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

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 Zd 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− ρ2) 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 β0i is the intercept for the i th group, β1 is the common slope for both groups, and εij is the measure error which follows a normal distribution [8]. The mean difference between two groups is the difference between two intercepts. Citation: Shan G, Ma C (2014) A Comment on Sample Size Calculation for Analysis of Covariance in Parallel Arm Studies. J Biomet Biostat 5: 184. doi:10.4172/2155-6180.1000184 J Biomet Biostat ISSN: 2155-6180 JBMBS, an open access journal Page 2 of 2 Volume 5 • Issue 1 • 1000184 threshold value is Fα, where Pr(F1,N−3 ≥ Fα)=α. Under the alternative, the test statistic follows a non-central F1,N−3,λ, distribution with the non-central parameter 2 2 / ε λ σ σ = b N [7], where 2 2 2 (1 ) , ε σ ρ σ = − 2 2 2 1 1 2 2 / ( ) / ( ) , σ μ μ μ μ = − + − b n N n N and 1 1 2 2 / / μ μ μ = + n N n N is the overall response outcome mean. The power of the study is then expressed as a probability of being greater than or equal to the threshold Fα in the non-central F distribution, Pr(F1,N−3,λ ≥ Fα). The required sample size is determined by increasing the sample size by one each time until the pre-specified power is reached. Method comparison We referred to the approach proposed by Borm et al. [5] as the B approach, and the other based on the F distribution as the F approach. Power is calculated as the percentage of trials with significant p-values using ANCOVA based on 10000 simulations. Calculated power is presented in Table 1 for α=0.05, β=0.2, σ=1, and μ2−μ1=0.5, and Table 2 for α=0.01, β=0.2, σ=1, and μ2−μ1=1. Sample size based on the F approach is calculated using PASS 12 [9]. As can be seen from both tables, the difference between the B approach and the F approach is negligible for small ρ values. The power of the B approach is much lower than the pre-specified power for large ρ values, as shown in Table 2, the power could be as low as 52%. Although the B approach and the F approach have similar performance when ρ is small, the F approach generally has more accurate sample size calculation as compared to the B approach. A parallel randomized clinical trial is illustrated for sample size calculation based on the B approach and the F approach. Patients with rheumatoid arthritis are randomized into one of the groups with or without leunomide [10]. The response outcome, the disease activity score, is measured before and after the treatment. The baseline measurement is considered as the covariate in the ANCOVA model. This example is also used by Borm et al. [5]. The standard deviation is estimated as σ=1.2. At a significance level of α=0.01 and 90% power, the sample size calculations based on the B approach to detect a mean difference of μ2−μ1=0.6 are 122, 86, and 46 as total sample sizes for ρ=0.7, 0.8, and 0.9, respectively. It needs total sample sizes of 126, 90, and 50 using the F approach. The sample size from the B approach is less than that from the F approach. The sample size from the B approach may not attain the pre-specified power of the study.