SMC-FCS With Multiple Incomplete Covariates

Why Multiple Incomplete Covariates Require Careful Updates

When several covariates are incomplete, an update to one covariate can change the substantive-model contribution of another. This is especially important when the analysis includes an interaction, a polynomial term, or another deterministic function of multiple covariates. An SMC-FCS sweep therefore has two linked tasks: update each incomplete covariate conditional on the current values of the others, and keep every derived term synchronized with its inputs.

Let $Y$ denote the outcome, $X=(X_1,\ldots,X_p)$ the covariates, and $Z$ any fully observed variables included in the imputation and substantive models. The substantive model specifies $f_Y(y\mid x,z;\psi)$, where $\psi$ denotes its parameters. For each $X_j$, a proposal conditional model specifies $f_j(x_j\mid x_{-j},z;\phi_j)$, where $x_{-j}$ denotes all covariate values other than $x_j$.

Definition: In an SMC-FCS update of $X_j$, with $x_{-j}$, $z$, and $y$ held fixed, the unnormalized target is $f_Y(y\mid x,z;\psi)f_j(x_j\mid x_{-j},z;\phi_j)$. If its integral over the allowable values of $x_j$ is finite and positive, the normalized update target is $$ \pi_j(x_j\mid x_{-j},z,y;\phi_j,\psi) = \frac{f_Y(y\mid x,z;\psi)f_j(x_j\mid x_{-j},z;\phi_j)} {\int f_Y(y\mid x_{-j},u,z;\psi)f_j(u\mid x_{-j},z;\phi_j)\,du}. $$

The expression is formed for one covariate at a time. In a sweep, the current values of the other covariates are used, including their latest imputations. If $X_j$ is observed for a person, that observed value remains fixed; the update applies to missing values of $X_j$. When parameters are uncertain, the algorithm also needs an appropriate procedure for estimating or drawing the proposal and substantive-model parameters. The update formula alone does not specify that parameter procedure.

One Sweep With Several Incomplete Covariates

Suppose $X_1$ and $X_2$ are both incomplete, and the substantive model includes $D=X_1X_2$. For a person whose $X_1$ is missing and $X_2$ is currently imputed, the $X_1$ target uses that current $X_2$. After $X_1$ is updated, $D$ must be recomputed before the $X_2$ update. The $X_2$ target then uses the updated $X_1$, not its value from the start of the sweep.

Set the current state.
Keep observed covariate values fixed and supply current values for missing covariates, including initial values at the first sweep.

Update one incomplete covariate.
For $X_j$, construct its proposal conditional using current $x_{-j}$ and $z$, and combine it with the substantive outcome likelihood to define the normalized target.

Refresh affected derived terms.
Recalculate every deterministic quantity whose inputs include the newly updated covariate. Do not treat a stale product, square, or threshold indicator as a separate freely varying value.

Continue through the covariates.
Use the latest available values when constructing the next covariate’s target; after a full sweep, begin the next sweep from the resulting state.

The order of updates is part of the algorithm’s specification. A fixed order is often convenient, but a different order changes which newly updated values are available within a sweep. Regardless of order, every update must condition on a clearly defined current state, and derived variables must always agree with their current inputs.

Worked Example: Updating a Continuous Covariate Under a Logistic Model

Consider a hypothetical person with $Y=1$ and current $X_2=1$. Suppose the substantive model is $\operatorname{logit}P(Y=1\mid X_1,X_2)=-1+0.8X_1+0.5X_2+0.4X_1X_2$. For this update, let the proposal for $X_1$, conditional on the current $X_2=1$, be $N(0,1)$. Assume the proposal has support wherever the substantive likelihood is positive, and the resulting target has a finite, positive normalizing integral.

At candidate value $x_1=0$, the linear predictor is $-1+0+0.5+0=-0.5$, so $P(Y=1\mid x_1=0,X_2=1)=1/(1+e^{0.5})\approx0.3775$. The proposal density is $f_1(0)=0.3989$, giving an unnormalized target density of $0.3989(0.3775)\approx0.1506$.

At candidate value $x_1=1$, the predictor is $-1+0.8+0.5+0.4=0.7$, so the outcome probability is $1/(1+e^{-0.7})\approx0.6682$. The proposal density is $f_1(1)=0.2420$, giving an unnormalized target density of $0.2420(0.6682)\approx0.1617$.

The ratio of these unnormalized densities is $0.1617/0.1506\approx1.074$. Thus the target density at $x_1=1$ is about 1.074 times its value at $x_1=0$. These are density values, not probabilities that $X_1$ equals those exact points. The example checks the proposal support and a finite normalizer is assumed; in an applied analysis, propriety of the target must be verified for the actual model and covariate domain.

Derived Terms Must Follow the Current Covariates

A term such as $D=X_1X_2$ is deterministic once $X_1$ and $X_2$ are specified. If either input changes, its value changes accordingly. The substantive likelihood should therefore be evaluated using the derived term implied by the candidate covariate value and the current values of the remaining inputs. This is not a third imputation model for $D$: it is recalculation of a deterministic function. This use of dependency relationships and passive variables follows the earlier material on Dependency Graphs for Multiple Derived Variables and Using Passive Variables as Imputation Predictors.

Worked Example: Keeping an Interaction Current During a Sweep

Suppose a hypothetical person has both covariates incomplete and the binary-outcome substantive model is $\operatorname{logit}P(Y=1\mid X_1,X_2)=-1+0.3X_1+0.2X_2+0.1X_1X_2$. At the beginning of a sweep, the current values are $X_1=2$ and $X_2=3$, so the interaction is $D=2(3)=6$. Assume the update for $X_1$ produces a draw of $1.5$, and the subsequent update for $X_2$ produces a draw of $2.4$. These are illustrative completed values, not deterministic outputs of the formula.

After updating $X_1$, hold the current $X_2=3$ fixed and recompute $D=1.5(3)=4.5$. The predictor at that intermediate state is $-1+0.3(1.5)+0.2(3)+0.1(4.5)=-1+0.45+0.60+0.45=0.50$.

After updating $X_2$, use the latest $X_1=1.5$ and set $D=1.5(2.4)=3.6$. The new predictor is $-1+0.3(1.5)+0.2(2.4)+0.1(3.6)=-1+0.45+0.48+0.36=0.29$. Carrying forward the stale value $D=6$ would instead give a predictor of $-1+0.45+0.48+0.60=0.53$, which does not correspond to the current covariate values. The interaction must be recalculated at each relevant update.

Compatibility: What the Product Does and Does Not Guarantee

It is important not to interpret the product of a proposal conditional and the outcome likelihood as a conditional density from a joint model automatically. If the proposal conditionals $f_j$ are compatible, meaning they are the full conditionals of a joint covariate distribution $f_X(x\mid z)$, then $f_Y(y\mid x,z;\psi)f_X(x\mid z)$ is a joint model for $Y$ and $X$ conditional on $z$. Conditioning this joint model on $x_{-j},z,y$ gives a conditional density for $x_j$ proportional to $f_Y(y\mid x,z;\psi)f_j(x_j\mid x_{-j},z;\phi_j)$.

Without compatibility, the product still defines the chosen normalized SMC-FCS update target whenever its normalizer exists. But it need not be the conditional density of $X_j$ under any coherent joint covariate distribution and outcome model. Normalizing each covariate-specific target does not establish that the targets are compatible with one another, nor does it prove that repeated updates have a stationary joint distribution with the intended interpretation.

Worked Example: A Pair of Incompatible Proposal Conditionals

Consider the hypothetical proposal models $X_1\mid X_2=x_2\sim N(0.8x_2,1)$ and $X_2\mid X_1=x_1\sim N(0.2x_1,1)$. Each is a proper normal conditional distribution. To be full conditionals of a single positive, sufficiently smooth joint density, their log densities must have matching mixed derivatives with respect to $x_1$ and $x_2$. Terms depending only on one variable disappear when taking that mixed derivative.

For the first conditional, the part of the log density involving both variables is $-\tfrac12(x_1-0.8x_2)^2$, whose mixed derivative is $0.8$. For the second, it is $-\tfrac12(x_2-0.2x_1)^2$, whose mixed derivative is $0.2$. Since $0.8\ne0.2$, these conditionals cannot both arise from the same joint density.

Multiplying each proposal conditional by a common substantive outcome likelihood and normalizing defines a target for each separate update, provided each normalizer is finite and positive. It does not remove the incompatibility: the mixed derivatives contributed by the common log outcome likelihood match, leaving the unequal proposal contributions. Thus one must not claim that the product is necessarily a coherent joint-model conditional in this example.

Compatibility is sufficient for the joint-model interpretation just described, but it is not a requirement for writing down each normalized SMC-FCS update target. In practice, the conditionals are working proposal models. Their adequacy, the behavior of the resulting chain, and the substantive analysis require assessment rather than an automatic guarantee from normalization.

Assumptions, Diagnostics, and Alternatives

For an SMC-FCS analysis to support the intended inference, the substantive model should represent the analysis of interest, and the proposal conditionals should adequately describe the covariate distributions needed for imputation. The conditional targets must be proper: the normalizing integral for each update must be finite and nonzero, and proposal support must cover values relevant under the substantive model. When likelihood-based inference treats the missingness mechanism as ignorable, the usual conditions include missing at random given the observed data used in the analysis and distinct missingness and model parameters. These assumptions do not make a misspecified covariate proposal correct.

With several incomplete covariates, check the behavior of the imputed values for each variable across iterations and chains, paying particular attention to trends, poor mixing, implausible ranges, and implausible combinations. Also check that deterministic terms exactly match their inputs in every completed data set. A stable trace alone does not establish that the proposal models are adequate, that the conditionals are compatible, or that the substantive model is correct.

Approach	How it handles multiple incomplete covariates	Important qualification
Ordinary FCS	Cycles through covariate-specific conditional models.	The resulting conditionals may not preserve a nonlinear or interaction-rich substantive model.
Passive imputation	Recalculates deterministic terms from their covariate inputs.	Correct identities alone do not ensure that the covariate updates are compatible with the substantive model.
SMC-FCS	Uses the substantive likelihood in each covariate-specific update target.	Each target needs a valid normalizer; incompatible proposals do not automatically define a coherent joint model.
Joint covariate model	Specifies one joint distribution for the covariates and can provide compatible conditionals.	It requires a defensible joint specification and may be difficult to formulate for complex covariates.

The main practical failures are stale derived terms, updates that use old rather than latest covariate values, proposal distributions with inadequate support, and improper targets. A further limitation is conceptual: SMC-FCS incorporates the specified substantive likelihood, but it cannot correct a wrongly specified substantive model or guarantee good finite-sample behavior merely because the updates are cycled. The particular method used to draw from a target is a separate computational issue.

Mistake: “Each proposal conditional is a valid density, so their product with the outcome likelihood must be a joint-model conditional.” Validity one conditional at a time does not imply compatibility across covariates. The product defines the selected normalized update target when proper; a coherent joint interpretation requires additional compatibility.
Mistake: “The interaction can be imputed once and then held fixed during the sweep.” An interaction is a deterministic function of its inputs. Recompute it whenever an input changes.
Mistake: “A stable chain proves the model is correct.” Stability is a computational diagnostic, not proof of compatibility, adequate support, or correct substantive specification.

Key takeaway: In a multiple-covariate SMC-FCS sweep, update each incomplete covariate using the latest values of the others, form its target from the substantive likelihood and its proposal conditional, and recalculate all dependent terms. Treat a coherent joint interpretation as justified only when the proposal conditionals are compatible.

Check Your Understanding

Answer each question, then compare with the solutions below.

If $X_1$ is updated while $X_2$ is held at its current value, what quantities enter the unnormalized SMC-FCS target for $X_1$?
Suppose current $X_1=2$, $X_2=4$, and $D=X_1X_2$. After an update changes $X_2$ to $3$, what is the new $D$, and why?
Do individually proper proposal conditionals necessarily define a joint covariate distribution? State the distinction relevant to SMC-FCS.
Name two checks particularly relevant when several covariates and their derived terms are imputed.

Solutions. (1) The target is proportional to $f_Y(y\mid x,z;\psi)f_1(x_1\mid x_2,\ldots,x_p,z;\phi_1)$, with all other covariates held at their current values. (2) $D=2(3)=6$; it must be recalculated because it is a deterministic function of the current inputs. (3) No. Each normalized target can define an update when its normalizer is finite and positive, but interpreting the collection as full conditionals under a coherent joint model requires compatible proposal conditionals. (4) Check mixing and plausible distributions for each imputed covariate, and verify that derived terms equal their functions of the completed covariate values.