A Compatibility Audit for Logistic Regression

Why Audit Imputation for a Logistic Interaction?

A logistic regression can represent an effect that changes across groups by including an interaction. If an analysis requires that interaction but the imputation models smooth it away, the completed data may not retain the relationship the analysis is intended to estimate. Conversely, including a product term in an analysis plan does not by itself make a set of chained conditional models compatible with that plan.

An audit asks whether the imputation specifications carry the information needed for the planned analysis. In this tutorial, the analysis has a binary outcome \(Y\), a continuous predictor \(X\), a binary effect modifier \(Z\), and an additional covariate \(C\). Some predictors may be missing. The aim is not to prove that every practical MICE specification is exactly compatible, but to identify important omissions and functional-form mismatches before fitting the analysis.

Write the Analysis Model Before Inspecting Imputation Models

The audit begins with the planned substantive model. For the variables above, suppose the analysis is

Here \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). The model assumes that the log odds are linear in the displayed terms, conditional on the other included terms. In particular, the \(X\)-by-\(Z\) interaction means the association between \(X\) and the log odds of \(Y=1\) differs between the two values of \(Z\). It does not assert that \(X\) causes \(Y\).

For a one-unit increase in \(X\), the conditional odds ratio is \(\exp(\beta_X)\) when \(Z=0\), and \(\exp(\beta_X+\beta_{XZ})\) when \(Z=1\), holding \(C\) fixed. These interpretations depend on the interaction being in the analysis model. A model with only \(X\) and \(Z\) main effects imposes a common odds ratio for \(X\) across the two groups.

Before auditing, state the assumptions relevant to the planned analysis: \(Y\) is coded consistently as binary; \(X\), \(Z\), and \(C\) are defined and measured consistently; the specified conditional logit form and interaction are scientifically and analytically intended; and the data provide enough information to estimate the terms. The imputation models also require defensible assumptions about the missingness and the conditional distributions used to generate missing values. An audit of model terms cannot establish those assumptions or repair a lack of information.

Three Core Checks for the Imputation Specifications

MICE fits a conditional model for each variable with missing values, using other variables as predictors. For this audit, inspect each such model against the substantive model rather than asking only whether every variable appears somewhere in the imputation system.

1. Outcome inclusion. When imputing an incomplete predictor, include the observed outcome \(Y\) as a predictor. The outcome contains information about the incomplete predictor that is relevant to the analysis. Omitting \(Y\) can weaken or distort the predictor–outcome association in the completed data. Including the observed outcome to impute predictors does not mean that the outcome is itself being imputed or used as an outcome predictor in the substantive analysis.
2. Interaction information. If \(X\) is incomplete and \(Z\) is observed, check whether the imputation model for \(X\) can represent the way its relationship to the other variables varies with \(Z\). A term involving \(Y\) and \(Z\), such as \(YZ\), can be important. If \(Z\) is incomplete, the corresponding relationship may require information involving \(Y\) and \(X\), such as \(YX\). The product \(XZ\) must also be represented appropriately in the analysis of each completed data set. Merely including \(Y\), \(X\), and \(Z\) as separate main effects in every imputation model may fail to preserve the substantive interaction.
3. Functional form. Check that the imputation models do not impose relationships that contradict those needed by the analysis. For example, if a continuous variable has a nonlinear term in the substantive model, a strictly linear imputation relationship may not preserve the relevant pattern. Likewise, check whether group-specific patterns implied by the interaction are represented. Matching variable names is not enough; the form of their relationships matters.

These are audit questions, not a universal recipe saying that one fixed list of product terms guarantees exact compatibility. The correct conditional distribution for a missing variable depends on the substantive model and on the joint distribution of the predictors. Ordinary chained conditional models can be convenient approximations, and their separate conditionals need not correspond to one coherent joint distribution.

Why an Interaction Can Induce a More Complex Conditional Model

A short derivation shows why the interaction audit cannot be reduced to a checklist of variable names. Let \(\eta_z\) be the linear predictor from the analysis model when \(Z=z\), at fixed \(X\) and \(C\). For binary \(Z\), Bayes' rule gives the odds of \(Z=1\) rather than \(Z=0\), conditional on \(Y\), \(X\), and \(C\):

Taking logarithms and using the logistic model, where \(P(Y=y\mid Z=z,X,C)=\exp(y\eta_z)/\{1+\exp(\eta_z)\}\), yields

The interaction makes \(\eta_1-\eta_0=\beta_Z+\beta_{XZ}X\), so the outcome information about \(Z\) depends on \(X\). The log terms also depend on the linear predictors. Thus, even with a simple logistic substantive model, the conditional distribution needed for imputing \(Z\) need not be a simple logit-linear model containing only a few main effects. The derivation justifies checking both interaction information and functional form; it also explains why a practical specification can be an approximation rather than an exact match.

Diagnostics, Alternatives, and Limitations

A practical audit should be followed by diagnostics of the generated imputations and of the analysis relationships. Look for signs that the model is not doing what the specification intended:

• Compare observed and imputed distributions of each incomplete variable, including distributions across \(Y\) and across \(Z\). A plausible overall distribution can still conceal a poor group-specific pattern.
• Check whether the relationship between \(X\) and the outcome differs across \(Z\) in the completed data in a way consistent with the intended interaction. The earlier material on stratified checks for imputed interactions provides a focused approach to those comparisons.
• Inspect convergence and stability of the chained updates, and check for implausible values or separation in a logistic conditional model. A chain that appears to stabilize does not establish compatibility or remove model misspecification.
• Compare the analysis estimates across imputations and assess whether the interaction estimate is sensitive to defensible changes in the imputation specification. Large changes can be a warning that the data provide limited information or that the imputation model matters substantially.

The main practical alternative to a basic main-effects chained specification is to enrich the conditional models with outcome and interaction information, while using suitable forms for continuous predictors. Another approach is a substantive-model-compatible imputation method, designed to respect the analysis model more directly. These approaches can be more demanding to specify and fit. Complete-case analysis avoids specifying predictor imputation models, but it discards incomplete records and can be biased or inefficient unless its own assumptions hold. No approach makes the missingness assumptions true merely by being selected.

The audit has limits. It cannot establish that the logistic analysis is scientifically adequate, that the missingness mechanism is ignorable, or that the sample contains enough information to estimate the interaction. Nor does one successful marginal diagnostic prove that all conditional relationships are preserved. With sparse data or rare outcome-modifier combinations, complex interaction terms may be weakly estimated or unstable. The appropriate response is to report the specification and its limitations, and to examine whether substantive conclusions depend on reasonable alternatives.