Does a relationship depend on something else? (moderation)

In brief

The relationship between X and Y is presumably not the same for everyone but depends on a third variable W, the moderator. Example: support (X) affects performance (Y) differently under high strain (W) than under low strain. Choose three columns: X and W numbers or columns with exactly two values, Y a number. Statable tests the interaction X·W and shows simple slopes of X at different values of W. Covariates are not yet possible.

What Statable calculates

• X and W are centered at the mean, and the product X·W is added. This gives two regressions, without and with the interaction; the data stay unchanged.
• ΔR² is the gain in explained variance due to the interaction (with F and p), f² the matching effect size.
• Simple slopes of X at W = mean − 1 SD, mean and mean + 1 SD (if W has only two values, at both) with standard error, t, p and 95% confidence interval.
• Johnson-Neyman: the values of W at which the slope of X becomes significant (only boundaries within the observed range of W).

Reading the result

• If the interaction is significant, the slope of X differs depending on W; the simple slopes show how, and the chart has one line per value of W.
• Because of the centering, the coefficients of X and W are conditional effects at the mean of the other variable. The interaction itself does not change through this.
• A significant interaction shows that the relationship differs by W - not that W causes it. Whether a variable is a moderator or a mediator is decided by theory.

Requirements and limits

• At least 10 people with all three answers; interactions are detected reliably only with many people (below 50 the result is very uncertain).
• Linear relationships, roughly equal spread of the residuals, independent people. Simple slopes outside the observed range of W are extrapolation.

Interaction: B = 1.73, 95% CI [1.19, 2.28], p < .001, ΔR² = .17, F(1, 142) = 39.66; simple slopes: M − 1 SD: b = 0.01, p = .977; M: b = 2.11, p < .001; M + 1 SD: b = 4.20, p < .001

For specialists

PSPP calculates the regressions (the centered helper columns exist only for the calculation); Statable calculates simple slopes and Johnson-Neyman from the coefficients and their covariance matrix. ΔR² corresponds to the test of the product term. Checked against R (lm on centered data, interactions::sim_slopes and johnson_neyman).

Typical examples

• Does study time have a stronger effect on performance when self-efficacy is high?
• Does the relationship between anxiety and performance depend on gender?