Why a Richer Imputation Model Can Be Appropriate
An analysis model is chosen to answer a substantive question; an imputation model has the additional task of using available information to predict missing values. These goals need not require identical formulas. A variable associated with an incomplete predictor, for example, may help predict that predictor even if it is not included in the final analysis. Likewise, a flexible term in an imputation model may help represent the data while leaving the conditional relationship used in the analysis unchanged.
The key is not that the imputation model must be a copy of the analysis model. It is that the imputation model must preserve the relationship the analysis is intended to estimate. A richer model can do this, but adding predictors or terms by itself does not establish that it does. The implied relationships must be checked.
For example, if the planned analysis models \(Y\) given \(X\), an auxiliary variable \(Z\) may help predict missing \(X\). It is possible for \(Z\) to appear in the conditional distribution of \(X\) given \(Y\) and \(Z\), while the marginal distribution of \(Y\) given \(X\) still has the specified form. Whether that happens depends on the joint model, not on the labels “auxiliary” or “flexible.”
The examples below assume the specified probability models are adequate for the hypothetical data, that the modeled variables and predictors are observed or otherwise available as required for imputation, and that the missingness assumptions used for inference are appropriate. Compatibility is a relationship between models; it does not by itself establish those assumptions, guarantee good finite-sample performance, or make an analysis model substantively appropriate.
Deriving the Compatibility Check
A direct way to check compatibility is to begin with a joint distribution, derive the conditional distribution needed for imputation, and then derive the analysis conditional distribution implied by that same joint model. The analysis conditional must match the planned analysis model. The imputation conditional may contain extra information, but it cannot imply a different target relationship.
In particular, a conditional model for an incomplete variable cannot always be chosen independently of the analysis model. Two separately specified regressions may each look reasonable but fail to be conditionals of any one joint distribution. Conversely, a conditional imputation model can look more complicated than the analysis model and still be compatible when both follow from a coherent joint specification.
Write the conditional distribution or regression relationship that the final analysis will use, including its link, predictors, and functional form.
Describe how extra predictors or flexible terms relate to the incomplete variable and the outcome. Do not assume their effect disappears simply because they are omitted from the final analysis.
Derive the distribution of the outcome given the analysis predictors from the same joint model. Confirm that it has the required form.
Worked Examples
Worked Example: A Normal Auxiliary Predictor With a Simple Analysis
Suppose \(X\) is incomplete, \(Z\) is fully observed, and the planned analysis is a linear regression of \(Y\) on \(X\). Consider this hypothetical joint construction:
The errors generating \(Z\) and \(Y\) are independent conditional on \(X,Z\), as implied by this factorization. To check the analysis relationship, first use \(Z\mid X\): its conditional mean is \(0.5X\) and its conditional variance is \(0.75\). Therefore,
Thus the marginal analysis model is \(Y\mid X\sim N(2+4X,7)\), a normal linear regression with intercept 2 and slope 4. The auxiliary variable has a direct role in the conditional model for \(Y\) given \(X,Z\), but after averaging over \(Z\mid X\), the stipulated analysis relationship is still linear in \(X\).
For imputation, derive \(X\mid Y,Z\). Given \(Z\), the prior distribution is \(X\mid Z\sim N(0.5Z,0.75)\). The likelihood from \(Y\mid X,Z\) has mean \(2+3X+2Z\) and variance 4. Combining the two normal densities gives conditional precision \(1/0.75+3^2/4=43/12\), so the conditional variance is \(12/43\). The conditional mean is
For a hypothetical record with \(Y=10\) and \(Z=1\), the imputation mean is \((90-18-10)/43=62/43=1.442\), and the conditional standard deviation is \(\sqrt{12/43}=0.5283\). An imputation draws \(X\) from this conditional distribution; it does not simply replace the missing value by 1.442. Here the richer imputation conditional uses both \(Y\) and \(Z\), while the analysis model remains the marginal regression of \(Y\) on \(X\).
Worked Example: A Flexible Outcome Relationship With an Auxiliary Predictor
Suppose the intended analysis is a quadratic regression. For hypothetical data, specify a joint model by
The second conditional does not depend on \(Z\). Consequently, the implied analysis model is exactly \(Y\mid X\sim N(1+2X+0.5X^2,1)\). The auxiliary variable helps predict \(X\), but this construction does not add a \(Z\) term to the analysis outcome relationship.
The compatible imputation distribution is obtained from the joint density:
For a hypothetical missing \(X\) with \(Y=4\) and \(Z=0\), the prior is \(X\mid Z=0\sim N(0,1)\). Omitting constants that do not depend on \(x\), its conditional density is proportional to
The quadratic term must be retained: replacing it with a linear predictor would no longer use the specified analysis relationship. To see the shape of the density, its unnormalized kernel at \(x=0\) is \(\exp(-4.5)\); at \(x=1\), it is \(\exp(-0.625)\); and at \(x=2\), it is \(\exp(-6.5)\). These are density-kernel values, not probabilities at those points. The conditional distribution is not generally normal, so a normal linear regression for \(X\) would not automatically generate draws from this compatible conditional. This example shows why preserving an analysis term can require a flexible imputation distribution rather than merely adding an auxiliary predictor to a simple regression.
Worked Example: An Auxiliary Predictor in a Logistic Analysis
Suppose both \(X\) and \(Y\) are binary, \(X\) is incomplete, and the planned analysis is logistic regression of \(Y\) on \(X\). In a hypothetical model, let \(P(Y=1\mid X=0)=0.10\) and \(P(Y=1\mid X=1)=0.25\). The outcome odds ratio comparing \(X=1\) with \(X=0\) is
Thus the analysis model has intercept \(\log(0.10/0.90)=-2.197\) and coefficient \(\log(3)=1.099\) for \(X\). Now let the auxiliary binary variable \(Z\) affect the distribution of \(X\): \(P(X=1\mid Z=0)=0.20\) and \(P(X=1\mid Z=1)=0.60\). Define the joint distribution by \(f(y\mid x)f(x\mid z)\). Since \(Y\mid X\) was specified first, the implied analysis model remains the stated logistic regression.
Bayes’ rule gives the odds for imputing \(X=1\) rather than \(X=0\):
For \(Y=1,Z=1\), the prior odds are \(0.60/0.40=1.5\), and the outcome likelihood ratio is \(0.25/0.10=2.5\). The posterior odds are \(1.5(2.5)=3.75\), giving \(P(X=1\mid Y=1,Z=1)=3.75/(1+3.75)=0.7895\). For \(Y=0,Z=1\), the likelihood ratio is \(0.75/0.90=0.8333\), so the posterior odds are \(1.5(0.8333)=1.25\), giving a probability of \(1.25/2.25=0.5556\). The imputation uses \(Z\) and \(Y\), while the outcome analysis retains its logistic relationship with \(X\).
What the Examples Establish—and What They Do Not
The examples use different ways to make the compatibility check visible. In the normal example, integrating over the auxiliary variable produced the analysis regression. In the quadratic example, factoring the joint density preserved the specified nonlinear outcome conditional, while yielding a non-normal imputation conditional. In the binary example, Bayes’ rule incorporated auxiliary information into imputation without changing the logistic outcome model.
These are demonstrations of a principle, not a guarantee that any added predictor is harmless. If \(Z\) is associated with the outcome in a way not represented by the proposed joint model, or if its relationship with \(X\) is misspecified, the implied analysis relationship may differ from the intended one. A term that appears only in an imputation regression can still change the distribution of completed data and therefore affect analysis estimates.
A useful comparison is between three choices:
| Approach | Potential benefit | Main compatibility concern |
|---|---|---|
| Use only analysis predictors in imputation | Simpler conditional models | May omit observed information useful for predicting missing values |
| Add auxiliary predictors or flexible terms | Can represent richer predictor relationships and use more observed information | Must check that the implied analysis conditional is preserved |
| Specify conditionals independently without checking their joint implications | May be convenient to fit | Conditionals may conflict or fail to correspond to a coherent joint distribution |
With MICE, several conditional models are fitted in sequence. A compatible set can be understood as conditionals arising from a common joint distribution. In practice, separately chosen conditional models need not have that property. Even when a coherent joint distribution is not explicitly constructed, the intended analysis terms should be represented appropriately in the imputation conditionals, and their implications should be examined. Compatibility is not the same as including every available variable, nor is it a substitute for checking whether the model describes the data reasonably.
Diagnostics, Limitations, and a Common Pitfall
To assess a richer imputation specification, first check that the analysis relationship is still represented: the outcome, its link, and the required transformations or interactions should not be lost from the imputation structure. Then examine whether the added predictor and flexible terms have plausible relationships to the incomplete variable and other variables. Compare observed and imputed values across relevant predictor groups, and inspect whether the imputed distributions and key analysis relationships look reasonable. Such checks can reveal obvious distortions, but they cannot prove compatibility or establish missingness assumptions.
Warning signs include imputed values that have implausible ranges, an auxiliary predictor that is ignored despite strong observed association with the incomplete variable, or completed data in which a required nonlinear or interaction pattern has been flattened. A poorly supported flexible model can also be unstable. The appropriate response is not to keep adding terms without limit: a larger model can make estimation less stable and may fit chance features. Model-size trade-offs and overfitting require deliberate attention.
Common mistake: “The final analysis is \(Y\) on \(X\), so the imputation model should contain only \(Y\) and \(X\); any extra predictor changes the analysis.” That conclusion is too strong. An auxiliary variable can inform \(X\)’s imputation while a coherent joint model still implies the intended \(Y\mid X\) relationship. The opposite mistake is equally important: “Adding predictors makes imputation better, so compatibility is automatic.” It is not. Derive or otherwise check the implied analysis conditional; predictor inclusion alone cannot establish that it is preserved.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In a compatible model, must every predictor in the imputation conditional also appear in the final analysis? Explain.
- In the normal example, calculate the analysis-model mean \(E(Y\mid X=2)\).
- In the logistic example, calculate \(P(X=1\mid Y=0,Z=0)\).
- Why does the quadratic example not justify imputing \(X\) from an ordinary normal linear regression on \(Y\) and \(Z\)?
Solutions. (1) No. An auxiliary predictor can help impute an incomplete variable while the joint model still implies the required analysis conditional; this must be checked. (2) The analysis mean is \(2+4(2)=10\), in the units of \(Y\). (3) For \(Z=0\), the prior odds are \(0.20/0.80=0.25\). For \(Y=0\), the likelihood ratio is \(0.75/0.90=0.8333\). The posterior odds are \(0.25(0.8333)=0.2083\), so the probability is \(0.2083/(1+0.2083)=0.1724\). (4) The compatible conditional density includes the quadratic outcome mean through the squared residual \([Y-(1+2X+0.5X^2)]^2\). It is not generally a normal linear regression conditional, so that simpler regression need not preserve the specified quadratic analysis relationship.