Sample Size Calculation to Evaluate Mediation Analysis

Mediation analysis is used to review the comparative change in the amount of strength of association of the primary predictor with the outcome after adjustment for the mediator. Mediation models are very widely used in social sciences and biomedical sciences. Before conducting mediation studies, researchers want to know the sample size (i.e. number of subjects) required for achieving the adequate power when testing for mediation. To the author’s knowledge, there is no any SAS ® procedure that produces sample size calculation for mediation analysis. The author presents a macro that implements the methodology for sample size calculation for mediation analysis written by Vittinghoff et al. (2009). It implements the methods to calculate sample sizes for the linear regression models. Very basic understanding of SAS is enough to use this macro. SAS Macro programming skill is not expected. When the macro is invoked, a series of windows will pop-up asking user to input required information. Output will be presented in SAS output window as well as in Microsoft word document. This macro is tested on Windows SAS 9.1 and above. INTRODUCTION Mediation analysis is used to review the comparative change in the amount of strength of association of the primary predictor with the outcome after adjustment for the mediator. Mediation models are very widely used in social sciences, especially in psychological studies. Before conducting mediation studies, researchers want to know the sample size (i.e. number of subjects) required for achieving the adequate power when testing for mediation. This problem of sample size calculation for testing the mediation effect of intermediate variable to the primary predictor is a common issue in epidemiologic and clinical research. To the author’s knowledge, there is no any SAS procedure or that produces sample size calculation for mediation analysis. The SAS macro presented in this paper will be used to calculate sample for testing mediation effect based on Vittinghoff et al. (2009). They proposed the method based on variance inflation factor in regression that provide the exact sample size for linear model and approximations for logistic, Poisson, and Cox models. In this paper, I will present sample size calculation for linear models. The following four cases will be considered: (a) continuous primary predictor, continuous mediator (b) binary primary predictor, continuous mediator (c) continuous primary predictor, binary mediator, and (d) binary primary predictor, binary mediator. The authors argued that their method are also applicable to detecting the independent effect of a primary predictor in the presence of substantial confounding The method of sample size calculation is based on the Wald tests as these tests are more common in practice even though these methods are less reliable in case of small samples and if the alternative hypothesis are far from null. SAMPLE SIZE CALCULATION FOR LINEAR MODEL Let Y be the response, X1 a primary predictor variable and X2 is a mediator variable. The simple linear regression model takes the form yi = β0 + β1 X1 + β2 X2 + εi , εi~N(0, σe ) (1) Where β1 is the estimator of primary predictor (X1) before the adjustment of mediator (X2), β2 is the regression coefficient for the mediator ( X2 ) and σe 2 is the variance of the random error term in the linear regression. If β1 ∗ is the estimator of the primary predictor (X1) before the adjustment of mediator ( X2 ) in the full model (1), then proportion of treatment effect explained (PTE), which assess the mediation as the relative change in the strength of association of the primary predictor with the outcome after adjustment for the mediator, is of the form