Conditioning on Baseline Covariates

Why Condition on Baseline Covariates?

Participants with different baseline characteristics can have different expected outcomes even when they are described by the same reference-arm model. A marginal reference prediction averages over the reference arm’s baseline mix. An individual prediction should instead use that participant’s baseline values. When pre-dropout outcomes are also used, the prediction should reflect both those values and the participant’s observed history.

This distinction matters in reference-based imputation. After dropout, a reference-based restriction specifies how the unobserved outcomes relate to the reference arm. Conditioning on baseline covariates makes that reference prediction relevant to the particular participant, rather than to an average participant in the reference group. It does not, by itself, make the post-dropout outcomes observed or remove the need to justify the reference-based assumption.

Definition: Let \(B\) denote a vector of baseline covariates and let \(Y_j\) denote the outcome at visit \(j\). A reference-arm predictive distribution conditional on baseline covariates is the distribution of the relevant outcomes among reference-arm participants with \(B=b\), written \(Y\mid B=b,\text{ reference arm}\). If observed outcome history \(Y_o=y_o\) is also used, the individual prediction is based on \(Y_m\mid B=b,Y_o=y_o,\text{ reference arm}\), where \(Y_m\) denotes the outcomes to be predicted.

Reference Predictions Conditional on Baseline

A convenient model specifies the reference-arm mean at each visit as a function of baseline covariates. For example, with a scalar covariate \(B\), a linear conditional mean at visit \(j\) is \(\mu_j^R(b)=\alpha_j^R+\beta_j^R b\). The superscript \(R\) identifies parameters for the reference arm. With several covariates, the linear predictor uses a vector of coefficients and may include prespecified nonlinear terms or interactions.

A marginal reference mean averages over the baseline distribution in the reference arm. It is generally not equal to the conditional mean for every baseline value. For a linear model, the marginal mean is \(\alpha_j^R+\beta_j^R E_R(B)\), whereas the mean for a participant with baseline value \(b\) is \(\alpha_j^R+\beta_j^R b\). Using the marginal mean for every participant discards the information in \(b\).

Worked Example: A Baseline Risk Marker Changes the Reference Mean

Suppose a hypothetical reference arm has a binary baseline risk marker \(B\), with \(B=1\) indicating an elevated marker. Its fitted reference-arm mean outcome at a particular post-dropout visit is 70 points for \(B=0\) and 60 points for \(B=1\). Suppose 40% of the reference arm has \(B=1\). Find the marginal reference mean and compare it with the conditional prediction for a participant with \(B=1\).

The marginal mean weights the two conditional means by their reference-arm proportions:

(1-0.40)(70) + (0.40)(60) = 42 + 24 = 66\text{ points}.

For a participant with \(B=1\), the conditional reference mean is 60 points, not 66 points. The marginal mean is 6 points higher because it averages over participants with both marker values. These are model-based means, not guarantees about any individual outcome. The calculation assumes that the stated conditional means and reference-arm proportions adequately describe the hypothetical reference group.

Conditioning on Baseline and Observed History

Often a participant has one or more observed outcomes before dropout. In a multivariate normal reference-arm model, the predictive distribution for missing outcomes can condition on both baseline covariates and those observed outcomes. For a fixed \(B=b\), partition the reference-arm outcome vector into observed components \(Y_o\) and components to predict \(Y_m\):

\begin{pmatrix}Y_o\\Y_m\end{pmatrix}\Bigm| B=b,\ R \sim N\left[ \begin{pmatrix}\mu_o^R(b)\\\mu_m^R(b)\end{pmatrix}, \begin{pmatrix} \Sigma_{oo}^R & \Sigma_{om}^R\\ \Sigma_{mo}^R & \Sigma_{mm}^R \end{pmatrix} \right].

Here \(R\) denotes the reference arm. The conditional predictive distribution is

Y_m\mid B=b,Y_o=y_o,R \sim N\left( \mu_m^R(b)+\Sigma_{mo}^R(\Sigma_{oo}^R)^{-1} \{y_o-\mu_o^R(b)\}, \ \Sigma_{mm}^R-\Sigma_{mo}^R(\Sigma_{oo}^R)^{-1}\Sigma_{om}^R \right).

The mean has two parts: the reference mean at the participant’s baseline values, and an adjustment based on how the observed history differs from its reference mean at those same baseline values. The covariance matrices describe the reference-arm association between observed and unobserved outcomes, conditional on baseline. This formula assumes the covariance block \(\Sigma_{oo}^R\) is invertible; in practice, the model and covariance estimate must also be appropriate for the observed visit pattern.

Justification: For fixed \(B=b\), subtract the conditional means and define \(E=Y_m-\mu_m^R(b)-\Sigma_{mo}^R(\Sigma_{oo}^R)^{-1}\{Y_o-\mu_o^R(b)\}\). The Gaussian covariance formula gives \(\operatorname{Cov}(E,Y_o\mid B=b,R)=\Sigma_{mo}^R-\Sigma_{mo}^R(\Sigma_{oo}^R)^{-1}\Sigma_{oo}^R=0\). Because \(E\) and \(Y_o\) are jointly normal, zero covariance implies independence. The variance of \(E\) is \(\Sigma_{mm}^R-\Sigma_{mo}^R(\Sigma_{oo}^R)^{-1}\Sigma_{om}^R\). Thus, conditioning on \(Y_o=y_o\) shifts the mean by the stated linear adjustment and leaves this conditional variance.

Worked Example: Predicting One Missing Visit Given Baseline and History

In a hypothetical reference-arm model, let \(B=1\) for an elevated baseline marker. The reference means at an observed visit and a later visit are \(\mu_o^R(b)=50+4b\) and \(\mu_m^R(b)=54+5b\), in points. The observed outcome for a participant with \(B=1\) is \(y_o=58\). Suppose the reference-arm variance at the observed visit is 16 points squared, the variance at the later visit is 25 points squared, and their covariance is 12 points squared. Calculate the conditional predictive mean and variance for the later outcome.

At \(b=1\), the two conditional means are \(\mu_o^R(1)=54\) and \(\mu_m^R(1)=59\) points. With one observed outcome, the coefficient for the history adjustment is the covariance divided by the observed-outcome variance: \(12/16=0.75\). The conditional mean is therefore

59 + 0.75(58-54) = 59 + 3 = 62\text{ points}.

The conditional variance is

25-\frac{12^2}{16} =25-9 =16\text{ points}^2.

The standard deviation is \(\sqrt{16}=4\) points. Under the specified Gaussian reference-arm model, the participant’s predictive distribution is \(N(62,16)\), conditional on the baseline marker and observed outcome. The calculation assumes the stated means, variances, and covariance are appropriate for this visit pair. It is a predictive distribution under the model, not a statement that the participant’s unobserved outcome equals 62.

Individual Predictions Need Not Match the Cohort Average

With a common covariance structure, baseline covariates affect both the reference means and the history adjustment. Consequently, two participants with the same observed outcome can have different predictions if their baseline values imply different expected observed and missing outcomes. The next example isolates this effect.

Worked Example: Same Observed History, Different Baseline Values

Suppose a hypothetical reference-arm model has one observed and one missing visit. Let \(B\) be a continuous baseline measure, and let the conditional means be \(\mu_o^R(b)=30+2b\) and \(\mu_m^R(b)=40+3b\), in points. Suppose the covariance adjustment coefficient is \(\Sigma_{mo}^R/\Sigma_{oo}^R=0.5\). Two participants have the same observed outcome, \(y_o=36\), but baseline values \(b=0\) and \(b=4\). Find their conditional predictions.

For the participant with \(b=0\), the reference observed mean is 30 and the missing-visit mean is 40. Thus,

40+0.5(36-30)=40+3=43\text{ points}.

For the participant with \(b=4\), the reference observed mean is \(30+2(4)=38\), while the missing-visit mean is \(40+3(4)=52\). The prediction is

52+0.5(36-38)=52-1=51\text{ points}.

The predictions differ by 8 points despite identical observed outcomes. For the first participant, 36 is 6 points above the reference observed mean; for the second, it is 2 points below that participant’s reference observed mean. The adjustment therefore depends on the history’s deviation from the baseline-specific reference mean, not just on the raw history value.

Assumptions, Diagnostics, and Alternatives

A covariate-conditioned prediction is only as defensible as the model and reference-based assumptions supporting it. In particular:

Conditional model adequacy. The reference-arm outcome means must describe how outcomes vary with baseline covariates, and the covariance model must represent the dependence between visits well enough for the conditional prediction. A Gaussian model adds distributional assumptions beyond a model for the mean alone.
Covariate support. The reference arm must contain enough information at baseline values relevant to the participants being imputed. Predictions far outside the reference arm’s observed covariate range rely heavily on extrapolation.
Reference-based assumption. Applying a reference-arm predictive distribution to outcomes after dropout in another arm is an assumption about the unobserved outcomes. Conditioning on baseline covariates does not make that assumption empirically verified.
Consistent conditioning. Baseline covariates must be measured and coded consistently, and the observed history used in the prediction must be the history specified by the model and imputation rule. Do not substitute a marginal reference mean after fitting a conditional model without accounting for the change.

Useful diagnostics include plotting reference-arm outcomes against continuous baseline covariates, checking residual patterns across covariate values and visits, and checking whether the model’s conditional means systematically miss observed outcomes in important subgroups. Inspect the covariate distributions in the reference arm and among participants requiring predictions. For a multivariate normal model, check that estimated covariance matrices are positive definite and that the required observed-visit covariance blocks are estimable. Sparse strata, extreme covariate values, implausible conditional variances, or predictions that change sharply under modest model changes are warnings to investigate.

A marginal reference prediction is simpler, but it gives the same reference mean to participants with different baseline profiles and may poorly represent individual predictions. Stratifying by categories of baseline covariates makes the conditional comparison explicit, but can leave small or empty strata and cannot automatically handle continuous covariates. A regression-based conditional model can use continuous information more efficiently, but depends on its functional form and may extrapolate. These are modeling choices, not ways to avoid stating the reference-based assumption.

A further limitation is that the displayed Gaussian formula treats the model parameters as fixed. In an application, estimated means and covariances are uncertain, and plug-in predictions do not include all that uncertainty. The formula explains the conditional distribution given model parameters; it does not establish that a complete analysis has accounted for parameter uncertainty or for the uncertainty inherent in the missing-data restriction.

Common Pitfall

Common mistake: “The reference-arm average is 66 points, so every participant who drops out should receive a prediction centered at 66.” This confuses a marginal mean, averaged over the reference arm’s baseline mix, with a baseline-conditioned prediction. If the participant has \(B=1\) in the first example, the specified conditional mean is 60 points before using any observed outcome history. If history is also used, the conditional mean may shift again. The appropriate prediction follows the stated conditioning set and model.

Key takeaway: Baseline covariates determine which reference-arm predictive distribution is relevant to a participant. When observed outcome history is included, prediction also adjusts for that history relative to its baseline-specific reference expectation.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

A reference-arm mean is 80 points for \(B=0\) and 68 points for \(B=1\). What is the conditional mean for a participant with \(B=1\), and why might it differ from the marginal reference mean?
In a one-observed-visit model, \(\mu_o^R(b)=20+b\), \(\mu_m^R(b)=30+2b\), the covariance is 6, the observed-outcome variance is 9, and \(y_o=27\). Find the conditional predictive mean when \(b=2\).
Why is a participant’s observed outcome adjusted relative to \(\mu_o^R(b)\), rather than relative to the marginal reference mean?
Name two diagnostics that could reveal that baseline conditioning is not working adequately.

Solutions. (1) The conditional mean is 68 points. It can differ from the marginal mean because that mean averages over the reference arm’s proportions with \(B=0\) and \(B=1\). (2) At \(b=2\), \(\mu_o^R=22\) and \(\mu_m^R=34\). The adjustment coefficient is \(6/9=2/3\), so the prediction is \(34+(2/3)(27-22)=34+10/3=37.33\) points. (3) The adjustment represents how unusual the observed history is relative to the reference expectation for that participant’s baseline values; a marginal mean ignores those values. (4) Examples include checking outcome residuals across baseline values, assessing reference-arm covariate overlap, and checking whether estimated covariance matrices are positive definite.