Why the Choice Is Not Simply “Passive or SMC-FCS”
Suppose an analysis uses a derived quantity such as \(X^2\) or \(X_1X_2\). That quantity is determined by its source variable or variables: it is not an independently observed variable with its own values to impute. But if a source is missing, its imputation model still matters. Imputing a source without respecting the substantive model can produce completed data in which the analysis relationship is poorly represented, even if the derived term is recalculated exactly afterward.
The practical distinction is therefore between how to maintain the identity and how to impute missing source values. Passive recalculation maintains the identity. SMC-FCS can be used to make imputations of incomplete sources compatible with the substantive model. In many applications, the appropriate workflow uses both: impute a source with SMC-FCS, then recalculate its derived term passively.
The decision depends on three questions: Is any source variable incomplete? What form does the substantive model take? And would the imputation models for incomplete sources preserve the conditional relationships implied by that substantive model? The earlier material on SMC-FCS established the relevant conditional-density idea; here the goal is to use it to choose a workflow.
Assumptions and a Decision Framework
As with other multiple-imputation procedures, the missingness and model assumptions matter. A common basis is missing at random (MAR): conditional on variables included in the imputation procedure, the probability that a value is missing does not depend on its unobserved value. The imputation models must also be adequately specified, include relevant observed information, and allow the observed data to inform the missing values. If these conditions fail, neither passive recalculation nor SMC-FCS alone guarantees valid inference.
For a missing source \(X\), the key compatibility issue can be expressed schematically. Let \(Z\) denote other covariates and \(Y\) the outcome. A source-only imputation model may draw from \(f(X\mid Z)\). But when the substantive model relates \(Y\) to \(X\), the distribution relevant to imputing a missing \(X\) generally also depends on \(Y\):
The right-hand side combines the covariate distribution with the substantive-model likelihood. SMC-FCS uses this substantive-model information when constructing conditional imputations. Passive recalculation, by itself, does not change the distribution from which a missing source was drawn; it only applies the deterministic function after source values are available.
Use the following distinctions when choosing an approach:
- All sources observed: there is no source value to impute. Calculate the derived term exactly; SMC-FCS is unnecessary for that term.
- Sources incomplete, relationship simple and adequately represented: impute the sources with appropriate compatible models, then recalculate the derived term passively. Passive recalculation is still essential for maintaining the identity.
- Sources incomplete, substantive model has a nonlinear effect or interaction that ordinary source-imputation models may not preserve: consider SMC-FCS for those source imputations, followed by passive recalculation.
- Derived term itself is imputed as if it were a separate variable: this is generally inappropriate when it must equal a known function of its sources. It can break the identity and create incoherent completed records.
This is a model-based decision, not a rule that every product term or square automatically requires SMC-FCS. If a conventional imputation specification adequately represents the relevant conditional distributions, it may be compatible. The concern is whether that adequacy is credible for the substantive model and missingness pattern.
Worked Choices
Worked Example: Complete Source for a Quadratic Predictor
In a hypothetical data set, a researcher models a health score using age squared. Age is complete for all participants. For one participant, age is 40 years. What should be done with the quadratic term, and is SMC-FCS needed for it?
The derived term is fixed by \(D=X^2\). For this participant,
Because age is observed, no imputation of age is required. Recalculate \(D\) exactly from age wherever the analysis data are assembled. There is no missing source value for SMC-FCS to draw, so SMC-FCS is not needed for this term. The substantive model may still require an appropriate analysis of the quadratic effect, but that is distinct from imputing missing covariates.
The identity is automatically satisfied: the recorded value of age determines the value of \(D\). Independently imputing or editing \(D\) would introduce a value inconsistent with the observed age.
Worked Example: Incomplete Source Under a Quadratic Substantive Model
Consider a hypothetical analysis with a continuous outcome and substantive model \(Y=4+X+0.5X^2+\varepsilon\), where \(\varepsilon\sim N(0,2^2)\). A participant has missing \(X\) and observed \(Y=5\). For illustration, suppose the covariate model gives \(X\sim N(0,1)\). Compare the unnormalized conditional-density contributions at candidate values \(X=0\) and \(X=1\).
Under the model, the outcome mean is \(\mu_Y(x)=4+x+0.5x^2\). The conditional density contribution is proportional to the covariate density times the outcome likelihood:
At \(x=0\), \(\mu_Y(0)=4\). The covariate density is approximately \(0.3989\), and the outcome density is \[ \frac{1}{2\sqrt{2\pi}}\exp(-1/8)\approx 0.1760. \] Thus the product is approximately \(0.3989(0.1760)=0.0702\).
At \(x=1\), \(\mu_Y(1)=4+1+0.5=5.5\). The covariate density is approximately \(0.2420\), and the outcome density is \[ \frac{1}{2\sqrt{2\pi}}\exp(-0.5^2/8)\approx 0.1933. \] The product is approximately \(0.2420(0.1933)=0.0468\). These are unnormalized contributions, not probabilities; normalizing requires the density over all possible \(x\). They show that the observed outcome changes the relative support for candidate source values.
For this incomplete source and nonlinear substantive model, SMC-FCS is a defensible choice if the ordinary imputation model would not preserve the outcome relationship. After drawing \(X\), calculate \(X^2\) passively. SMC-FCS addresses how \(X\) is imputed; passive recalculation ensures the quadratic term equals the square of the imputed value.
Worked Example: Incomplete Sources for an Interaction
In a hypothetical regression, the substantive model includes \(X_1X_2\). Both \(X_1\) and \(X_2\) have missing values. For one record, imputation produces \(X_1=2\) units and \(X_2=3\) units. Decide what to calculate and what model choice to consider.
The interaction must be recalculated as \[ D=X_1X_2=(2\text{ units})(3\text{ units})=6\text{ units}^2. \] It should not be imputed independently: an independently drawn value such as \(D=8\text{ units}^2\) would contradict the source values.
The interaction in the substantive model makes the source-imputation specification important. If ordinary conditional imputation models do not adequately represent the relationships implied by the interaction and the outcome, consider SMC-FCS for the incomplete sources. Then recalculate \(D\) for each completed record. If the sources are complete, or if the chosen source-imputation models are justified as compatible with the substantive model, passive recalculation remains necessary but SMC-FCS need not be selected automatically.
Why the Conditional Distribution Matters
The displayed conditional-density relation gives a short justification for the distinction. Suppose the covariate model is \(f(X\mid Z)\) and the substantive model is \(f(Y\mid X,Z)\). The joint density factors as
Conditioning on the observed \(Y\) and \(Z\) gives
because \(f(Y\mid Z)\) is constant with respect to \(X\). Thus, when imputing a missing \(X\), the outcome likelihood can change which values of \(X\) are plausible. A source imputation that ignores this information can be incompatible with the substantive model. Recomputing \(X^2\) or \(X_1X_2\) afterward cannot repair that mismatch: it changes the derived column, not the distribution of imputed source values.
This argument assumes the specified covariate and substantive models are suitable and that the conditional density can be sampled adequately. SMC-FCS is not a correction for a misspecified substantive model, omitted predictors of missingness, or unsupported extrapolation. The model and data must still provide credible information about missing values.
Comparison, Diagnostics, and Limits
| Question | Passive recalculation | SMC-FCS for source variables |
|---|---|---|
| What does it do? | Computes the deterministic function from current source values. | Uses the substantive model when imputing incomplete sources. |
| Does it impute a missing source? | No. | Yes, when that source is incomplete and included in the procedure. |
| Does it guarantee the derived identity? | Yes, if recalculated after source values change. | Not by itself; recalculate the derived term after the source draw. |
| When is it especially relevant? | Whenever a derived term is used and its sources have current values. | When incomplete sources and the substantive model make ordinary imputation compatibility doubtful. |
Useful checks include confirming that every completed value of a derived term equals the function of its sources, that terms are recalculated after every relevant source update, and that the imputation specification includes appropriate observed predictors. A mismatch in the deterministic identity signals a calculation or workflow problem. Plausible identities alone, however, do not establish that source imputations are compatible with the analysis model.
There are also practical limits. SMC-FCS can be computationally demanding, and the conditional distribution may be difficult to sample, especially with multiple incomplete covariates or complex substantive models. Poor proposal behavior or inadequate exploration can undermine the intended procedure. Conversely, passive recalculation is simple but does not solve an incompatible source-imputation model. The choice also cannot establish MAR from the observed data; sensitivity to plausible departures from MAR is a separate issue.
Common Mistakes
- Wrong reasoning: “I recalculated \(X^2\) after imputation, so the quadratic analysis is compatible.” Recalculation guarantees \(D=X^2\), not that missing \(X\) values were drawn from a distribution compatible with the outcome model. Consider SMC-FCS when the ordinary source model does not preserve that relationship.
- Wrong reasoning: “The model contains an interaction, so SMC-FCS is always mandatory.” The model form alone does not determine the answer. Check source missingness and whether the selected imputation models adequately represent the substantive relationships.
- Wrong reasoning: “SMC-FCS replaces passive recalculation.” SMC-FCS addresses source imputation; the deterministic term should still be recalculated from the completed sources.
- Wrong reasoning: “The derived term is missing, so I should impute it as a separate variable.” If it is a deterministic function, calculate it from its sources. Independent imputation can break the identity.
Check Your Understanding
Answer each question, then compare with the solutions.
- A complete variable \(X\) is used through \(X^2\). What method is needed to obtain the term, and why?
- A missing \(X\) is used in a quadratic outcome model. Explain in one sentence why passive recalculation alone may be insufficient.
- A record has imputed source values \(X_1=4\) and \(X_2=1.5\). What is the product term, and what are its units if the sources are measured in kilograms and metres?
- True or false: using SMC-FCS means an interaction term should be independently imputed.
Solutions. (1) Recalculate passively: \(X^2\) is determined by the observed \(X\), so there is no missing source to impute. (2) Recalculation enforces the square identity but does not make the imputed \(X\) values reflect the outcome likelihood implied by the substantive model. (3) \(X_1X_2=(4\text{ kg})(1.5\text{ m})=6\text{ kg}\cdot\text{m}\); calculate it from the current source values. (4) False. SMC-FCS concerns imputation of incomplete sources; calculate the interaction from those sources to preserve the deterministic identity.