Sample size determination for confidence intervals of interaction effects in moderated multiple regression with continuous predictor and moderator variables

Abstract

Moderated multiple regression (MMR) has been widely employed to analyze the interaction or moderating effects in behavior and related disciplines of social science. Much of the methodological literature in the context of MMR concerns statistical power and sample size calculations of hypothesis tests for detecting moderator variables. Notably, interval estimation is a distinct and more informative alternative to significance testing for inference purposes. To facilitate the practice of reporting confidence intervals in MMR analyses, the present article presents two approaches to sample size determinations for precise interval estimation of interaction effects between continuous moderator and predictor variables. One approach provides the necessary sample size so that the designated interval for the least squares estimator of moderating effects attains the specified coverage probability. The other gives the sample size required to ensure, with a given tolerance probability, that a confidence interval of moderating effects with a desired confidence coefficient will be within a specified range. Numerical examples and simulation results are presented to illustrate the usefulness and advantages of the proposed methods that account for the embedded randomness and distributional characteristic of the moderator and predictor variables.

The impact of ignoring random features of predictor and moderator variables on sample size for precise interval estimation of interaction effects

“…The other approach gives the sample size required to ensure, with a given tolerance probability, that a confidence interval of moderating effects will be within a specified range. The vital discrepancies between the conditional and unconditional settings in power and precision analyses are also closely evaluated in Shieh (2009Shieh ( , 2010. The results reveal substantial detrimental effects of failing to account for the randomness of predictor and moderator variables.…”

“…The aim of this article is to contribute to the design of MMR study by illustrating how the specification of inherent features associated with continuous predictor and moderator variables in sample size calculations influences the resulting precision of confidence intervals for the prescribed two interval width appraisals in the inference of an interactive effect. Consequently, the presentation here complements the work of Shieh (2010) with distinct precision criteria in sample size determinations.…”

“…In view of the relatively scarce description of interval estimation in MMR, it is prudent to contribute to the documentation and examination of confidence intervals in different perspectives. The precision considerations in Shieh (2010) may not be the only criteria of practical importance. Just as in the instance of Kelley and Maxwell, two other useful principles related to the statistical properties of a confidence interval deal with the control of expected width and tolerance probability of interval width within a designated value.…”

“…In the context of MMR, the distinct formulations of fixed and random modeling of the simple interaction models were especially emphasized in Shieh (2009Shieh ( , 2010. Specifically, Shieh (2009) considered the power calculation and sample size determination for significance tests of moderating effects.…”

“…The procedure takes account of the critical factors of strength of moderator effect, magnitude of error variation, and distributional property of predictor and moderator variables. On the other hand, Shieh (2010) incorporated the random nature of continuous moderator and predictor variables into two approaches to sample size computation for precise interval estimation of interaction effects. One approach provides the necessary sample size so that the designated interval for the least squares estimator of moderating effects attains the specified coverage probability.…”