Why the Substantive Model Must Inform Imputation
When an analysis includes a derived term, such as \(X^2\) or \(XZ\), that term must agree with its inputs in every completed data set. Passive imputation preserves this algebraic identity by recomputing the derived term whenever an input is imputed. But algebraic consistency alone does not ensure that the imputed values of \(X\) preserve the relationship between \(X\) and the outcome assumed by the analysis model.
Substantive-model-compatible fully conditional specification (SMC-FCS) addresses this problem by using the substantive outcome model when updating incomplete covariates. The outcome model is not merely fitted after imputation: its likelihood contribution helps determine which candidate covariate values are plausible, in light of the observed outcome. This is especially important when the model contains nonlinear or interaction terms.
Here \(Y\) is the outcome, \(X_j\) is a covariate being updated, \(X_{-j}\) denotes the other covariates, and \(Z\) denotes additional fully observed predictors. The substantive model specifies the form of \(f_Y\), for example a normal linear regression or a logistic regression. Compatibility does not mean that every modeling choice is automatically correct, nor that a sequence of conditional updates necessarily corresponds to a unique joint distribution. The result depends on the covariate models, the substantive model, and implementation.
How the Outcome Model Guides a Covariate Update
Suppose a row has an observed outcome \(y\), some observed covariates, and a missing value of \(X_j\). A working covariate model proposes plausible values of \(X_j\) given the other covariates. SMC-FCS also evaluates how well each candidate value accords with the observed outcome under the substantive model. In generic form, the update has a target proportional to
where \(q\) is the working covariate model and \(\psi\) denotes its parameters. The proportionality hides a normalizing constant: the expression must be normalized over the allowable values of \(X_j\) to define a probability distribution or density. In practical SMC-FCS algorithms, the construction and sampling method must respect the support and form of both models. The next step in the course develops the conditional density more fully; the essential principle here is that the substantive-model likelihood changes the relative plausibility of candidate imputations.
This principle can be justified by comparing two candidate values \(x_a\) and \(x_b\). Their relative target weight is
The second ratio is the outcome-model likelihood ratio. If the covariate proposal gives the two candidates equal weight, the outcome likelihood ratio alone determines their relative target weight. If the proposal weights differ, both factors matter. Thus an observed outcome can favor one candidate value of a missing covariate over another, even when the covariate-only model regards them as equally plausible.
For a derived term, its value is recalculated from the candidate covariate value before evaluating the substantive model. For example, when the model includes \(X^2\), candidate \(x\) contributes \(x^2\); it is not valid to impute \(X\) and \(X^2\) independently and then allow them to disagree. This combines the algebraic consistency emphasized in the earlier material on passive imputation with outcome-informed covariate updating.
Worked Likelihood Comparisons
Worked Example: A Quadratic Outcome Model
Hypothetical data. Suppose a continuous outcome follows the substantive model \(Y\mid X=x\sim N(\mu(x),1.5^2)\), with \(\mu(x)=2+0.5x+0.4x^2\). One record has observed \(y=5.5\) and missing \(X\). Compare candidate values \(x=1\) and \(x=3\), assuming for this comparison that the covariate proposal gives them equal weight.
At \(x=1\), the model mean is \(2+0.5(1)+0.4(1^2)=2.9\). At \(x=3\), it is \(2+0.5(3)+0.4(3^2)=2+1.5+3.6=7.1\). The ratio of outcome likelihoods for \(x=3\) versus \(x=1\) is
The observed outcome therefore gives \(x=3\) about 2.54 times the outcome-model weight of \(x=1\). This is a likelihood comparison, not by itself a posterior probability or a complete imputation probability. If the proposal weights are unequal, their ratio must also be included.
Worked Example: An Interaction Term
Hypothetical data. Let the substantive model be \(Y\mid X,Z\sim N(\mu,1^2)\), where \(\mu=1+0.8X+0.6Z+0.5XZ\). A record has observed \(Z=2\) and \(y=4.0\), but \(X\) is missing. Compare \(X=1\) with \(X=3\), again treating the proposal weights as equal for this likelihood comparison.
At \(X=1\), the product is \(XZ=2\), and the fitted mean is \(1+0.8(1)+0.6(2)+0.5(2)=4.0\). At \(X=3\), the product is \(XZ=6\), and the fitted mean is \(1+0.8(3)+0.6(2)+0.5(6)=7.6\). Therefore,
The outcome-model likelihood strongly favors \(X=1\) over \(X=3\) for this record. The interaction is evaluated using the same candidate \(X\) and observed \(Z\); it cannot be treated as an independent predictor with a value unrelated to \(XZ\).
Worked Example: A Logistic Substantive Model
Hypothetical data. Consider a binary outcome with \(\operatorname{logit}\{P(Y=1\mid X,Z)\}=-2+0.7X-0.15X^2+0.4Z\). For an observed \(Z=1\), compare candidate values \(X=2\) and \(X=0\) for a record whose observed outcome is \(Y=1\).
At \(X=2\), the linear predictor is \(-2+0.7(2)-0.15(4)+0.4(1)=-2+1.4-0.6+0.4=-0.8\). Hence
At \(X=0\), the linear predictor is \(-2+0-0+0.4=-1.6\), giving \(P(Y=1\mid X=0,Z=1)=\exp(-1.6)/(1+\exp(-1.6))\approx0.1680\). Because the observed outcome is \(Y=1\), the likelihood ratio is \(0.3100/0.1680\approx1.845\). Thus the outcome likelihood favors \(X=2\) over \(X=0\) by a factor of about 1.845, before accounting for any difference in proposal weights.
Assumptions, Diagnostics, and Failure Modes
SMC-FCS does not remove the assumptions required for multiple imputation. In particular, inference generally relies on a suitable missing-at-random assumption conditional on the variables included in the imputation procedure, together with adequate modeling of the relevant conditional distributions. The substantive outcome model must also be an appropriate representation of the intended analysis: its link, functional form, interactions, and outcome distribution must match the planned analysis. Covariate models must support plausible values over the relevant range and respect such features as bounds, discreteness, and structural constraints.
The imputation and analysis must use consistent definitions of the variables. A product, polynomial, or other deterministic term should be reconstructed from its inputs for each updated or completed record. Parameter uncertainty in the imputation models should be represented according to the chosen implementation; repeatedly generating values from fixed fitted parameters can understate uncertainty. Standard multiple-imputation inference also requires enough imputations for the desired precision and appropriate pooling of the completed-data estimates.
Useful checks include comparing observed and imputed covariate distributions, by outcome and by important predictors; checking whether draws respect support and domain restrictions; and verifying that every derived term equals the function of its inputs. Monitor chains or repeated runs for unstable behavior and inspect whether the imputed values produce plausible outcome-model predictions. These checks cannot prove that the missingness assumption or models are correct, but they can reveal obvious failures.
- Outcome model omitted from the update. A covariate-only imputation may give equal weight to values that imply very different outcome likelihoods. Adding the derived term passively afterward does not make the covariate draw outcome-informed.
- Derived terms imputed separately. This can create records with \(X^2\ne X\cdot X\) or \(XZ\) unequal to the product of the recorded inputs. Recompute deterministic terms from the updated covariates.
- Proposal model has poor support. If plausible values are rarely or never proposed, outcome weighting cannot recover them. Revise the covariate model and examine the support of its draws.
- Substantive-model misspecification. SMC-FCS targets compatibility with the specified model, not necessarily with the true data-generating process. A wrong functional form can produce systematically inappropriate imputations.
- Confusing a likelihood ratio with an imputation probability. The likelihood contribution is only one factor in the update; proposal weights and normalization also matter.
How This Differs From the Main Alternatives
| Approach | What informs the covariate imputation? | Main concern when the analysis has derived terms |
|---|---|---|
| Ordinary FCS with separate conditional models | Working conditional models for each incomplete variable | Those models may not preserve the substantive outcome relationship or its nonlinear structure. |
| Passive imputation | The input covariates are imputed; derived terms are recalculated deterministically | Algebraic identities are preserved, but outcome information need not influence the covariate draws in a substantive-model-compatible way. |
| SMC-FCS | Working covariate models together with the substantive outcome-model contribution | Requires careful specification and implementation; compatibility with a misspecified substantive model is not protection against bias. |
The choice is not simply between “correct” and “incorrect” methods. Passive imputation is useful for maintaining deterministic identities and may be adequate in suitable settings. SMC-FCS is motivated when ordinary conditional models may fail to reflect important features of the intended substantive model. It can be computationally more demanding, sensitive to model choices, and difficult when the covariate proposal has limited support. The introductory material on passive imputation establishes why deterministic terms must be recomputed; SMC-FCS adds the distinct requirement that covariate draws account for the substantive outcome model.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In one sentence, state what information SMC-FCS adds to an incomplete-covariate update.
- A normal outcome model has equal proposal weights for two candidate values. Their outcome likelihoods are 0.12 and 0.03, respectively. What is the likelihood ratio favoring the first candidate, and what does it not represent?
- Why does passive recomputation of \(X^2\) not, by itself, ensure that imputed \(X\) values reflect a quadratic outcome relationship?
- Name two model or data conditions that should be checked before relying on SMC-FCS imputations.
Solutions. (1) SMC-FCS uses the substantive outcome-model contribution, as well as a covariate model, to guide the relative plausibility of candidate values for an incomplete covariate. (2) The likelihood ratio is \(0.12/0.03=4\). The first candidate has four times the outcome likelihood of the second, but this is not a normalized imputation probability; proposal weights and the normalizing constant also matter. (3) Passive recomputation guarantees that each recorded square equals the square of its imputed input, but a covariate draw generated without the outcome likelihood may still fail to preserve the relationship specified by the quadratic outcome model. (4) Examples include checking that the substantive outcome model is appropriate, that the covariate model supports plausible values, that imputations obey variable bounds, and that missingness assumptions are defensible given included variables.