x18 <- ciTest(dat, set=c("log.t", "CSBt"))
print(ci.dumpx6x18)
## Testing log.t _|_ CSBt
## Statistic (DEV): 34.811 df: 1 p-value: 0.0000 method: CHISQ
# V.esc and CSBt
mqgx12x18 <- glm(y ~ x12*x18 + x6 + x10 + x14)
summary(mqgx12x18)
##
## Call:
## glm(formula = y ~ x12 * x18 + x6 + x10 + x14)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.786144 0.232251 -7.691 7.92e-12 ***
## x12 0.218292 0.004236 51.527 < 2e-16 ***
## x18 -0.513986 0.030320 -16.952 < 2e-16 ***
## x6 0.608129 0.040641 14.964 < 2e-16 ***
## x10 1.316115 0.063063 20.870 < 2e-16 ***
## x14 1.164030 0.122462 9.505 7.29e-16 ***
## x12:x18 0.003507 0.000796 4.405 2.53e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.02284832)
##
## Null deviance: 1296.3894 on 112 degrees of freedom
## Residual deviance: 2.4219 on 106 degrees of freedom
## AIC: -97.559
##
## Number of Fisher Scoring iterations: 2
# With a p-value of 2.53e-05, the interaction between escape velocity and central surface
# brightness is highly significant for velocity dispersion.
# An increase in the interaction leads to an increase in velocity dispersion
# of 0.03507 Km/s
# CORRELATION INDEX:
cor(x12, x18) # confirms a strong negative correlation
## [1] -0.7282425
# CONFIDENCE INTERVAL:
# The test statistic (DEV) is 85.399 and a p-value very close to 0 suggest
# a strong dependency between the two variables
ci.dumpx12x18 <- ciTest(dat, set=c("V.esc", "CSBt"))
print(ci.dumpx12x18)
## Testing V.esc _|_ CSBt
## Statistic (DEV): 85.399 df: 1 p-value: 0.0000 method: CHISQ
# CORRELATION RESULT ANALYSIS:
# It is evident that central surface brightness is influenced by absolute magnitude,
# central relaxation time, and central escape velocity!
# UPDATING THE MODEL EXPRESSING THE IDENTIFIED INTERACTIONS BETWEEN VARIABLES:
mqgf <- glm(y ~ x6*x18 + x10*x18 + x12*x18 + x14)
print(summary(mqgf))
##
## Call:
## glm(formula = y ~ x6 * x18 + x10 * x18 + x12 * x18 + x14)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.554126 0.999568 -2.555 0.01206 *
## x6 0.808217 0.104794 7.712 7.77e-12 ***
## x18 -0.362439 0.114506 -3.165 0.00203 **
## x10 1.480536 0.110841 13.357 < 2e-16 ***
## x12 0.238598 0.008667 27.531 < 2e-16 ***
## x14 0.996829 0.137669 7.241 8.00e-11 ***
## x6:x18 -0.023628 0.010148 -2.328 0.02183 *
## x18:x10 -0.027620 0.012746 -2.167 0.03252 *
## x18:x12 0.001511 0.001076 1.404 0.16340
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.02174587)
##
## Null deviance: 1296.3894 on 112 degrees of freedom
## Residual deviance: 2.2616 on 104 degrees of freedom
## AIC: -101.3
##
## Number of Fisher Scoring iterations: 2
# ANALYSIS OF THE NEW MODEL:
# The previous glm model, which does not include interaction terms, already
# shows good consistency with the data, as indicated by the low AIC and high
# significance of the coefficients.
# However, the addition of interaction terms has led to some improvements,
# as indicated by an even lower AIC in the second model.
# The interaction x12:x18 is not significant!
# PROCEEDING WITH THE REMOVAL OF SUPERFLUOUS INTERACTION
# REMOVING the interaction between x12 and x18
mqgf <- glm(y ~ x6*x18 + x10*x18 + x12 + x14)
print(summary(mqgf))
##
## Call:
## glm(formula = y ~ x6 * x18 + x10 * x18 + x12 + x14)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.115921 0.920163 -3.386 0.000998 ***
## x6 0.852553 0.100381 8.493 1.43e-13 ***
## x18 -0.291760 0.103315 -2.824 0.005676 **
## x10 1.556646 0.097119 16.028 < 2e-16 ***
## x12 0.248964 0.004556 54.645 < 2e-16 ***
## x14 1.027439 0.136557 7.524 1.90e-11 ***
## x6:x18 -0.031114 0.008673 -3.587 0.000509 ***
## x18:x10 -0.038621 0.010098 -3.825 0.000223 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.02194682)
##
## Null deviance: 1296.3894 on 112 degrees of freedom
## Residual deviance: 2.3044 on 105 degrees of freedom
## AIC: -101.18
##
## Number of Fisher Scoring iterations: 2
# The resulting model is further improved and the significance of the
# variables and interactions has increased.
# There is better model quality in terms of balance between fit
# and complexity (More balanced)
# CORRELATION GRAPHS BETWEEN VARIABLES AND CENTRAL VELOCITY DISPERSION:
plot(x6, y)
plot(x10, y)
plot(x12, y)
plot(x14, y)
plot(x18, y)
# FROM THESE GRAPHS, IT IS IMMEDIATELY VISIBLE THAT THE CENTRAL ESCAPE VELOCITY
# IS CLOSELY CORRELATED TO THE CENTRAL VELOCITY DISPERSION
# AS THE TWO QUANTITIES INCREASE LINEARLY
# CORRELATION INDEX:
cor(x12, y) # Extreme positive correlation
## [1] 0.9939332
# PHASE 7:
# MULTIVARIATE ANALYSIS BASED ON GRAPHICAL MODELS:
# Searching for a formula for the independent model (BIC Forward), based on the
# general linear model identified as final
mmug <- cmod(~.^1, marginal =c("Mv","log.t", "V.esc","E.B.V", "CSBt"), data = dat, fit = TRUE)
mmug_bic <- stepwise(mmug, direction = "forward", k=log(sum(dat))) # BIC forward
plot(mmug_bic)
print(formula(mmug_bic))
## ~E.B.V * Mv * CSBt * V.esc * log.t
print(ciTest(dat, set = c("Mv","log.t", "V.esc","E.B.V", "CSBt")))
## Testing Mv _|_ log.t | V.esc E.B.V CSBt
## Statistic (DEV): 127.832 df: 1 p-value: 0.0000 method: CHISQ
set.frame <- data.frame(
Mv = dat$Mv,
log.t = dat$log.t,
V.esc = dat$V.esc,
E.B.V = dat$E.B.V,
CSBt = dat$CSBt
)
print(cor(set.frame))
## Mv log.t V.esc E.B.V CSBt
## Mv 1.00000000 -0.08430665 -0.8085964 0.05965678 0.6554871
## log.t -0.08430665 1.00000000 -0.3096412 -0.39958765 0.5149092
## V.esc -0.80859641 -0.30964120 1.0000000 0.20965656 -0.7282425
## E.B.V 0.05965678 -0.39958765 0.2096566 1.00000000 0.2083057
## CSBt 0.65548711 0.51490924 -0.7282425 0.20830574 1.0000000
# Based on the correlation matrix, removing less significant dependencies (values close to zero)
# Discarding weak correlations |corr| < 0.5
mmug_bic <- update(mmug_bic, list(dedge=~Mv * E.B.V + log.t * Mv + V.esc * log.t + E.B.V * log.t + V.esc * E.B.V + E.B.V * CSBt))
plot(mmug_bic)
print(formula(mmug_bic))
## ~E.B.V + log.t * CSBt + Mv * CSBt * V.esc
# Found formula:
# ~E.B.V + log.t * CSBt + Mv * CSBt * V.esc
# Creating a Bayesian network
mbn <- hc(set.frame)
plot(mbn)
# In this case too, the same correlations between
# the variables of the final model with their respective directions have been identified.