Why Summarize the Association Parameter?
A joint model links a subject’s longitudinal biomarker trajectory to that subject’s event hazard. Once the model has been fitted, the association parameter is rarely the whole scientific answer. Readers also need to know how uncertain it is, how much posterior support there is for either direction of association, and what the estimate implies for a specified biomarker contrast on the hazard-ratio scale.
These are different questions. A posterior distribution may put nearly all its mass above zero, giving strong evidence about direction, while still being spread across values that imply substantially different hazard ratios. Conversely, a narrow posterior near zero may indicate that large associations are unlikely, even if its sign is not well determined. A single estimate cannot communicate both features.
Parameter and Posterior Summaries
Consider a shared-parameter joint model in which the event hazard includes the latent biomarker value \(m_i(t)\) through a term \(\alpha m_i(t)\) on the log-hazard scale. For example, the hazard may be written as \(h_i(t)=h_0(t)\exp\{\gamma^\mathsf{T}w_i+\alpha m_i(t)\}\), where \(h_0(t)\) is the baseline hazard and \(w_i\) denotes other covariates. In this parameterization, \(\alpha\) is the log-hazard association per one-unit increase in the latent biomarker, holding the other terms fixed.
The posterior distribution of \(\alpha\), written \(p(\alpha\mid D)\), describes uncertainty about this parameter conditional on the specified joint model, its priors, and the observed data \(D\). It is a marginal posterior: uncertainty in other model parameters is integrated out. The summaries below describe this distribution; they do not establish that changing the biomarker would cause a change in event risk.
An equal-tail \(100(1-\gamma)\%\) posterior interval uses the lower and upper posterior quantiles at probabilities \(\gamma/2\) and \(1-\gamma/2\). Define the quantile convention \(Q(u)=\inf\{a:F_\alpha(a)\geq u\}\), where \(F_\alpha\) is the posterior cumulative distribution function. When the posterior is continuous, the interval from \(Q(\gamma/2)\) to \(Q(1-\gamma/2)\) contains posterior probability \(1-\gamma\). If the posterior has atoms, the closed interval can contain more than \(1-\gamma\); the actual included posterior mass should then be reported or otherwise clarified.
A posterior interval is not a confidence interval interpreted through repeated-sampling coverage. Its probability statement is conditional on the model, priors, and observed data. A posterior probability such as \(P(\alpha>0\mid D)\) is also not the probability that the model is correct or that the null hypothesis is true.
Worked Example: Summarizing Direction and Magnitude
Suppose, for a hypothetical joint-model analysis, the marginal posterior is \(\alpha\mid D\sim N(0.42,0.18^2)\), with \(\alpha\) measured per one-unit increase in the latent biomarker. Find a 95% equal-tail posterior interval and the posterior probability of a positive association.
Interval. For a normal posterior, the 2.5th and 97.5th percentiles are approximately the mean plus or minus 1.96 posterior standard deviations:
Because this posterior is continuous, the interval contains 95% posterior probability. Conditional on the model, priors, and data, the posterior probability that \(\alpha\) is between 0.0672 and 0.7728 is 0.95.
Direction. Standardizing the value zero gives \(z=(0-0.42)/0.18=-2.333\). Therefore,
The posterior assigns about 99.02% probability to a positive association. The interval and sign probability tell related but distinct things: the interval gives a range of plausible magnitudes, whereas the sign probability quantifies posterior support for the direction.
Intervals When the Posterior Has Atoms
Quantile summaries require care when the posterior is not continuous. Atoms can arise, for example, from a prior that assigns positive probability to particular parameter values. With the quantile convention above, an endpoint can itself carry posterior mass. Consequently, a closed quantile interval can include more probability than its nominal level.
Worked Example: An Equal-Tail Interval with Point Masses
Suppose a hypothetical posterior for \(\alpha\) assigns probability 0.10 to \(-0.2\), 0.20 to \(0\), 0.40 to \(0.3\), and 0.30 to \(0.8\). Find the posterior mean, the probability of a positive association, and the 95% equal-tail quantile interval. Then calculate the probability actually contained in that closed interval.
Posterior mean. Weight each possible value by its posterior probability:
Direction. Positive values are \(0.3\) and \(0.8\), so \(P(\alpha>0\mid D)=0.40+0.30=0.70\). The probability of a nonnegative association is different: \(P(\alpha\geq 0\mid D)=0.20+0.40+0.30=0.90\), because the atom at zero is included.
Quantile interval and its mass. The cumulative probabilities at \(-0.2,0,0.3,0.8\) are 0.10, 0.30, 0.70, and 1.00. Thus \(Q(0.025)=-0.2\) and \(Q(0.975)=0.8\). The closed interval \([-0.2,0.8]\) contains all four atoms, so its posterior probability is 1.00, not 0.95. Reporting “95% equal-tail quantile interval” describes how the endpoints were selected; when atoms occur, report the actual included probability as well.
An ordinary nonrandomized set cannot always have exactly the nominal posterior probability when mass comes in indivisible atoms. Do not silently describe a quantile interval as having exactly its nominal probability in that situation. State the endpoint convention and the probability the reported interval actually contains.
From Association Parameters to Hazard Ratios
Because \(\alpha\) multiplies the latent biomarker on the log-hazard scale, comparing two biomarker values separated by \(\Delta\) units gives the conditional hazard ratio
This comparison holds the other terms in the hazard model fixed. For \(\Delta=1\), a positive \(\alpha\) corresponds to a hazard ratio above 1; a negative \(\alpha\) corresponds to a hazard ratio below 1. The units matter: if the biomarker is rescaled, the numerical value of \(\alpha\) changes, although a correctly specified contrast describes the same comparison.
For a positive \(\Delta\), exponentiation preserves ordering. If a continuous posterior interval for \(\alpha\) is \([L,U]\), the corresponding interval for \(HR(\Delta)\) is \([\exp(\Delta L),\exp(\Delta U)]\). Its posterior probability is unchanged by this one-to-one transformation. If \(\Delta<0\), the ordering reverses and the transformed interval is \([\exp(\Delta U),\exp(\Delta L)]\). For posterior probabilities, \(P\{HR(\Delta)>1\mid D\}=P(\alpha>0\mid D)\) when \(\Delta>0\); the inequality reverses when \(\Delta<0\). At \(\Delta=0\), the hazard ratio is always 1.
Worked Example: Transforming a Posterior Interval for a Two-Unit Contrast
Suppose a hypothetical posterior is \(\alpha\mid D\sim N(0.30,0.25^2)\), where \(\alpha\) is per one biomarker unit. Calculate the 95% posterior interval for the hazard ratio comparing latent biomarker values two units apart, and find the probability that this hazard ratio exceeds 1.
First obtain the interval for \(\alpha\):
For a two-unit contrast, \(\Delta=2\), so \(HR(2)=\exp(2\alpha)\). Since \(\Delta\) is positive, exponentiate twice each endpoint without reversing their order:
This interval is wide and includes 1, so the posterior allows both lower and higher hazards for the two-unit contrast. The posterior probability the hazard ratio exceeds 1 is
Thus the posterior assigns about 88.49% probability to a hazard ratio above 1, despite substantial uncertainty in its magnitude. The posterior median hazard ratio is \(\exp\{2(0.30)\}\approx1.822\). This is not the posterior mean hazard ratio: for a normal \(\alpha\), the latter is \(\exp\{2(0.30)+2^2(0.25^2)/2\}=\exp(0.725)\approx2.064\). Nonlinear transformations do not generally commute with averaging.
How to Report and Check the Summaries
A useful report makes clear which feature of the posterior each number summarizes. The posterior mean or median describes a central value; a credible interval describes uncertainty in magnitude; and a sign probability describes uncertainty in direction. For a hazard ratio, state the biomarker contrast and the covariate-conditional interpretation. For example: “For a two-unit increase in the latent biomarker, the posterior median conditional hazard ratio was 1.82, with a 95% equal-tail posterior interval from 0.684 to 4.854; the posterior probability that the association was positive was 0.885.” This wording does not imply a causal effect.
When summaries are calculated from posterior simulation draws, inspect whether the draws adequately represent the posterior before relying on fine distinctions. Poor mixing, strong autocorrelation, or too few effective draws can make quantiles and tail probabilities unstable. Monte Carlo error is especially relevant for probabilities close to zero or one and for extreme quantiles. Compare summaries across adequately explored chains or longer runs, and report enough precision to reflect the simulation’s reliability rather than printing many unsupported digits.
A useful alternative to the equal-tail interval is a highest-posterior-density region, which seeks a region containing the requested posterior mass while favoring higher-density values. The two summaries answer somewhat different questions and can differ for skewed or multimodal posteriors. An equal-tail interval has a direct quantile definition; a highest-density region may be disconnected and its shape can depend on the density representation. Neither removes the need to examine posterior shape or, when relevant, point masses. Frequentist standard errors and likelihood-based intervals from maximum-likelihood analysis answer questions under a different inferential framework; they should not be relabeled as posterior intervals.
Common reporting failures include giving only the sign probability and omitting the plausible magnitudes, reporting a hazard ratio without specifying the biomarker contrast, and exponentiating a posterior mean while calling it the posterior mean hazard ratio. Another error is to interpret \(P(\alpha>0\mid D)\) as the probability that a null hypothesis is true or as evidence of causation. The correction is to state the model-conditional posterior probability precisely and report the interval or transformed summary needed to describe magnitude.
Check Your Understanding
Answer each question, then compare with the solutions.
- A continuous posterior for \(\alpha\) has \(P(\alpha>0\mid D)=0.96\), but its 95% interval is \([-0.03,0.70]\). Is this inconsistent? Explain.
- If the posterior median of \(\alpha\) is 0.20 per biomarker unit, what is the corresponding median hazard ratio for a 3-unit increase?
- Why can a closed equal-tail quantile interval from a posterior with atoms contain more than its nominal probability?
- For a positive biomarker contrast, what does \(P\{HR(\Delta)>1\mid D\}\) equal in the log-linear association model?
- A report gives \(\exp\{E(\alpha\mid D)\}\) as the posterior mean hazard ratio. What is the potential error?
Solutions. (1) It is possible. The interval and sign probability summarize different posterior features; a small amount of posterior mass below zero can put the lower 2.5% quantile below zero while most mass remains positive. (2) For \(\Delta=3\), the transformed median is \(\exp(3(0.20))=\exp(0.60)\approx1.822\). (3) A quantile endpoint may itself carry positive posterior mass, and a closed interval includes that endpoint mass; calculate and report the actual included probability. (4) For \(\Delta>0\), \(HR(\Delta)>1\) exactly when \(\alpha>0\), so the probability is \(P(\alpha>0\mid D)\). (5) Exponentiation is nonlinear, so in general \(\exp\{E(\alpha\mid D)\}\neq E\{\exp(\alpha)\mid D\}\). Compute the transformed posterior mean from transformed draws or use an appropriate distributional calculation.