Bayesian Logistic Regression Models for Credit Scoring by Gregg Webster

Classification 
 
Logistic regression models a binary outcome. The objective is often to classify.
In order to perform classification, Hosmer and Lemeshow (2000) explain that a cut-off 
point, 
, must be defined. The estimated probabilities from the logistic regression model 
are compared to this cut-off point. If the estimated probability exceeds 
, we let the 
derived variable be equal to 1; if the estimated probability is less than 
, we let the derived 
variable be equal to 0.
Classification tables, according to Hosmer and Lemeshow (2000) are a good way to 
summarize the results of a fitted logistic regression model. The outcome variable 
is cross 
classified with a dichotomous variable whose values are derived from the estimated 
logistic probabilities. 
The predictive performance of the logistic regression model is probably the best way to 
assess the fit of the model. The way to do this will be to split a data set into a training and a 
test set. The model will be estimated on the training set and its performance will be tested 
on a test set with a certain cut-off probability, 
. A classification table will then be 
established and the error rate of the model can be used as a measure of how well the model 
fits (Table 3.1).
 
 

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Table 3.1
Classification table of the predictive performance of the logistic regression 
model.
Predicted 
Good 
Bad 
Actual 
Good 
Bad 
From Table 3.1, a number of facts can be established.
-
represents the number of applicants in the test set. 
-
is the number of applicants classified as bad. This is the number of applicants who 
were rejected in their application for credit.
-
is the number of applicants classified as good. This is the number of applicants who 
were accepted in their application for credit.
-
is the number of applicants correctly classified as good and 
is the number of 
applicants correctly classified as bad. 
-
is the number of applicants classified as bad but are in fact good. This number represents 
missed out profits for the financial institution. 
-
is the number of applicants classified as good but are in fact bad. This number represents 
bad debts and losses in income for the financial institution.
-
The total error probability of the classification is 
( ) ( )
. This value 
must be small. It also gives an indication of the goodness-of-fit of the model.
-
For the applicants that will be accepted by the financial institution, the error probability is 
( )
. This is the error rate realized by the bank. Thus, it is very important that 
is 
as small as possible.
-
A cut-off probability 
, needs to be found which minimizes the classification error.
The choice of the cut-off probability 
, is often a subjective choice. For lower 
, more 
applicants will be classified as bad. For higher 
, the more applicants will be classified as 
good. A lower cut-off probability means that the financial institution is more risk averse as 
opposed to one with a higher cut-off probability. Because the error rate realized by the 
financial institution is greatly affected by how large 
is, it is important that the error 
among the bad applicants is minimized as well as the total error.

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The optimal cut-off probability can be found by using a validation set. The classification 
error can be determined for different cut-off probabilities. The cut-off probability which 
gives the lowest classification error on the validation set will be chosen and used in further 
analysis.