The Cost of Naive Imputation to a Derived Estimand: An Exact Law, with the Square as Lead Case

Multiple imputation is validated for variables; applied researchers report functions ofthem: a product, a square, a difference of correlations. An imputation model fitted tothe columns cannot carry a relationship it does not contain, so the filled-in rowsarrive stripped of it. The direction of the resulting bias is established. For thenonlinear terms studied here we are not aware of a published expression for its size asa function of the missing rate, and the literature we searched reaches that size bysimulation, one design at a time. We derive it, as one law: under missingness completelyat random the bias is the filled-in rows' failure to satisfy the analysis's own normalequations, converted into a coefficient error by the moments the analysis sees. The lawreaches any derived term entering those equations, and not a ratio-and-transformestimand such as a difference of correlations. For an outcome regressed on a standardnormal predictor and its square, with the predictor missing completely at random at rateπ, the attenuation of the squared term under naive imputation is an exact rationalfunction of the design and π. Proportionality to π is its first-order term, accurate toone part in a thousand in our baseline design and to one part in ten where the linearcoefficient is large. One minus that first-order coefficient factors, with the squaredlinear coefficient a common factor of every term: the filled-in rows preserve a squareonly through the linear term. The ratio of attenuation to missing fraction runs from0.570 to 1.000 across nine designs, reaching 1.000 where the outcome carries no linearinformation about the incomplete predictor, and exceeding one in other designs, wherethe loss outruns the missing fraction. Outrunning the missing fraction is not reversingthe coefficient: a global bound proves that the completed squared term stays strictlybetween zero and its true value at every rate below one, so its sign never flips. Undera logistic missing-at-random mechanism the attenuation reduces exactly to elevenselection moments, which numerical integration (quadrature) evaluates; no elementaryformula for them is derived, and no impossibility is proved. Nominal 95% coverage fallsfrom 0.912 at N = 50 to 0.003 at N = 800 with 40% missing. Of four strategies compared,only substantive-model-compatible imputation was insensitive to the mechanism, and afitted version using no true parameters — the substantive model and the covariate modelboth estimated — recovers it to within three Monte Carlo standard errors.