Algebraic Consistency Is Not Model Compatibility
A passive term is recalculated from its defining variables, so a completed data set can satisfy an exact identity such as \(D=X^2\) in every row. That is valuable: the analysis will not use a square that disagrees with its underlying covariate. But the identity says nothing by itself about whether the imputed values of \(X\) have a plausible distribution given the outcome. When the substantive model is nonlinear in \(X\), that distinction matters: the outcome likelihood can change which values of a missing \(X\) are plausible.
The issue is not a failure of arithmetic. It is a mismatch between two statistical tasks. Passive recalculation enforces the relation between variables; a compatible imputation must also represent the information about the incomplete covariate carried by the observed outcome under the substantive model.
For clarity, let \(Y\) be an outcome, \(X\) an incomplete covariate, and \(Z\) other observed covariates. Suppose the substantive model specifies \(p(Y\mid X,Z)\), and a covariate model specifies \(p(X\mid Z)\). Under an ignorable missingness mechanism and the stated models, the conditional distribution needed to impute a missing \(X\), given observed \(Y\) and \(Z\), is proportional to their product:
For discrete \(X\), the integral in the denominator is replaced by a sum over its possible values. The denominator normalizes the distribution so its probabilities sum to one. The essential point is the likelihood factor \(p(Y\mid X,Z)\): when the outcome model is nonlinear in \(X\), it can change the shape of the conditional distribution for \(X\). Recomputing \(g(X)\) after drawing \(X\) does not insert that likelihood factor into the draw.
A Short Derivation of the Conditional Target
The conditional distribution follows directly from Bayes' rule. Starting from the factorization \(p(Y,X\mid Z)=p(Y\mid X,Z)p(X\mid Z)\), condition on the observed \(Y\) and \(Z\). The joint density in the numerator is divided by its marginal density for \(Y\) given \(Z\):
Thus an imputation that draws from \(p(X\mid Z)\) alone has the correct target only when the likelihood does not change the distribution of \(X\) given \(Y,Z\), for example in a special case where \(Y\) provides no information about \(X\) conditional on \(Z\). A passive term cannot generally make that condition true. It can represent \(g(X)\) perfectly while leaving the conditional distribution of \(X\) unchanged from the one used to draw it.
Worked Example: A Correct Square From an Uninformative Draw
Suppose a hypothetical record has an unobserved \(X\), and the imputation procedure draws \(X=1.7\) from a model that uses \(Z\) but does not condition on the observed outcome \(Y\). A passive variable is defined as \(D=X^2\).
Recalculation. The derived value is \(D=(1.7)^2=2.89\). The pair \((X,D)=(1.7,2.89)\) satisfies the defining identity exactly.
What this verifies. The analysis will not use a square inconsistent with the imputed \(X\). This is the complete algebraic check.
What it does not verify. If the specified outcome model makes \(Y\) more likely at some values of \(X\), the observed \(Y\) should affect the distribution from which missing \(X\) is drawn. The calculation \(1.7^2=2.89\) uses neither \(Y\) nor its likelihood. It therefore cannot establish that \(1.7\) was drawn from the appropriate conditional distribution.
How a Nonlinear Outcome Likelihood Changes Imputations
The difference can be quantified in a small hypothetical example. Suppose \(X\) takes values 0, 1, and 2, with \(P(X=0)=0.25\), \(P(X=1)=0.50\), and \(P(X=2)=0.25\). Let the substantive model be logistic regression with a linear and a squared effect:
Assume there are no additional covariates for this calculation, the stated covariate distribution and outcome model are correct, and missingness is ignorable. The square \(D=X^2\) can be passively recalculated, but the conditional distribution of \(X\) must also account for the observed outcome.
Worked Example: Conditional Probabilities Under a Quadratic Logistic Model
First calculate the outcome probabilities at each possible \(X\). The linear predictors are \(-1\), \(-0.1\), and \(1.8\), respectively. Applying the logistic function gives \(P(Y=1\mid X=0)=0.2689\), \(P(Y=1\mid X=1)=0.4750\), and \(P(Y=1\mid X=2)=0.8581\).
For a record with observed \(Y=1\), multiply each likelihood by its prior probability for \(X\). The unnormalized weights are
Their sum is \(0.5192\), so the conditional probabilities are \(0.06723/0.5192=0.1295\), \(0.2375/0.5192=0.4574\), and \(0.2145/0.5192=0.4131\). In particular, \(P(X=2\mid Y=1)=0.4131\), not its covariate-model probability of 0.25.
For a record with \(Y=0\), use the failure probabilities. The unnormalized weights are \(0.25(0.7311)=0.1828\), \(0.50(0.5250)=0.2625\), and \(0.25(0.1419)=0.03546\), with sum \(0.4807\). Normalizing gives conditional probabilities \(0.3802\), \(0.5460\), and \(0.07377\), respectively.
After drawing from either conditional distribution, passive recalculation gives \(D=0\), \(1\), or \(4\) for \(X=0\), \(1\), or \(2\). That guarantees \(D=X^2\), but only a draw that reflects the outcome likelihood has the conditional probabilities just calculated.
This example also illustrates why simply adding a passive term to an imputation system is not a general solution. A derived term can be useful as a predictor in models for other incomplete variables. But when \(X\) itself is missing, the imputation distribution for \(X\) must account for the outcome information in the form specified by the substantive model. A term that is a deterministic function of \(X\) does not independently supply that missing likelihood contribution.
A Continuous Example and a Practical Diagnostic
The same argument applies to continuous covariates. Suppose, hypothetically, that \(X\sim N(0,1)\) and the outcome follows a logistic model with linear predictor \(-1+0.8X^2\). For an observed \(Y=1\), the conditional density is proportional to the standard normal density times \(\operatorname{expit}(-1+0.8x^2)\). It is not simply the standard normal density, even though the derived square can always be recalculated.
Worked Example: Comparing Relative Densities at Two Covariate Values
Compare the conditional density at \(x=2\) with that at \(x=0\), given \(Y=1\). Under the \(N(0,1)\) covariate model, the ratio of prior densities is
The outcome probabilities are \(\operatorname{expit}(-1)=0.2689\) at \(x=0\) and \(\operatorname{expit}(2.2)=0.9002\) at \(x=2\). Their likelihood ratio for \(Y=1\) is \(0.9002/0.2689=3.348\). Therefore the ratio of conditional densities is
These are relative density values at two points, not probabilities that a continuous variable equals either point. They show that the outcome likelihood substantially changes the relative weighting: drawing from the covariate model alone would retain the ratio \(0.1353\), whereas conditioning on \(Y=1\) gives a ratio of about \(0.453\). Recalculating \(X^2\) changes neither ratio.
A useful diagnostic question is therefore not only, “Does every derived value equal its defining function?” Also ask, “Does the imputation distribution for each incomplete covariate reflect the observed outcome through the specified substantive likelihood?” In practice, inspect the imputation model specification and compare its implied conditional draws with the target implied by the substantive model. For simple discrete examples, the target can be calculated by hand as above. For more complex models, use model-appropriate diagnostics and sensitivity analyses rather than treating a perfect identity check as evidence of compatibility.
Alternatives, Limitations, and Common Errors
| Approach | What it ensures or models | Key limitation for this issue |
|---|---|---|
| Passive recalculation | Recomputes a derived variable from its inputs in each completed data set. | Does not by itself make the draw for an incomplete input conditional on the substantive outcome likelihood. |
| Ordinary chained-equation imputation | Fits conditional imputation models for incomplete variables using selected predictors. | A convenient conditional model can still fail to represent the conditional distribution implied by a nonlinear substantive model. |
| SMC-FCS | Targets covariate imputation distributions that incorporate the substantive-model likelihood. | Requires suitable model specification and computational implementation; it is not a guarantee against misspecification or poor diagnostics. |
The earlier material on the SMC-FCS Conditional Density gives the conditional-density framework; the point here is why deterministic recalculation alone does not achieve it. SMC-FCS is a principled alternative when compatibility with the substantive model is needed. It does not remove the need to specify the covariate distribution and substantive model appropriately, assess convergence and draw behavior, or examine sensitivity to modeling choices.
- Common mistake: “The passive square equals \(X^2\) in every imputed data set, so the imputation respects the quadratic analysis model.” Correction: The equality checks the derived variable only. Compatibility also concerns how the imputed \(X\) values depend on \(Y\).
- Common mistake: “Including the outcome as a predictor always makes an ordinary imputation model compatible.” Correction: Including \(Y\) can be important, but a chosen conditional-model form may still not reproduce the distribution implied by a nonlinear substantive likelihood.
- Common mistake: “A derived term is an extra independent measurement.” Correction: If \(D=g(X)\) deterministically, it contains no independent measurement beyond \(X\). Recalculation preserves the relationship; it does not add information about the missing \(X\).
The main limitation is that the conditional target depends on the assumed covariate and substantive models. If either is misspecified, an imputation method designed to align them can still yield unsuitable imputations. Conversely, passive recalculation remains useful and often necessary for ensuring that derived terms are internally consistent. The appropriate conclusion is not to avoid passive variables, but to avoid treating their consistency as proof that the imputation model preserves the analysis model.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions below.
- If \(D=X^2\) is recalculated after each imputed value of \(X\), what property is guaranteed, and what property is not guaranteed?
- For a discrete \(X\), write the unnormalized conditional weight for a candidate value \(x\), given observed \(Y\) and \(Z\).
- In the quadratic logistic example, why does the conditional probability \(P(X=2\mid Y=1)\) differ from \(P(X=2)\)?
- For continuous \(X\), why is a ratio of density values at two points not the probability that \(X\) equals either point?
- Name one role for passive recalculation and one reason it is not a substitute for SMC-FCS when compatibility is needed.
Solutions. (1) It guarantees the identity \(D=g(X)\), but not that the imputation distribution for \(X\) conditions correctly on \(Y\). (2) The unnormalized weight is \(p(Y\mid X=x,Z)p(X=x\mid Z)\); divide by the sum of these weights over possible \(x\) to normalize. (3) The likelihood of \(Y=1\) is larger at \(X=2\) under the specified model, so conditioning on \(Y=1\) reweights the covariate probabilities. (4) A continuous variable assigns probability through intervals; a density value is not a point probability. (5) Passive recalculation keeps a derived term consistent with its inputs. It does not itself incorporate the substantive likelihood into the draw of a missing covariate, which is the compatibility issue SMC-FCS addresses.