Correcting Exposure-Outcome Bias Under Adaptive Dosing

Why Adaptive Dosing Can Bias an Exposure-Outcome Analysis

An adaptive dosing rule uses patient information collected during the trial to choose later doses. That information may predict the outcome as well as influence dose assignment. For example, an early biomarker associated with poor prognosis may prompt a dose increase. A comparison of outcomes by achieved exposure can then mix the effect of exposure with the prognostic differences that helped determine it. The observed exposure-outcome association need not equal the causal effect of changing exposure.

The distinction between assigned dose and achieved exposure matters. Assigned dose is a treatment decision; exposure is the resulting drug concentration or another pharmacometric quantity, influenced by dose and patient characteristics such as clearance. A randomized dose comparison may support an intention-to-treat contrast, but a direct analysis of achieved exposure is not automatically randomized. This tutorial concerns adjustment for measured information that drives dosing and predicts outcomes; it does not make unmeasured confounding disappear.

The target must be stated before adjustment. It might be the mean outcome under a specified exposure regimen compared with another regimen, or the change in expected outcome per unit increase in exposure under a defined model. These are not necessarily the same as the effect of assignment to a dose. For longitudinal questions, a regimen specifies the sequence or rule for doses over time, rather than just a final cumulative exposure.

A Simple Confounding Derivation

Suppose \(U\) is a measured prognostic history variable, \(X\) is exposure, and \(Y\) is a continuous outcome. Consider the linear outcome model

The conditional-mean assumption is explicit: it justifies interpreting the specified conditional mean as \(\mathbb{E}(Y\mid X,U)\). If \(U\) is omitted and \(Y\) is regressed only on \(X\), the population slope is

To obtain this expression, substitute the outcome model into the covariance. The intercept contributes zero covariance, and the conditional-mean assumption implies \(\operatorname{Cov}(X,\varepsilon)=0\). Thus \(\operatorname{Cov}(X,Y)=\beta_X\operatorname{Var}(X)+\beta_U\operatorname{Cov}(X,U)\). Dividing by \(\operatorname{Var}(X)\) gives the displayed slope. If history predicts both exposure and outcome, the second term is generally nonzero. Adjustment for \(U\) removes this particular omitted-variable contribution under the specified model, but only if the relevant confounders are measured and the model is adequate.

Outcome Modeling and Standardization

For measured history, one approach is to model the outcome conditional on exposure and history, then average predicted outcomes over a common target distribution of history. For a binary outcome, an outcome model might specify \(m(x,u)=P(Y=1\mid X=x,U=u)\). Standardized risk under exposure level \(x\) is then estimated by averaging \(m(x,U_i)\) over the trial participants, rather than comparing the participants who happened to receive each exposure. In the preceding example, the equal-stratum target distribution yields the two standardized rates \(0.375\) and \(0.625\).

For multiple decision times, the outcome model must account for the relevant history and the exposure sequence. A joint likelihood can represent the data-generation process through components for evolving histories, dose or exposure, and outcome, for example \(p(L_1)\prod_{t=1}^T \left\{p(A_t,X_t\mid \text{past},L_t)\,p(L_{t+1}\mid \text{past},L_t,A_t,X_t)\right\}\,p(Y\mid \text{history and exposure sequence})\), with the terminal history term adapted to the trial’s observation schedule. The outcome component estimates the outcome relation conditional on measured history; the exposure and history components describe how the observed treatment paths arose. A joint likelihood is not itself a guarantee of causal correction. The variables, conditional dependencies, and outcome model must be appropriate, and causal contrasts require the same substantive assumptions as other adjustment methods.

Sequential Inverse-Probability Weighting

An alternative is to reweight observed treatment histories so that, in the weighted analysis, measured history is less associated with the treatment decisions. Let \(H_t\) denote all measured information available immediately before decision \(t\), including relevant past doses, exposures, and covariates. For discrete dose decisions, the treatment probability at time \(t\) is \(P(A_t=a_t\mid H_t)\). A basic unstabilized weight for an observed sequence is the inverse of its probability under the observed decision process:

In practice, stabilized weights often replace the numerator 1 with probabilities conditional on a smaller set of variables or past treatment, chosen to improve stability while preserving the target contrast. For continuous exposure, probabilities are replaced by exposure densities; a weight uses the density under the target exposure assignment divided by the density under the observed assignment, conditional on history. The exposure model must match the variable being analyzed: a dose-assignment probability is not automatically a valid model for achieved concentration.

The rationale is that histories that make an observed treatment path unlikely receive greater weight, compensating for their underrepresentation in that treatment path. This constructs a pseudo-population for the measured history, not a literal randomized sample. For a causal interpretation, required conditions include consistency (the observed outcome under a received regimen equals the corresponding potential outcome), sequential exchangeability (given measured history at each decision, treatment is independent of relevant future potential outcomes), positivity (each regimen being compared has nonzero probability for histories in the target population), and adequate models for the treatment probabilities or densities. Interference and outcome measurement must also be addressed as appropriate to the trial.

Diagnostics, Alternatives, and Failure Modes

Weighting and outcome modeling target the same broad problem but rely on different statistical models. Outcome modeling predicts outcomes conditional on exposure and history and then standardizes to a target history distribution. Weighting models exposure or dose assignment and uses the resulting probabilities to construct a weighted comparison. A correctly specified outcome model can work well when exposure and history predict the outcome in a tractable way; weighting makes the treatment assignment process explicit, but can be unstable when some observed decisions are rare. Neither approach repairs unmeasured confounding merely by being called a joint model or a weighted analysis.

Diagnostics should examine whether the assumed adjustment is plausible and whether the data support the contrast. For weighting, inspect the distribution of weights, their range, and the effective sample size; compare measured-history distributions across exposure groups before and after weighting. Extreme weights often indicate near-violations of positivity or an assignment model that predicts probabilities close to zero or one. Truncating weights can reduce variance but changes the estimator and may introduce bias, so the truncation rule and its consequences should be reported. For outcome models, inspect fit, residual patterns where applicable, and whether predictions are extrapolating beyond observed exposure-history combinations. Compare adjusted estimates from reasonable specifications, without treating agreement as proof of no bias.

A common mistake is to say, “The dose algorithm was adaptive, so dose is randomized conditional on the rule; therefore the crude exposure-outcome comparison is unbiased.” Adaptation does not ensure this. If dose depended on a prognostic biomarker, exposure groups can differ on prognosis; moreover, achieved exposure can depend on patient pharmacokinetics even when assigned dose was randomized. The correction is to define the intended causal contrast, identify the information available before each decision, and use an adjustment method whose assumptions and diagnostics are stated. If important confounders were not measured, or the data contain no relevant exposure overlap, the causal effect may not be identified from these data alone.