The data for this exercise are from a study of the frequencies of urinary tract infections in n = 98 men infected with the HIV virus. CD4+ cell counts were also measured. They are used as an indication of how well the immune system is working in people infected with HIV. CD4+ counts are reported as the number of cells per cubic millimeter of blood. Normal levels of CD4+ counts typically range from 500 to 1500 cells per cubic millimeter of blood. In general, lower CD4+ counts are an indication of progression of HIV and a weakening immune system. As a result of a weakening immune system, people are less likely to resist other infections.

The data shown in the following table were collected at Utrecht University Hospital in the Netherlands and reported by Morel and Neerchal (2012).

Note that different subjects were exposed to different follow-up times.

Consequently, you will need to use the square root of the follow-up time as an offset. Use proc genmod to complete the following exercises.

a. Write an appropriate Poisson regression model to relate the expected number of urinary tract infections per month of follow-up time to the CD4+ cell counts. In the context of this study, explain what each parameter in your model represents.

b. Use genmod to compute estimates of the model parameters and corresponding standard errors. Be sure to include the square root of the follow-up time in the offset option to adjust for variation in follow-up times.

c. Test the null hypothesis that the coefficient for the CD4+ count variable is zero, against a one-sided alternative that the coefficient is positive. Use a 0.05 significance level. State your conclusion.

d. Estimate the increase in risk of a urinary tract infection, as measured by the odds ratio, for a decrease of 100 in the CD4+ cell count. Report a standard error for your estimate.

e. Check for overdispersion by looking at the value of the Pearson goodness-of-fit statistic divided by its degrees of freedom. Discuss what you see.

f. Fit a negative binomial regression model to these data, and report the estimates of the regression coefficients and their standard errors.

Also report the estimates of the variance inflation parameter for the negative binomial distribution.

g. Based on the negative binomial model, is there a significant relationship between the expected rates of urinary tract infections per month of follow-up time and levels of CD4+ counts? Provide a statistical test or confidence interval to support your conclusion.