Parameter Draws for Reference-Based Multiple Imputation

Why Draw Parameters Before Drawing Outcomes?

A reference-based imputation is a predictive distribution, not just a formula for filling in a missing value. Its conditional mean and covariance depend on quantities estimated from the trial, such as reference-arm means and covariance matrices. If those quantities are uncertain, an imputation procedure that holds them fixed leaves out a source of variation.

The distinction is between outcome uncertainty given parameters and parameter uncertainty. Even if the model parameters were known, a missing outcome would vary around its predictive mean. In addition, the estimated mean and covariance used to form that prediction vary across samples. A parameter-draw procedure represents both sources by first drawing plausible parameters and then drawing missing outcomes conditional on those parameters.

The reference-based rules established earlier determine which arm's behavior, mean structure, and covariance structure apply to a participant's missing outcomes. Parameter draws do not change that scientific rule. They carry uncertainty in its estimated ingredients through the rule and into the imputations.

Definition: A parameter draw is a random draw of model parameters from a specified posterior distribution or an appropriate sampling distribution based on the observed data. A predictive outcome draw is then drawn from the missing-outcome distribution conditional on that parameter draw and the participant's observed history. In reference-based multiple imputation, the reference rule is applied to the drawn parameters before the predictive outcome is generated.

Joint Draws Under a Multivariate Normal Model

Let $Y_i$ denote a participant's vector of repeated outcomes, with $T$ visits. A basic arm-specific model is $Y_i\mid\mu,\Sigma\sim N(\mu,\Sigma)$, where $\mu$ contains the visit means and $\Sigma$ contains the variances and covariances across visits. A practical analysis may use separate arm means, a common covariance, covariates, or stratification; the parameter draws must follow the fitted model actually used for the trial.

One standard way to describe parameter uncertainty for a multivariate normal distribution is the normal-inverse-Wishart model. With $\Sigma$ positive definite, take $\Sigma\sim IW(\nu_0,\Lambda_0)$ and $\mu\mid\Sigma\sim N(\mu_0,\Sigma/\kappa_0)$. Here $\kappa_0>0$, and the inverse-Wishart distribution has degrees of freedom $\nu_0>T-1$ and scale matrix $\Lambda_0$. Given complete vectors $Y_1,\ldots,Y_n$, define $\bar{Y}$ as their vector mean and $S=\sum_i(Y_i-\bar{Y})(Y_i-\bar{Y})^\mathsf{T}$. The posterior is also normal-inverse-Wishart:

\begin{aligned} \kappa_n &= \kappa_0+n, \qquad \nu_n=\nu_0+n,\\ \mu_n &= \frac{\kappa_0\mu_0+n\bar{Y}}{\kappa_n},\\ \Lambda_n &= \Lambda_0+S+ \frac{\kappa_0n}{\kappa_n}(\bar{Y}-\mu_0)(\bar{Y}-\mu_0)^\mathsf{T},\\ \Sigma\mid Y &\sim IW(\nu_n,\Lambda_n), \qquad \mu\mid\Sigma,Y\sim N(\mu_n,\Sigma/\kappa_n). \end{aligned}

A posterior draw therefore draws $\Sigma$ first and then draws $\mu$ conditional on that same $\Sigma$. The dependence matters: a mean draw has covariance $\Sigma/\kappa_n$, so drawing a mean independently of the covariance generally does not represent this posterior. In a multi-arm analysis, draw the arm means and covariance parameters from the joint posterior or sampling distribution implied by that analysis. Do not use the single-group formulas above as though they automatically described every multi-arm model.

For each draw, apply the reference-based specification to obtain the relevant mean vector and covariance matrix for the participant's missing pattern. Denote the resulting quantities by $\mu^{RB}$ and $\Sigma^{RB}$. These can depend on drawn parameters from one or more arms. The notation means that the reference rule has been applied; it does not assert that every reference-based method uses the same parameter mapping.

Conditional normal result: Partition a participant's outcome vector into observed components $Y_O$ and missing components $Y_M$. Under a multivariate normal model, conditional on a parameter draw, $$ Y_M\mid Y_O=y_O,\mu^{RB},\Sigma^{RB} \sim N\left( \mu_M^{RB}+\Sigma_{MO}^{RB}(\Sigma_{OO}^{RB})^{-1}(y_O-\mu_O^{RB}), \ \Sigma_{MM}^{RB}-\Sigma_{MO}^{RB}(\Sigma_{OO}^{RB})^{-1}\Sigma_{OM}^{RB} \right). $$ The conditional mean uses the observed history; the conditional covariance is the Schur complement. The observed block must be invertible for this expression as written.

Why Parameter Draws Add Uncertainty

Write $\theta$ for all parameters needed by a reference-based conditional model, and $Y_M$ for a missing outcome or vector. The predictive distribution averages over parameter uncertainty:

\operatorname{Var}(Y_M\mid Y_{\mathrm{obs}}) = E\!\left[\operatorname{Var}(Y_M\mid Y_{\mathrm{obs}},\theta)\mid Y_{\mathrm{obs}}\right] + \operatorname{Var}\!\left(E[Y_M\mid Y_{\mathrm{obs}},\theta]\mid Y_{\mathrm{obs}}\right).

The first term is average outcome variation conditional on parameters. The second is variation in the conditional mean across parameter draws. A plug-in procedure evaluates the conditional distribution at one estimate $\hat{\theta}$; it includes outcome variation at that estimate but does not represent the second term. This variance identity follows by writing $Y_M-E(Y_M\mid Y_{\mathrm{obs}})$ as the sum of $Y_M-E(Y_M\mid Y_{\mathrm{obs}},\theta)$ and $E(Y_M\mid Y_{\mathrm{obs}},\theta)-E(Y_M\mid Y_{\mathrm{obs}})$. The cross-product has expectation zero conditional on $Y_{\mathrm{obs}}$, because the first term has conditional mean zero given $Y_{\mathrm{obs}}$ and $\theta$. Taking variances gives the two terms shown.

Worked Example: A Mean and Variance Draw in One Visit

Suppose four hypothetical reference-arm outcomes, in points, are $8,10,12,10$. For illustration, use the univariate normal model with prior $\mu\mid\sigma^2\sim N(10,\sigma^2)$ and $\sigma^2\sim\text{Inv-}\chi^2(4,4)$. This notation uses prior degrees of freedom 4 and prior scale squared 4. The sample mean is $10$, the sample variance is $8/3$, and the posterior parameters are $\kappa_n=5$, $\mu_n=10$, and $\nu_n=8$. The posterior scale squared is

s_n^2= \frac{4(4)+(4-1)(8/3)+(1)(4/5)(10-10)^2}{8} =\frac{24}{8}=3.

Thus $\sigma^2\mid Y$ has the scaled inverse-chi-square distribution with 8 degrees of freedom and scale squared 3, and $\mu\mid\sigma^2,Y\sim N(10,\sigma^2/5)$. Suppose a simulated chi-square draw is $10$, and a subsequent standard normal draw for the mean is $1.2$. Then

\sigma^{2*}=\frac{8(3)}{10}=2.4,\qquad \mu^*=10+1.2\sqrt{\frac{2.4}{5}}=10.831.

If the next standard normal draw for a missing outcome is $-0.5$, then $Y_M^*=\mu^*+\sqrt{\sigma^{2*}}(-0.5)=10.831-0.775=10.056$ points. The hypothetical random-number draws illustrate the steps, not observed data. Both the estimated mean and variance varied before the missing outcome was generated.

Conditional Draws Across Visits

When earlier outcomes are observed, a missing visit is not generally drawn from its marginal distribution. The observed history shifts its conditional mean, and the covariance between observed and missing visits affects the conditional variance. Both effects must be recalculated for each parameter draw. This is especially important when applying a reference-based rule: use the rule to construct the parameters for the participant's missing pattern, then condition on the actual observed history.

Worked Example: Updating a Conditional Prediction for a Parameter Draw

Consider two hypothetical visits. Under one reference-based parameter draw, let the relevant mean vector be $(10.2,11.4)$ points and covariance matrix be

\Sigma^{RB*}= \begin{pmatrix} 4.84&3.30\\ 3.30&10.89 \end{pmatrix}.

The matrix is positive definite: its first diagonal entry is positive and its determinant is $4.84(10.89)-3.30^2=41.8176>0$. Suppose visit 1 is observed as $9.5$ points and visit 2 is missing. The conditional mean and variance are

\begin{aligned} E(Y_2\mid Y_1=9.5,\theta^*)&=11.4+\frac{3.30}{4.84}(9.5-10.2)=10.923,\\ \operatorname{Var}(Y_2\mid Y_1,\theta^*)&=10.89-\frac{3.30^2}{4.84}=8.640. \end{aligned}

If the predictive standard normal draw is $0.5$, the imputed visit-2 value is $10.923+0.5\sqrt{8.640}=12.392$ points. A new parameter draw would generally give a different conditional mean and variance, even for the same observed value $9.5$.

Applying a Reference Rule Draw by Draw

A useful implementation principle is to separate parameter uncertainty from the rule that defines the post-dropout distribution. For each imputed data set, draw the required model parameters jointly, construct the pattern-specific reference-based mean and covariance, and draw missing outcomes from the resulting conditional distribution. If a participant has several missing visits, draw the missing vector jointly or use a sequential factorization that is mathematically equivalent to that joint distribution.

For example, suppose a hypothetical participant has no post-baseline outcome observed and a prespecified J2R rule assigns the post-dropout outcome vector the reference-arm predictive distribution, conditional on the participant's baseline covariates. A draw of the reference mean and covariance is used for that vector; the treatment-arm point estimates are not substituted for those post-dropout parameters. The precise mapping depends on the selected reference rule and model. It should be stated explicitly and applied consistently to every draw.

Worked Example: Drawing a Two-Visit Reference-Based Vector

Consider a hypothetical participant whose baseline covariates are fixed, with both follow-up visits missing after dropout. Under the illustrative reference-based rule for this example, their post-dropout vector follows the reference predictive distribution conditional on those covariates. One joint draw gives mean $(11.4,9.6)$ points and covariance

\Sigma^{RB*}= \begin{pmatrix} 10&3\\ 3&15 \end{pmatrix}.

A Cholesky factor is $L=\begin{pmatrix}\sqrt{10}&0\\3/\sqrt{10}&\sqrt{14.1}\end{pmatrix}$, since $LL^\mathsf{T}=\Sigma^{RB*}$. For independent standard normal draws $z_1=0.5$ and $z_2=-1$, generate $Y_M^*=\mu^{RB*}+Lz$:

\begin{aligned} Y_2^*&=11.4+\sqrt{10}(0.5)=12.981,\\ Y_3^*&=9.6+\frac{3}{\sqrt{10}}(0.5)+\sqrt{14.1}(-1)=6.319. \end{aligned}

The calculation carries the drawn covariance into the association between visits; it does not generate the visits as independent values. In a full multiple-imputation run, the parameters and normal variates are redrawn for each imputed data set. The numerical vector here is one hypothetical realization, not a fixed prediction for the participant.

Diagnostics, Alternatives, and Limitations

Parameter draws are only as credible as the model and the data informing them. Check that covariance draws are positive definite, that their scales and correlations are plausible, and that simulated means and missing outcomes are not dominated by implausible extremes. Compare the distribution of draws across arms and dropout patterns, and inspect whether a small reference arm produces unusually broad or unstable predictions. For simulation-based posterior draws, assess chain mixing and convergence; for other sampling methods, check that the method reproduces the intended sampling distribution. Also verify that repeated-visit draws preserve the specified covariance structure and that the reference rule changes parameters only where intended.

Common alternatives have distinct trade-offs. A plug-in imputation uses fitted parameters as fixed and is simpler, but omits uncertainty in those estimates. A nonparametric bootstrap can represent sampling variation by resampling participants and refitting the model, but it must respect trial design features such as randomization strata and may be unstable with small arms or sparse dropout patterns. A Bayesian posterior draw represents uncertainty under the chosen likelihood and prior; its conclusions can depend on those modeling choices, particularly when data are sparse. These approaches are not interchangeable without considering the estimand, model, and inferential procedure.

A common mistake is to draw missing outcomes repeatedly from $N(\hat{\mu}^{RB},\hat{\Sigma}^{RB})$ and call the resulting imputations “proper” because each outcome is random. That adds residual outcome variation but fixes the uncertain parameters. The correction is to draw the relevant parameters first, apply the reference-based rule to that draw, and then draw the missing outcomes conditionally. Parameter draws do not, by themselves, establish that the model is correctly specified, that a reference assumption is clinically appropriate, or that the final inferential procedure has calibrated uncertainty. Those questions require separate scrutiny.

Key takeaway: A reference-based imputation should reflect uncertainty in both the predictive parameters and the missing outcome. Draw means and covariances from the fitted model's joint uncertainty distribution, apply the prespecified reference rule to each draw, and then generate the missing vector conditional on the participant's observed history.

Check Your Understanding

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

What two sources of variation are represented by drawing parameters before drawing a missing outcome?
In a conditional normal model, what role does $\Sigma_{MO}(\Sigma_{OO})^{-1}$ play in the conditional mean?
Why is it generally inappropriate to draw $\mu$ independently of $\Sigma$ under the normal-inverse-Wishart model described here?
A plug-in procedure draws outcomes randomly but holds estimated means and covariances fixed. Which component of predictive variance does it omit?
Give two diagnostics for detecting problematic parameter or predictive draws in a reference-based analysis.

Solutions. (1) It represents outcome variation conditional on parameters and uncertainty in the parameters themselves. (2) It weights the observed residual $y_O-\mu_O$ by the covariance between missing and observed outcomes, adjusted for the observed-outcome covariance; this shifts the conditional mean. (3) The posterior specifies $\mu\mid\Sigma,Y$ with covariance $\Sigma/\kappa_n$, so the mean's uncertainty depends on the drawn covariance. (4) It omits variation across parameter values in the conditional predictive mean, the second term in the variance decomposition. (5) Examples include checking covariance draws for positive definiteness and plausible correlations, assessing simulation-chain convergence, and inspecting predictive tails for implausible extremes or instability in small reference arms.