# Power analysis for moderator effect in regression with two continuous predictors

Related to an earlier question on power analysis for multiple regression, a social science researcher asked me about power analysis for moderator regression (i.e., an interaction effect). The researcher asked me:

I seem to recall that power of tests for moderation with two continuous predictor variables is low - do you know the minimum sample size requirement in this context?

From the context, it can further be assumed that this is an observational study (not an experimental study) and that the dependent variable is continuous.

### Question

• What advice would you give regarding calculating the minimum sample size required?
• Are there any caveats that you would present?

If I had to do this, I would use a simulation approach. This would involve making assumptions about the regression coefficients, predictor distributions, correlation between predictors, and error variance (with help from the researcher), generating data sets using the assumed model, and seeing what proportion of these give a significant p-value for the interaction. Then use trial and error to find the minimum sample size giving the required power.

• 's sounds like a good plan. I'd add that the power problem would stem not just from having continuous predictors being multiplied, but from having "social science" (i.e., error-prone) variables. Limited reliability gets compounded when multiplying two such variables. – rolando2 Apr 30 '11 at 0:27

Assuming that the IV (X) and the Moderator (M) are continuous variables, and your research question is: Is the relationship between X and Y moderated by M? Your regression model would have 3 predictors X, M, and their (centered) interaction (X*M).

If you run the analysis using GPower (http://gpower.hhu.de/) you would set it up using the following parameters.

F tests - Linear multiple regression: Fixed model, R² deviation from zero Analysis: A priori: Compute required sample size Input: Effect size f² = 0.15 α err prob = 0.05 Power (1-β err prob) = 0.80 Number of predictors = 3 Output: Noncentrality parameter λ = 11.5500000 Critical F = 2.7300187 Numerator df = 3 Denominator df = 73 Total sample size = 77 Actual power = 0.8017655

You could vary the effect size, f2 to small .02, medium .15, or large .35.
In my above example f2 was set to .15.

Alpha should be set to .05, and power (1-B err prob) should be set to .80