حجم نمونه مورد نیاز برای تشخیص فعل و انفعالات بین دو باینری اثرات ثابت در یک مدل رگرسیون خطی اثرات مخلوط

Mixed-effects linear regression models have become more widely used for analysis of repeatedly measured outcomes in clinical trials over the past decade. There are formulae and tables for estimating sample sizes required to detect the main effects of treatment and the treatment by time interactions for those models. A formula is proposed to estimate the sample size required to detect an interaction between two binary variables in a factorial design with repeated measures of a continuous outcome. The formula is based, in part, on the fact that the variance of an interaction is fourfold that of the main effect. A simulation study examines the statistical power associated with the resulting sample sizes in a mixed-effects linear regression model with a random intercept. The simulation varies the magnitude (ΔΔ) of the standardized main effects and interactions, the intraclass correlation coefficient (ρρ), and the number (k)(k) of repeated measures within-subject. The results of the simulation study verify that the sample size required to detect a 2×2 interaction in a mixed-effects linear regression model is fourfold that to detect a main effect of the same magnitude.

The mixed-effects linear regression model (Harville, 1977 and Laird and Ware, 1982) is widely used in observational studies and randomized controlled clinical trials (RCT) in which there are repeated measures over time. In designing a study, the Ethical Guidelines of the American Statistical Association (1999) advise statisticians to provide informed recommendations for sample size such that a research protocol will neither propose an inadequate nor an excessive number of subjects to detect a scientifically noteworthy result with acceptable statistical power. Several authors have examined the sample sizes required to detect the main effects and interaction of treatment and time in longitudinal studies with repeated measures (e.g., Hsieh (1988), Rochon (1991), Overall and Doyle (1994), Hedeker et al. (1999), Raudenbush and Liu (2001) and Diggle et al. (2002)). Yet a study that is designed to detect the main effect of treatment will not have sufficient power to detect the interaction between two binary fixed effects. In a 2×2 factorial fixed-effects ANOVA with equal cell sizes and an assumption of independence among observations, for instance, the sample size required to detect an interaction is four times that for a main effect of the same magnitude (Fleiss, 1986). However, we are not aware of formulae to estimate the sample size needed to detect an interaction between two binary fixed effects in a mixed-effects linear regression model for analysis of repeatedly measured correlated data. The objective of this manuscript is to examine the sample size required to detect a 2×2 interaction of two binary fixed effects in mixed-effects linear regression analyses. The model, described in detail in Section 2, also incorporates a time-varying covariate, but that covariate does not interact with group membership. We sought to determine if, as with the fixed-effects factorial ANOVA, the sample size needed to detect an interaction in a repeated measures design is fourfold that of a main effect. A formula for the sample size required to detect an interaction is presented below. A simulation study then examines the statistical power of the resulting sample sizes to detect interactions of various magnitudes in a 2×2 factorial design with repeated measures of a continuous outcome.