Analyzing a Randomized Cancer Prevention Trial with a Missing Binary Outcome, an Auxiliary Variable, and All-or-None Compliance
Baker, Stuart G., Journal of the American Statistical Association
The Prostate Cancer Prevention Trial is a randomized chemoprevention trial designed to compare the effect of daily finasteride versus placebo on prostate cancer determined by biopsy. Investigators have scheduled a biopsy at the end of the trial in 7 years or following a positive prostate-specific antigen (PSA) on annual screening. The analysis will need to adjust for two likely complications. First, some subjects will not receive a biopsy, depending in part on whether or not they had a positive PSA. The indicator of positive PSA is called an auxiliary variable, which is a variable observed after randomization and prior to outcome. Second, starting soon after randomization, some subjects randomized to finasteride will stop taking their tablets, and some subjects randomized to placebo will obtain finasteride outside of the trial. This type of noncompliance is called all-or-none. To adjust for these complications, we formulate the appropriate likelihoods and obtain closed-form maximum likelihood estimates and va riances. Without these adjustments, estimates may be biased, two-sided type I errors above nominal levels, and coverage of confidence intervals below nominal levels.
KEY WORDS: Finasteride; Nonignorable missing data; Prostate Cancer Prevention Trial; Randomized trials.
The Prostate Cancer Prevention Trial (PCPT) is a chemoprevention trial in which 18,000 men age 55 and older were randomized to either daily finasteride or placebo tablets for 7 years. The primary goal is to determine whether subjects assigned to finasteride have a different prevalence of prostate cancer determined by biopsy than subjects randomized to placebo. (For more details of the study design, see Feigl et al. 1994.) The trial, which involves 222 sites in the United States, opened in 1993, and recruitment was completed in 1996. Researchers from the Southwest Oncology Group are coordinating the study, which also involves researchers from the Eastern Cooperative Oncology Group and the Cancer and Leukemia Group B. The study is funded by the National Cancer Institute. Merck and Co., Inc. is providing both the finasteride and the placebo at no charge.
Because the outcome of the study is prostate cancer determined by biopsy, investigators scheduled all subjects to receive a biopsy at the end of the trial. To increase support for the trial among urologists, investigators also scheduled an annual screening test for prostate-specific antigen (PSA) with a biopsy for those testing positive. The investigators defined a positive PSA in the placebo group as a PSA level above 4.0 ng/mL. Because finasteride lowers PSA levels, the investigators defined a positive PSA in the finasteride group as a PSA above a specific level such that the same percentage is positive as in the placebo group. We say that PSA is an auxiliary variable, because it is observed after randomization and prior to endpoint.
Feigl et al. (1994) anticipated that 23% of the 18,000 subjects will not receive a biopsy due to loss to follow-up or refusal. One analytic challenge is to adjust for missing outcome data when biopsy depends on the auxiliary variable.
In Sections 2, 3, and 4 I discuss missing-data adjustments under an intent-to-treat (ITT) analysis. In Section 2 I discuss model B, a basic adjustment in which biopsy depends only on assigned treatment. In Section 3 I present model A, an adjustment using the auxiliary variable, and in Section 4 I generalize model A to model AM, which allows a missing auxiliary variable.
Feigl et al. (1994) also anticipated that 5% of subjects in the placebo arm will receive finasteride outside the trial and that 14% of subjects in the finasteride arm will not take their tablets. (Investigators are monitoring finasteride use via serum dihydrotestosterone levels). A second analytic challenge is to adjust for this all-or-none compliance (ANC) by extending the models of Angrist, Imbens, and Rubin (1996), Baker (1998a, 1998b), Baker and Lindeman (1994), Frangakis and Rubin (1998), and Sheiner and Rubin (1995). …