A Comment on Sample Size Calculation for Analysis of Covariance in Parallel Arm Studies

Abstract

Introduction Randomized clinical trials are commonly used to confirm the efficacy of a new treatment. There are several advantages for using randomization in clinical trials, such as selection bias reduction and increased comparability among groups with potential confounding factors [1]. Balanced studies are often conducted to maximize the power of a study for a given total sample size. Sample size calculation plays a very important role in clinical trials. It has been studied for many years and achieved significant progress [2-4]. As far as we know, sample size calculation approaches for analysis of covariance (ANCOVA) are very limited. Recently, Borm et al. [5] proposed a simple sample size calculation closed form for ANCOVA with one covariate which is considered as a baseline of the response outcome. Based on the sample size from a two sample t-test and the correlation between response outcomes and covariate values, they show that this formula has accurate sample size calculation. The other method is based on a ratio of mean squares [6,7] where the null distribution follows a F distribution and the alternative is a non-central F distribution. There is no systematic comparison between these two approaches. We reviewed two existing sample size calculation approaches for ANCOVA with one covariate in Section 2. In Section 3, we compare the two approaches using exact simulation studies and an example from a randomized study is used to illustrate these two approaches. Section 4 is given to discussion. Methods Suppose that Y ij be the j th response outcome for the i th group, i=1, 2; j=1,2,…,n i , and X ij be the associated covariate. We consider the first group as the control, and the second group as the treatment group in this article. The covariate can be viewed as the baseline for the output. The regression model for the relationship between Y and X within the i th group is given as Y ij =β 0i +β 1 X ij +ε ij , Borm et al. [5] proposed a simple sample size calculation for ANCOVA by multiplying the number of subjects for the two sample t-test by a design factor. The factor here is 1−ρ 2 , where ρ is the correlation coefficient between the outcome and the covariate. Sample size calculation for the two-sample t-test is based on response outcomes. Given a significance level of α, a pre-specified power 1−β, the mean difference between the treatment group and the control of μ 2 −μ 1 , and a common standard deviation of response outcome σ, sample size per group is calculated as n=2σ 2 (Z 1−α/2 +Z 1−β) 2 /(μ 2 −μ 1) 2 , N=2(n + 1)(1−ρ 2) , = b w MS T MS where MS b is the mean square between groups, and MS w is the mean square within the group [7]. Under the null hypothesis with no Abstract We compare two sample size calculation approaches for analysis of covariance with one covariate. Exact simulation studies are conducted to compare the sample size calculation based on an approach by Borm et al. (2007) (referred to as the B approach) and an exact approach (referred to as the F approach). Although the B approach and the F approach have similar performance when the correlation coefficient is small, the F approach generally has a more accurate sample size calculation as compared to the B approach. Therefore, the F approach for sample size calculation is generally recommended for use in practice. where Z d is the d−th percentile of a standard normal distribution. Borm et al. [5] showed that the total sample size for the ANCOVA N=2n(1− ρ 2) may not be accurate enough for small sample settings to retain the pre-specified power. They provided some power plots to show that power with this sample size formula is generally smaller than 1−β for small sample settings. For this reason, they proposed to be used as the sample size by adding one subject for each group in the sample size calculation. They claimed that this sample size is accurate for all sample sizes. The second method is an exact approach based on a ratio of mean squares, where β 0i is the intercept for the i th group, β 1 is the common slope for both groups, and ε ij is the measure error which follows a normal distribution [8]. The mean difference between two groups is the difference between two intercepts.