Consequences of Delayed Treatment Effects on Analysis of Time-to-Event Endpoints
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Consequences of Delayed Treatment Effects on Analysis of Time-to-Event Endpoints
Gil D. Fine, PhD Senior Director, Statistics and Data Management, SuperGen Inc., Dublin, California
Key Words Survival analysis; Log rank; Weighted test; Power; Pancreatic cancer Correspondence Address Gil D. Fine, PhD, SuperGen Inc., 4140 Dublin Boulevard, Suite 200, Dublin, CA 94568 (e-mail: [email protected]).
The assumption of proportional hazard ratios is implicit in certain analyses of time-to-event endpoints such as Cox regression. Other statistical analyses, such as the nonparametric logrank test, possess some desirable properties only under the proportional hazards model. Data models for delayed effects of treatment on time-to-event endpoints such as survival violate the proportional hazards assumption.
INTRODUCTION Consider a parallel-design clinical trial in which subjects are randomly assigned to one of k treatment groups. Because subjects are comparable at baseline, it is assumed that, without intervention, the hazard for each subject will follow the same function h(t) throughout the study. Subjects in group 1 receive placebo or no treatment, and their hazard function remains h1(t) = h(t); subjects in group j (j = 2, . . . , k) receive active treatment that changes their hazard function to hj(t). Generally, it is assumed that treatments have an immediate effect on the hazard, but the delayed treatment effect model (DTEM) postulates subjects’ survival obeys the baseline hazard until t = εj. Thus, under the DTEM, survival in group j is characterized by the two-part hazard function h(t) h j (t) = h *j (t)
0 ≤ t < εj εj ≤ t < ∞
where h*j(t) ≤ h(t) for t ≥ εj describes the benefit of treatment j ( j = 2, . . . , k) compared to placebo. The time delay εj until treatment has an effect on the hazard could be a random variable, but here it is modeled as a constant. This model appears to fit data collected from patients with advanced-stage pancreatic carcinoma during a trial of supportive care versus an oral test agent as third-line therapy (1). It is hypothesized that, because patients’ prognoses
Fleming and Harrington’s G ρ,γ class of weighted log-rank tests, a new option in SAS 9.1, is appropriate to test against a broad range of alternative hypotheses, including delayed treatment effects. A model for delayed treatment effects is proposed, and it is demonstrated that weighted log-rank tests are more powerful under this model than the standard unweighted log-rank test.
are so poor owing to metastases, several weeks or months of treatment are required to overcome the imminent risk of death from their disease. Indeed, death often precedes the time required for a treatment to manifest its effect. Figure 1 illustrates survival curves that may be expected in this setting. Of course, the data model and survival analysis methods presented here may be used to describe the time until the occurrence of any well-defined event, not just death. Statistical analysis of survival data from such a trial may be accomplished usin
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