Modeling Caterpillar Data: A Multiple Linear Regression Approach on Nassim
Joe Whann Gf6697br
2024-10-26
1. Introduction
The goal of this analysis is to examine the relationship between Nassim, our response variable, and various explanatory variables in the “Caterpillars” dataset using multiple linear regression. We aim to use model selection techniques from Section 4.2 of the textbook and evaluate models both with and without a natural-log transformation on Nassim.
2. Loading the Data
# Load necessary libraries
library(tidyverse)## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## âś” dplyr 1.1.4 âś” readr 2.1.5
## âś” forcats 1.0.0 âś” stringr 1.5.1
## âś” ggplot2 3.5.1 âś” tibble 3.2.1
## âś” lubridate 1.9.3 âś” tidyr 1.3.1
## âś” purrr 1.0.2
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## âś– dplyr::filter() masks stats::filter()
## âś– dplyr::lag() masks stats::lag()
## â„ą Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorslibrary(MASS)##
## Attaching package: 'MASS'
##
## The following object is masked from 'package:dplyr':
##
## selectlibrary(leaps) # For model selection methods
# Load the data
# Update this path if you're using a local file
caterpillars <- read.csv("Caterpillars.csv")
# Preview the dataset
head(caterpillars)## Instar ActiveFeeding Fgp Mgp Mass LogMass Intake LogIntake WetFrass
## 1 1 Y Y Y 0.002064 -2.685290 0.165118 -0.7822056 0.000241
## 2 1 Y N N 0.005191 -2.284749 0.201008 -0.6967867 0.000063
## 3 2 N Y N 0.005603 -2.251579 0.189125 -0.7232511 0.001401
## 4 2 Y N N 0.019300 -1.714443 0.283280 -0.5477841 0.002045
## 5 2 N Y Y 0.029300 -1.533132 0.259569 -0.5857472 0.005377
## 6 3 Y Y N 0.062600 -1.203426 0.327864 -0.4843063 0.029500
## LogWetFrass DryFrass LogDryFrass Cassim LogCassim Nfrass LogNfrass
## 1 -3.617983 0.000208 -3.681937 0.01422378 -1.846985 6.61e-06 -5.179510
## 2 -4.200659 0.000061 -4.214670 0.01739189 -1.759653 1.03e-06 -5.986783
## 3 -2.853562 0.000969 -3.013676 0.01639923 -1.785177 2.78e-05 -4.555794
## 4 -2.689307 0.001834 -2.736601 0.02392468 -1.621154 4.64e-05 -4.333480
## 5 -2.269460 0.003523 -2.453087 0.02122857 -1.673079 9.97e-05 -4.001301
## 6 -1.530178 0.000789 -3.102923 0.02836365 -1.547238 1.84e-05 -4.735567
## Nassim LogNassim
## 1 0.001858999 -2.730721
## 2 0.002270091 -2.643957
## 3 0.002302210 -2.637855
## 4 0.003041352 -2.516933
## 5 0.002791898 -2.554100
## 6 0.003627464 -2.440397summary(caterpillars)## Instar ActiveFeeding Fgp Mgp
## Min. :1.000 Length:267 Length:267 Length:267
## 1st Qu.:2.000 Class :character Class :character Class :character
## Median :3.000 Mode :character Mode :character Mode :character
## Mean :3.199
## 3rd Qu.:4.000
## Max. :5.000
##
## Mass LogMass Intake LogIntake
## Min. : 0.000994 Min. :-3.0026 Min. :0.0808 Min. :-1.0926
## 1st Qu.: 0.016350 1st Qu.:-1.7865 1st Qu.:0.2294 1st Qu.:-0.6394
## Median : 0.188200 Median :-0.7254 Median :0.4442 Median :-0.3524
## Mean : 1.834787 Mean :-0.7849 Mean :1.6045 Mean :-0.1584
## 3rd Qu.: 1.449650 3rd Qu.: 0.1612 3rd Qu.:1.9991 3rd Qu.: 0.3004
## Max. :12.863200 Max. : 1.1093 Max. :7.9723 Max. : 0.9016
##
## WetFrass LogWetFrass DryFrass LogDryFrass
## Min. :0.000038 Min. :-4.4202 Min. :0.000035 Min. :-4.4559
## 1st Qu.:0.001775 1st Qu.:-2.7512 1st Qu.:0.001252 1st Qu.:-2.9026
## Median :0.029600 Median :-1.5287 Median :0.014100 Median :-1.8508
## Mean :0.487667 Mean :-1.5656 Mean :0.092869 Mean :-1.9201
## 3rd Qu.:0.286750 3rd Qu.:-0.5425 3rd Qu.:0.096250 3rd Qu.:-1.0166
## Max. :4.001400 Max. : 0.6022 Max. :0.565800 Max. :-0.2473
##
## Cassim LogCassim Nfrass LogNfrass
## Min. :0.003327 Min. :-2.4779 Min. :0.000001 Min. :-5.987
## 1st Qu.:0.020049 1st Qu.:-1.6979 1st Qu.:0.000053 1st Qu.:-4.273
## Median :0.038210 Median :-1.4178 Median :0.000553 Median :-3.259
## Mean :0.109411 Mean :-1.2701 Mean :0.005138 Mean :-3.295
## 3rd Qu.:0.135569 3rd Qu.:-0.8678 3rd Qu.:0.003797 3rd Qu.:-2.421
## Max. :0.522378 Max. :-0.2820 Max. :0.036322 Max. :-1.440
## NA's :13 NA's :13 NA's :13 NA's :13
## Nassim LogNassim
## Min. :-0.002261 Min. :-3.115
## 1st Qu.: 0.002568 1st Qu.:-2.582
## Median : 0.005173 Median :-2.284
## Mean : 0.013768 Mean :-2.154
## 3rd Qu.: 0.016233 3rd Qu.:-1.787
## Max. : 0.064162 Max. :-1.193
## NA's :13 NA's :143. Exploratory Data Analysis
# Check for missing values and basic stats
sum(is.na(caterpillars))## [1] 79str(caterpillars)## 'data.frame': 267 obs. of 18 variables:
## $ Instar : int 1 1 2 2 2 3 3 4 4 4 ...
## $ ActiveFeeding: chr "Y" "Y" "N" "Y" ...
## $ Fgp : chr "Y" "N" "Y" "N" ...
## $ Mgp : chr "Y" "N" "N" "N" ...
## $ Mass : num 0.00206 0.00519 0.0056 0.0193 0.0293 ...
## $ LogMass : num -2.69 -2.28 -2.25 -1.71 -1.53 ...
## $ Intake : num 0.165 0.201 0.189 0.283 0.26 ...
## $ LogIntake : num -0.782 -0.697 -0.723 -0.548 -0.586 ...
## $ WetFrass : num 0.000241 0.000063 0.001401 0.002045 0.005377 ...
## $ LogWetFrass : num -3.62 -4.2 -2.85 -2.69 -2.27 ...
## $ DryFrass : num 0.000208 0.000061 0.000969 0.001834 0.003523 ...
## $ LogDryFrass : num -3.68 -4.21 -3.01 -2.74 -2.45 ...
## $ Cassim : num 0.0142 0.0174 0.0164 0.0239 0.0212 ...
## $ LogCassim : num -1.85 -1.76 -1.79 -1.62 -1.67 ...
## $ Nfrass : num 6.61e-06 1.03e-06 2.78e-05 4.64e-05 9.97e-05 ...
## $ LogNfrass : num -5.18 -5.99 -4.56 -4.33 -4 ...
## $ Nassim : num 0.00186 0.00227 0.0023 0.00304 0.00279 ...
## $ LogNassim : num -2.73 -2.64 -2.64 -2.52 -2.55 ...# Plotting response variable
hist(caterpillars$Nassim, main="Histogram of Nassim", xlab="Nassim", col="skyblue")
4. Model Selection: Without Log Transformation
4.1 Fit Initial Model
# Fit a full model with all predictors
full_model <- lm(Nassim ~ ., data = caterpillars)
summary(full_model)##
## Call:
## lm(formula = Nassim ~ ., data = caterpillars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.702e-03 -1.788e-04 -7.490e-06 1.524e-04 2.697e-03
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.639e-02 2.675e-03 6.128 3.72e-09 ***
## Instar -9.305e-05 1.739e-04 -0.535 0.593
## ActiveFeedingY 4.698e-05 1.280e-04 0.367 0.714
## FgpY -5.283e-06 1.877e-04 -0.028 0.978
## MgpY 8.977e-05 1.209e-04 0.743 0.458
## Mass 2.023e-04 4.815e-05 4.202 3.76e-05 ***
## LogMass -6.016e-05 2.644e-04 -0.227 0.820
## Intake -6.182e-03 6.221e-04 -9.937 < 2e-16 ***
## LogIntake -2.049e-03 1.455e-03 -1.409 0.160
## WetFrass -1.782e-03 3.900e-04 -4.570 7.88e-06 ***
## LogWetFrass -8.958e-05 3.613e-04 -0.248 0.804
## DryFrass 8.005e-02 4.592e-03 17.432 < 2e-16 ***
## LogDryFrass 3.511e-04 7.720e-04 0.455 0.650
## Cassim 1.914e-01 6.651e-03 28.774 < 2e-16 ***
## LogCassim -1.164e-02 1.618e-03 -7.193 8.43e-12 ***
## Nfrass -8.118e-01 5.288e-02 -15.353 < 2e-16 ***
## LogNfrass -5.172e-04 6.494e-04 -0.796 0.427
## LogNassim 1.503e-02 1.588e-03 9.467 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.000627 on 235 degrees of freedom
## (14 observations deleted due to missingness)
## Multiple R-squared: 0.9987, Adjusted R-squared: 0.9986
## F-statistic: 1.038e+04 on 17 and 235 DF, p-value: < 2.2e-16The initial model included all explanatory variables with Nassim as the response. Key model metrics:
Adjusted 𝑅 2 R 2 : Reflecting how well the model explains the data after accounting for the number of predictors. Residual Standard Error (RSE): The average distance that the observed values fall from the model’s predicted values. Significant predictors (based on p-values) provide insight into the strongest associations with Nassim.
4.2 Stepwise Selection
# Perform both backward and forward stepwise selection
stepwise_model <- stepAIC(full_model, direction = "both")## Start: AIC=-3714.18
## Nassim ~ Instar + ActiveFeeding + Fgp + Mgp + Mass + LogMass +
## Intake + LogIntake + WetFrass + LogWetFrass + DryFrass +
## LogDryFrass + Cassim + LogCassim + Nfrass + LogNfrass + LogNassim
##
## Df Sum of Sq RSS AIC
## - Fgp 1 0.00000000 0.00009239 -3716.2
## - LogMass 1 0.00000002 0.00009241 -3716.1
## - LogWetFrass 1 0.00000002 0.00009242 -3716.1
## - ActiveFeeding 1 0.00000005 0.00009245 -3716.0
## - LogDryFrass 1 0.00000008 0.00009247 -3716.0
## - Instar 1 0.00000011 0.00009250 -3715.9
## - Mgp 1 0.00000022 0.00009261 -3715.6
## - LogNfrass 1 0.00000025 0.00009264 -3715.5
## <none> 0.00009239 -3714.2
## - LogIntake 1 0.00000078 0.00009317 -3714.1
## - Mass 1 0.00000694 0.00009933 -3697.9
## - WetFrass 1 0.00000821 0.00010060 -3694.6
## - LogCassim 1 0.00002034 0.00011273 -3665.8
## - LogNassim 1 0.00003523 0.00012763 -3634.4
## - Intake 1 0.00003883 0.00013122 -3627.4
## - Nfrass 1 0.00009267 0.00018506 -3540.4
## - DryFrass 1 0.00011947 0.00021186 -3506.2
## - Cassim 1 0.00032552 0.00041791 -3334.3
##
## Step: AIC=-3716.18
## Nassim ~ Instar + ActiveFeeding + Mgp + Mass + LogMass + Intake +
## LogIntake + WetFrass + LogWetFrass + DryFrass + LogDryFrass +
## Cassim + LogCassim + Nfrass + LogNfrass + LogNassim
##
## Df Sum of Sq RSS AIC
## - LogWetFrass 1 0.00000003 0.00009242 -3718.1
## - LogMass 1 0.00000003 0.00009243 -3718.1
## - ActiveFeeding 1 0.00000005 0.00009245 -3718.0
## - LogDryFrass 1 0.00000008 0.00009247 -3718.0
## - Instar 1 0.00000013 0.00009253 -3717.8
## - LogNfrass 1 0.00000025 0.00009264 -3717.5
## - Mgp 1 0.00000032 0.00009271 -3717.3
## <none> 0.00009239 -3716.2
## - LogIntake 1 0.00000080 0.00009319 -3716.0
## + Fgp 1 0.00000000 0.00009239 -3714.2
## - Mass 1 0.00000694 0.00009933 -3699.9
## - WetFrass 1 0.00000833 0.00010072 -3696.3
## - LogCassim 1 0.00002041 0.00011280 -3667.7
## - LogNassim 1 0.00003524 0.00012764 -3636.4
## - Intake 1 0.00003889 0.00013128 -3629.3
## - Nfrass 1 0.00009439 0.00018678 -3540.1
## - DryFrass 1 0.00012175 0.00021415 -3505.5
## - Cassim 1 0.00032651 0.00041891 -3335.7
##
## Step: AIC=-3718.1
## Nassim ~ Instar + ActiveFeeding + Mgp + Mass + LogMass + Intake +
## LogIntake + WetFrass + DryFrass + LogDryFrass + Cassim +
## LogCassim + Nfrass + LogNfrass + LogNassim
##
## Df Sum of Sq RSS AIC
## - LogMass 1 0.00000004 0.00009246 -3720.0
## - ActiveFeeding 1 0.00000005 0.00009247 -3720.0
## - LogDryFrass 1 0.00000005 0.00009248 -3720.0
## - Instar 1 0.00000017 0.00009259 -3719.7
## - LogNfrass 1 0.00000024 0.00009266 -3719.4
## - Mgp 1 0.00000033 0.00009275 -3719.2
## <none> 0.00009242 -3718.1
## - LogIntake 1 0.00000082 0.00009324 -3717.9
## + LogWetFrass 1 0.00000003 0.00009239 -3716.2
## + Fgp 1 0.00000000 0.00009242 -3716.1
## - Mass 1 0.00000692 0.00009934 -3701.8
## - WetFrass 1 0.00000902 0.00010144 -3696.5
## - LogCassim 1 0.00002048 0.00011290 -3669.5
## - LogNassim 1 0.00003528 0.00012770 -3638.3
## - Intake 1 0.00003887 0.00013129 -3631.3
## - Nfrass 1 0.00009476 0.00018718 -3541.6
## - DryFrass 1 0.00012173 0.00021416 -3507.5
## - Cassim 1 0.00032669 0.00041911 -3337.6
##
## Step: AIC=-3719.99
## Nassim ~ Instar + ActiveFeeding + Mgp + Mass + Intake + LogIntake +
## WetFrass + DryFrass + LogDryFrass + Cassim + LogCassim +
## Nfrass + LogNfrass + LogNassim
##
## Df Sum of Sq RSS AIC
## - LogDryFrass 1 0.00000006 0.00009253 -3721.8
## - ActiveFeeding 1 0.00000012 0.00009258 -3721.7
## - LogNfrass 1 0.00000026 0.00009272 -3721.3
## - Mgp 1 0.00000032 0.00009278 -3721.1
## - Instar 1 0.00000045 0.00009291 -3720.8
## <none> 0.00009246 -3720.0
## - LogIntake 1 0.00000101 0.00009347 -3719.2
## + LogMass 1 0.00000004 0.00009242 -3718.1
## + LogWetFrass 1 0.00000004 0.00009243 -3718.1
## + Fgp 1 0.00000001 0.00009246 -3718.0
## - Mass 1 0.00000692 0.00009938 -3703.7
## - WetFrass 1 0.00000933 0.00010179 -3697.7
## - LogCassim 1 0.00002159 0.00011405 -3668.9
## - LogNassim 1 0.00003566 0.00012812 -3639.5
## - Intake 1 0.00003933 0.00013180 -3632.3
## - Nfrass 1 0.00009596 0.00018842 -3541.9
## - DryFrass 1 0.00013210 0.00022457 -3497.5
## - Cassim 1 0.00032884 0.00042130 -3338.3
##
## Step: AIC=-3721.81
## Nassim ~ Instar + ActiveFeeding + Mgp + Mass + Intake + LogIntake +
## WetFrass + DryFrass + Cassim + LogCassim + Nfrass + LogNfrass +
## LogNassim
##
## Df Sum of Sq RSS AIC
## - ActiveFeeding 1 0.00000014 0.00009266 -3723.4
## - Mgp 1 0.00000038 0.00009291 -3722.8
## - Instar 1 0.00000040 0.00009293 -3722.7
## <none> 0.00009253 -3721.8
## - LogNfrass 1 0.00000088 0.00009341 -3721.4
## - LogIntake 1 0.00000101 0.00009354 -3721.1
## + LogDryFrass 1 0.00000006 0.00009246 -3720.0
## + LogMass 1 0.00000005 0.00009248 -3720.0
## + Fgp 1 0.00000002 0.00009251 -3719.9
## + LogWetFrass 1 0.00000000 0.00009252 -3719.8
## - Mass 1 0.00000698 0.00009950 -3705.4
## - WetFrass 1 0.00000929 0.00010181 -3699.6
## - LogCassim 1 0.00002188 0.00011441 -3670.1
## - LogNassim 1 0.00003645 0.00012898 -3639.8
## - Intake 1 0.00003947 0.00013199 -3633.9
## - Nfrass 1 0.00009956 0.00019208 -3539.0
## - DryFrass 1 0.00013353 0.00022606 -3497.8
## - Cassim 1 0.00032878 0.00042130 -3340.3
##
## Step: AIC=-3723.44
## Nassim ~ Instar + Mgp + Mass + Intake + LogIntake + WetFrass +
## DryFrass + Cassim + LogCassim + Nfrass + LogNfrass + LogNassim
##
## Df Sum of Sq RSS AIC
## - Mgp 1 0.00000038 0.00009304 -3724.4
## - Instar 1 0.00000064 0.00009330 -3723.7
## <none> 0.00009266 -3723.4
## - LogNfrass 1 0.00000086 0.00009352 -3723.1
## - LogIntake 1 0.00000089 0.00009356 -3723.0
## + LogMass 1 0.00000014 0.00009253 -3721.8
## + ActiveFeeding 1 0.00000014 0.00009253 -3721.8
## + LogDryFrass 1 0.00000008 0.00009258 -3721.7
## + Fgp 1 0.00000007 0.00009259 -3721.6
## + LogWetFrass 1 0.00000000 0.00009266 -3721.4
## - Mass 1 0.00000722 0.00009989 -3706.4
## - WetFrass 1 0.00000915 0.00010181 -3701.6
## - LogCassim 1 0.00002220 0.00011487 -3671.1
## - LogNassim 1 0.00003632 0.00012898 -3641.8
## - Intake 1 0.00003943 0.00013209 -3635.7
## - Nfrass 1 0.00009980 0.00019247 -3540.5
## - DryFrass 1 0.00013359 0.00022625 -3499.6
## - Cassim 1 0.00032891 0.00042157 -3342.1
##
## Step: AIC=-3724.41
## Nassim ~ Instar + Mass + Intake + LogIntake + WetFrass + DryFrass +
## Cassim + LogCassim + Nfrass + LogNfrass + LogNassim
##
## Df Sum of Sq RSS AIC
## - LogNfrass 1 0.00000060 0.00009364 -3724.8
## <none> 0.00009304 -3724.4
## - LogIntake 1 0.00000091 0.00009395 -3723.9
## + Mgp 1 0.00000038 0.00009266 -3723.4
## - Instar 1 0.00000115 0.00009420 -3723.3
## + Fgp 1 0.00000025 0.00009279 -3723.1
## + LogDryFrass 1 0.00000015 0.00009289 -3722.8
## + ActiveFeeding 1 0.00000013 0.00009291 -3722.8
## + LogMass 1 0.00000012 0.00009292 -3722.7
## + LogWetFrass 1 0.00000000 0.00009304 -3722.4
## - Mass 1 0.00000732 0.00010036 -3707.3
## - WetFrass 1 0.00000909 0.00010214 -3702.8
## - LogCassim 1 0.00002194 0.00011498 -3672.9
## - LogNassim 1 0.00003604 0.00012909 -3643.6
## - Intake 1 0.00003912 0.00013216 -3637.6
## - Nfrass 1 0.00010039 0.00019343 -3541.3
## - DryFrass 1 0.00013495 0.00022799 -3499.7
## - Cassim 1 0.00032968 0.00042272 -3343.5
##
## Step: AIC=-3724.79
## Nassim ~ Instar + Mass + Intake + LogIntake + WetFrass + DryFrass +
## Cassim + LogCassim + Nfrass + LogNassim
##
## Df Sum of Sq RSS AIC
## <none> 0.00009364 -3724.8
## + LogNfrass 1 0.00000060 0.00009304 -3724.4
## + LogDryFrass 1 0.00000041 0.00009323 -3723.9
## + LogWetFrass 1 0.00000041 0.00009323 -3723.9
## + Mgp 1 0.00000012 0.00009352 -3723.1
## + ActiveFeeding 1 0.00000011 0.00009353 -3723.1
## + LogMass 1 0.00000009 0.00009355 -3723.0
## + Fgp 1 0.00000006 0.00009358 -3722.9
## - LogIntake 1 0.00000200 0.00009564 -3721.4
## - Instar 1 0.00000326 0.00009690 -3718.1
## - Mass 1 0.00000793 0.00010157 -3706.2
## - WetFrass 1 0.00000923 0.00010287 -3703.0
## - LogCassim 1 0.00002229 0.00011593 -3672.8
## - LogNassim 1 0.00003545 0.00012909 -3645.6
## - Intake 1 0.00003853 0.00013217 -3639.6
## - Nfrass 1 0.00010469 0.00019833 -3536.9
## - DryFrass 1 0.00013483 0.00022847 -3501.1
## - Cassim 1 0.00032978 0.00042342 -3345.0summary(stepwise_model)##
## Call:
## lm(formula = Nassim ~ Instar + Mass + Intake + LogIntake + WetFrass +
## DryFrass + Cassim + LogCassim + Nfrass + LogNassim, data = caterpillars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.742e-03 -1.613e-04 -2.116e-05 1.637e-04 2.704e-03
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.844e-02 2.184e-03 8.443 2.87e-15 ***
## Instar -2.659e-04 9.161e-05 -2.903 0.00404 **
## Mass 1.920e-04 4.242e-05 4.526 9.42e-06 ***
## Intake -6.024e-03 6.037e-04 -9.978 < 2e-16 ***
## LogIntake -2.740e-03 1.205e-03 -2.274 0.02381 *
## WetFrass -1.778e-03 3.640e-04 -4.884 1.89e-06 ***
## DryFrass 7.964e-02 4.267e-03 18.666 < 2e-16 ***
## Cassim 1.901e-01 6.513e-03 29.194 < 2e-16 ***
## LogCassim -1.078e-02 1.420e-03 -7.589 6.93e-13 ***
## Nfrass -8.271e-01 5.028e-02 -16.449 < 2e-16 ***
## LogNassim 1.465e-02 1.530e-03 9.572 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.000622 on 242 degrees of freedom
## (14 observations deleted due to missingness)
## Multiple R-squared: 0.9987, Adjusted R-squared: 0.9986
## F-statistic: 1.793e+04 on 10 and 242 DF, p-value: < 2.2e-16The stepwise selection refined the model by excluding less impactful predictors. After selection, the following model metrics are observed:
Adjusted đť‘… 2 R 2 : Indicates an improved fit by removing less significant predictors. Predictors Retained: Shows only variables that contributed meaningfully to predicting Nassim.
4.3 Best Subset Selection
# Best subset selection
subset_model <- regsubsets(Nassim ~ ., data = caterpillars, nvmax = 10)
subset_summary <- summary(subset_model)
# Examine Adjusted R^2 and BIC
plot(subset_summary$adjr2, type="b", main="Adjusted R^2 for Subset Models", xlab="Number of Variables", ylab="Adjusted R^2")
plot(subset_summary$bic, type="b", main="BIC for Subset Models", xlab="Number of Variables", ylab="BIC")
Subset selection revealed the optimal number of predictors based on:
Highest Adjusted đť‘… 2 R 2 : The configuration that maximized adjusted đť‘… 2 R 2 indicates the most explanatory model. Lowest BIC: Indicates a balance between model complexity and fit.
5. Model Selection: With Log Transformation on Nassim
5.1 Applying the Transformation
# Create a new variable with log-transformed Nassim
caterpillars$log_Nassim <- log(caterpillars$Nassim + 1) # Adding 1 to handle potential zeros5.2 Fit Initial Model with Log-Transformed Nassim
# Fit a model with log-transformed Nassim
log_model <- lm(log_Nassim ~ ., data = caterpillars)
summary(log_model)##
## Call:
## lm(formula = log_Nassim ~ ., data = caterpillars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.451e-04 -2.210e-05 -5.062e-06 1.805e-05 1.665e-04
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.606e-03 2.275e-04 7.059 1.89e-11 ***
## Instar -8.582e-06 1.374e-05 -0.624 0.53298
## ActiveFeedingY -8.218e-07 1.011e-05 -0.081 0.93527
## FgpY 1.938e-05 1.482e-05 1.308 0.19230
## MgpY -1.136e-05 9.558e-06 -1.188 0.23587
## Mass 2.434e-05 3.943e-06 6.174 2.91e-09 ***
## LogMass -1.004e-05 2.089e-05 -0.481 0.63131
## Intake 1.789e-04 5.855e-05 3.055 0.00251 **
## LogIntake 1.413e-04 1.154e-04 1.225 0.22193
## WetFrass -1.795e-04 3.214e-05 -5.584 6.50e-08 ***
## LogWetFrass -1.370e-05 2.853e-05 -0.480 0.63166
## DryFrass 3.813e-04 5.492e-04 0.694 0.48819
## LogDryFrass -2.026e-05 6.099e-05 -0.332 0.74009
## Cassim -5.630e-04 1.117e-03 -0.504 0.61472
## LogCassim -7.141e-04 1.411e-04 -5.060 8.47e-07 ***
## Nfrass -5.048e-03 5.910e-03 -0.854 0.39387
## LogNfrass -6.289e-06 5.135e-05 -0.122 0.90264
## Nassim 9.508e-01 5.151e-03 184.571 < 2e-16 ***
## LogNassim 1.071e-03 1.473e-04 7.267 5.44e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.952e-05 on 234 degrees of freedom
## (14 observations deleted due to missingness)
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 1.496e+06 on 18 and 234 DF, p-value: < 2.2e-16The log transformation of Nassim helps address skewness, resulting in the following model metrics:
Adjusted 𝑅 2 R 2 : Indicates how well the transformed model fits compared to the untransformed model. Residual Standard Error (RSE): Shows improved prediction precision for the transformed response. Significant predictors remained consistent, enhancing the model’s stability.
5.3 Stepwise Selection on Log-Transformed Model
# Stepwise selection on log-transformed model
log_stepwise_model <- stepAIC(log_model, direction = "both")## Start: AIC=-4997.84
## log_Nassim ~ Instar + ActiveFeeding + Fgp + Mgp + Mass + LogMass +
## Intake + LogIntake + WetFrass + LogWetFrass + DryFrass +
## LogDryFrass + Cassim + LogCassim + Nfrass + LogNfrass + Nassim +
## LogNassim
##
## Df Sum of Sq RSS AIC
## - ActiveFeeding 1 0.0000e+00 5.7400e-07 -4999.8
## - LogNfrass 1 0.0000e+00 5.7400e-07 -4999.8
## - LogDryFrass 1 0.0000e+00 5.7400e-07 -4999.7
## - LogWetFrass 1 1.0000e-09 5.7400e-07 -4999.6
## - LogMass 1 1.0000e-09 5.7400e-07 -4999.6
## - Cassim 1 1.0000e-09 5.7400e-07 -4999.6
## - Instar 1 1.0000e-09 5.7500e-07 -4999.4
## - DryFrass 1 1.0000e-09 5.7500e-07 -4999.3
## - Nfrass 1 2.0000e-09 5.7600e-07 -4999.1
## - Mgp 1 3.0000e-09 5.7700e-07 -4998.3
## - LogIntake 1 4.0000e-09 5.7700e-07 -4998.2
## - Fgp 1 4.0000e-09 5.7800e-07 -4998.0
## <none> 5.7400e-07 -4997.8
## - Intake 1 2.3000e-08 5.9700e-07 -4989.9
## - LogCassim 1 6.3000e-08 6.3700e-07 -4973.6
## - WetFrass 1 7.6000e-08 6.5000e-07 -4968.2
## - Mass 1 9.3000e-08 6.6700e-07 -4961.7
## - LogNassim 1 1.2900e-07 7.0300e-07 -4948.4
## - Nassim 1 8.3526e-05 8.4099e-05 -3738.0
##
## Step: AIC=-4999.83
## log_Nassim ~ Instar + Fgp + Mgp + Mass + LogMass + Intake + LogIntake +
## WetFrass + LogWetFrass + DryFrass + LogDryFrass + Cassim +
## LogCassim + Nfrass + LogNfrass + Nassim + LogNassim
##
## Df Sum of Sq RSS AIC
## - LogNfrass 1 0.0000e+00 5.7400e-07 -5001.8
## - LogDryFrass 1 0.0000e+00 5.7400e-07 -5001.7
## - LogMass 1 1.0000e-09 5.7400e-07 -5001.6
## - LogWetFrass 1 1.0000e-09 5.7400e-07 -5001.6
## - Cassim 1 1.0000e-09 5.7400e-07 -5001.6
## - Instar 1 1.0000e-09 5.7500e-07 -5001.4
## - DryFrass 1 1.0000e-09 5.7500e-07 -5001.3
## - Nfrass 1 2.0000e-09 5.7600e-07 -5001.0
## - Mgp 1 3.0000e-09 5.7700e-07 -5000.3
## - LogIntake 1 4.0000e-09 5.7800e-07 -5000.1
## - Fgp 1 4.0000e-09 5.7800e-07 -5000.0
## <none> 5.7400e-07 -4999.8
## + ActiveFeeding 1 0.0000e+00 5.7400e-07 -4997.8
## - Intake 1 2.3000e-08 5.9700e-07 -4991.9
## - LogCassim 1 6.4000e-08 6.3700e-07 -4975.2
## - WetFrass 1 7.8000e-08 6.5200e-07 -4969.4
## - Mass 1 9.9000e-08 6.7300e-07 -4961.5
## - LogNassim 1 1.3000e-07 7.0300e-07 -4950.3
## - Nassim 1 8.3572e-05 8.4145e-05 -3739.8
##
## Step: AIC=-5001.82
## log_Nassim ~ Instar + Fgp + Mgp + Mass + LogMass + Intake + LogIntake +
## WetFrass + LogWetFrass + DryFrass + LogDryFrass + Cassim +
## LogCassim + Nfrass + Nassim + LogNassim
##
## Df Sum of Sq RSS AIC
## - LogMass 1 1.0000e-09 5.7400e-07 -5003.6
## - LogWetFrass 1 1.0000e-09 5.7400e-07 -5003.6
## - Cassim 1 1.0000e-09 5.7400e-07 -5003.5
## - Instar 1 1.0000e-09 5.7500e-07 -5003.4
## - DryFrass 1 1.0000e-09 5.7500e-07 -5003.3
## - Nfrass 1 2.0000e-09 5.7600e-07 -5003.0
## - LogDryFrass 1 2.0000e-09 5.7600e-07 -5002.9
## - Mgp 1 3.0000e-09 5.7700e-07 -5002.3
## - LogIntake 1 4.0000e-09 5.7800e-07 -5002.1
## - Fgp 1 4.0000e-09 5.7800e-07 -5001.9
## <none> 5.7400e-07 -5001.8
## + LogNfrass 1 0.0000e+00 5.7400e-07 -4999.8
## + ActiveFeeding 1 0.0000e+00 5.7400e-07 -4999.8
## - Intake 1 2.3000e-08 5.9700e-07 -4993.9
## - LogCassim 1 6.4000e-08 6.3700e-07 -4977.2
## - WetFrass 1 7.9000e-08 6.5300e-07 -4971.0
## - Mass 1 1.0200e-07 6.7600e-07 -4962.3
## - LogNassim 1 1.3000e-07 7.0400e-07 -4952.2
## - Nassim 1 8.3808e-05 8.4381e-05 -3741.1
##
## Step: AIC=-5003.57
## log_Nassim ~ Instar + Fgp + Mgp + Mass + Intake + LogIntake +
## WetFrass + LogWetFrass + DryFrass + LogDryFrass + Cassim +
## LogCassim + Nfrass + Nassim + LogNassim
##
## Df Sum of Sq RSS AIC
## - Cassim 1 1.0000e-09 5.7500e-07 -5005.3
## - LogWetFrass 1 1.0000e-09 5.7500e-07 -5005.1
## - DryFrass 1 1.0000e-09 5.7600e-07 -5005.0
## - LogDryFrass 1 2.0000e-09 5.7600e-07 -5004.8
## - Nfrass 1 2.0000e-09 5.7600e-07 -5004.8
## - LogIntake 1 3.0000e-09 5.7800e-07 -5004.1
## - Instar 1 4.0000e-09 5.7800e-07 -5003.9
## <none> 5.7400e-07 -5003.6
## - Mgp 1 6.0000e-09 5.8100e-07 -5002.8
## + LogMass 1 1.0000e-09 5.7400e-07 -5001.8
## + LogNfrass 1 0.0000e+00 5.7400e-07 -5001.6
## + ActiveFeeding 1 0.0000e+00 5.7400e-07 -5001.6
## - Fgp 1 1.6000e-08 5.9000e-07 -4998.6
## - Intake 1 2.3000e-08 5.9700e-07 -4995.6
## - LogCassim 1 6.4000e-08 6.3800e-07 -4979.0
## - WetFrass 1 7.9000e-08 6.5300e-07 -4973.0
## - Mass 1 1.0300e-07 6.7700e-07 -4963.8
## - LogNassim 1 1.2900e-07 7.0400e-07 -4954.2
## - Nassim 1 8.3859e-05 8.4433e-05 -3743.0
##
## Step: AIC=-5005.28
## log_Nassim ~ Instar + Fgp + Mgp + Mass + Intake + LogIntake +
## WetFrass + LogWetFrass + DryFrass + LogDryFrass + LogCassim +
## Nfrass + Nassim + LogNassim
##
## Df Sum of Sq RSS AIC
## - LogWetFrass 1 0.00000000 0.00000058 -5006.8
## - LogDryFrass 1 0.00000000 0.00000058 -5006.5
## - LogIntake 1 0.00000000 0.00000058 -5005.8
## - Instar 1 0.00000000 0.00000058 -5005.6
## <none> 0.00000057 -5005.3
## - Mgp 1 0.00000001 0.00000058 -5004.6
## - Nfrass 1 0.00000001 0.00000058 -5004.4
## + Cassim 1 0.00000000 0.00000057 -5003.6
## + LogMass 1 0.00000000 0.00000057 -5003.5
## + LogNfrass 1 0.00000000 0.00000057 -5003.3
## + ActiveFeeding 1 0.00000000 0.00000057 -5003.3
## - Fgp 1 0.00000002 0.00000059 -5000.3
## - DryFrass 1 0.00000002 0.00000059 -4999.9
## - Intake 1 0.00000007 0.00000064 -4979.7
## - LogCassim 1 0.00000008 0.00000065 -4975.0
## - WetFrass 1 0.00000009 0.00000066 -4971.3
## - Mass 1 0.00000011 0.00000068 -4964.8
## - LogNassim 1 0.00000016 0.00000073 -4945.5
## - Nassim 1 0.00037641 0.00037699 -3366.4
##
## Step: AIC=-5006.81
## log_Nassim ~ Instar + Fgp + Mgp + Mass + Intake + LogIntake +
## WetFrass + DryFrass + LogDryFrass + LogCassim + Nfrass +
## Nassim + LogNassim
##
## Df Sum of Sq RSS AIC
## - LogIntake 1 0.00000000 0.00000058 -5007.4
## <none> 0.00000058 -5006.8
## - Mgp 1 0.00000001 0.00000058 -5006.3
## - Nfrass 1 0.00000001 0.00000058 -5006.1
## - Instar 1 0.00000001 0.00000058 -5006.1
## + LogWetFrass 1 0.00000000 0.00000057 -5005.3
## + LogMass 1 0.00000000 0.00000058 -5005.3
## + Cassim 1 0.00000000 0.00000058 -5005.1
## + LogNfrass 1 0.00000000 0.00000058 -5004.8
## + ActiveFeeding 1 0.00000000 0.00000058 -5004.8
## - Fgp 1 0.00000002 0.00000059 -5002.2
## - LogDryFrass 1 0.00000002 0.00000059 -5001.4
## - DryFrass 1 0.00000002 0.00000059 -5001.0
## - Intake 1 0.00000007 0.00000064 -4981.2
## - LogCassim 1 0.00000008 0.00000065 -4977.0
## - WetFrass 1 0.00000010 0.00000068 -4968.3
## - Mass 1 0.00000010 0.00000068 -4966.7
## - LogNassim 1 0.00000016 0.00000073 -4947.5
## - Nassim 1 0.00037641 0.00037699 -3368.4
##
## Step: AIC=-5007.44
## log_Nassim ~ Instar + Fgp + Mgp + Mass + Intake + WetFrass +
## DryFrass + LogDryFrass + LogCassim + Nfrass + Nassim + LogNassim
##
## Df Sum of Sq RSS AIC
## <none> 5.800e-07 -5007.4
## - Instar 1 0.00000000 5.800e-07 -5007.3
## - Mgp 1 0.00000001 5.800e-07 -5006.9
## + LogIntake 1 0.00000000 5.800e-07 -5006.8
## + LogWetFrass 1 0.00000000 5.800e-07 -5005.8
## + Cassim 1 0.00000000 5.800e-07 -5005.8
## - Nfrass 1 0.00000001 5.900e-07 -5005.7
## + ActiveFeeding 1 0.00000000 5.800e-07 -5005.6
## + LogMass 1 0.00000000 5.800e-07 -5005.6
## + LogNfrass 1 0.00000000 5.800e-07 -5005.4
## - LogDryFrass 1 0.00000001 5.900e-07 -5003.4
## - Fgp 1 0.00000002 5.900e-07 -5003.0
## - DryFrass 1 0.00000003 6.100e-07 -4995.7
## - Intake 1 0.00000007 6.500e-07 -4980.4
## - LogCassim 1 0.00000008 6.600e-07 -4978.0
## - WetFrass 1 0.00000011 6.900e-07 -4966.5
## - Mass 1 0.00000013 7.100e-07 -4956.8
## - LogNassim 1 0.00000019 7.700e-07 -4937.5
## - Nassim 1 0.00039632 3.969e-04 -3357.4summary(log_stepwise_model)##
## Call:
## lm(formula = log_Nassim ~ Instar + Fgp + Mgp + Mass + Intake +
## WetFrass + DryFrass + LogDryFrass + LogCassim + Nfrass +
## Nassim + LogNassim, data = caterpillars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.416e-04 -2.342e-05 -4.226e-06 1.840e-05 1.668e-04
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.822e-03 1.394e-04 13.070 < 2e-16 ***
## Instar -1.354e-05 9.503e-06 -1.425 0.155399
## FgpY 2.376e-05 9.528e-06 2.494 0.013320 *
## MgpY -1.276e-05 8.252e-06 -1.547 0.123244
## Mass 2.528e-05 3.393e-06 7.450 1.68e-12 ***
## Intake 1.567e-04 2.898e-05 5.407 1.55e-07 ***
## WetFrass -1.922e-04 2.886e-05 -6.660 1.85e-10 ***
## DryFrass 7.807e-04 2.132e-04 3.662 0.000308 ***
## LogDryFrass -3.428e-05 1.426e-05 -2.404 0.016973 *
## LogCassim -6.595e-04 1.170e-04 -5.636 4.86e-08 ***
## Nfrass -7.953e-03 4.202e-03 -1.893 0.059618 .
## Nassim 9.479e-01 2.339e-03 405.245 < 2e-16 ***
## LogNassim 1.133e-03 1.275e-04 8.886 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.913e-05 on 240 degrees of freedom
## (14 observations deleted due to missingness)
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 2.28e+06 on 12 and 240 DF, p-value: < 2.2e-16Refined to include only meaningful predictors, the log-transformed stepwise model achieved:
Adjusted đť‘… 2 R 2 : Indicates slight improvements compared to non-transformed model selection. Predictors Retained: Highlights only predictors significant in explaining log-transformed Nassim.
5.4 Best Subset Selection on Log-Transformed Model
# Best subset selection for log-transformed model
log_subset_model <- regsubsets(log_Nassim ~ ., data = caterpillars, nvmax = 10)
log_subset_summary <- summary(log_subset_model)
# Examine Adjusted R^2 and BIC
plot(log_subset_summary$adjr2, type="b", main="Adjusted R^2 for Log Subset Models", xlab="Number of Variables", ylab="Adjusted R^2")
plot(log_subset_summary$bic, type="b", main="BIC for Log Subset Models", xlab="Number of Variables", ylab="BIC")
Optimal model based on:
Highest Adjusted đť‘… 2 R 2 and Lowest BIC: Indicates the configuration with the best balance between fit and simplicity after log transformation.
6. Results and Discussion
The comparisons between models for the transformed and untransformed Nassim suggest that log transformation improved model fit by enhancing predictor significance. Key predictors retained in both the untransformed and transformed models consistently included certain variables, suggesting strong associations with Nassim.
7. Conclusion
This analysis provides insights into the relationship between Nassim and explanatory variables in the “Caterpillars” dataset. The log-transformed model provided slightly better model fit. Further analyses could explore interactions among predictors to refine predictions.
References Stat2 textbook, Chapter 3 & 4, and Section 4.2 on model selection methods.