Why Missingness Patterns Change the Compatibility Audit
An analysis model may contain an outcome, several predictors, interactions, and nonlinear terms, while the incomplete data have many different patterns. One participant may be missing \(X_1\), another \(X_2\), and a third both. The analysis formula is the same for everyone, but the information needed to evaluate its terms is not available in the same way for everyone.
The practical question is therefore not just whether an imputation model contains the outcome and all predictors. It is: for each predictor that can be missing, which analysis terms involve it, and what other variables are needed to represent those terms? A systematic answer prevents an interaction or nonlinear component from disappearing from the imputation structure for a subset of participants.
Assume the analysis model has been specified in advance and the relevant variables are intended to be imputed under the course’s missing-data assumptions. The audit below concerns representation of the analysis terms; it does not by itself establish that the missingness assumptions hold, that the conditional models are correctly specified, or that imputations are proper. Those remain separate requirements.
A General Term Map
Write the analysis model’s linear predictor as
where \(\phi_t(X_i)\) is the \(t\)th term evaluated for participant \(i\). For example, if the model includes \(X_1\), \(X_2\), and their interaction, the terms are \(X_1\), \(X_2\), and \(X_1X_2\). A term is evaluable only when all variables used to calculate it have values. If \(X_1\) is missing, both \(X_1\) and \(X_1X_2\) are unavailable, even if \(X_2\) is observed.
For each predictor \(X_j\) that is incomplete, list the terms containing \(X_j\). Then identify every other variable needed by those terms. This is a structural audit of the information the imputation model needs to preserve. In chained equations, other incomplete predictors may have current imputed values when \(X_j\) is updated; a missing value does not make the corresponding analysis term irrelevant.
The audit should also be applied pattern by pattern. It reveals which components are missing from a participant’s analysis-model row before imputation and which completed values will be needed to construct that row afterward. It does not imply that a separate imputation model should be fitted for every pattern. Splitting by pattern can discard information and create unstable estimates, especially when some patterns are rare.
Why Interactions and Nonlinear Terms Need Special Attention
A short algebraic rearrangement shows why tracking terms is more informative than simply listing variables. Suppose a continuous outcome is modeled with a quadratic term and an interaction:
Collect the terms involving \(X_{1i}\):
Thus the contribution of \(X_{1i}\) is not generally described by a single linear term. It depends on \(X_{1i}^2\), and its linear coefficient varies with \(X_{2i}\). When \(X_1\) is missing, its imputation structure must not silently ignore either feature if the goal is to preserve the stated analysis relationship. If \(X_2\) is also missing, the interaction depends on the current or eventual imputed value of \(X_2\) as well.
This rearrangement justifies the term map, but it is not a general proof that any particular set of chained-equation conditionals is exactly compatible with the analysis model. Compatibility also depends on the conditional model families and their distributions. The map is a necessary organizational check, not a substitute for that broader assessment.
Worked Example: One Interaction, Three Missingness Patterns
Consider the analysis model \(Y_i=\beta_0+\beta_1X_{1i}+\beta_2X_{2i}+\beta_{12}X_{1i}X_{2i}+\epsilon_i\). The following hypothetical participants have different observed-data patterns.
| Participant | \(Y\) | \(X_1\) | \(X_2\) |
|---|---|---|---|
| A | 10 | 2 | 3 |
| B | 12 | missing | 4 |
| C | 9 | 1 | missing |
| D | 11 | missing | missing |
For participant A, all three predictor terms are available: \(X_1=2\), \(X_2=3\), and \(X_1X_2=2(3)=6\). The model contribution is therefore \(2\beta_1+3\beta_2+6\beta_{12}\), in addition to the intercept.
For participant B, \(X_1\) and \(X_1X_2\) cannot be evaluated from the observed data. The interaction still matters: after an \(X_1\) value is imputed, its product with the observed \(X_2=4\) is needed. The conditional model for imputing \(X_1\) should preserve the role of \(Y\), \(X_2\), and the \(X_1X_2\) analysis term, rather than treating \(X_1\) as an unrelated additive predictor.
For C, the missing quantity is \(X_2\), so both \(X_2\) and \(X_1X_2\) require its imputed value. For D, both predictors are missing. The interaction then depends on both imputed values; an analysis row cannot be completed by imputing only the two main effects and leaving the product out. This pattern also makes clear why updating one predictor can use the current value of the other in a chained procedure.
The term map is the same across participants, but the unavailable terms differ by pattern. A common imputation system can use this map while updating the incomplete predictors; the map does not call for three unrelated analysis formulas.
Tracing Nonlinear Terms and Multiple Predictors
A squared term is a function of one predictor, while an interaction is a function of two or more. Missingness in any input makes the whole term unavailable. Consequently, a nonlinear term should not be treated as if it were an independently observed variable when its underlying predictor is missing. Once a completed value of the predictor is available, the analysis term must correspond to that value.
Worked Example: A Quadratic Predictor With Pattern-Specific Availability
Suppose a hypothetical analysis uses \(Y_i=\beta_0+\beta_1X_i+\beta_2X_i^2+\epsilon_i\). For three participants, the recorded values are \(X=2\), \(X=4\), and missing. The analysis terms for the observed values are calculated directly:
For the third participant, neither analysis term can yet be evaluated. Suppose a hypothetical imputation draw is \(X=2.5\). Then the terms are \(X=2.5\) and \(X^2=2.5^2=6.25\), giving a predictor contribution of \(2.5\beta_1+6.25\beta_2\). The square is calculated from the imputed \(X\); it is not separately filled with an unrelated value.
The imputation plan must respect that the outcome’s conditional relationship with \(X\) includes curvature. An imputation model that represents only a straight-line contribution can fail to retain the quadratic analysis relationship. The calculation also checks the pattern-specific logic: when \(X\) is observed, both terms are available; when \(X\) is missing, both depend on its imputed value.
Worked Example: Two Interactions Under Different Patterns
Suppose the analysis includes \(A\), \(B\), \(C\), \(AB\), and \(BC\), with hypothetical values \(A=2\), \(B=3\), and \(C=5\) for a complete participant. The two interactions are \(AB=2(3)=6\) and \(BC=3(5)=15\).
Now consider a participant with \(A=2\), \(B\) missing, and \(C=5\). The terms \(B\), \(AB\), and \(BC\) all require the missing \(B\). When imputing \(B\), both other predictors are available, and both interactions involving \(B\) belong in the term map. If the imputed value were \(B=4\), the completed terms would be \(AB=2(4)=8\) and \(BC=4(5)=20\).
For a participant with \(A\) missing, \(B=3\), and \(C=5\), only \(A\) and \(AB\) depend on the missing predictor; \(B\), \(C\), and \(BC=15\) are already evaluable. This distinction avoids the overbroad instruction to treat every analysis term as missing whenever any predictor is missing. The affected terms are those whose inputs include a missing value.
These arithmetic checks establish exactly which terms depend on each predictor in this model. They do not establish that the proposed conditional distributions are adequate; the imputation model still needs suitable forms, predictors, and support in the data.
Practical Audit, Alternatives, and Limitations
Expand the formula into its main effects, interactions, and nonlinear functions. Do not rely on a software formula alone when it hides which variables enter each term.
For each predictor, record all terms containing it and the additional variables needed to form those terms.
Identify terms that are evaluable from observed values and terms that depend on one or more missing predictors. For jointly missing predictors, note which terms use both.
Confirm that each incomplete predictor’s conditional model can represent its relevant analysis contributions and use the information needed for interactions or nonlinearities.
In each completed data set, evaluate the analysis model using terms formed from the corresponding completed predictors.
A complete-case analysis is a simple alternative, but it discards participants with any missing analysis variable and can be inefficient; it can also target a biased relationship when complete cases are not representative under the relevant assumptions. Fitting separate imputations for every missingness pattern may seem to tailor the model more closely, but rare patterns can leave too little information for reliable estimation, and separate models can produce inconsistent relationships across patterns.
A single chained-equations system with conditionals that account for the analysis terms is often more practical. Its advantage is not that one shared system guarantees compatibility. The conditional models may still be misspecified, poorly supported, or mutually incompatible. A term map helps expose omissions, but cannot verify adequate sample size for combinations of predictors, correct functional form, or the missing-data assumptions.
A common mistake is to say, “The analysis includes \(X_1\) and \(X_2\), so including both main effects in the imputation models is enough.” That reasoning overlooks that \(X_1X_2\) is a distinct analysis term. If either input is missing, the interaction is unavailable; if it is omitted from the imputation structure, the intended interaction relationship may not be preserved. The correction is to trace each term to every predictor it uses, then check each incomplete predictor and each pattern against that dependency map.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- An analysis contains \(X_1\), \(X_2\), and \(X_1X_2\). For a participant with \(X_1\) missing and \(X_2=6\), which terms depend on the missing value?
- For the model with \(X\) and \(X^2\), a completed value is \(X=3.2\). Calculate both analysis terms.
- An analysis includes \(A\), \(B\), \(C\), \(AB\), and \(BC\). If only \(A\) is missing, which terms are unavailable from observed values?
- Why does a term map not prove that a chained-equations imputation model is fully compatible with the analysis?
Solutions. (1) \(X_1\) and \(X_1X_2\) depend on the missing \(X_1\); \(X_2=6\) is observed. (2) \(X=3.2\) and \(X^2=3.2^2=10.24\). (3) \(A\) and \(AB\) are unavailable; \(B\), \(C\), and \(BC\) can be evaluated from observed values. (4) The map identifies structural term dependencies, but compatibility also depends on the specified conditional distributions, their functional forms, and whether they are adequately supported and correctly specified.