Why Use a Checklist?
A regression formula is more than a list of variables: its terms specify the relationships the analysis will estimate. An imputation model that overlooks one of those relationships can weaken or distort the analysis even when each imputed variable has a plausible marginal distribution. A practical audit makes the intended analysis explicit, then asks whether the imputation models retain the information and structure needed for its reported result.
The audit is a disciplined screening tool, not a guarantee. Passing it does not establish that the missingness assumptions are appropriate, that every conditional model is correctly specified, or that the regression itself is scientifically sound. It helps locate avoidable mismatches before fitting and reporting the analysis.
Write \(Y\) for the outcome and \(X_1,\ldots,X_p\) for covariates. The analysis formula might, for example, be \(Y\) regressed on \(X_1\), \(X_2\), their product \(X_1X_2\), and a squared term \(X_1^2\). In chained equations, each incomplete variable has a conditional imputation model. The audit asks whether those models account for the outcome and the relevant analysis relationships—not merely whether they contain columns with the same names as the regression formula.
The Audit: From Formula to Imputation Models
First write down the exact regression formula and the quantity to be reported. Then inspect each incomplete variable’s conditional model. The checklist below is intended to be applied separately to each such model; a single pass through the predictor list can miss important differences.
Record the analysis outcome, all terms in the regression formula, and the reported estimand: for example, a coefficient, a contrast at a specified covariate value, or a population-standardized contrast.
For each incomplete covariate, identify its imputation model. Check whether the outcome is available to that model and, if the outcome is itself incomplete, whether the plan appropriately represents its role in the analysis.
Review the analysis model for interactions, nonlinear terms, stratification, and other structured relationships. Ask whether the relevant conditional imputation model can represent them.
Check that the covariate distribution, contrasts, and any averaging or standardization used for reporting are preserved by the imputation and analysis plan.
Inspect observed-data patterns and imputed relationships, note sparse or unsupported regions, and document unresolved concerns. A reassuring diagnostic cannot establish compatibility on its own.
For an incomplete covariate, the outcome is often an important predictor of the covariate’s missing values. Leaving it out can erase observed outcome–covariate association and attenuate an estimated regression relationship. This is a practical warning, not a rule that simply inserting \(Y\) makes an imputation model adequate. The conditional model must also be suitable for the variable’s type and must represent the relevant relationships.
Next inspect each analysis term. An interaction means that one covariate’s relationship with the outcome depends on another covariate. A nonlinear term means that the relationship changes across the covariate’s range. A conditional imputation model that assumes only simple additive linear relationships may fail to preserve either feature. The necessary conditional form depends on the joint model and variable types; copying the analysis formula verbatim into every imputation model is not a universal solution.
Why an Interaction Changes the Audit
A short derivation shows why outcome inclusion alone may not be enough. Suppose, for illustration, that \(X\) and \(Y\) are jointly normal conditional on a fully observed \(Z\), and that
Under these assumptions, the conditional covariance between \(X\) and \(Y\) at a fixed \(Z\) is \(\tau^2(\beta_X+\beta_{XZ}Z)\). The conditional variance of \(Y\) is \(\tau^2(\beta_X+\beta_{XZ}Z)^2+\sigma^2\). The normal conditional-distribution formula therefore gives
The coefficient multiplying the outcome residual generally varies with \(Z\). Thus a simple additive model for \(X\) using \(Y\) and \(Z\) need not represent the conditional distribution implied by this joint model. The derivation relies on the stated normal and linear conditional assumptions; it is not a universal formula for every imputation problem. Its practical lesson is that an analysis interaction should trigger a check of the corresponding conditional relationships, not a mechanical “include \(Y\)” tick.
Worked Example: Auditing an Interaction
Suppose a continuous outcome \(Y\) is analyzed using \(Y=10+2X+3Z+XZ+\varepsilon\), where \(X\) is incomplete, \(Z\) is observed, and \(\varepsilon\) has mean zero conditional on \(X,Z\). The analysis reports how the association between \(X\) and \(Y\) changes with \(Z\). Consider whether an imputation model for \(X\) that includes \(Y\) and \(Z\) only as additive terms has passed the audit.
Check the analysis relationship. At fixed \(Z=z\), the coefficient of \(X\), and hence the conditional mean change in \(Y\) per one-unit increase in \(X\), is \(2+z\).
Substitute two values. At \(z=0\), this change is \(2+0=2\) outcome units per unit of \(X\). At \(z=2\), it is \(2+2=4\) outcome units per unit of \(X\). The analysis therefore specifies different \(X\)–\(Y\) relationships at the two values of \(Z\).
Audit decision. An additive imputation model \(X\) on \(Y\) and \(Z\) forces the modeled relationship with \(Y\) to be constant across \(Z\), so it does not automatically preserve the interaction structure. The audit flags this plan for revision or stronger justification. It does not prescribe one universal replacement model: the appropriate conditional specification depends on the variables’ distributions and the joint structure being represented.
Check Nonlinear Terms and the Reported Estimand
A nonlinear term can create a similar mismatch. If an analysis uses \(X^2\), a simple linear conditional relationship involving \(Y\) and \(X\) may not preserve the outcome information associated with curvature. The audit should identify the nonlinear term and check whether the imputation model can represent the relevant relationship over the observed range. The presence of \(X^2\) in the analysis formula does not by itself tell you that a particular transformation must be inserted into every imputation model.
The estimand—the specific quantity the analysis is intended to report—also matters. A regression coefficient, a contrast at a fixed covariate value, and an average contrast over a target population are not interchangeable. For a model with interactions, a coefficient for the exposure may describe the contrast only at the reference value of a modifier. A reported population average may additionally depend on the distribution used for averaging. An audit that checks terms but not the final target can miss a mismatch between what was modeled and what is claimed.
Worked Example: From an Interaction Formula to a Standardized Contrast
In a hypothetical study, let \(A\) be a binary exposure and \(S\) a binary subgroup indicator. Suppose the fitted conditional mean is \(E(Y\mid A,S)=10+4A+2S-3AS\). The intended report is the average exposure contrast in a target population where 70% have \(S=0\) and 30% have \(S=1\). Assume these weights correctly describe the target population and the model is used for the stated standardization.
Find the subgroup contrasts. At \(S=0\), the contrast for changing \(A\) from 0 to 1 is \(4-3(0)=4\) outcome units. At \(S=1\), it is \(4-3(1)=1\) outcome unit.
Average using the target weights. The standardized contrast is
Thus the reported target-population contrast is 3.1 outcome units under the stated model and weights. The coefficient 4 is instead the contrast at \(S=0\); reporting it as the population-average contrast would not match the estimand. In the audit, check that the interaction and the target weights are represented and that the imputation plan retains the relevant subgroup–outcome relationships. This calculation does not show that any particular imputation model will do so.
Diagnostics, Alternatives, and Limits
After the formula-based audit, use diagnostics to look for evidence that the planned conditional models are behaving poorly. Compare observed and imputed distributions, but also compare relationships: for example, whether imputed values retain outcome associations across moderator levels or across the range of a nonlinear predictor. Review convergence and trace behavior, sparse cells, implausible predictions, and whether the data contain support for the interactions or curvature the analysis requires. These checks can reveal problems; good-looking plots cannot prove that unobserved relationships are correct.
The audit also clarifies what alternatives do and do not solve. A predictor-selection approach that keeps only strong marginal predictors may omit a weak marginal term that is important through an interaction. A very large imputation model may include many analysis terms but become unstable or overfit when data are sparse. An analysis based only on complete cases avoids specifying imputation models, but changes the information used and does not automatically answer the same inferential question. A more explicitly analysis-aligned imputation method may be considered when ordinary conditional specifications cannot adequately represent the target structure; choosing among methods requires assumptions and technical assessment beyond a checklist.
The main limitations are consequential. This audit cannot establish the missingness assumptions, identify relationships that the data do not support, or ensure correct extrapolation into regions with few observed cases. It cannot decide whether the analysis formula is scientifically appropriate. Nor does a matching list of terms prove that a set of conditional models corresponds to a coherent joint distribution. Record the target, each risk identified, the checks performed, and any remaining uncertainty.
Common Mistake
Correction: Variable inclusion is only the first check. The imputation model may still impose an additive relationship when the analysis includes an interaction, or a linear relationship when the analysis allows curvature. It may also fail to preserve the target-population weighting needed for the reported estimand. Audit relationships and target, not just names.
Worked Example: Screening a Nonlinear Analysis
Suppose the analysis model is \(E(Y\mid X,Z)=5+3X-0.5X^2+2Z\), where \(X\) is incomplete and \(Z\) is observed. The outcome is measured in outcome units, and \(X\) is measured in its original units. Check what an additive linear imputation model for \(X\) using only \(Y\) and \(Z\) would fail to address.
Identify the nonlinear analysis term. The model includes \(-0.5X^2\), so the relationship of the conditional mean to \(X\) is curved rather than a straight line.
Check the implied change across \(X\). The derivative of the specified conditional mean with respect to \(X\) is \(3-X\). At \(X=1\), it is \(3-1=2\) outcome units per unit of \(X\); at \(X=4\), it is \(3-4=-1\) outcome unit per unit of \(X\). The direction of the local relationship differs across these values.
Audit decision. A simple additive linear imputation model does not automatically account for this changing relationship. Flag it for a conditional-model assessment and check imputed relationships across the observed range of \(X\). This derivative describes the stated analysis mean; it does not establish a unique conditional model for imputing \(X\).
Putting the Checklist to Work
Use the audit as a compact record rather than as a numerical score. A “yes” for outcome inclusion does not cancel a “no” for interactions or estimand alignment. If a term cannot be represented reliably because its data are too sparse, document that limitation instead of treating a software fit as evidence that the problem has disappeared.
- Formula: Is the exact analysis formula written down, including interactions and nonlinear terms?
- Outcome: Is the outcome appropriately available to the imputation model for each incomplete covariate?
- Relationships: Can the conditional models represent the important covariate–outcome relationships, not merely additive main effects?
- Estimand: Does the imputation-and-analysis plan support the particular coefficient, contrast, or standardized quantity being reported?
- Evidence and limits: Have diagnostics, sparse regions, and unresolved assumptions been considered and documented?
Check Your Understanding
Answer each question, then compare with the solutions below.
- An analysis includes an exposure-by-subgroup interaction. What should the audit check beyond whether exposure and subgroup appear as predictors in the imputation model?
- A model’s exposure coefficient is a contrast at the reference subgroup. Is it automatically the target-population average contrast? Explain.
- Why can outcome inclusion be necessary for imputing an incomplete covariate but still fail to establish compatibility?
- In a model with mean \(6+2X-0.25X^2\), calculate the derivative with respect to \(X\) at \(X=2\) and \(X=6\). What does the change indicate for an audit?
Solutions. (1) Check whether the conditional imputation models can preserve the outcome–exposure relationship as it varies by subgroup; additive inclusion alone may not do so. (2) No. With an interaction, that coefficient generally describes the contrast at the reference subgroup. A population average requires subgroup-specific contrasts and appropriate target-population weights. (3) Inclusion can preserve an important outcome association, but the model may still omit interactions or curvature, use an unsuitable conditional form, or fail to support the estimand. (4) The derivative is \(2-0.5X\). At \(X=2\), it is \(2-1=1\); at \(X=6\), it is \(2-3=-1\), in outcome units per unit of \(X\). The relationship changes direction, so a purely linear conditional imputation relationship deserves scrutiny.