Mapping Dose-Response Predictions to Decision Outcomes

From Model Output to Decision-Facing Outcomes

A dose-response model does not always report its predictions in the units needed to describe outcomes at a dose. A fitted model may produce a linear predictor, a log mean, or a hazard ratio, while a decision maker needs a probability of response, an average outcome in clinical units, or an absolute event risk over a specified period. Translating between these quantities is part of interpreting the model, not a cosmetic change in presentation.

The key distinction is between a model scale, on which a model is specified or estimated, and a decision-facing outcome scale, on which the predicted outcome is expressed. A link function connects the two. In addition, predictions may be conditional on particular covariates or summarized over a population. Those summaries answer different questions.

The estimand defines which outcome, population, and conditions the prediction is intended to describe. Before translating a model output, therefore, make sure that the prediction target matches the estimand: an individual’s conditional response, a population average, and a risk by a fixed time are not interchangeable. A transformation cannot repair a mismatch in the outcome or population being predicted.

Model Scales, Link Functions, and Prediction Targets

For a generalized linear model, a typical form is \(g\{E(Y\mid d,x)\}=\eta(d,x)\), where \(Y\) is the outcome, \(d\) is dose, \(x\) denotes covariates, \(g\) is a link function, and \(\eta\) is the linear predictor. The outcome-scale prediction is obtained by applying the inverse link \(g^{-1}\): \(E(Y\mid d,x)=g^{-1}\{\eta(d,x)\}\). The exact interpretation depends on the outcome and model. For example, in a binary-outcome logistic model the inverse link gives a probability; in a log-link model it gives a conditional mean.

A prediction should also name its time horizon when the outcome changes over time, and its covariate condition when it is conditional. If the objective is a population summary, the conditional predictions generally need to be averaged over the target population’s covariate distribution. Because the inverse link is generally nonlinear, evaluating the model at average covariates is not usually the same as averaging outcome-scale predictions.

For a binary outcome, probability and risk are often used for the predicted proportion experiencing the outcome. When risk is reported, specify the event and, for a time-dependent outcome, the time period. A probability of response by the end of treatment and a hazard ratio during treatment are different quantities even if they come from related models.

Transforming Log-Scale Predictions Carefully

A log-scale model can make positive, right-skewed outcomes easier to model, but exponentiating its predicted log mean does not generally produce the arithmetic mean on the original scale. If \(\ln(Y)\mid d,x\sim N\{\mu(d,x),\sigma^2\}\), then \(Y\) has a lognormal distribution. Its median is \(\exp\{\mu(d,x)\}\), while its arithmetic mean is \(\exp\{\mu(d,x)+\sigma^2/2\}\). Thus, the appropriate back-transformation depends on the quantity the decision requires.

The mean formula follows directly from the normal moment-generating result. If \(Z=\ln(Y)\sim N(\mu,\sigma^2)\), then \(Y=\exp(Z)\), and \(E(Y)=E\{\exp(Z)\}=\exp(\mu+\sigma^2/2)\). The variance term matters: since \(\sigma^2\) is nonnegative, the arithmetic mean is at least as large as the median, and is larger whenever \(\sigma^2>0\).

Absolute Risk from Time-to-Event Predictions

In a proportional-hazards model, a hazard ratio describes the relative instantaneous event rate among individuals still event-free. It is not an absolute probability. To report an absolute event risk by a fixed time, a baseline cumulative hazard and an appropriate survival-model prediction are needed. In the absence of competing risks, survival through time \(t\) is \(S(t\mid d,x)=\exp\{-H(t\mid d,x)\}\), so the probability of the event by \(t\) is \(1-S(t\mid d,x)\).

Conditional Predictions and Population Summaries

A model may give a prediction at a covariate profile \(x\), but a dose decision may concern a target population with varying covariates. A standardized population prediction averages outcome-scale predictions over that population. For strata \(x_1,\ldots,x_K\) with target-population weights \(w_1,\ldots,w_K\), where \(w_k\geq0\) and \(\sum_k w_k=1\), the standardized outcome is \(\sum_k w_k m(d,x_k)\). The weights should represent the intended population, not automatically the sample composition if that composition differs from the target.

Uncertainty, Diagnostics, and Common Pitfalls

A predicted value is not known without uncertainty. Parameter uncertainty, residual variability, and—when relevant—variation in covariates can affect a dose-specific quantity. Uncertainty intervals should be calculated for the quantity being reported, respecting the model and its prediction target. For a nonlinear transformation, simply transforming the endpoints of a symmetric interval on the model scale may not give an appropriate interval on the outcome scale. Simulation from the fitted parameter distribution, followed by transformation and summarization, is one general way to propagate parameter uncertainty. A predictive distribution for an individual outcome includes additional outcome variability and is not the same as an interval for a population mean or response probability.

Useful checks include confirming that probabilities lie between 0 and 1, risks refer to a stated horizon, predicted continuous outcomes have sensible units and ranges, and group weights sum to one. Examine whether the dose lies within the range supporting the data, whether covariate profiles are represented, and whether the model’s dose-response pattern is plausible. Calibration, residual, and model-fit diagnostics should be considered for the outcome and model being used. Implausible transformed predictions can signal extrapolation, a link-scale misunderstanding, or a model that does not describe the data adequately.

Common mistake: “The logistic model predicts 0.7, so the response probability is 70%,” when 0.7 is actually the linear predictor. The correction is to identify the model’s link and apply its inverse; for the logit link, the probability is \(\exp(0.7)/\{1+\exp(0.7)\}=0.6682\). Similarly, a hazard ratio is not an absolute risk, and exponentiating a log-scale mean may yield a median rather than an arithmetic mean. Always label the quantity, scale, units, population, and time horizon.

Alternatives depend on the intended summary. A conditional prediction describes a specified covariate profile; standardization summarizes predictions over a target covariate distribution. A mean on the original scale and a geometric mean answer different questions for skewed outcomes. A hazard ratio summarizes relative instantaneous rates, whereas an absolute risk over a horizon is often easier to interpret in event counts. None is universally preferable: select and report the quantity that corresponds to the estimand and decision context. These transformations and summaries describe outcomes; combining outcomes into an explicit preference or tradeoff requires an additional decision framework.