Graphical displays for clarifying how allocation ratio affects total sample size for the two sample logrank test

Abstract

For time-to-event data, the power of the two sample logrank test for the comparison of two treatment groups can be greatly influenced by the ratio of the number of patients in each of the treatment groups. Despite the possible loss of power, unequal allocations may be of interest due to a need to collect more data on one of the groups or to considerations related to the acceptability of the treatments to patients. Investigators pursuing such designs may be interested in the cost of the unbalanced design relative to a balanced design with respect to the total number of patients required for the study. We present graphical displays to illustrate the sample size adjustment factor, or ratio of the sample size required by an unequal allocation compared to the sample size required by a balanced allocation, for various survival rates, treatment hazards ratios, and sample size allocation ratios. These graphical displays conveniently summarize information in the literature and provide a useful tool for planning sample sizes for the two sample logrank test. SUMMARYFor time-to-event data, the power of the two sample logrank test for the comparison of two treatment groups can be greatly influenced by the ratio of the number of patients in each of the treatment groups.Despite the possible loss of power, unequal allocations may be of interest due to a need to collect more data on one of the groups or to considerations related to the acceptability of the treatments to patients.Investigators pursuing such designs may be interested in the cost of the unbalanced design relative to a balanced design with respect to the total number of patients required for the study. We present graphical displays to illustrate the sample size adjustment factor, or ratio of the sample size required by an unequal allocation compared to the sample size required by a balanced allocation, for various survival rates, treatment hazards ratios, and sample size allocation ratios. These graphical displays conveniently summarize information in the literature and provide a useful tool for planning sample sizes for the two sample logrank test.

Statistical Considerations for an Adaptive Design for a Serious Rare Disease

“…Hence, the study was powered for the ITT population while protecting the power in seropositive cohort (Table 2). A total of 67 adjudication committee (AC)–determined relapses are required (Equation 1) 13,14 to test the primary null hypotheses with 90% power (β = 0.1) and α = 0.05 (2-sided),…”

A multistage analysis strategy for a clinical trial to assess successively more stringent criteria for a primary endpoint with a low event rate

“…Also, for clinical trials with low event rates and minimal losses to follow‐up, φ is essentially the same as the planned ratio of the sample sizes (in terms of numbers of patients) for the new treatment and placebo groups. In this regard, φ = 1 would typically apply to a clinical trial to exclude excess risk of an adverse event such as MACE, whereas φ ≥ 2 would typically apply to a vaccine trial to show vaccine efficacy ≥ 75 % , with the corresponding rationales being the production of similar numbers of events for the two treatments under the alternative hypothesis for which the specified power applies (see Saville et al ).…”

“…The expression for the needed number of events d * to address the logrank test of H o : θ ≥ θ o versus H A : θ ≤ θ A , where θ A < θ o denote hazard ratios, is shown in for φ = 1 as based on the ad hoc modification of expressions in Saville et al and Freedman where ; in this regard, θ 蜧 = θ o when θ A = 1, and θ 蜧 = (1 ∕ θ A ) when θ o = 1.…”