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Introduction As part of its efforts to increase customer loyalty and reduce churn, a telecommunications company is interested in modeling the "time to churn" in order to determine the factors that are associated with customers who are quick to switch to another service. To this end, a random sample of customers is selected and their time spent as customers, whether they are still active customers, and various demographic fields are pulled from the database for use in a Cox Regression loyalty analysis.
Analysis
We will be examining the potential influences on churn of several key candidate predictors: age; marital status; education; employment status (retired vs. still working); gender; length of time at current address; length of time with current employer; and customer category. Some candidate predictors that we will test in the churn model are quantitative variables such as age or length of time at current address. Other possible predictors (e.g., marital status) are categorical variables, because they cannot be measured on a quantitative scale. The following categorical variable codings are a useful reference for interpreting the regression coefficients for categorical covariates, particularly dichotomous variables:
In this particular analysis, by default, the reference category is the last category of a categorical covariate. Thus, for example, even though Married customers have variable values of 1 in the data file, they are coded as 0 for the purposes of the regression. The Cox Regression model-building process takes place in two blocks. In the first, a forward stepwise algorithm is employed. The omnibus tests are measures of how well the model performs. (The chi-square change from previous step is the difference between the -2 log-likelihood of the model at the previous step and the current step.) Here is the summary table of output from the model-generation process, followed by an explanation and discussion:
If the step was to add a variable, the inclusion makes sense if the significance of the change is less than 0.05. If the step was to remove a variable, the exclusion makes sense if the significance of the change is greater than 0.10. In the first three steps, AGE, EMPLOY, and ADDRESS are added to the model. In the fourth step, AGE is removed from the model, likely because the variation in time to churn that is explained by AGE is also explained by EMPLOY and ADDRESS; thus, when these variables are added to the model, AGE is no longer necessary. Finally, MARITAL is added in the fifth step. The final model for Block 1 includes MARITAL, ADDRESS, and EMPLOY. H ere is a table of predictive model coefficients, followed by an explanation and discussion:
The value of Exp(B) for MARITAL means that the churn hazard for an unmarried customer is 1.556 times that of a married customer. (Recall from the categorical variable codings that unmarried = 1 for the regression.) The value of Exp(B) for ADDRESS means that the churn hazard is reduced by 100%-(100%x0.941)=5.9% for each year (on a compounded basis) that a customer has lived at the same address. A more useful computational formula for calculating this involves raising the Exp(B) to a power equal to the number of years at current address. For example, the churn hazard for a customer who has lived at the same address for five years is reduced by 100%-(100%x(0.941^5))=26.2%. [Note that in this formula the ^ symbol represents raising a number to a power.] Likewise, the value of Exp(B) for EMPLOY means that the churn hazard is reduced by 100%-(100%x0.920)=8.0% for each year (on a compounded basis) that a customer has worked for the same employer. Using the aforementioned alternative computational formula, the churn hazard for a customer who has worked for the same employer for three years is reduced by 100%-(100%x(0.920^3))=22.1%. Now we move to the second phase of the Cox Regression model-building process (“Block 2”), where we add customer Category as a categorical predictor and then examine its influence on churn. Here is the next table of output, followed by explanation and discussion: 
The change from previous step and change from previous block both report the effect of adding customer category to the model selected in Block 1. Since the significance value of the change is less than 0.05, we can be confident that customer category contributes to the model. Next comes the table of predictive model coefficients, followed by explanation and discussion:
Block 2: Method = Enter
The Cox Regression coefficients for the first three levels of CUSTCAT are relative to the reference category, which corresponds to Total service customers. The regression coefficient for the first category, corresponding to Basic service customers, suggests that the hazard for Basic service customers is 1.129 times that of Total service customers. However, the significance value for this coefficient is greater than 0.10, so any observed difference between these customer categories could be due to chance. By contrast, the significance values for the second and third categories, corresponding to E-service and Plus service customers, are less than 0.05, which means they are statistically different from the Total service customers. The regression coefficients suggest that the hazard for E-service customers is 0.563 times that of Total service customers, and the hazard for Plus service customers is 0.518 times that of Total service customers. Below is a graphical representation of the “survival” or loyalty function generated from the model. The basic survival curve is a visual display of the model-predicted time to churn for the "average" customer. The horizontal axis shows the time to event. The vertical axis shows the probability of survival. Thus, any point on the survival curve shows the probability that the "average" customer will remain a customer past that time. Past 55 months the survival curve becomes less smooth. There are fewer customers who have been with the company for that long, so there is less information available, and thus the curve is blocky.
The plot of the survival curves gives a visual representation of the effect of customer category, which is shown in the graph below:
From the above graph we can see that Total service and Basic service customers have lower survival curves because, as we have learned from their regression coefficients, they are more likely to have shorter times to churn. The basic hazard curve, shown below, is a visual display of the cumulative model-predicted potential to churn for the "average" customer:
The horizontal axis shows the time to event. The vertical axis shows the cumulative hazard, equal to the negative log of the survival probability. Beyond 55 months, the hazard curve, like the survival curve, becomes less smooth, for the reason stated previously. The plot of the hazard gives a visual representation of the effect of customer category:
Total service and Basic service customers have higher hazard curves because, as we have learned from their regression coefficients, they have a greater potential to churn.
Summary and Conclusions We have found a suitable Cox Regression model for predicting time to customer churn. The use of separate blocks for fitting the model has allowed us to guarantee that customer category would be in the final model, while still taking advantage of the stepwise techniques for choosing the other variables in the model. To create this model, we included customer category in the second block. [Alternatively, the addition of customer category to the model could have taken place in the first block, and the stepwise methods to choose the other variables in the second block.] We have discovered that marital status, length of time at current address, and length of time with current employer are all significant influences on time to churn, as is customer category. By understanding these influences, we can identify customers who are most likely to defect at any given point in the customer relationship. This makes it possible for us to target these vulnerable customers with timely outreach efforts aimed at maintaining loyalty. Back to Marketing Analytics page
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