Model Averaging Across Association Links

Why Average Across Association Links?

In a joint model, the association link specifies how a subject’s longitudinal biomarker history enters the event hazard. Plausible choices can include a current-value link, a threshold link, or a link involving a slope. These choices need not produce identical estimates or predictions, even when they are fitted to the same data. Selecting one link gives a result conditional on that choice; reporting several separate results shows sensitivity but does not provide one combined estimate.

Model averaging provides a principled summary over a prespecified set of plausible links. It assigns each fitted model a nonnegative weight summing to one, then averages a target that has the same interpretation under every model. The weights do not establish that the candidate links are all scientifically correct. Rather, they describe how the chosen averaging procedure distributes support across the candidates.

Definition: Let $M_1,\ldots,M_K$ denote candidate joint models that differ in their association link, and let $w_k\geq 0$ with $\sum_{k=1}^{K}w_k=1$ be their weights. For a common scalar target $\theta$, let $\hat{\theta}_k$ be its estimate under $M_k$. The model-averaged estimate is $$ \hat{\theta}_{\mathrm{MA}}=\sum_{k=1}^{K}w_k\hat{\theta}_k. $$ For a prediction, such as an event probability by a specified time for a specified covariate and biomarker history, the same formula applies to the model-specific predictions.

The requirement of a common target is essential. A coefficient multiplying current biomarker value in one model and a coefficient multiplying biomarker slope in another are not estimates of the same quantity. They should not be averaged merely because both are called association parameters. A more comparable target might be the predicted event probability by a fixed horizon for the same specified patient profile, or the log hazard contrast at a fixed time between two specified biomarker histories.

Choosing Weights and Defining the Target

Weights must be chosen and interpreted according to a stated procedure. One familiar information-criterion choice is the Akaike-weight construction. If $AIC_k$ is the Akaike information criterion for model $M_k$, define $\Delta_k=AIC_k-\min_j AIC_j$, then set

w_k=\frac{\exp(-\Delta_k/2)} {\sum_{j=1}^{K}\exp(-\Delta_j/2)}.

These weights are normalized so they add to one. They are a relative weighting device based on the candidate set and the criterion; they are not posterior probabilities that the models are true. Bayesian model averaging instead uses posterior model probabilities, which require specified prior model probabilities and marginal likelihoods. The two weighting schemes can lead to different results and have different interpretations.

Before calculating any average, state the target precisely: its scale, time point or horizon, covariate and biomarker profile, and any conditioning information. For example, “event probability by 24 months for a profile with a specified longitudinal history” is a target that each candidate model can predict. A model’s association coefficient alone may not be comparable if its link function or units differ.

Worked Example: Information-Criterion Weights and a Common Prediction

Suppose three candidate association links have AIC differences $\Delta_1=0$, $\Delta_2=2$, and $\Delta_3=4$. For the same hypothetical patient profile, all three models estimate the probability of an event by 24 months. Their predictions are 0.12, 0.16, and 0.20. Calculate the AIC weights and the model-averaged prediction.

The unnormalized weights are $1$, $\exp(-1)=0.367879$, and $\exp(-2)=0.135335$. Their sum is $1.503215$, so the normalized weights are approximately $w_1=0.6652$, $w_2=0.2447$, and $w_3=0.0900$. They sum to one, up to rounding.

The weighted prediction is

\hat{q}_{\mathrm{MA}} =(0.66524096)(0.12)+(0.24472847)(0.16) +(0.09003057)(0.20) =0.1369916\approx 0.1370.

Thus the model-averaged predicted event probability by 24 months is about 13.7% for this hypothetical profile. This is a weighted prediction over the three specified links, not a claim that the event probability is known exactly or that the weights are probabilities that each link is true.

Uncertainty Within and Between Links

A weighted point estimate alone hides an important distinction. Each model has uncertainty in its own estimate or prediction, and the models may disagree with one another. For a model-averaged distribution, let $\mu_k$ be the model-specific mean of the target and $V_k$ its variance under model $M_k$. The total variance separates into the average of the within-model variances and the variance among the model-specific means:

\operatorname{Var}(\theta) =\sum_{k=1}^{K}w_k V_k +\sum_{k=1}^{K}w_k(\mu_k-\mu_{\mathrm{MA}})^2, \qquad \mu_{\mathrm{MA}}=\sum_{k=1}^{K}w_k\mu_k.

The first term is within-model uncertainty; the second is between-model uncertainty, reflecting differences among candidate links. The identity follows by adding and subtracting $\mu_k$ inside the squared deviation:

\begin{aligned} \operatorname{Var}(\theta) &=\sum_k w_k E\{(\theta-\mu_{\mathrm{MA}})^2\mid M_k\}\\ &=\sum_k w_k\left[ E\{(\theta-\mu_k)^2\mid M_k\} +(\mu_k-\mu_{\mathrm{MA}})^2 \right]. \end{aligned}

The cross-product vanishes because $E(\theta-\mu_k\mid M_k)=0$. This is the law of total variance. It applies directly when the weights describe a mixture distribution, as in Bayesian model averaging. With information-criterion weights, the same decomposition is a useful model-averaging uncertainty summary, but it does not automatically guarantee exact frequentist coverage. In a frequentist analysis, the sampling variance of a weighted sum of estimates from the same data also depends on their covariances; model averaging does not make the fitted estimates independent.

Worked Example: Within-Link and Between-Link Uncertainty

For a hypothetical 12-month event-probability prediction, suppose the three model weights are 0.6897, 0.2537, and 0.0566. The model-specific predictions are 0.15, 0.21, and 0.32. Suppose their within-model standard errors for the predicted probabilities are 0.04, 0.05, and 0.07. Calculate the averaged prediction and the mixture variance decomposition.

Keeping full precision through the weighted sum,

\hat{q}_{\mathrm{MA}} =(0.6897)(0.15)+(0.2537)(0.21)+(0.0566)(0.32) =0.174844\approx 0.1748.

The average within-model variance is

$$ (0.6897)(0.04^2)+(0.2537)(0.05^2)+(0.0566)(0.07^2) =0.00201511. $$

The between-model component uses deviations from 0.174844:

\begin{aligned} &(0.6897)(0.15-0.174844)^2 +(0.2537)(0.21-0.174844)^2\\ &\qquad +(0.0566)(0.32-0.174844)^2 =0.00193184. \end{aligned}

Therefore the total mixture variance is $0.00201511+0.00193184=0.00394695$, with standard deviation $\sqrt{0.00394695}=0.06282$, or about 6.28 percentage points. The between-model component is substantial: disagreement among the links contributes nearly as much variance as the within-model uncertainty. This standard deviation summarizes the specified mixture uncertainty; it should not be mistaken for a universal frequentist standard error or for an interval with guaranteed coverage.

Scale Matters When Averaging

Averages depend on the scale. Suppose each model gives a log hazard ratio for the same fixed-time contrast between two biomarker histories. Averaging those log hazard ratios and exponentiating gives a weighted geometric average of the hazard ratios. Averaging the hazard ratios directly gives a weighted arithmetic average. These are generally different quantities.

Worked Example: Averaging a Log Hazard Contrast

Two hypothetical association-link models estimate log hazard contrasts of 0.20 and 0.80 for the same fixed-time comparison, with weights 0.5 and 0.5. Compare the exponentiated average log contrast with the average of the two hazard ratios.

The weighted log contrast is $0.5(0.20)+0.5(0.80)=0.50$. Its exponent is $\exp(0.50)=1.6487$. The model-specific hazard ratios are $\exp(0.20)=1.2214$ and $\exp(0.80)=2.2255$; their arithmetic average is $(1.2214+2.2255)/2=1.7235$.

Thus 1.6487 and 1.7235 answer different averaging questions. The analysis should specify whether its target is the average log contrast, interpreted after exponentiation, or the average hazard ratio. Neither operation is automatically preferable; consistency with the scientific estimand is the criterion.

Assumptions, Diagnostics, and Alternatives

Model averaging does not repair a poorly specified candidate set. Its interpretation depends on several conditions:

Candidate links are scientifically plausible. The set should represent reasonable alternatives, not every conceivable link or only the link favored by one initial fit.
The target is comparable. Every model must estimate or predict the same defined quantity on the same scale and for the same profile, time, and conditioning information.
Weights are justified and documented. State whether they come from an information criterion, posterior model probabilities, or another specified rule. Results can be sensitive to the candidate set and weighting procedure.
Each model is adequately fitted. A weighted combination of unstable or misspecified models can still be misleading. Convergence, implausible parameter estimates, and poor fit require attention before averaging.

Useful diagnostics include a table of model-specific estimates or predictions, their weights, and their within-model standard errors or variances. Check whether one model receives nearly all the weight, whether small changes in the candidate set alter the result, and whether a low-weight model has an extreme prediction that materially increases between-model variance. Large disagreement is itself informative: a narrow within-model standard error does not remove uncertainty about which association link is appropriate.

The main alternative is to select a single link, for example by an information criterion, and report inference conditional on that selection. This is simpler but can understate uncertainty from link choice. Another alternative is to report separate sensitivity results for each link without combining them. That preserves the differences transparently but does not yield one averaged target. Model averaging lies between these approaches: it summarizes results using explicit weights, while its validity remains dependent on the candidate set, target comparability, and uncertainty procedure.

A common mistake is to say, “The model with the smallest AIC is the best, so use its estimate and ignore the rest,” or to average association coefficients that have different meanings. The correction is to distinguish model selection from model averaging, define a common target first, and then report both the weighted result and the contribution of between-link disagreement. If a common target cannot be stated, averaging is not meaningful.

Key takeaway: Model averaging summarizes uncertainty across a prespecified set of association links only when the target is common across models. Report the weighting rule, the averaged result, and how much uncertainty arises within models versus from differences among links.

Check Your Understanding

Answer the questions, then compare your reasoning with the solutions.

Why is it generally invalid to average a current-value association coefficient with a slope association coefficient?
Two models have equal weights and predict a 6-month event probability of 0.10 and 0.18 for the same profile. What is the model-averaged prediction?
What does the between-model variance component represent?
Explain why AIC weights should not be described as probabilities that the candidate models are true.

Solutions. (1) The coefficients multiply different quantities and therefore have different interpretations and units; they are not estimates of a common target. A shared prediction or a carefully defined common contrast could instead be averaged. (2) The prediction is $0.5(0.10)+0.5(0.18)=0.14$, or 14% by 6 months for that profile. (3) It measures the weighted dispersion of the model-specific target means around their weighted mean, so it reflects disagreement attributable to link choice. (4) AIC weights are normalized relative support values derived from an information criterion within the candidate set; unlike posterior model probabilities, they are not calculated from specified prior model probabilities and marginal likelihoods.