Tests of Simple Slopes in Multiple Regression Models with an Interaction: Comparison of Four Approaches

Yu Liu, Stephen West, Roy Levy, Leona S. Aiken

Research output: Contribution to journalArticlepeer-review

38 Scopus citations


In multiple regression researchers often follow up significant tests of the interaction between continuous predictors X and Z with tests of the simple slope of Y on X at different sample-estimated values of the moderator Z (e.g., ±1 SD from the mean of Z). We show analytically that when X and Z are randomly sampled from the population, the variance expression of the simple slope at sample-estimated values of Z differs from the traditional variance expression obtained when the values of X and Z are fixed. A simulation study using randomly sampled predictors compared four approaches: (a) the Aiken and West (1991) test of simple slopes at fixed population values of Z, (b) the Aiken and West test at sample-estimated values of Z, (c) a 95% percentile bootstrap confidence interval approach, and (d) a fully Bayesian approach with diffuse priors. The results showed that approach (b) led to inflated Type 1 error rates and 95% confidence intervals with inadequate coverage rates, whereas other approaches maintained acceptable Type 1 error rates and adequate coverage of confidence intervals. Approach (c) had asymmetric rejection rates at small sample sizes. We used an empirical data set to illustrate these approaches.

Original languageEnglish (US)
Pages (from-to)1-20
Number of pages20
JournalMultivariate Behavioral Research
StateAccepted/In press - Apr 27 2017


  • Bayesian
  • bootstrap
  • interaction
  • multiple regression
  • simple slope

ASJC Scopus subject areas

  • Statistics and Probability
  • Experimental and Cognitive Psychology
  • Arts and Humanities (miscellaneous)


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