Likelihood Ratio Test

A logistic regression is said to provide a better fit to the data if it demonstrates an improvement over a model with fewer predictors. This is performed using the likelihood ratio test, which compares the likelihood of the data under the full model against the likelihood of the data under a model with fewer predictors. Removing predictor variables from a model will almost always make the model fit less well (i.e. a model will have a lower log likelihood), but it is necessary to test whether the observed difference in model fit is statistically significant. Given that H0 holds that the reduced model is true, a p-value for the overall model fit statistic that is less than 0.05 would compel us to reject the null hypothesis. It would provide evidence against the reduced model in favor of the current model. The likelihood ratio test can be performed in R using the lrtest() function from the lmtest package or using the anova() function in base.

anova(mod_fit_one, mod_fit_two, test ="Chisq")

library(lmtest)
lrtest(mod_fit_one, mod_fit_two)

Pseudo R^2

Unlike linear regression with ordinary least squares estimation, there is no R2 statistic which explains the proportion of variance in the dependent variable that is explained by the predictors. However, there are a number of pseudo R2 metrics that could be of value. Most notable is McFadden’s R2, which is defined as 1−[ln(LM)/ln(L0)] where ln(LM) is the log likelihood value for the fitted model and ln(L0) is the log likelihood for the null model with only an intercept as a predictor. The measure ranges from 0 to just under 1, with values closer to zero indicating that the model has no predictive power.

library(pscl)
pR2(mod_fit_one)  # look for 'McFadden'
##           llh       llhNull            G2      McFadden          r2ML 
## -344.42107079 -366.51858123   44.19502089    0.06029029    0.07101099 
##          r2CU 
##    0.10068486

Hosmer-Lemeshow Test

Another approch to determining the goodness of fit is through the Homer-Lemeshow statistics, which is computed on data after the observations have been segmented into groups based on having similar predicted probabilities. It examines whether the observed proportions of events are similar to the predicted probabilities of occurence in subgroups of the data set using a pearson chi square test. Small values with large p-values indicate a good fit to the data while large values with p-values below 0.05 indicate a poor fit. The null hypothesis holds that the model fits the data and in the below example we would reject H0.

library(MKmisc)
HLgof.test(fit = fitted(mod_fit_one), obs = training$Class)
library(ResourceSelection)
hoslem.test(training$Class, fitted(mod_fit_one), g=10)

Wald Test

A wald test is used to evaluate the statistical significance of each coefficient in the model and is calculated by taking the ratio of the square of the regression coefficient to the square of the standard error of the coefficient. The idea is to test the hypothesis that the coefficient of an independent variable in the model is significantly different from zero. If the test fails to reject the null hypothesis, this suggests that removing the variable from the model will not substantially harm the fit of that model.

library(survey)
regTermTest(mod_fit_one, "ForeignWorker")
## Wald test for ForeignWorker
##  in glm(formula = Class ~ Age + ForeignWorker + Property.RealEstate + 
##     Housing.Own + CreditHistory.Critical, family = "binomial", 
##     data = training)
## F =  0.949388  on  1  and  594  df: p= 0.33027
regTermTest(mod_fit_one, "CreditHistory.Critical")
## Wald test for CreditHistory.Critical
##  in glm(formula = Class ~ Age + ForeignWorker + Property.RealEstate + 
##     Housing.Own + CreditHistory.Critical, family = "binomial", 
##     data = training)
## F =  16.67828  on  1  and  594  df: p= 5.0357e-05

Variable Importance

To assess the relative importance of individual predictors in the model, we can also look at the absolute value of the t-statistic for each model parameter. This technique is utilized by the varImp function in the caret package for general and generalized linear models.

varImp(mod_fit)
## glm variable importance
## 
##                        Overall
## CreditHistory.Critical  100.00
## Property.RealEstate      57.53
## Housing.Own              50.73
## Age                      22.04
## ForeignWorker             0.00

Classification Rate

When developing models for prediction, the most critical metric regards how well the model does in predicting the target variable on out of sample observations. The process involves using the model estimates to predict values on the training set. Afterwards, we will compared the predicted target variable versus the observed values for each observation. In the example below, you’ll notice that our model accurately predicted 67 of the observations in the testing set.

pred = predict(mod_fit, newdata=testing)
accuracy <- table(pred, testing[,"Class"])
sum(diag(accuracy))/sum(accuracy)
## [1] 0.705
pred = predict(mod_fit, newdata=testing)
confusionMatrix(data=pred, testing$Class)

ROC Curve

The receiving operating characteristic is a measure of classifier performance. Using the proportion of positive data points that are correctly considered as positive and the proportion of negative data points that are mistakenly considered as positive, we generate a graphic that shows the trade off between the rate at which you can correctly predict something with the rate of incorrectly predicting something. Ultimately, we’re concerned about the area under the ROC curve, or AUROC. That metric ranges from 0.50 to 1.00, and values above 0.80 indicate that the model does a good job in discriminating between the two categories which comprise our target variable. Bear in mind that ROC curves can examine both target-x-predictor pairings and target-x-model performance. An example of both are presented below.

library(pROC)
# Compute AUC for predicting Class with the variable CreditHistory.Critical
f1 = roc(Class ~ CreditHistory.Critical, data=training) 
plot(f1, col="red")
## 
## Call:
## roc.formula(formula = Class ~ CreditHistory.Critical, data = training)
## 
## Data: CreditHistory.Critical in 180 controls (Class Bad) < 420 cases (Class Good).
## Area under the curve: 0.5944
library(ROCR)
# Compute AUC for predicting Class with the model
prob <- predict(mod_fit_one, newdata=testing, type="response")
pred <- prediction(prob, testing$Class)
perf <- performance(pred, measure = "tpr", x.measure = "fpr")
plot(perf)
auc <- performance(pred, measure = "auc")
auc <- [email protected][[1]]
auc
## [1] 0.6540625

K-Fold Cross Validation

When evaluating models, we often want to assess how well it performs in predicting the target variable on different subsets of the data. One such technique for doing this is k-fold cross-validation, which partitions the data into k equally sized segments (called ‘folds’). One fold is held out for validation while the other k-1 folds are used to train the model and then used to predict the target variable in our testing data. This process is repeated k times, with the performance of each model in predicting the hold-out set being tracked using a performance metric such as accuracy. The most common variation of cross validation is 10-fold cross-validation.

ctrl <- trainControl(method = "repeatedcv", number = 10, savePredictions = TRUE)

mod_fit <- train(Class ~ Age + ForeignWorker + Property.RealEstate + Housing.Own + 
                   CreditHistory.Critical,  data=GermanCredit, method="glm", family="binomial",
                 trControl = ctrl, tuneLength = 5)

pred = predict(mod_fit, newdata=testing)
confusionMatrix(data=pred, testing$Class)