A study was conducted on a sample of 53 patients presenting with prostate cancer who had also undergone a laparotomy to ascertain the extent of nodal involvement (Collett 1991). The result of the laparotomy is a binary response variable, where 0 signifies the absence of, and 1 the presence of, nodal involvement. The purpose of the study was to determine variables that could be used to forecast whether the cancer has spread to the lymph nodes. Five predictor variables were considered, each measurable without surgery. The following printout provides information for fitting two logistic models based on these data. The five predictor variables were age of patient at diagnosis, level of serum acid phosphatase, result of an X-ray examination (0 = negative, 1 = positive), size of the tumor as determined by a rectal exam (0 = small, 1 = large), and summary of the pathological grade of the tumor as determined from a biopsy (0 = less serious, 1 = more serious).
a. Which method of estimation do you think was used to obtain estimates of parameters for both models—conditional or unconditional ML estimation? Explain briefly.
b. For model I, test the null hypothesis of no effect of X-ray status on response. State the null hypothesis in terms of an odds ratio parameter; give the formula for the test statistic; state the distribution of the test statistic under the null hypothesis; and, finally, carry out the test, using the computer output information for Model I. Is the test significant?
Model I

"The deviance statistic is an LR statistic that compares a current model of interest to the baseline model containing as many parameters as there are data points. The difference in deviance statistics obtained for two (hierarchically ordered) models being compared is equivalent to the difference in log likelihood statistics for each model. Thus, an LR test can equivalently be carried out by using differences in deviance statistics. See Section 24.5 for further details about deviances.

c. Using the printout data for Model I, compute the point estimate and 95% confidence interval for the odds ratio for the effect of X-ray status on response for a person of age 50, with phosphatase acid level of 0.50, tsize equal to 0, and tgrad equal to 0.
d. State the logit form of Model II given in the accompanying computer printout.
e. Using the results for Model II, give an expression for the estimated odds ratio that describes the effect of X-ray status on the response, controlling for age, phosphatase acid level, tsize, and tgrad. Using this expression, compute the estimated odds ratio of the effect of X-ray status on the response for a person of age 50, with phosphatase acid level of 0.50, tsize equal to 0, and tgrad equal to 0.
f. For Model II, give an expression for the estimated variance of the estimated adjusted odds ratio relating X-ray status to response for a person of age 50, with phosphatase acid level of 0.50, tsize equal to 0, and tgrad equal to 0. Write this expression for the estimated variance in terms of estimated variances and covariances obtained from the variance-covariance matrix of regression parameter estimates.
g. Using your answer to part (f), give an expression for a 95% confidence interval for the odds ratio relating X-ray status to response for a person of age 50, with phosphatase acid level of 0.50, tsize equal to 0, and tgrad equal to 0.
h. For Model II, carry out a "chunk" test for the combined interaction of X-ray status with each of the variables age, phosphatase acid level, tsize, and tgrad. State the null hypothesis in terms of one or more model coefficients; give the formula for the test statistic and its distribution and degrees of freedom under the null hypothesis; and report the /'-value. Is the test significant?
i. If you had to choose between Model I and Model II, which would you pick as the "better" model? Explain.
Assume that the following model has been defined as the initial model to be considered in a backward-elimination strategy to obtain a "best" model:
logit[pr(y = 1)] = α + β1,(xray) + β2(age) + β3(acid) + β4(tsize) + β5(tgrad) + β6(age × acid) + β7(xray × age) + β8(xray × acid) + β9(xray × tsize) + /β10(xray × tgrad) + β11(xray × age × acid)
j. For the above model—and considering the variable xray to be the only "exposure" variable of interest, with the variables age, acid, tsize, and tgrad considered for control—which /3's are coefficients of (potential) effect modifiers? Explain briefly.
k. Assume that the only interaction term found significant is the product term xray × age. What variables are left in the model at the end of the interaction assessment stage?
1. Based on the (reduced) model described in part (k) (where the only significant interaction term is xray × age), what expression for the odds ratio describes the effect of xray on nodal involvement status?
m. Suppose that, as a result of confounding and precision assessment, the variables age × acid, tgrad, and tsize are dropped from the model described in part (k). What is your final model, and what expression for the odds ratio describes the effect of xray on nodal involvement status?