Model Averaging for Competing Exposure-Response Curves

Why Average Competing Exposure-Response Curves?

A dose-ranging dataset can support more than one plausible exposure-response shape. Over a limited exposure range, a straight line may describe the observed trend well even when the underlying response eventually plateaus. An Emax curve may represent saturation, while a sigmoid Emax curve allows a more gradual or more abrupt transition. If the data do not clearly distinguish these shapes, choosing one curve and treating it as certain can make predictions—and decisions based on them—overconfident.

Model averaging addresses this uncertainty by assigning relative weights to a prespecified set of candidate models and combining their predictions. It does not make a poor candidate set adequate, nor does it identify a uniquely true curve. Its value is that uncertainty about which candidate best represents the data can be carried into the prediction.

Candidate Curves and Assumptions

Let \(x\) denote exposure in mg/L, and let \(R\) denote the mean response in response points at a prespecified assessment time. The three candidate mean curves are:

Here \(E_0\) is the response at zero exposure; \(\beta\) is the linear slope in response points per mg/L; \(E_{\max}\) is the maximum increase above \(E_0\); \(EC_{50}\) is the exposure producing half of that increase; and \(h\) is the sigmoid shape parameter. The Emax and sigmoid Emax forms assume nonnegative exposure and positive \(EC_{50}\); the sigmoid form also requires positive \(h\). For the increasing-response case considered here, \(\beta\) and \(E_{\max}\) are positive. A decreasing effect can instead be represented with a signed effect parameter and interpreted consistently with the endpoint.

For the worked comparison, assume the same independent response observations are fitted under all three models, with normally distributed residuals of common variance within each model. The candidate-specific residual variance is estimated by maximum likelihood. This setup makes the Gaussian likelihood and its AICc comparable across candidates. The usual AICc calculation also requires \(n>k+1\), where \(k\) counts all estimated parameters, including residual variance. The analysis additionally assumes that the candidates are scientifically plausible, use the same response data and likelihood basis, and have been fitted successfully. Model weights cannot repair dependence, an unsuitable residual distribution, or an omitted exposure-response feature.

AICc Weights: Relative Support, Not Certainty

For Gaussian models fitted to the same \(n\) responses, with residual sum of squares \(RSS\), a convenient AICc expression—up to a constant shared by the candidates—is:

The first term rewards fit, while the remaining terms penalize model complexity, with an additional small-sample correction. The correction matters particularly when \(n\) is not large relative to \(k\). Let \(\Delta_j=AICc_j-\min_{\ell}AICc_\ell\). The relative likelihood and normalized Akaike weight are:

The weight is a relative measure within the candidate set and under the stated likelihood and AICc approximation. It is not the probability that a model is true, nor a probability that its assumptions hold.

From Weights to Predictions and Uncertainty

The model-averaged prediction uses each candidate’s prediction at the same target exposure and outcome scale. For the three predictions in the first example, use the rounded weights from the AICc comparison:

This point prediction is a weighted compromise, not a claim that any candidate has that curve shape. Its interpretation is conditional on the chosen candidates, the data, and the weighting method. Model averaging is most useful when several curves remain plausible and their predictions differ meaningfully at exposures relevant to the application.

The variance decomposition follows the law of total variance. Imagine first selecting a candidate \(M_j\) with probability \(w_j\), then drawing a prediction around that model’s conditional mean. The overall mean is \(\sum_jw_j\hat{R}_j\). Expanding the squared deviation around that mean gives:

where \(v_j\) is the conditional prediction variance under candidate \(j\). To see why the cross-term vanishes, write a draw as \(\hat{R}_j+\epsilon\), where the conditional mean of \(\epsilon\) is zero. The cross-term is proportional to \(\sum_jw_j(\hat{R}_j-\bar{R})E(\epsilon\mid M_j,\text{data})=0\). This decomposition is useful only if the \(v_j\) values refer to compatible prediction targets and include the intended sources of within-model uncertainty.

Diagnostics, Alternatives, and Limitations

AICc weights should be calculated only after checking that the candidate fits are meaningful. Inspect residual patterns against exposure and fitted response, and examine whether any dose group or influential observation drives the ranking. The earlier material on diagnosing PD model misspecification from residual patterns is relevant here: structured residuals are evidence that a candidate may be inadequate, regardless of its relative AICc. Also check convergence, parameter plausibility, and whether estimates such as \(EC_{50}\) or \(h\) are weakly identified. A sharply curved model can fit nearly as well as a simpler curve while having unstable parameters and highly uncertain predictions outside the observed range.

A common mistake is to say, “Emax has the largest weight, so Emax is the true model; I will report only its prediction.” The correction is that the largest weight indicates the greatest relative support among these candidates, not certainty. Here Emax’s weight is only about 0.562, and the candidate predictions at 8.00 mg/L range from 11.60 to 13.12 points. The model-averaged prediction of about 12.32 points makes the remaining model-selection uncertainty visible rather than discarding it.

Model averaging differs from selecting the single minimum-AICc model: selection uses one curve, whereas averaging combines predictions. AICc is preferable to uncorrected AIC when the sample is modest relative to model complexity; neither criterion makes weights posterior probabilities. Bayesian model averaging is a distinct approach that uses prior model probabilities and marginal likelihoods. Predictive validation can also compare out-of-sample performance, but it answers a different question and can be unstable with limited data. These approaches are not interchangeable, and each depends on a defensible candidate set and suitable data.

Finally, averaging does not remove uncertainty shared by all candidates. If all three curves omit a delayed effect, a plateau beyond the studied range, or a relevant covariate, their average may still be biased. It is especially risky to average predictions far beyond observed exposures, where the linear curve can continue increasing while Emax curves approach a plateau. Report the target exposure, outcome units, candidate set, weighting criterion, and the extent of extrapolation. Treat an averaged result as one piece of evidence, not as an automatic dose recommendation.