Predictor correlations and statistical power

Setting a correlation tells MCPower to generate two predictors that move together rather than independently, the way real predictors usually do (income and education, age and experience). This cuts both ways for power: shared variance gives each predictor less unique variance, so a main effect needs a larger sample to reach the same power — but it makes their product more variable, so an interaction often needs a smaller one. Only the strength \(|r|\) matters, not the sign.

Why correlation cuts both ways

The effect of correlated predictors on power depends on what you are testing:

• Main effects — correlation reduces power. Shared variance means less unique variance per predictor, so standard errors grow.
• Interaction effects — correlation increases power. Correlated predictors produce a more variable product term, and a more variable interaction is easier to detect.

MCPower generates correlated predictors and then transforms each to its target distribution while preserving the correlation structure.

A correlation you set yourself is treated as a latent (Gaussian) correlation — exact for normally-distributed predictors. When MCPower instead measures a correlation from uploaded data (partial mode), it targets the rank (Spearman) correlation of your data, not the Pearson one, because redrawing each predictor through its empirical shape preserves rank order rather than linear scale.

The matrix must be consistent (PSD)

MCPower validates every correlation you supply before any simulation runs, and rejects impossible input on three levels:

• Each pair must be in range. A correlation must lie in \([-1, 1]\); a stray corr(a, b)=1.5 fails with correlation pair (a, b): value must be in [-1, 1], got 1.5.
• A full matrix must be well-formed. If you pass a matrix it must be square, symmetric, and have an all-1s diagonal — MCPower errors loudly rather than silently coercing a malformed one.
• The whole structure must be consistent. Even when every pair is individually in range, the combination can still be impossible. A valid matrix is positive semi-definite (PSD) — i.e. the correlations could actually co-occur in real data. A non-PSD matrix fails with correlation matrix is not positive semi-definite.

How much does it matter?

Main effects — required sample size grows with r

Interactions — required sample size shrinks with r

| Correlation \(|r|\) | Multiplier vs. \(r = 0\) | |---|---| | 0.00 | 1.00× | | 0.30 | 0.82× | | 0.50 | 0.76× |

Three correlated predictors — the impact compounds

Multipliers are illustrative, from simulations of two- and three-predictor models with small effects. The relationship — not the absolute sample size — is what carries across designs.

Typical correlations by domain

Binary and factor predictors can't be correlated directly

Correlation is continuous-only by design. Binary predictors and factor predictors (3+ levels) are generated from their marginals, so naming either one in a correlation pair is rejected. When you need correlated categorical and continuous variables, simulate from empirical data, which preserves the joint structure of your dataset directly.

Correlations below ~0.30 have negligible impact; above ~0.50 the sample-size increase becomes substantial. Use scenario analysis to see how a noisy or uncertain correlation structure moves your power.