Averaging Parameters Across Models with Different Interpretations

Why the Quantity Must Come Before the Average

Candidate PK/PD models can describe the same exposure-response question with different mathematical structures. One model might use an Emax curve, another a linear relationship, and another a nonlinear link for a binary response. Their coefficients need not represent the same scientific quantity. Averaging coefficients merely because they occupy similar positions in model output can therefore produce a number with no coherent interpretation.

The solution is to define the scientific target first. Each candidate model can then be used to estimate that same target, even if the model-specific calculations differ. Examples include a response contrast between two specified exposures, a predicted response at a specified time and dose, or a population-average difference under a defined covariate distribution. The target must specify the outcome scale, exposure or dosing conditions, time point, population, and contrast as needed for the question.

The functions \(g_k\) can differ. What must be shared is the meaning of \(\psi_k\), not the formula used to calculate it. For example, an Emax model and a linear model have different response equations, but both can estimate the change in mean response from exposure \(C=0\) to \(C=2\) mg/L at the same time point and in the same population.

In Bayesian model averaging, let \(w_k=P(M_k\mid y)\) be the posterior probability of model \(M_k\) given data \(y\). The weights are nonnegative and sum to one. For a common estimand, the posterior distribution is the weighted mixture of its model-conditional posterior distributions. This tutorial focuses on estimating and quantifying uncertainty in that shared quantity, rather than on averaging raw model parameters.

When Parameters Do—and Do Not—Match

A parameter can be averaged across models only when it has the same definition, scale, units, and scientific interpretation in each model. Even then, an average may answer a different question from the one of interest. A shared name in software output is not evidence of a shared estimand.

These coefficients are not interchangeable. For instance, \(E_{\max}\) has response units, an Emax \(EC_{50}\) has exposure units, and a linear slope has response units per exposure unit. A logistic coefficient is measured on a log-odds scale, whereas a probit coefficient is measured on a latent normal scale. Their numerical values cannot be averaged as if they estimated the same parameter.

Uncertainty Includes Variation Within and Between Models

A model-averaged point estimate alone does not describe uncertainty. Each model has uncertainty about the target conditional on that model, and the target may also differ across models. Bayesian model averaging includes both sources through the law of total variance.

Let \(\mu_k=E(\psi\mid y,M_k)\) be the posterior mean of the common estimand under model \(M_k\), and let \(V_k=\operatorname{Var}(\psi\mid y,M_k)\) be its conditional posterior variance. The overall posterior mean is the weighted mean of the conditional means. The overall variance is the weighted average of the conditional variances plus the weighted variance of the conditional means:

The first term is within-model uncertainty. The second is between-model uncertainty: it increases when models with substantial posterior weight imply different values for the shared estimand. The identity follows by writing the squared deviation from the overall mean as a deviation from the model-specific mean plus the model-specific mean’s deviation from the overall mean. The cross-product has conditional expectation zero because \(E(\psi-\mu_k\mid y,M_k)=0\).

Compute Nonlinear Targets on Their Shared Scale

A common estimand may require transforming model-specific coefficients before averaging. This matters particularly when candidate models use different link functions. For a binary outcome, a probability difference is on the response scale; a logistic coefficient and a probit coefficient are not. Compute the probability under each model at each specified exposure, form the probability difference within that model, and then average those differences.

Practical Checks, Alternatives, and Limitations

Before averaging, verify that the target is operationally identical across models. Confirm the outcome scale, exposure contrast, time point, population, covariate distribution, and handling of baseline response. If a model cannot estimate the target without an unsupported extrapolation, a numerical value from that model does not become comparable merely because its units match. Check that derived posterior quantities are finite and that numerical integration or simulation is sufficiently accurate.

For posterior computation, the target should be calculated from joint parameter draws, not by separately combining parameter summaries. In general, \(E\{g(\theta)\mid y,M_k\}\) is not equal to \(g\{E(\theta\mid y,M_k)\}\) when \(g\) is nonlinear. For each draw, compute the model-specific target; then combine draws according to the posterior model probabilities. This preserves parameter dependence and propagates uncertainty through the transformation.

A useful diagnostic is to report the model-conditional target estimates alongside the averaged estimate and weights. Large differences between conditional estimates indicate that structural uncertainty matters for the target. Also check Monte Carlo error, sensitivity to the candidate set and prior model probabilities, and whether the target is stable over scientifically relevant exposure or time values. Posterior model weights sum to one for the specified candidate set; they do not account for a scientifically important model that was omitted.

The main alternatives answer different questions. Selecting a single model reports inference conditional on that selection and does not include between-model uncertainty. Model averaging is appropriate when the candidate models represent plausible uncertainty and the target can be defined across them. Averaging predictions at a particular condition is one form of averaging a shared quantity; averaging a parameter is justified only when that parameter itself has a common interpretation. No averaging method makes incompatible quantities commensurate, repairs poor diagnostics, or guarantees reliable extrapolation.

A common mistake is to say, “Each model has an exposure coefficient, so I will average those coefficients.” That reasoning is wrong when the coefficients use different scales or encode different response shapes. The correction is to define a shared scientific target, calculate it within each model on the same scale, and average those model-specific target distributions using the stated model weights. If no scientifically defensible common target exists, do not force a parameter average.