Different analyses of recurrent events data:

The bladder cancer data listed in Wei, Lin, and Weissfeld (1989) is used in Example 54.8/49.8 of SAS to  illustrate different analyses of  recurrent events data using different models. The data consist of 86 patients with superficial bladder tumors, which were removed when the patients entered the study. Of these patients, 48 were randomized into the placebo group, and 38 were randomized into the thiotepa group. Many patients had multiple recurrences of  tumors during the study, and new tumors were removed at each visit. The data set contains the first four recurrences of the tumor for each patient, and each recurrence time was measured from the patient's entry time into the study. The data consist of the following eight variables:

• Trt, treatment group (1 =placebo and 2=thiotepa)
• Time, follow-up time
• Number, number of initial tumors
• Size, initial tumor size
• Tl, T2, T3, and T4, times of the four potential recurrences of the bladder tumor. A patient with only two recurrences has missing values in T3  and T4.

Write the formulae of the proportional intensity and the proportional mean models estimated using appropriate SAS procedures (define the relevant variables, functions, parameters and identify the formulas for two models) using the covariate x (x=O for placebo and x=l for thiotepa). What are the variance estimates for these two regression estimates (specify their numerical values and the names of the estimators )?

Interpret the main results of the analyses of bladder cancer data under these two models. Are the key modeling assumptions of these analyses are justified (provide relevant plots with explanations)?