Patient Covariates In Joint Dose-Response Models

Why Include Baseline Covariates?

In a dose-finding study, patients may differ at baseline in ways that predict efficacy, toxicity, or both. A prognostic covariate can improve predictions by accounting for these differences. The key is to be precise about what adjustment does: it allows predicted joint outcomes to vary with baseline characteristics, but it does not, by itself, say that a covariate changes the dose response.

This tutorial uses a multinomial model for the joint efficacy-toxicity outcome. It represents the dependence between the two outcomes directly, without assuming that efficacy and toxicity are independent given dose. The model is a way to estimate outcome probabilities conditional on dose and baseline characteristics. Whether those predictions support a dose decision remains a separate design question.

A Joint Model With Covariates

Suppose efficacy \(E\) and toxicity \(T\) are binary outcomes assessed over prespecified windows, with their definitions and handling of intercurrent events set by the trial estimand. Their joint outcome has four mutually exclusive categories: \(00\) (neither efficacy nor toxicity), \(10\) (efficacy without toxicity), \(01\) (toxicity without efficacy), and \(11\) (both). Use \(00\) as the reference category. Let \(x\) denote a coded dose value and \(z\) a baseline covariate. For a binary covariate, \(z=0\) and \(z=1\) can label two baseline states; the same model form can use a continuous covariate.

For each nonreference category \(k\), specify a multinomial logit:

Here \(\alpha_k\) is the category-specific intercept, \(\beta_k\) is the dose slope when \(z=0\), \(\gamma_k\) is the covariate contrast when \(x=0\), and \(\delta_k\) is the dose-by-covariate interaction. The terms \(\gamma_k z\) adjust for a covariate association; they do not make the dose slope vary. That change in slope is represented by \(\delta_k xz\).

The model assumes one observed joint category per patient; categories are exhaustive and mutually exclusive. Conditional on assigned dose and the included baseline covariates, patients’ outcomes are modeled as independent draws from this four-category distribution. The specified linear predictor must be a reasonable description of dose and covariate effects on the log-odds scale over the doses and covariate values being used. The model does not require efficacy and toxicity to be conditionally independent: their association is represented by the joint category probabilities. A continuous dose term assumes a linear log-odds trend unless the model is extended, for example with prespecified nonlinear terms.

The three logits determine probabilities through normalization. Set \(\eta_{00}=0\); then each category probability is its exponentiated linear predictor divided by the sum across all four categories:

and \(P(Y=00\mid x,z)\) is the reciprocal of that same denominator. The normalization ensures that the four probabilities sum to one. Marginal efficacy and toxicity probabilities are obtained by adding the relevant joint-category probabilities: \(P(E=1\mid x,z)=P(Y=10\mid x,z)+P(Y=11\mid x,z)\), and \(P(T=1\mid x,z)=P(Y=01\mid x,z)+P(Y=11\mid x,z)\).

Adjustment Is Not Interaction

The distinction can be derived directly from the model. For category \(k\), compare dose \(x_1\) with dose \(x_0\) at a fixed covariate value \(z\). Subtracting the two logits gives:

The intercept and covariate main effect cancel because they do not change between the two dose values. Thus, for a one-unit dose contrast, the log-odds contrast is \(\beta_k\) when \(z=0\), and \(\beta_k+\delta_k\) when \(z=1\). If \(\delta_k=0\), the dose contrast is the same across covariate values on this log-odds scale, even though the predicted probabilities can still differ by covariate value. Conversely, a nonzero interaction describes a difference in log-odds dose contrasts. It does not automatically imply a particular difference in marginal efficacy or toxicity probabilities, because the four probabilities share a normalization denominator.

Calculate Covariate-Specific Joint Predictions

The coefficients do not directly give category probabilities. First calculate all three logits for the dose and covariate values of interest, exponentiate them, and normalize. The following hypothetical coefficients will be used for worked predictions:

From Patient Profiles to a Target Population

A covariate-specific prediction answers a question about a specified profile. A different question is what the model predicts for a population with a stated covariate distribution. One standardization approach calculates predictions at each covariate value and averages them using the target population’s covariate proportions. It is important to use the same target distribution when comparing doses; otherwise, apparent differences may partly reflect different patient mixes.

Choosing a Covariate Model and Checking Its Use

Three approaches answer related but distinct questions. An unadjusted model omits baseline covariates and estimates a dose-outcome relationship averaged over the covariate mix represented in its data. An adjusted model includes covariate main effects but sets \(\delta_k=0\), allowing different baseline outcome distributions while imposing a common dose slope for each category on the log-odds scale. An interaction model estimates \(\delta_k\) terms and permits those category-specific dose slopes to vary with the covariate. A stratified analysis instead estimates dose patterns separately within covariate groups; it can be transparent but may be imprecise when data are sparse. These approaches are not interchangeable, and the choice should match the scientific question and available information.

Adjustment alone does not establish effect modification. Nor does an interaction on the log-odds scale guarantee an interaction of the same form on the risk-difference scale. If the goal is to compare risks or jointly defined outcomes, calculate the predicted probabilities at the relevant profiles or standardize them to an explicit target distribution. A covariate association is not a causal effect of changing that characteristic. In a randomized dose study, randomization supports causal comparisons of assigned doses under the trial’s assumptions; covariate-specific causal interpretations still require appropriate design, outcome definitions, and adequate data.

Useful checks begin with the data supporting the prediction. Tabulate joint outcome counts across doses and important covariate ranges. Very few observations in a category, empty dose-covariate combinations, or complete separation can make coefficients unstable or effectively unidentified. Check whether the assumed dose trend is plausible across the studied range and whether predicted probabilities are sensible at the dose-covariate combinations that matter. If a covariate is continuous, inspect whether a linear term on the logit scale is defensible rather than assuming that coding it as a number guarantees linearity. Predictions far outside the observed covariate or dose support are extrapolations, even when the software returns a precise-looking number.

The model also depends on consistent outcome ascertainment and on the specified joint endpoint. If efficacy or toxicity definitions, assessment windows, or handling of intercurrent events differ across patients, the fitted probabilities may not represent a common estimand. Missing outcomes require the approach established for the estimand and analysis; simply treating unobserved joint outcomes as one of the four observed categories is not justified. Finally, estimated probabilities carry uncertainty. Point predictions alone can conceal substantial uncertainty, particularly for rare joint categories or complex interaction models.