SMC-FCS With Logistic Substantive Models

Why the Logistic Outcome Model Must Enter the Imputation

Suppose a binary outcome $Y$ is analyzed using logistic regression, but a continuous covariate $X$ is partly missing. The substantive model may include a quadratic term such as $X^2$, or an interaction such as $XZ$ with another covariate $Z$. In either case, the association between $X$ and the outcome is not represented by a simple linear effect of $X$ on the log-odds. An imputation model that ignores this structure can generate values of $X$ that are poorly aligned with the specified outcome model.

Substantive-model-compatible fully conditional specification (SMC-FCS) addresses this issue by combining a model for the incomplete covariate with the likelihood from the substantive outcome model. The target distribution for an imputed $X$ therefore reflects both how $X$ is distributed given the other covariates and how the observed outcome is related to $X$ under the logistic model. This tutorial focuses on one incomplete continuous covariate and an observed binary outcome; it does not cover the extension to multiple incomplete covariates.

Definition: Let $Y$ be a binary outcome, $X$ an incomplete continuous covariate, and $Z$ a fully observed covariate or vector of covariates. SMC-FCS targets the conditional density $$ f(x\mid y,z) \propto f(x\mid z;\alpha)\,P(Y=y\mid X=x,Z=z;\beta), $$ where $f(x\mid z;\alpha)$ is a covariate model and the second factor is the outcome likelihood under the specified substantive model. The proportionality constant is chosen so that the density integrates to one over the allowed values of $x$.

The expression is a target distribution, not a claim that $X$ is generated by the outcome. It is a conditional distribution used to impute missing values while preserving the chosen joint-model structure. The earlier material on the SMC-FCS conditional density establishes the general construction; here the important feature is the form of the likelihood for a logistic substantive model.

Logistic Likelihood in the Conditional Density

For a binary outcome, specify the substantive model as

\operatorname{logit}\{P(Y=1\mid X=x,Z=z)\}=\eta(x,z;\beta), \qquad P(Y=1\mid x,z)=\frac{\exp\{\eta(x,z;\beta)\}}{1+\exp\{\eta(x,z;\beta)\}}.

For an observed outcome $y\in\{0,1\}$, its Bernoulli likelihood contribution is

P(Y=y\mid x,z;\beta) = \left\{\frac{\exp(\eta(x,z;\beta))}{1+\exp(\eta(x,z;\beta))}\right\}^{y} \left\{\frac{1}{1+\exp(\eta(x,z;\beta))}\right\}^{1-y}.

Thus, when $y=1$, the likelihood factor is the predicted probability of the event; when $y=0$, it is the predicted probability of no event. In either case, the likelihood reweights the covariate distribution $f(x\mid z;\alpha)$. A candidate value of $x$ receives more weight when it is plausible under the covariate model and makes the observed outcome plausible under the substantive model.

Key result: If the logistic predictor contains a quadratic or interaction term, that term must be evaluated at each candidate $x$ when calculating $P(Y=y\mid x,z;\beta)$. The resulting target density need not belong to the same simple family as $f(x\mid z;\alpha)$. For example, a normal covariate model multiplied by a logistic likelihood generally does not produce a normal conditional density.

Why the Likelihood Weighting Follows

The target form follows directly from the factorization of the joint conditional distribution of $X$ and $Y$ given $Z$. If the covariate model is $f(x\mid z;\alpha)$ and the outcome model is $P(y\mid x,z;\beta)$, then

f(x,y\mid z;\alpha,\beta) = f(x\mid z;\alpha)P(y\mid x,z;\beta).

Condition on the observed $y$ and $z$. Dividing this joint density by the marginal probability of $y$ given $z$ gives

f(x\mid y,z;\alpha,\beta) = \frac{f(x\mid z;\alpha)P(y\mid x,z;\beta)} {\int f(u\mid z;\alpha)P(y\mid u,z;\beta)\,du}.

The denominator is the normalizing constant: it sums or integrates the numerator over all possible covariate values. Since it does not depend on $x$, the numerator gives the proportionality statement in the definition. This derivation also explains why the observed outcome belongs in the imputation conditional distribution. Omitting its likelihood would discard information about $X$ carried by $Y$.

Worked Logistic Models with Interaction and Curvature

Worked Example: Likelihood Weighting with an Interaction

Consider hypothetical data with binary outcome $Y$, continuous covariate $X$, and fully observed $Z$. Let the substantive model be

\operatorname{logit}\{P(Y=1\mid x,z)\} =-1+0.7x+0.5z+0.8xz.

Suppose $X\mid Z=z\sim N(0.2+0.3z,1.2^2)$. For a case with $z=1$ and observed $y=1$, the covariate model is $X\mid Z=1\sim N(0.5,1.44)$. Compare candidate values $x=0$ and $x=1$.

At $x=0$, the linear predictor is $\eta=-1+0+0.5+0=-0.5$, so the event probability is $1/(1+\exp(0.5))=0.3775$. The normal density is

f(0\mid z=1) = \frac{1}{1.2\sqrt{2\pi}} \exp\left\{-\frac{(0-0.5)^2}{2(1.44)}\right\} \approx 0.305.

The unnormalized target density at zero is therefore $0.305(0.3775)\approx 0.115$. At $x=1$, $\eta=-1+0.7+0.5+0.8=1.0$, giving event probability $0.7311$. The normal density is again about $0.305$, because zero and one are equally distant from the mean $0.5$. The unnormalized target density is $0.305(0.7311)\approx 0.223$. Thus the target density at $x=1$ is about $0.223/0.115=1.94$ times its value at $x=0$. The interaction changes how the outcome likelihood weights $X$ at $z=1$.

Worked Example: A Quadratic Logistic Effect

For a second hypothetical model, let $X\sim N(0,1)$, and suppose

\operatorname{logit}\{P(Y=1\mid X=x)\}=-1+1.2x-0.8x^2.

For an observed event $y=1$, compare the unnormalized target density at $x=-1,0,1$. The standard normal densities are approximately $0.2420, 0.3989, 0.2420$, respectively. At $x=-1$, $\eta=-1-1.2-0.8=-3$, so $P(Y=1\mid x=-1)=0.04743$, and the product is $0.2420(0.04743)=0.01148$. At $x=0$, $\eta=-1$, giving probability $0.2689$ and product $0.3989(0.2689)=0.1073$. At $x=1$, $\eta=-1+1.2-0.8=-0.6$, giving probability $0.3543$ and product $0.2420(0.3543)=0.08574$.

These are pointwise density values, not probabilities at three points and not a normalized discrete distribution. They show how the quadratic outcome model reshapes the covariate density for an observed event. In particular, the outcome likelihood is much smaller at $x=-1$ than at $x=0$, even though the covariate model assigns equal density to $-1$ and $1$.

Worked Example: The Interaction Changes the Conditional Odds Ratio

Suppose a logistic model has predictor $\eta(x,z)=-0.6+0.4x+0.7z+0.9xz$. For a one-unit increase in $X$, holding $Z=z$ fixed, the change in log-odds is $0.4+0.9z$. Therefore the conditional odds ratio is $\exp(0.4+0.9z)$. At $z=0$, it is $\exp(0.4)=1.492$. At $z=1$, it is $\exp(1.3)=3.669$.

This calculation checks the interpretation of the interaction: the odds ratio for a one-unit increase in $X$ differs across $Z$. In SMC-FCS, this same interaction must appear inside $\eta(x,z)$ when constructing the likelihood weight. Using only the main effect $0.4x$ would incorrectly apply the $z=0$ relationship at every value of $Z$.

Assumptions and Practical Implementation

The construction relies on several modeling and data conditions. First, the logistic substantive model must be a suitable representation of the intended analysis, including its link, covariates, nonlinear terms, and interactions. Second, the covariate model $f(x\mid z;\alpha)$ must adequately describe the distribution of $X$ given the fully observed covariates used in that model. A normal model, for example, requires a plausible continuous support and an adequate conditional mean and variance specification; if $X$ is bounded or markedly skewed, an unbounded normal model may be unsuitable.

For imputing missing $X$, $Y$ and the relevant $Z$ values must be observed, or their missingness must be handled in a broader model. The missingness assumption for $X$ is typically missing at random conditional on the variables used in the imputation, such as $Y$ and $Z$. This means that after conditioning on those observed variables, the probability that $X$ is missing does not depend on its unobserved value. The assumption cannot be verified from observed data alone. There must also be adequate overlap: the observed data need to support the covariate and outcome relationships across the combinations of $Y$ and $Z$ that occur.

In practice, SMC-FCS cycles through imputation models while preserving the specified substantive-model likelihood in the target for the incomplete covariate. Model parameters are treated as uncertain rather than fixed at single estimates, so imputations reflect uncertainty in both the covariate and outcome models. The particular computational sampler used to draw from the target is an implementation choice; regardless of sampler, it must target the likelihood-weighted conditional density, not merely the covariate proposal distribution.

Specify the analysis model.
Write the logistic predictor with every planned nonlinear term and interaction, and retain the observed outcome and covariates needed to evaluate it.

Specify the covariate model.
Choose $f(x\mid z;\alpha)$ to represent the incomplete continuous covariate conditional on the available covariates, with a plausible support and distributional form.

Form the target density.
For each missing $X$, multiply its covariate density by the Bernoulli likelihood for that person's observed $Y$, evaluating all terms in the logistic predictor at each candidate $x$.

Draw imputations and repeat.
Use an appropriate stochastic procedure to draw from the target, update model uncertainty, and repeat the chained procedure to create multiple completed data sets.

Diagnostics, Alternatives, and Limitations

A standard fully conditional specification model for $X$ that regresses $X$ on $Y$ and $Z$ may be easy to fit, but a simple linear form can be incompatible with a substantive model containing $X^2$ or $XZ$. Merely computing a passive term after imputing $X$ ensures that the term equals the transformation of the imputed value; it does not ensure that the distribution of $X$ was shaped by the outcome likelihood in the intended nonlinear model. SMC-FCS instead builds that likelihood into the target. This distinction is particularly important when the nonlinear or interaction effect is central to the analysis.

Useful checks include comparing observed and imputed $X$ distributions within outcome and covariate strata, checking whether imputed values remain within scientifically plausible support, and examining trace plots and between-chain mixing for the imputation procedure. Also check whether the fitted logistic model exhibits extreme fitted probabilities or separation, since these can make likelihood weighting unstable or highly concentrated. Compare the substantive-model estimates across imputations and assess whether a small number of imputations or poor mixing is driving the results. These diagnostics can reveal problems, but agreement between observed and imputed distributions does not prove that the missingness assumption or either model is correct.

SMC-FCS does not protect against misspecification. If the logistic model omits an important nonlinear term, or if the covariate distribution is poorly modeled, the imputation can still be biased for the intended analysis. Severe outcome-model separation, sparse combinations of $Y$ and $Z$, or weak support for parts of the covariate range can also make the target difficult to estimate or sample. The method is more computationally demanding than fitting a simple regression imputation model, and sensitivity to plausible alternative model specifications remains important. Its purpose is compatibility with the stated substantive model, not automatic correction of all missing-data problems.

Common pitfall: “I can impute $X$ from a linear model, then calculate $X^2$ or $XZ$, so the logistic analysis will be compatible.” The derived terms will be algebraically correct, but that alone does not make the imputation distribution compatible with the nonlinear or interaction likelihood. The correction is to include the specified substantive-model likelihood in the target conditional density for $X$.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

Write the SMC-FCS target density for a missing continuous $X$ when $Y$ is observed and binary.
In a logistic model with predictor $\eta=-0.5+0.6x+0.8xz$, what is the conditional odds ratio for a one-unit increase in $X$ when $z=1$?
If $y=0$, which logistic likelihood factor weights a candidate $x$: $P(Y=1\mid x,z)$ or $P(Y=0\mid x,z)$?
Why does calculating $X^2$ after a linear imputation of $X$ not, by itself, guarantee compatibility with a quadratic logistic analysis?

Solutions. (1) $f(x\mid y,z)\propto f(x\mid z;\alpha)P(Y=y\mid x,z;\beta)$, normalized by integrating over the possible $x$ values. (2) The log-odds change is $0.6+0.8(1)=1.4$, so the odds ratio is $\exp(1.4)\approx4.055$. (3) Use $P(Y=0\mid x,z)=1-P(Y=1\mid x,z)$, the probability of no event under the logistic model. (4) The transformation guarantees only that the completed value of $X^2$ equals the square of $X$; it does not make the imputation distribution for $X$ reflect the outcome likelihood containing $X^2$.