Multiple Imputation for Missing Phenotypes and Covariates

Why Multiple Imputation Matters in Genetic Association

An association analysis can lose information when a phenotype or an adjustment covariate is missing. An analysis restricted to participants with complete records may also be biased if inclusion in that subset is related to the phenotype, genotype, or covariates. Multiple imputation offers a way to represent uncertainty about missing values rather than filling each gap with a single fixed prediction.

The central procedure is to create several completed datasets, fit the same planned association model to each one, and combine the resulting estimates and variances. The separate imputations are not independent studies: they are alternative plausible completions of the same observed data. Their variation is part of the uncertainty that must be included in the final inference.

Constructing Imputations for Association Analysis

Before imputing, specify the association model and its target parameter. For a single-variant analysis, that model commonly includes the phenotype, the tested variant’s genotype or dosage, and the prespecified covariates. The imputation models should include these variables as appropriate: when a covariate is missing, the phenotype and tested genotype can be informative predictors of it; when the phenotype is missing, genotype and covariates can help predict the phenotype. Omitting the tested genotype when imputing a missing phenotype or covariate can distort the genotype–phenotype relationship that the analysis is meant to estimate.

Predictors of missingness or of the missing values can also improve imputations. Depending on the study, these may include ancestry principal components, recruitment site, assay batch, or other observed variables. Their inclusion should be scientifically justified and should not replace the planned adjustment set by an unprincipled, data-driven search. If the analysis includes a nonlinear term or an interaction, the imputation strategy should preserve that structure where relevant; imputing only main effects and later analyzing an interaction can make the imputation and analysis models incompatible.

A common implementation is fully conditional specification, also called chained equations. Each variable with missing entries is assigned a conditional model given the other variables. The procedure cycles through the variables, updating missing values by draws from the corresponding conditional distributions. Model choice follows the variable’s scale and support: for example, a normal linear model may suit a roughly continuous covariate, logistic regression a binary covariate, and multinomial or ordinal regression a categorical covariate. A model should respect meaningful bounds and categories; a normal model is generally not suitable for a binary variable or a count that cannot be negative.

The conditional models should be stochastic and include uncertainty in their fitted parameters as well as residual variation. A deterministic prediction substituted repeatedly would not generate the between-imputation uncertainty needed for valid pooling. Chained equations are a practical strategy, but their use does not guarantee that all conditional models correspond to a coherent joint distribution. Check convergence and plausibility of the generated values, and use models appropriate to the study design and analysis.

Rubin’s Rules for Pooling Estimates

Let \(Q_i\) be the estimate of the target parameter from completed dataset \(i\), and let \(U_i\) be its estimated variance from that dataset, for \(i=1,\ldots,m\). For example, \(Q_i\) may be a genotype coefficient and \(U_i\) its squared standard error. Rubin’s rules combine the \(m\) estimates and variances as follows.

Here \(\bar{Q}\) is the pooled estimate, \(\bar{U}\) is the average within-imputation variance, and \(B\) is the between-imputation variance. The total variance and pooled standard error are

The factor \(1+1/m\) includes the finite number of imputations: \(B\) reflects variation in estimates across plausible completed datasets, and \(B/m\) accounts for the uncertainty from estimating the pooled result using only \(m\) imputations. Consequently, pooling only the mean coefficient and its average within-imputation variance omits an important source of uncertainty whenever \(B\) is nonzero.

For a scalar estimate, a commonly used large-sample degrees-of-freedom approximation is

When \(B=0\), this expression is treated as having infinite degrees of freedom. Otherwise, a confidence interval is \(\bar{Q}\pm t_{1-\alpha/2,df}\sqrt{T}\), where \(t_{1-\alpha/2,df}\) is the corresponding quantile of the \(t\) distribution. This is a scalar, large-sample pooling rule; more elaborate finite-sample adjustments may be appropriate when complete-data degrees of freedom are limited.

The variance rule follows the law of total variance in a simple way. Conditional on a completed dataset, the analysis has estimated variance \(U_i\), representing sampling uncertainty within that completion. Across completions, the estimates vary because the missing values are uncertain, measured by \(B\). If the mean estimate is based on only \(m\) draws, its Monte Carlo contribution is approximately \(B/m\). Adding the average within variance and both between-imputation components gives \(T=\bar{U}+B+B/m\).

Pooling Binary-Trait Association Results

For logistic association, pool the regression coefficient on the log-odds scale and its variance—not odds ratios or p-values. If \(Q_i\) is the genotype coefficient, the pooled odds ratio per allele is \(\exp(\bar{Q})\). Exponentiate the endpoints of the pooled coefficient interval to obtain an interval for that odds ratio. This procedure keeps the pooling calculations on the approximately symmetric coefficient scale.

Practical Checks and Common Pitfalls

A defensible workflow keeps the imputation and association analyses aligned. Specify the target association model first, select conditional imputation models suited to each incomplete variable, generate multiple stochastic completed datasets, and fit the same analysis model with the same coding and covariate set in every dataset. Then pool the coefficient and variance using Rubin’s rules and report the number of imputations, imputation-model choices, and the pooled uncertainty.

• Do not omit the tested genotype or phenotype automatically. They can predict missing covariates and phenotypes and help preserve the association being estimated.
• Do not use a single deterministic fill. A mean, mode, or predicted value substituted once fails to represent uncertainty in the missing data.
• Do not pool p-values or odds ratios directly. Pool the model coefficient and its variance; transform the pooled coefficient and interval only afterward when needed.
• Do not assume MAR just because multiple imputation was used. The method depends on its assumptions and models; imputation alone cannot demonstrate that missingness is ignorable.