Multiple Imputation for Causal Effects

Why Multiple Imputation Is Used in Causal Analysis

When outcomes or confounders are missing, analyzing only records with complete data can change the population being compared and can induce selection bias. Multiple imputation addresses this by replacing each missing value with several plausible values, creating \(m\) completed datasets. The same causal analysis is then performed in every dataset, and the estimates and their uncertainty are combined.

The goal is not to make missing values known. Rather, the variation across completed datasets represents uncertainty about those values. A causal effect estimate must still rely on assumptions about confounding, positivity, and missingness; imputation does not by itself make an observational comparison causal.

Specify the Causal Target and Required Assumptions

Let \(A\) denote treatment, \(Y\) the outcome, and \(C\) measured baseline confounders. For a binary treatment, one common target is the average treatment effect (ATE), \(\psi=E[Y(1)-Y(0)]\), where \(Y(a)\) is the potential outcome under treatment level \(a\). Under consistency, conditional exchangeability \(Y(a)\perp A\mid C\), and positivity \(P(A=a\mid C=c)>0\) for covariate patterns in the target population, this estimand can be identified by standardization:

Missing data add assumptions. A common working condition is missing at random (MAR): conditional on variables observed for a person, the probability a value is missing does not depend on its unobserved value. For example, an outcome may be MAR if its missingness depends on treatment, observed baseline severity, and observed auxiliary variables, but not on the unobserved outcome after conditioning on those variables. MAR is an assumption about the data-generating and observation processes, not a conclusion established merely by fitting an imputation model.

A valid analysis also requires imputation models that adequately represent the relevant conditional distributions and preserve the relationships needed by the causal analysis. Imputation should reflect uncertainty in model parameters and missing values; simply replacing each value by a fitted conditional mean is generally not proper multiple imputation. The causal identification assumptions and the missing-data assumptions do different work: the former identify the effect in the absence of missingness, while the latter support recovering the necessary information despite missing values.

Constructing Imputations for Causal Variables

The imputation model should include treatment, outcome, confounders, and useful observed auxiliary variables that predict missingness or the missing values. If a confounder is missing, treatment and outcome information can be important predictors of that confounder. If the outcome is missing, treatment and the confounders used in the causal analysis should generally be included. Leaving out treatment or important confounders can weaken the association structure needed to estimate the causal contrast.

The model also needs to preserve the form of the substantive analysis. For example, if the outcome model includes a treatment-by-confounder interaction or a nonlinear covariate effect, an imputation procedure that imposes incompatible relationships can distort the resulting treatment contrast. Categorical variables need models suitable for categorical values; continuous variables require distributions appropriate to their support and shape. With several variables incomplete, imputations may be generated jointly or through a sequence of conditional models. In either case, the resulting completed datasets should represent uncertainty rather than a single “best guess.”

Once each dataset is complete, apply the same estimand, target population, and analysis procedure in every dataset. For an ATE, that might mean fitting the same outcome regression and standardizing predicted outcomes to the same covariate distribution in each dataset. Use a variance estimator \(U_i\) that corresponds to that completed-data estimator. Do not average raw imputations first and analyze only the averaged data: that discards the between-imputation uncertainty.

Rubin’s Rules and Why They Have This Form

The within-imputation component, \(\bar{U}\), reflects sampling uncertainty that would remain if the completed data were observed. The between-imputation component, \(B\), estimates how much the effect estimate varies across imputations because the missing values are uncertain. The factor \(1+1/m\) incorporates the additional finite-imputation contribution to the variance of the pooled estimate. In particular, \(B\) is the sample variance across the \(m\) estimates; it is not, on average, a downward-biased estimate of the underlying imputation-distribution variance.

The variance formula follows from a simple decomposition. If the complete-data estimate has conditional variance represented by \(U_i\), uncertainty in the missing data contributes variation in the conditional estimate across imputations. Averaging \(m\) imputed estimates leaves finite simulation uncertainty, represented by \(B/m\), in addition to the between-imputation component \(B\). Combining the within-data variance and this uncertainty yields:

For inference, a common large-complete-data-sample approximation uses a \(t\) distribution with degrees of freedom

A confidence interval is then \(\bar{Q}\pm t_{0.975,\nu}\sqrt{T}\) for a 95% interval. When \(B=0\), this expression for degrees of freedom is not used as written; uncertainty from imputation is absent in the observed set of imputations, and inference needs an appropriate complete-data or finite-sample degrees-of-freedom treatment. In small samples, refinements that account for finite complete-data degrees of freedom may be preferable.

Common Pitfalls and Practical Interpretation

Multiple imputation is useful because it carries missing-value uncertainty into the causal estimate, but it is not a universal correction for missing data. Its conclusions depend on the imputation model, the MAR assumption, and the causal identification assumptions. If missingness depends on an unobserved value even after conditioning on observed information, standard MAR imputations may not represent the missing-data process. That concern requires additional assumptions and analyses, rather than a claim that pooling has resolved it.

• Imputing without treatment or outcome information. A confounder imputation model that ignores relevant treatment or outcome associations can fail to preserve the structures needed for adjustment. Include variables that are substantively relevant and observed.
• Using an incompatible model. An imputation model that omits a substantive interaction or imposes an unsuitable distribution can distort the causal contrast. Align imputation and analysis models as far as practicable.
• Treating imputed values as observed truth. A completed dataset is one plausible version of the missing data, not a recovered record. Use all \(m\) analyses and include between-imputation uncertainty.
• Assuming MAR because a model was fitted. MAR is not testable from observed data alone in general. A well-fitted model does not prove the assumption.
• Changing the estimand across imputations. Use the same target population, causal contrast, and analysis procedure in all completed datasets so that the \(Q_i\) estimates the same quantity.

The central practical rule is to define the causal estimand first, specify an imputation strategy compatible with its analysis, estimate that same contrast in every completed dataset, and pool both estimates and uncertainty. The resulting confidence interval describes sampling and imputation uncertainty conditional on the modeling and identification assumptions; it does not include uncertainty about whether those assumptions are true.