Why Inspect Endpoints Together?
A prior can imply plausible values for each clinical endpoint separately and still imply implausible combinations when the endpoints are considered together. For example, a prior might allow a moderate benefit on each of two outcomes, but assign too much probability to large benefits on both at once—or too little to a benefit on one outcome accompanied by harm on the other. Checking the marginal distributions alone will not reveal these joint implications.
A joint prior predictive check simulates values for related endpoints together, preserving the dependence specified in the model. The aim is to inspect patterns that matter clinically: whether endpoints tend to move together, whether discordant patterns are possible, and how often combinations judged extreme or incompatible with clinical knowledge appear. The check is about the consequences of the prior and outcome model before trial data are used.
For two endpoints, write their parameters as \(\theta_1\) and \(\theta_2\), and their simulated trial outcomes or summaries as \(Y_1\) and \(Y_2\). These symbols could represent, for example, treatment contrasts for two continuous endpoints, or endpoint-specific event probabilities. Their precise interpretation depends on the outcome model. In every case, it is important to distinguish dependence between \(\theta_1\) and \(\theta_2\) in the prior from dependence between \(Y_1\) and \(Y_2\) in the sampling model.
A Correlated Prior for Two Endpoints
A common starting point for continuous endpoint parameters is a bivariate normal prior. Let the prior means be \(\mu_1\) and \(\mu_2\), the prior standard deviations be \(\tau_1\) and \(\tau_2\), and the prior correlation be \(\rho\). Then the covariance between the parameters is \(\rho\tau_1\tau_2\).
The sign of \(\rho\) describes the direction of linear association under this prior: positive values make high values of one parameter tend to accompany high values of the other; negative values make high values of one tend to accompany low values of the other. A correlation of zero means the parameters are independent under this bivariate normal prior. Zero correlation does not generally imply independence for other joint distributions.
The prior needs to be clinically and statistically justified, not selected only because a correlation is convenient to specify. The scales \(\tau_1\) and \(\tau_2\) must use the units of their respective parameters; \(\rho\) is unitless. For more than two parameters, the full covariance matrix must be positive semidefinite, so its entries cannot be chosen independently of one another.
Worked Example: Interpreting a Correlated Prior
Suppose two continuous treatment effects, measured in their respective endpoint units, have prior means zero, prior standard deviations \(0.20\) and \(0.30\), and prior correlation \(0.60\). Find their prior covariance and describe the conditional distribution of the second effect when the first is \(0.20\).
The covariance is \(\rho\tau_1\tau_2=(0.60)(0.20)(0.30)=0.0360\). Thus the covariance matrix has diagonal entries \(0.20^2=0.0400\) and \(0.30^2=0.0900\), and off-diagonal entries \(0.0360\). The positive correlation encodes a tendency for treatment effects to be high or low together; it does not say that the effects are equal or that one causes the other.
For a bivariate normal distribution, the conditional distribution is \(\theta_2\mid\theta_1=x\sim N\big(\mu_2+\rho(\tau_2/\tau_1)(x-\mu_1),\ \tau_2^2(1-\rho^2)\big)\), where the second argument is the variance. At \(x=0.20\), the conditional mean is \(0+(0.60)(0.30/0.20)(0.20-0)=0.180\). The conditional variance is \(0.30^2(1-0.60^2)=0.0900(0.640)=0.0576\), giving a conditional standard deviation of \(\sqrt{0.0576}=0.240\). The positive prior association therefore shifts the conditional mean of endpoint 2 upward when endpoint 1 is above its prior mean, while leaving substantial uncertainty about endpoint 2.
From Parameter Dependence to Predictive Dependence
The joint prior on parameters is only one source of dependence in predicted outcomes. Outcomes can also be associated because they are measured on the same patients, share trial-level features, or are linked by the sampling model. A simulation must represent whichever sources of dependence the specified model includes; it must not add or remove dependence by accident.
One simple model illustrates the distinction. Suppose \(Y_j\) is a trial estimate for endpoint \(j\), with \(Y_j\mid\theta_j\sim N(\theta_j,s_j^2)\) for \(j=1,2\), and suppose the two estimates are conditionally independent given \(\theta_1,\theta_2\). Here \(s_j\) is the sampling standard error assumed known for this illustration. The marginal predictive variance combines prior uncertainty and sampling variance. The covariance between predictive estimates is the prior covariance because conditional sampling covariance is zero.
This is an application of the variance and covariance decomposition into variation between parameter values and variation conditional on those values. If the endpoints have conditional sampling covariance \(c\) instead, the predictive covariance becomes \(\rho\tau_1\tau_2+c\). Thus, a simulated correlation among outcomes need not equal the prior correlation among parameters.
Worked Example: Predictive Correlation After Sampling Variation
Use the prior in the previous example, and suppose the two endpoint estimates have sampling standard errors \(s_1=0.10\) and \(s_2=0.15\). Assume the estimates are conditionally independent given the endpoint parameters. Calculate their marginal predictive standard deviations and correlation.
The predictive variances are \(\operatorname{Var}(Y_1)=0.20^2+0.10^2=0.0400+0.0100=0.0500\) and \(\operatorname{Var}(Y_2)=0.30^2+0.15^2=0.0900+0.0225=0.1125\). The predictive standard deviations are therefore \(\sqrt{0.0500}=0.2236\) and \(\sqrt{0.1125}=0.3354\).
Conditional independence means the predictive covariance is the prior covariance, \(0.0360\). Hence the predictive correlation is \(0.0360/\{\sqrt{0.0500}\sqrt{0.1125}\}=0.480\), to three significant figures. Sampling variation has increased each marginal variance without adding covariance, so the predictive correlation is smaller than the prior parameter correlation of \(0.60\). This calculation depends on the stated conditional-independence assumption; it would change if the two endpoint estimates were conditionally associated.
Simulating and Inspecting Endpoint Pairs
For each simulation replicate, draw parameters jointly from their prior, then draw outcomes from the model conditional on those parameters. Retain each pair together. Independently shuffling the two endpoint columns after simulation would destroy their joint structure and answer a different question. Before interpreting the generated pairs, check that the assumed endpoint models, units, and any conditional dependence match the intended trial.
Generate \((\theta_1,\theta_2)\) from the specified joint prior, not from two separate marginal priors if dependence is intended.
Draw endpoint outcomes or summaries from the sampling model, including any within-patient or other conditional association the model specifies.
Use a scatterplot and clinically interpretable regions or thresholds to see whether the joint patterns are plausible.
Investigate surprising concordant or discordant patterns and check whether they arise from the prior, the sampling model, or both.
A scatterplot is useful because it displays the simulated pairs rather than only separate endpoint distributions. Add reference lines for clinically meaningful values when those values are defined. Look for an unexpected slope or shape, clusters in implausible regions, and combinations near the plot boundaries that the marginal summaries might conceal. For discrete or binary endpoints, a table of joint outcome categories can be more informative than a scatterplot.
Worked Example: Checking Jointly Favorable and Discordant Effects
Return to the prior with means zero, standard deviations \(0.20\) and \(0.30\), and correlation \(0.60\). Suppose clinicians define effects greater than \(0.20\) for endpoint 1 and \(0.30\) for endpoint 2 as favorable thresholds. These are hypothetical thresholds in the respective endpoint units. In an illustrative simulation of 100,000 joint parameter draws, 10,060 draws exceed both thresholds and 235 have endpoint 1 above its threshold while endpoint 2 is below the negative of its threshold. Calculate and interpret the two simulated proportions.
The proportion exceeding both favorable thresholds is \(10{,}060/100{,}000=0.1006\), or \(10.06\%\). The proportion with the specified discordant pattern is \(235/100{,}000=0.00235\), or \(0.235\%\). Each draw preserves the correlation between the effects. These are Monte Carlo estimates of prior probabilities under the specified model, not probabilities that the thresholds are true after observing trial data.
The small simulated proportion for this particular discordant pattern should not automatically be judged reassuring: whether it is too small depends on clinical knowledge about how often benefit on one endpoint could accompany harm on the other. Similarly, the probability of exceeding both favorable thresholds is meaningful only in relation to the intended prior beliefs. The purpose of the check is to make such implications visible for clinical review.
What the Check Can and Cannot Establish
A useful comparison is between the correlated prior and an independent prior that has the same marginal distributions. The independent prior leaves each endpoint's prior mean and variance unchanged, but sets the covariance to zero. Comparing their simulated pairs isolates the effect of the dependence assumption more clearly than comparing priors that also differ in scale or location. With positive correlation, the correlated model will generally produce more pairs in which the two parameters are simultaneously high or simultaneously low, and fewer pairs in which one is high while the other is low. The extent of this change depends on the distribution and on the regions being examined.
An independent prior may be appropriate when independence is defensible, but it is not a neutral default: it is still a substantive joint assumption. A multivariate normal prior is convenient and makes correlation interpretation direct, yet it imposes a particular joint shape. If the clinical relationship is nonlinear, asymmetric, or has unusual joint tail behavior, a different joint distribution or a copula-based construction may be more appropriate. Matching the marginal priors and a single correlation does not guarantee that two models have the same probabilities of extreme combinations.
The check is limited by the assumptions and knowledge supplied to it. Simulation can expose implications of a model, but cannot establish that the prior relationship is clinically correct. Finite simulation also adds Monte Carlo error: rare combinations may not appear in a modest number of draws, and a zero count does not prove that the model assigns them probability zero. If a pattern of interest is rare, increase the number of draws or calculate its probability with a suitable method. Such estimates remain assessments of the specified model, not substitutes for clinical judgment.
Common Mistakes and Diagnostics
- Checking only the marginals. Plausible distributions for each endpoint separately do not ensure plausible pairs. Inspect joint plots or clinically defined joint regions.
- Equating parameter correlation with outcome correlation. Sampling variation can change the correlation of predicted outcomes. State whether the correlation refers to prior parameters, patient outcomes, or trial summaries.
- Simulating endpoints independently by convenience. Separate draws erase intended prior dependence. Conversely, applying the same random variation to both endpoints without a model justification can introduce spurious dependence.
- Interpreting correlation as a deterministic rule. A positive prior correlation does not mean every high value of one endpoint must pair with a high value of the other. The conditional distribution still has uncertainty.
- Relying on correlation to summarize all joint behavior. Correlation captures linear association, not every feature of a joint distribution. Inspect the shape and the regions that matter clinically.
- Ignoring units or endpoint direction. A positive correlation has a different clinical meaning if one endpoint is coded so that larger values indicate harm. Clarify scales and whether larger values are favorable before interpreting simulated pairs.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solution.
- A bivariate normal prior has correlation \(-0.4\). What does this say about the two parameters, and what does it not say? Solution: Values above the mean of one parameter tend to accompany values below the mean of the other. The association is not deterministic, and it does not establish a causal relationship.
- Two predictive endpoint summaries have covariance \(0.02\), including \(0.01\) of conditional sampling covariance. What is their prior parameter covariance under the covariance decomposition? Solution: Predictive covariance is prior parameter covariance plus expected conditional sampling covariance. Thus the prior parameter covariance is \(0.02-0.01=0.01\).
- Why can two priors with identical marginal distributions yield different clinical implications? Solution: Their dependence structures can assign different probabilities to joint patterns, such as both endpoints being favorable or one being favorable while the other is harmful.
- A simulation generates each endpoint from its marginal prior independently, although the intended prior specifies positive correlation. What is wrong? Solution: The procedure preserves the marginals but removes the specified dependence, so its paired predictions do not come from the intended joint prior.
- A simulated run contains no pairs in a clinically concerning region. What conclusion is justified? Solution: Only that no such pair appeared in that finite run. The model may still assign the region positive probability; more draws or a probability calculation may be needed to assess a rare event.