Question 1 & 2: Develop the Model & Assess Predictor
Significance
Step 1: Install and load required libraries
#install.packages("readxl")
#install.packages("Hmisc")
#install.packages("pscl")
#if(!require(pROC)) install.packages("pROC")
library(readxl) #allows us to import excel files
library(Hmisc) #allows us to call the correlation function
##
## Attaching package: 'Hmisc'
## The following objects are masked from 'package:base':
##
## format.pval, units
library(pscl) #allows us to call the psuedo R-square package to evaluate our model
## 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 area under the curve (AUC) packages to get the plot and AUC score
## Type 'citation("pROC")' for a citation.
##
## Attaching package: 'pROC'
## The following objects are masked from 'package:stats':
##
## cov, smooth, var
Step 2: Import & clean the data
tire_df <- read_excel("Class Exercise 15_TireRatings.xlsx")
tir_df <- subset(tire_df, select = -c(Tire)) #drop irrelevant column
Step 3: Summarize the data
head(tir_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 the features are presented in the table below
Variable | Definition
------------- | -------------
1. Wet | A customer buys a tire with a Wet Tire Performance Rating
2. Noise | A customer buys a tire with a Noise Tire Performance Rating
3. Buy_Again | A customer buys a Wet/Noise Tire Again (0: Probabily and 1: Definitely)
summary(tir_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 average Wet Tire Purchase is 7.75
Step 5: Feature selection (i.e., correlation analysis)
corr <- rcorr(as.matrix(tir_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 not significant with the target variable, There is multicollinearity in the data
Step 5: Build the logistic regression model
model <- glm(Purchase ~ Wet + Noise, data = tir_df, family = binomial)
summary(model)
##
## Call:
## glm(formula = Purchase ~ Wet + Noise, family = binomial, data = tir_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 not significant (p-value > 0.05)
Question 3: Overall Model Significance
Likelihood Ratio Test
# Fit a null model
null_model <- glm(Purchase ~ 1, data = tir_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 Wet and Noise as predictors in our LR model does not significantly predict the likelihood of the customer purchasing the tires again, relative to a model that predicts Purchase 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 0.705 means that our LR model explains about 70.5% of the variability in the outcome relative to the model with no predictors. This is considered a good fit, where values above 0.2 to 0.4 are often seen as indicative of a useful model.
Area Under the Curve (AUC)
The Area Under the Curve (AUC) score represents the ability of the model to correctly classify customers who purchase Wet and Noise Performance rated tires
# Compute ROC Curve and the AUC score
roc_curve <- roc(tir_df$Purchase, fitted(model))
## Setting levels: control = 0, case = 1
## Setting direction: controls < cases
plot(roc_curve)
auc(roc_curve)
## Area under the curve: 0.9741
Interpretation: An AUC score of 0.97 indicates that the LR model has a high level of accuracy in predictingcustomers purchasing tires
Question 6: Odds Ratio
# Extract the coefficients
coefficients <- summary(model)$coefficients
# Calculate the odd ratio for "Noise"
odds_ratio_noise <- exp(coefficients["Noise" , "Estimate"])
odds_ratio_noise
## [1] 6.148919
Interpretation: The odd ratio = 6.14 > 1 and this indicates that Noise Performance rating is associated with higher odds of customers for the tire purchased compared to not purchasing the tires