Bigger Sample, Worse Coverage: Pooled Imputation and the Difference Between Two Correlations

Researchers routinely ask whether two variables are more strongly related in one groupthan another, and one variable is often incomplete. Standard guidance recommendsmultiple imputation, but a common main-effects-only default forces a common slope on ananalysis that lets groups differ: the imputation and analysis models are uncongenial. Westudy the difference of Fisher-z transformed correlations, ζ = z(ρ₁) − z(ρ₂), when theincomplete variable is missing at random on the outcome, comparing full data, listwisedeletion, pooled and group-wise imputation across N = 40–320 and 20%/40% missingness,with Monte Carlo standard errors throughout. Pooled imputation attenuates ζ by a factorwe derive in closed form and verify against simulation: the observed cases carry eachgroup's own association while the imputed cases carry the pooled slope, which cancels inthe difference. Because the bias is a fixed fraction of the estimand while the standarderror shrinks at 1/√N, nominal 95% coverage at 40% missing falls from 0.992 at N = 40 to0.703 at N = 320, and power halves relative to group-wise imputation. The failure isone-directional shrinkage in the standardised designs we study; groups that share both aconditional slope and a residual variance are a separate, exact case, in which pooledimputation moves nothing. When they differ in scale, pooled imputation manufactures adifference from a true null, with rejection rates reaching 0.309. In iris, a realspecies difference detected at p = .025 with complete data survives, at 40% missing, innone of 2000 replications after default imputation.