A Worked SMC-FCS Example With a Quadratic Effect

Why a Quadratic Effect Needs Care

When an outcome model includes a squared predictor, the outcome mean can bend rather than change at a constant rate. For a missing predictor value, this means that covariate values on different sides of the curve’s turning point can imply similar fitted outcomes. An imputation that ignores the outcome model may therefore produce covariate draws that do not adequately reflect the information in the observed outcome.

This tutorial works through one SMC-FCS update for a missing continuous predictor \(X\), given an observed continuous outcome \(Y\) and observed covariate \(Z\). The example uses a normal proposal model for \(X\mid Z\) and a normal substantive model for \(Y\mid X,Z\). The numerical calculations illustrate the conditional draw; in an analysis, model parameters are also updated or drawn as part of the imputation procedure.

Definition: In this example, the proposal density \(q(x\mid z;\gamma)\) models the predictor distribution given observed covariates. The substantive density \(f(y\mid x,z;\beta)\) is the outcome model used for the scientific analysis. SMC-FCS draws a missing \(X\) from a conditional density proportional to their product: \(q(x\mid z;\gamma)f(y\mid x,z;\beta)\).

The assumed substantive model is

Y\mid X=x,Z=z \sim N\!\left(\beta_0+\beta_1x+\beta_2x^2+\beta_z z,\ \sigma^2\right).

The variance is \(\sigma^2\), and the conditional mean is \(m(x,z)=\beta_0+\beta_1x+\beta_2x^2+\beta_z z\). The proposal model in the worked calculations will be \(X\mid Z=z\sim N(\gamma_0+\gamma_z z,\tau^2)\). These models are assumptions for the example, not consequences of using SMC-FCS. The outcome model must represent the intended analysis, and the proposal must be a reasonable model for the predictor distribution with support wherever the target conditional density has support.

The Conditional Draw and Its Adjustment

Suppose \(y\) and \(z\) are observed for a record whose \(x\) is missing. Conditional on the current model parameters, the SMC-FCS target density for that missing value is

p(x\mid y,z;\beta,\gamma) = \frac{q(x\mid z;\gamma)f(y\mid x,z;\beta)} {\int q(u\mid z;\gamma)f(y\mid u,z;\beta)\,du}.

The denominator makes the density integrate to one. It does not depend on the candidate value \(x\), so the target is proportional to the proposal density multiplied by the outcome likelihood. The outcome likelihood is not itself a covariate density: using it alone would omit the predictor information represented by \(q(x\mid z;\gamma)\).

A direct way to sample from this target is rejection sampling. Draw a candidate \(x^\ast\) from \(q\), then accept it with probability \(f(y\mid x^\ast,z;\beta)/M\), where \(M\) is a finite upper bound on \(f(y\mid x,z;\beta)\) over the proposal’s support. Rejected candidates are discarded and a new proposal is drawn. For the normal outcome model, the likelihood as a function of \(x\) is

f(y\mid x,z;\beta) = \frac{1}{\sqrt{2\pi}\sigma} \exp\!\left\{-\frac{[y-m(x,z)]^2}{2\sigma^2}\right\}.

If some value of \(x\) makes \(m(x,z)=y\), a convenient envelope is the maximum normal density \(M=1/(\sqrt{2\pi}\sigma)\). The acceptance probability then simplifies to \(\exp\{-[y-m(x^\ast,z)]^2/(2\sigma^2)\}\). If no value of \(x\) can make the mean equal \(y\), this simplified expression remains a valid, possibly loose, acceptance probability, but its maximum over \(x\) is below one. One can instead use the actual maximum likelihood over the proposal’s support as \(M\), provided that maximum is correctly calculated.

Justification: A proposal value is accepted with probability \(f(y\mid x,z;\beta)/M\). Thus the density of accepted values is proportional to \(q(x\mid z;\gamma)f(y\mid x,z;\beta)/M\). Since \(M\) is constant in \(x\), normalizing this density gives exactly the SMC-FCS target above. This requires a valid envelope \(M\geq \sup_x f(y\mid x,z;\beta)\) and proposal support wherever the target is positive.

Worked Examples

Worked Example: A Proposal Draw and Its Acceptance

Consider hypothetical model values \(\beta_0=8\), \(\beta_1=1\), \(\beta_2=0.5\), \(\beta_z=0\), and \(\sigma=2\). For this record, the observed outcome is \(y=10\) and \(z=0\). The proposal is \(X\mid Z=0\sim N(0,4)\). Since the quadratic mean can equal 10, use \(M=1/(\sqrt{2\pi}\,2)\), so the acceptance probability is \(\exp\{-[10-m(x^\ast,0)]^2/8\}\).

First suppose the proposal gives \(x^\ast=-3.5\), with an auxiliary uniform draw \(U=0.98\). The fitted mean and acceptance probability are

m(-3.5,0)=8+(-3.5)+0.5(-3.5)^2=10.625, \qquad a(-3.5)=\exp\!\left\{-\frac{(10-10.625)^2}{8}\right\} =\exp(-0.04883)=0.9523.

Because \(U=0.98\) is greater than \(0.9523\), reject this proposal. Now suppose the next proposal is \(x^\ast=0\) and \(U=0.40\). Then

m(0,0)=8, \qquad a(0)=\exp\!\left\{-\frac{(10-8)^2}{8}\right\} =\exp(-0.5)=0.6065.

Since \(0.40<0.6065\), accept \(x=0\) as this record’s imputed value for the current update. This single draw is an illustration, not an estimate of the record’s most likely value. Across repeated updates, the accepted values follow the target density, which reflects both the proposal distribution and the outcome likelihood.

Worked Example: Two Covariate Values Can Fit the Outcome

Use the same substantive model and observed \(y=10,z=0\). Values of \(x\) that give a fitted mean of 10 solve

8+x+0.5x^2=10 \quad\Longleftrightarrow\quad x^2+2x-4=0 \quad\Longrightarrow\quad x=-1\pm\sqrt{5}.

The two solutions are approximately \(x=1.236\) and \(x=-3.236\). At either value the outcome likelihood reaches its maximum, because the residual \(y-m(x,0)\) is zero. However, the proposal model \(N(0,4)\) does not give the two points equal density. Its density ratio at these values is

\frac{q(1.236\mid 0)}{q(-3.236\mid 0)} = \exp\!\left\{ -\frac{1.236^2-(-3.236)^2}{2(4)} \right\} \approx \exp(1.118)=3.06.

Thus the proposal density at the positive solution is about 3.06 times its density at the negative solution, even though the outcome likelihood is the same at both. This is a comparison of densities at two points, not a ratio of the probabilities of two “branches”; those probabilities depend on the density over intervals around the solutions. The example shows why a quadratic outcome term can create separated regions of good outcome fit, while the predictor model still influences how often each region is represented.

Worked Example: When the Outcome Mean Cannot Reach the Observation

Keep the same quadratic mean but suppose the observed outcome is \(y=6\), with \(z=0\) and \(\sigma=2\). Completing the square gives \(m(x,0)=7.5+0.5(x+1)^2\), so the smallest possible fitted mean is 7.5, attained at \(x=-1\). The observation 6 is below every possible mean. The largest likelihood over \(x\) is therefore attained at \(x=-1\), and the valid acceptance probability relative to that maximum is

a(x^\ast)= \exp\!\left\{ -\frac{[6-m(x^\ast,0)]^2-(6-7.5)^2}{8} \right\}.

For a proposal \(x^\ast=0\), the fitted mean is \(m(0,0)=8\). Substitution gives

a(0)= \exp\!\left\{-\frac{(6-8)^2-(6-7.5)^2}{8}\right\} =\exp\!\left(-\frac{1.75}{8}\right) =0.8035.

With a hypothetical uniform draw \(U=0.75\), this proposal is accepted because \(0.75<0.8035\). Using the global normal-density maximum as the envelope would also be valid, but would give the smaller acceptance probability \(\exp(-4/8)=0.6065\) for this proposal. The tighter envelope improves acceptance efficiency without changing the target distribution.

Assumptions, Implementation, and Checks

The calculations above condition on parameter values. In a full imputation procedure, uncertainty in the proposal and substantive-model parameters must also be represented according to the implementation’s parameter-updating scheme; repeatedly drawing \(X\) while treating all fitted parameters as known would generally understate uncertainty. The normal proposal and normal substantive outcome model are substantive modeling choices. Their adequacy cannot be guaranteed just because rejection sampling is implemented correctly.

For the worked target to be meaningful, the outcome model must be the model intended for the analysis, including its quadratic term, and the predictor model must describe \(X\) conditional on relevant observed information. For likelihood-based inference with missing data, the missingness mechanism must also be ignorable under the usual conditions: missingness is missing at random conditional on included observed data, and missingness-model parameters are distinct from the data-model parameters. These conditions do not make misspecified covariate or outcome models correct.

In an implementation, use the current candidate value to calculate \(x^2\) inside the substantive mean. After an \(X\) draw is accepted, calculate the passive squared term from that accepted value; do not impute \(X^2\) as an unrelated random variable. If multiple covariates are incomplete, the conditional update must also respect the current values and models for the other covariates, as appropriate to the specified SMC-FCS procedure.

Several checks are useful while developing or reviewing the calculation:

Confirm that the mean used in the likelihood is \(\beta_0+\beta_1x+\beta_2x^2+\beta_z z\), with the same coefficients and quadratic form as the substantive analysis.
Check the residual, variance, and sign in the exponent. A larger squared residual must yield a smaller likelihood adjustment.
Verify that the envelope bounds the likelihood over the full proposal support. An envelope that is too small can produce acceptance probabilities above one and an invalid draw.
Check that the proposal can generate values throughout the target’s support. A proposal that excludes a region with positive target density cannot produce the correct conditional draw.
Watch for extremely low acceptance. It may signal a poor proposal, a very concentrated outcome likelihood, or a model/data conflict; it can also make computation inefficient even when the envelope is valid.

A passive-only imputation that predicts \(X\) without using the observed outcome may fail to preserve the quadratic substantive relationship. Conversely, using only the outcome likelihood as a weighting rule neglects the covariate distribution and is not the target conditional density. SMC-FCS combines these pieces. It does not, however, protect against a wrong quadratic specification, insufficient proposal support, an incorrect likelihood envelope, or poor parameter estimation.

Common Pitfall

A tempting argument is: “The outcome is best matched at the roots of the quadratic equation, so I should impute only one of those roots.” This is incorrect. The roots identify points with maximal outcome likelihood, not all values with positive conditional probability. The SMC-FCS target is a continuous density proportional to the proposal density times the outcome likelihood. Values near either root can receive substantial density, and the relative frequency of the regions depends on both modeled components. Restricting imputations to the exact roots would produce a discrete distribution rather than the specified continuous target.

Key takeaway: For a quadratic substantive effect, an SMC-FCS proposal is adjusted by the outcome likelihood evaluated at the proposed covariate value. Rejection sampling gives the desired conditional draw only when the likelihood envelope and proposal support are valid and the specified models are appropriate.

Check Your Understanding

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

For a normal outcome model, what factor adjusts a proposal draw, and what role does the envelope \(M\) play?
Under the worked model with \(y=10,z=0,\sigma=2\), calculate the acceptance probability for a proposal \(x^\ast=1\).
Why do the two roots that give \(m(x,0)=10\) not imply that the imputed value must equal one of those roots exactly?
For \(y=6\) in the third worked example, why is \(x=-1\) the value at which the likelihood is maximized?
Name two conditions needed for the rejection-sampling draw to target the stated conditional density correctly.

Solutions. (1) The acceptance probability is \(f(y\mid x^\ast,z;\beta)/M\), where \(M\) must be at least the maximum likelihood over the proposal support; it keeps the probability at or below one. (2) Here \(m(1,0)=8+1+0.5=9.5\), so the probability is \(\exp\{-(10-9.5)^2/8\}=\exp(-0.03125)=0.9692\). (3) The roots are points of maximum outcome likelihood, but the target is a continuous density proportional to proposal density times likelihood, so nearby values can also be drawn. (4) Since \(m(x,0)=7.5+0.5(x+1)^2\), its minimum is 7.5 at \(x=-1\), the attainable mean closest to \(y=6\); that minimizes the squared residual and maximizes the likelihood. (5) Examples include using a valid likelihood envelope and a proposal with support wherever the target is positive. The substantive and proposal models must also be specified as intended for the analysis.