Question 1 & 2: Develope the Model & Assess Predictor
Significance
Step 1: Install and load required libraries
#install.packages("readxl")
#install.packages("Hmisc")
#install.packages("pscl")
#install.packages("pROC")
library("readxl") #allows us to import excel files
library("Hmisc") #allows us to call the correlation function
## Warning: package 'Hmisc' was built under R version 4.3.3
##
## Attaching package: 'Hmisc'
## The following objects are masked from 'package:base':
##
## format.pval, units
library("pscl") #allows us to call the pseudo R-squared to evaluate our model
## Warning: package 'pscl' was built under R version 4.3.3
## Classes and Methods for R originally developed in the
## Political Science Computational Laboratory
## Department of Political Science
## Stanford University (2002-2015),
## by and under the direction of Simon Jackman.
## hurdle and zeroinfl functions by Achim Zeileis.
library("pROC") #allows us to run the AUC package to get the plot and AUC score
## Warning: package 'pROC' was built under R version 4.3.3
## Type 'citation("pROC")' for a citation.
##
## Attaching package: 'pROC'
## The following objects are masked from 'package:stats':
##
## cov, smooth, var
Step 2: Import and clean the data
tireratings_df <- read_excel(file.choose())
tire_df <- subset(tireratings_df, select = -c(Tire)) #drop irrelevant column
Step 3: Summarize the data
head(tire_df)
## # A tibble: 6 × 4
## Wet Noise Buy_Again Purchase
## <dbl> <dbl> <dbl> <dbl>
## 1 8 7.2 6.1 0
## 2 8 7.2 6.6 1
## 3 7.6 7.5 6.9 1
## 4 6.6 5.4 6.6 0
## 5 5.8 6.3 4 0
## 6 6.3 5.7 4.5 0
Data Description: A description of some of the features are presented in the table below.
Variable | Definition
------------- | -------------
1. Wet | The average ratings for each tire's wet traction performance.
2. Noise | The average ratings for the noise level generated by each tire.
3. Buy-Again | The average of the buy-again responses on a 10-point scale.
4. Purchase | The respondants averages of the buy-again response (1: if the value of the Buy-Again variable is 7 or greater, 0: if the value of the Buy-Again variable is less than 7)
summary(tire_df)
## Wet Noise Buy_Again Purchase
## Min. :4.300 Min. :3.600 Min. :1.400 Min. :0.0000
## 1st Qu.:6.450 1st Qu.:6.000 1st Qu.:3.850 1st Qu.:0.0000
## Median :7.750 Median :7.100 Median :6.150 Median :0.0000
## Mean :7.315 Mean :6.903 Mean :5.657 Mean :0.4412
## 3rd Qu.:8.225 3rd Qu.:7.925 3rd Qu.:7.400 3rd Qu.:1.0000
## Max. :9.200 Max. :8.900 Max. :8.900 Max. :1.0000
#Interpretation: The median wet traction performance is 7.75, with a median of 7.1 for noise level generated.
Step 4: Feature selection (i.e., correlation analysis)
corr <- rcorr(as.matrix(tire_df))
corr
## Wet Noise Buy_Again Purchase
## Wet 1.00 0.76 0.91 0.74
## Noise 0.76 1.00 0.83 0.72
## Buy_Again 0.91 0.83 1.00 0.83
## Purchase 0.74 0.72 0.83 1.00
##
## n= 68
##
##
## P
## Wet Noise Buy_Again Purchase
## Wet 0 0 0
## Noise 0 0 0
## Buy_Again 0 0 0
## Purchase 0 0 0
Interpretation: All the predictors are significant with the target variable (i.e., Purchase). There's no multicollinearity in the data.
Step 5: Build the logistic regression
model <- glm(Purchase ~ Wet + Noise, data = tire_df, family = binomial)
summary(model)
##
## Call:
## glm(formula = Purchase ~ Wet + Noise, family = binomial, data = tire_df)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -39.4982 12.4779 -3.165 0.00155 **
## Wet 3.3745 1.2641 2.670 0.00760 **
## Noise 1.8163 0.8312 2.185 0.02887 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 93.325 on 67 degrees of freedom
## Residual deviance: 27.530 on 65 degrees of freedom
## AIC: 33.53
##
## Number of Fisher Scoring iterations: 8
Interpretation: All the independent variables were significant (p-value < 0.05)
Question 3: Overall Model Significance
Likelihood Ratio Test
# Fit a null model
null_model <- glm(Purchase ~ 1, data = tire_df, family = binomial)
# Perform likelihood ratio test
anova(null_model, model, test = "Chisq")
## Analysis of Deviance Table
##
## Model 1: Purchase ~ 1
## Model 2: Purchase ~ Wet + Noise
## Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1 67 93.325
## 2 65 27.530 2 65.795 5.162e-15 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Interpretation: The inclusion of Wet and Noise as predictors in out LR model does indeed significantly predict the likelihood of respondents buying the tire again, relative to a model that predicts purchases based solely on the mean of observed outcomes (i.e., null model).
Pseudo-R-Squared
pR2(model)
## fitting null model for pseudo-r2
## llh llhNull G2 McFadden r2ML r2CU
## -13.7649516 -46.6623284 65.7947536 0.7050093 0.6199946 0.8305269
Interpretation: A McFadden R-squared of 0.71 means that our LR model explains about 71% of the variability in the ourcome relative to a model with no predictors. This is considered an average fit, where values above 0.2 to 0.4 are often seen as indicative of a useful model.
Area Under the Curve (AUC)
# Compute ROC Curve and the AUC score
roc_curve <- roc(tire_df$Purchase, fitted(model))
## Setting levels: control = 0, case = 1
## Setting direction: controls < cases
plot(roc_curve)
auc(roc_curve)
## Area under the curve: 0.9741
Interpretation: An AUC score of 0.97 indicates that the LR model has a high level of accuracy in predicting respondant purchases.
Question 6: Odds Ratio
# Extract the coefficients
coefficients <- summary(model)$coefficients
# Calculate the odd ratio for 'Noise'
odds_ratio_noise <- exp(coefficients["Noise", "Estimate"])
round((odds_ratio_noise),2)
## [1] 6.15
Interpretation: The odd ratio = 6.15 > 1 and this indicates that noise is associated with higher odds of respondants purchasing again compared to not purchasing again.