As is the case in linear regression, effective use of diagnostic statistics depends on our ability to interpret and understand the values of the statistics. The purpose of this problem is to provide a...

As is the case in linear regression, effective use of diagnostic statistics depends on our ability to interpret and understand the values of the statistics. The purpose of this problem is to provide a few structured examples to examine the effect on the fitted logistic regression model and diagnostic statistics when data are moved away from the model (i.e., poorer fit), and also toward the model (i.e., better fit). Table 5.23 lists values of the independent variable x, and seven different columns of the outcome variable y, labeled “Model”. All models fit in this problem use the given values of x for the covariate. Different models are fit using the seven different columns for the outcome variable. The data for the column labeled “Model 0” are constructed to represent a “typical” realization when the logistic regression model is correct. In the columns labeled “Model 1” to “Model 3” we have changed some of the y values away from the original model. Namely some cases with small values of x have had y changed from 0 to 1 and others with large values of x have had the y values changed from 1 to 0. For models labeled “Model −1” and “Model −2” we have

moved the y values in the direction of the model. That is, we have changed y from 1 to 0 for some small values of x and have changed y from 0 to 1 for some large values of x. Fit the six logistic regression models for the data in columns “Model −2” to “Model 3”. Compute for each fitted model the values of the leverage, h, the change in chi-square, X2, and the influence diagnostic, bˆ. Plot each of these versus the fitted values, predicted logistic probabilities. Compare the plots over the various models. Do the statistics pick out poorly fit and influential cases? How do the estimated coefficients change relative to Model 0? Fit “Model −i”. What happens and why? Refer to the discussion in Section 4.5 on complete separation.