Why Examine the Association Prior?
In a joint model, the association parameter connects a participant’s longitudinal trajectory to their event hazard. When event information is limited, the data may not pin down this parameter precisely, so its posterior can depend appreciably on the prior. Comparing fits under defensible alternative priors shows whether an apparent association is supported robustly by the data or relies on a particular prior choice.
Prior sensitivity is a controlled comparison, not a search for the prior that produces a preferred conclusion. The data, likelihood, parameterization, and all other priors should remain fixed while the association-parameter prior changes. Before interpreting any difference, the fits must also be computationally reliable: poor mixing or substantial Monte Carlo error can make posterior summaries look different when the underlying posteriors are not.
Specify the Association and Its Scale
For illustration, consider a shared-current-value association model with hazard
Here \(h_i(t\mid b_i)\) is participant \(i\)’s event hazard at time \(t\), conditional on their random effects \(b_i\); \(h_0(t)\) is the baseline hazard; \(\boldsymbol{x}_i\) denotes survival-model covariates; and \(m_i(t)\) is the longitudinal model’s fitted marker value at time \(t\). The association parameter \(\alpha\) is a log-hazard ratio per one-unit increase in \(m_i(t)\), holding the other terms fixed. Thus \(\exp(\alpha)\) is the conditional hazard ratio for that increase. This is an association, not a causal effect.
A prior scale is interpretable only in relation to the marker’s units. If the marker is measured in milligrams per liter, a prior on \(\alpha\) describes log-hazard change per milligram per liter. Rescaling the marker changes the numerical value of \(\alpha\) and therefore requires a corresponding change to its prior. If the marker is standardized, the coefficient instead describes the change per one standard deviation. State the scale explicitly and use the same scale in every refit.
For a concrete prior-calibration check, suppose \(m_i(t)\) is measured in units for which a one-unit change is clinically interpretable. A prior \(\alpha\sim N(0,0.25^2)\) places approximately 95% of prior probability between \(-0.49\) and \(0.49\), since \(1.96(0.25)=0.49\). On the hazard-ratio scale this is approximately \([\exp(-0.49),\exp(0.49)]=[0.613,1.632]\). By contrast, \(\alpha\sim N(0,0.75^2)\) gives an approximate 95% hazard-ratio range of \([\exp(-1.47),\exp(1.47)]=[0.230,4.350]\). Whether either range is plausible depends on the marker and scientific setting; the wider prior is not automatically a better sensitivity choice merely because it is less concentrated.
Assumptions for a Fair Comparison
The interpretation of a sensitivity comparison depends on several conditions. First, every fit must use the same observed longitudinal and event data, the same joint likelihood, and the same association definition. The joint model itself must be appropriate for the data structure; for example, in a shared-random-effects formulation, the longitudinal and event processes are assumed conditionally independent given the shared random effects and modeled covariates. Second, the alternative priors must be proper, expressed on the same parameter scale, and defensible before looking at which one yields the most attractive result.
Third, the fits must be computed well enough to compare posterior summaries. Use the chain diagnostics developed in the earlier material on MCMC convergence, including trace plots, between-chain agreement, effective sample size, and Monte Carlo error. If a posterior interval changes by less than the Monte Carlo uncertainty, the apparent difference is not persuasive evidence of prior sensitivity. Conversely, convergence does not establish that a prior is scientifically reasonable; that requires calibration on the parameter and hazard-ratio scales.
Hold the baseline-hazard specification and its prior fixed during this exercise. Changing them at the same time would make it difficult to attribute differences in \(\alpha\) to its prior. Posterior predictive checks, from the earlier material on posterior predictive checks for joint models, can help assess model-data discrepancies, but they do not replace a controlled prior comparison.
A Normal Approximation That Shows the Mechanism
A useful calculation isolates how a prior can affect an uncertain association estimate. Suppose the likelihood for \(\alpha\) is adequately approximated by a normal summary likelihood with estimate \(\hat{\alpha}\) and standard error \(s_\alpha\):
If the prior is \(\alpha\sim N(0,\tau^2)\), multiplying likelihood and prior gives a normal posterior. Collecting the coefficients of \(\alpha^2\) and \(\alpha\) in the exponent yields
The posterior mean is a precision-weighted average of the likelihood estimate and prior mean zero. When the likelihood is imprecise relative to the prior, the posterior mean is pulled more toward zero. This calculation is a normal approximation for explanation, not a substitute for refitting the full joint model: in the actual model, the marginal likelihood for \(\alpha\) need not be normal.
Worked Example: A Concentrated, Zero-Centered Prior
Suppose a hypothetical joint-model fit gives an approximate association estimate \(\hat{\alpha}=0.12\) per marker unit with standard error \(s_\alpha=0.30\). Use the prior \(\alpha\sim N(0,0.25^2)\). This summary calculation assumes the stated normal likelihood approximation is reasonable; it does not claim that an actual joint-model posterior must be exactly normal.
The likelihood precision is \(1/0.30^2=11.1111\), and the prior precision is \(1/0.25^2=16\). Therefore,
The posterior standard deviation is \(\sqrt{0.0368852}=0.1921\). Under this normal posterior approximation, a 95% credible interval is
Exponentiating gives a posterior median hazard ratio of \(\exp(0.04918)=1.050\), with a 95% credible interval approximately \([\exp(-0.327),\exp(0.426)]=[0.721,1.531]\) per one-unit increase in the marker. The interval includes 1, so these results do not establish a direction of association at the 95% interval level. The interval is a posterior probability statement conditional on this model and prior, not a frequentist confidence interval.
Worked Example: A Wider Prior and a Changed Interval
Keep the same hypothetical likelihood summary, \(\hat{\alpha}=0.12\) and \(s_\alpha=0.30\), but use the wider prior \(\alpha\sim N(0,0.75^2)\). The likelihood and data summary have not changed; only the association prior has changed.
The prior precision is \(1/0.75^2=1.7778\). Thus the posterior variance and mean are
The posterior standard deviation is \(\sqrt{0.0775862}=0.2785\), and the approximate 95% credible interval is
On the hazard-ratio scale, the posterior median is \(\exp(0.10345)=1.109\), and the interval is approximately \([\exp(-0.442),\exp(0.649)]=[0.643,1.914]\). Both priors produce intervals that include a hazard ratio of 1, but the wider prior yields a wider posterior interval and a median farther from 1. Neither result justifies claiming a clear positive association. The posterior values should be calculated from unrounded quantities before reporting rounded endpoints.
Compare Full-Model Fits, Not Just One Summary
In an applied analysis, refit the full joint model under each prespecified association prior. A useful comparison includes the posterior median or mean of \(\alpha\), a credible interval, the corresponding hazard-ratio interval, and a probability tied to a stated scientific threshold. Posterior draws can be transformed individually: if \(\alpha^{(1)},\ldots,\alpha^{(M)}\) are draws, then \(\exp(\alpha^{(1)}),\ldots,\exp(\alpha^{(M)})\) are draws from the posterior hazard-ratio distribution. Quantiles of those transformed draws give its credible interval. For a positive association, \(P(\alpha>0\mid\text{data})\) is also \(P(\exp(\alpha)>1\mid\text{data})\).
A scientifically meaningful threshold can be more informative than simply asking whether an interval includes zero. For instance, a hazard ratio above 1.2 per marker unit corresponds to \(\alpha>\log(1.2)=0.1823\). Compare the posterior probability of that event across priors if 1.2 is a defensible threshold in context. Do not select a threshold after seeing which one makes the results look decisive.
Worked Example: Comparing Two Hypothetical Full-Model Fits
Suppose two converged full-model fits use the same hypothetical data and all the same model components, but different defensible priors for \(\alpha\). Posterior summaries from the first fit are median \(0.18\), 95% credible interval \([-0.05,0.42]\), and \(P(\alpha>0\mid\text{data})=0.93\). The second fit gives median \(0.10\), interval \([-0.12,0.34]\), and probability 0.80. These are illustrative MCMC summaries, not data from a real study.
For the first fit, exponentiating the median and interval gives a median hazard ratio \(\exp(0.18)=1.197\) and interval \([\exp(-0.05),\exp(0.42)]=[0.951,1.522]\). For the second, the corresponding summaries are \(\exp(0.10)=1.105\) and \([\exp(-0.12),\exp(0.34)]=[0.887,1.405]\).
Both 95% intervals include a hazard ratio of 1, and both posterior probabilities for \(\alpha>0\) exceed 0.5. Yet the strength of evidence for a positive association is less under the second prior, and neither interval excludes modest negative associations. A balanced conclusion is that the direction is more often positive in the posterior than negative, but the evidence and estimated magnitude vary with the prior; these fits do not establish a precise positive association. The comparison is interpretable only after checking that Monte Carlo error is small relative to the differences shown.
Diagnostics, Alternatives, and Limitations
Look first for a fair computational comparison. Check that all chains mix adequately for \(\alpha\), that effective sample sizes support stable interval endpoints and posterior probabilities, and that Monte Carlo errors are small. Compare the fitted models’ data, likelihood, marker units, covariates, and non-association priors. A changed dataset, altered baseline hazard, or changed random-effects structure is not an association-prior sensitivity analysis. If an apparent change is small relative to Monte Carlo error, report the estimates as computationally indistinguishable rather than as evidence of sensitivity.
The main alternative is to report a single primary prior and its posterior, possibly alongside a prior predictive assessment. That may be adequate when prior information is strong and the choice is well justified, but it does not show how conclusions respond to plausible alternatives. A frequentist estimate can provide another point of comparison, but it does not answer the Bayesian question of how posterior inference changes across priors. It is not a replacement for holding the joint model fixed and varying only the association prior.
Sensitivity analysis has limits. If every candidate prior is implausible, the comparison does not become valid merely because it is systematic. If all candidate priors yield similar posteriors, that is reassuring about sensitivity to those choices but does not prove the model is correct or the parameter is well identified. If conclusions differ substantially, report that dependence clearly; do not select the most favorable fit. The next step is to justify the prior scale, inspect the information in the data, and describe the range of conclusions supported under reasonable assumptions.
Common Mistake
State what one unit of the longitudinal marker means and what \(\exp(\alpha)\) represents.
Translate candidate prior ranges to hazard ratios and justify their plausibility before comparing results.
Keep the data, likelihood, parameterization, and all other priors fixed; verify convergence and Monte Carlo precision.
Summarize association and hazard-ratio posteriors, relevant posterior probabilities, and whether the substantive conclusion changes.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions below.
- If the marker is rescaled from units to tenths of a unit, should the same numerical prior for \(\alpha\) be used?
- Under a normal likelihood summary, what happens to the posterior mean as a zero-centered prior becomes more concentrated?
- A 95% posterior interval for \(\alpha\) is \([-0.20,0.10]\). What is the corresponding hazard-ratio interval, approximately, and does it include 1?
- Two prior fits differ slightly in posterior median, but the difference is smaller than the Monte Carlo error. Is this persuasive evidence of prior sensitivity?
Solutions. (1) No. The coefficient is now per tenth of the original unit, so its numerical scale and prior must be transformed consistently; otherwise the prior no longer represents the same beliefs about hazard ratios. (2) It is pulled closer to zero, because the prior contributes more precision centered at zero. (3) Exponentiate the endpoints: \([\exp(-0.20),\exp(0.10)]=[0.819,1.105]\), which includes 1. (4) No. The apparent difference is not distinguishable from simulation noise; obtain more precise computation or report that the summaries are computationally indistinguishable.