Priors for Correlated Clinical Endpoints

Why Specify Dependence Between Clinical Effects?

A trial may assess two outcomes that matter to patients, such as symptom improvement and avoidance of hospitalization. A prior for each treatment effect separately describes what is believed about that endpoint before the trial data are observed. But the two beliefs may also be related: a treatment thought likely to improve symptoms might also be thought likely to reduce hospitalizations. To represent that belief, a joint prior must specify both marginal distributions and how the effects vary together.

This tutorial uses a bivariate normal prior as a direct way to express that dependence. The correlation describes prior dependence between the two treatment effects; it is not automatically the correlation between patient-level outcomes, and it does not establish that improvement in one outcome causes improvement in the other. The effects must first be coded so that their directions have a clear clinical meaning.

Definition: Let $\theta_1$ and $\theta_2$ be two treatment effects, each coded so that a larger value means a more favorable treatment effect. A bivariate normal prior specifies $$ \begin{pmatrix}\theta_1\\\theta_2\end{pmatrix} \sim N_2\left[ \begin{pmatrix}\mu_1\\\mu_2\end{pmatrix}, \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2\\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix} \right]. $$ Here $\mu_j$ and $\sigma_j$ are the prior mean and standard deviation for endpoint $j$, and $\rho$ is the prior correlation between the two effects.

The diagonal entries of the covariance matrix are the marginal variances. The off-diagonal entries are the covariance, which has units equal to the product of the two effect scales. Thus, $\rho$ is dimensionless, but $\rho\sigma_1\sigma_2$ is not. The marginal priors remain $\theta_1\sim N(\mu_1,\sigma_1^2)$ and $\theta_2\sim N(\mu_2,\sigma_2^2)$, whatever value of $\rho$ is chosen.

For a nondegenerate bivariate normal distribution, both marginal standard deviations must be positive and $-1<\rho<1$. Equivalently, its covariance matrix must be positive definite. At $\rho=1$ or $\rho=-1$, the distribution is degenerate: one effect is an exact linear function of the other, so the prior assigns probability only to a line in the two-dimensional effect space.

Constructing a Joint Prior

Specify the endpoint effects and their directions before choosing the correlation. For example, let $\theta_1$ be mean improvement in a symptom score, measured in points, where a positive value means greater improvement. A log hazard ratio is usually oriented the other way: a negative log hazard ratio indicates benefit. One can instead define $\theta_2=-\log(HR)$, so that positive values also indicate benefit. The marginal means and standard deviations must be expressed on the chosen scales, and the correlation must refer to those coded effects.

A correlation is a substantive prior assumption, not a mathematical convenience to add after the fact. Positive $\rho$ means that high values of one effect tend to occur with high values of the other under the prior. Negative $\rho$ means that high values of one tend to occur with low values of the other. Zero correlation means no linear association; under the bivariate normal model it also implies prior independence. In other joint distributions, zero correlation does not in general imply independence.

Worked Example: Building the Covariance Matrix

Suppose a hypothetical trial has two benefit-coded treatment effects. Prior beliefs are $\theta_1\sim N(0.10,0.20^2)$ symptom-score points and $\theta_2\sim N(0.05,0.15^2)$ on a second clinical scale. Suppose the prior correlation is $\rho=0.60$. Construct the joint prior covariance matrix.

The covariance is the correlation multiplied by both marginal standard deviations: $$ \operatorname{Cov}(\theta_1,\theta_2) =\rho\sigma_1\sigma_2 =(0.60)(0.20)(0.15)=0.018. $$ The marginal variances are $0.20^2=0.040$ and $0.15^2=0.0225$. Therefore, $$ \begin{pmatrix}\theta_1\\\theta_2\end{pmatrix} \sim N_2\left[ \begin{pmatrix}0.10\\0.05\end{pmatrix}, \begin{pmatrix}0.040&0.018\\0.018&0.0225\end{pmatrix} \right]. $$ The off-diagonal entries have units of symptom-score points multiplied by units of the second effect. The matrix is valid: its determinant is $0.040(0.0225)-0.018^2=0.000576>0$, and its diagonal entries are positive. The two marginal priors are unchanged by adding the correlation.

How Correlation Changes Beliefs About Joint Benefit

A common prior question is whether both treatment effects are beneficial. With benefit-coded effects, that event is $\{\theta_1>0,\theta_2>0\}$. Its prior probability depends on both marginal beliefs and their dependence. Correlation alone cannot determine this probability: the marginal means, standard deviations, and clinical benefit thresholds matter too.

One useful special case has zero prior means and a benefit threshold of zero on each scale. For standardized bivariate normal variables with correlation $\rho$, the joint probability is $$ P(\theta_1>0,\theta_2>0)=\frac{1}{4}+\frac{\arcsin(\rho)}{2\pi}. $$ This result makes the role of dependence transparent: with fixed, symmetric marginals, positive correlation increases the chance that both effects fall on the beneficial side, while negative correlation decreases it.

Justification: Write the standardized effects as $Z_1$ and $Z_2=\rho Z_1+\sqrt{1-\rho^2}Z_0$, where $Z_1$ and $Z_0$ are independent standard normal variables. The pair $(Z_1,Z_0)$ has a rotationally symmetric density, so its direction is uniform around a circle. The conditions $Z_1>0$ and $Z_2>0$ select a wedge with angle $\pi/2+\arcsin(\rho)$. Dividing that angle by the full angle $2\pi$ gives the stated probability.

Worked Example: Comparing Joint Benefit Probabilities

Suppose two hypothetical benefit-coded effects have zero prior means and positive marginal standard deviations. Compare the prior probability that both effects are beneficial when their correlation is $-0.50$, $0$, or $0.80$. Each marginal probability of benefit is $0.50$.

For $\rho=-0.50$, $\arcsin(-0.50)=-\pi/6$, so $$ P(\theta_1>0,\theta_2>0)=\frac14-\frac{1}{12}=\frac16=0.1667. $$ For $\rho=0$, the probability is $1/4=0.2500$. For $\rho=0.80$, $\arcsin(0.80)\approx0.9273$, giving $$ P(\theta_1>0,\theta_2>0) =\frac14+\frac{0.9273}{2\pi} \approx0.3976. $$ Thus, the same marginal probability of benefit on each endpoint is compatible with markedly different prior probabilities of simultaneous benefit. These calculations apply to zero-centered normal marginals and zero thresholds; they should not be used as a general formula when the prior means or clinical thresholds differ.

Conditional Beliefs and Direction of Coding

The joint prior also gives a conditional distribution for one effect given the other. For the bivariate normal model, $$ \theta_2\mid\theta_1=x \sim N\left( \mu_2+\rho\frac{\sigma_2}{\sigma_1}(x-\mu_1), \ \sigma_2^2(1-\rho^2) \right). $$ The conditional mean shows how a value of $\theta_1$ shifts the prior expectation of $\theta_2$; the conditional variance shows that learning $\theta_1$ reduces uncertainty about $\theta_2$ when $\rho\ne0$. This is a statement about the specified prior distribution, not a claim that observing a clinical outcome reveals the true treatment effect without uncertainty.

Worked Example: Conditional Prior for the Second Effect

Use the joint prior from the covariance-matrix example. What distribution does it assign to $\theta_2$ when $\theta_1=0.30$ symptom-score points?

The conditional mean is $$ 0.05+(0.60)\frac{0.15}{0.20}(0.30-0.10) =0.05+0.45(0.20)=0.14. $$ The conditional variance and standard deviation are $$ 0.15^2(1-0.60^2)=0.0225(0.64)=0.0144, \qquad \sqrt{0.0144}=0.12. $$ Therefore $\theta_2\mid\theta_1=0.30\sim N(0.14,0.12^2)$, in units of the second clinical effect. In this hypothetical prior, a value of $\theta_1$ above its mean shifts the conditional mean of $\theta_2$ upward because the correlation is positive. The conditional probability that $\theta_2>0$ is approximately $\Phi(0.14/0.12)=\Phi(1.167)=0.878$.

Coding affects the interpretation of the sign. If both effects are multiplied by $-1$, their correlation is unchanged. If only one is multiplied by $-1$, the correlation changes sign. For instance, if both raw log hazard ratios are positively correlated, defining one benefit effect as minus its log hazard ratio while leaving the other as the raw log hazard ratio reverses the correlation between the coded quantities. State the coding explicitly before interpreting $\rho$.

Assumptions, Diagnostics, and Alternatives

The bivariate normal prior assumes a continuous joint distribution summarized by means, variances, and a linear correlation. It also has unbounded tails and imposes a particular dependence pattern across the full range of effects. Before using it, check that the chosen scales are meaningful, each marginal prior is plausible, the covariance matrix is positive definite, and the correlation reflects defensible prior beliefs. Examine joint contours or simulated prior draws: they can reveal whether the specified dependence implies implausible combinations of effects.

Do not substitute correlation between observed endpoints in patients for correlation between treatment effects without a model that justifies the connection. Patient-level outcome association and uncertainty about treatment effects are different quantities. Similarly, a correlation should not be described as the probability that both endpoints benefit. The joint probability of benefit is a separate quantity calculated from the full joint prior.

The main simple alternative is an independent prior, formed by setting $\rho=0$. It is easy to specify, but independence is itself an assumption and may misrepresent beliefs about coordinated effects. A bivariate normal prior allows a concise correlation parameter, but can be unsuitable when plausible dependence is nonlinear, tails are important, or the marginals are not normal. More flexible joint distributions, such as models built from copulas, can separate marginal distributions from dependence structure, but require additional assumptions and specification. Whichever approach is used, report the marginal priors and the dependence assumptions, not just a correlation value.

Common pitfall: “The endpoints are positively correlated in patients, so the treatment-effect prior correlation must be positive.” This does not follow. Patient-level association concerns outcomes within individuals; prior effect correlation concerns uncertainty about two treatment effects across plausible states of knowledge. Specify the latter from an explicit model or substantive prior judgment rather than equating the two.

Worked Example: Reversing One Effect’s Direction

Suppose a hypothetical prior is specified for two raw log hazard ratios, $L_1$ and $L_2$, with standard deviations $0.20$ and $0.15$, and correlation $0.60$. For endpoint 1 define benefit as $\theta_1=-L_1$, but for endpoint 2 leave the effect as $\theta_2=L_2$. Find the correlation and covariance of the benefit-coded effects.

Multiplying one variable by $-1$ reverses the covariance: $$ \operatorname{Cov}(\theta_1,\theta_2) =\operatorname{Cov}(-L_1,L_2) =-\operatorname{Cov}(L_1,L_2). $$ The raw covariance is $(0.60)(0.20)(0.15)=0.018$, so the coded covariance is $-0.018$. The standard deviations do not change, and therefore the coded correlation is $$ \frac{-0.018}{(0.20)(0.15)}=-0.60. $$ The negative sign is not evidence of antagonistic clinical effects; it results from the chosen direction of coding. To interpret dependence as a tendency toward simultaneous benefit, both effects should first be oriented so that benefit corresponds to larger values.

Check Your Understanding

Answer each question, then compare with the solutions below.

If $\sigma_1=0.10$, $\sigma_2=0.30$, and $\rho=-0.40$, what is the covariance?
For zero-mean normal effects with $\rho=0.50$, calculate the probability that both are positive.
What does $\rho=0$ imply about independence under a bivariate normal prior? Does the same implication hold for every joint distribution?
If one treatment effect is multiplied by $-1$, what happens to its correlation with the other effect?

Solutions. (1) $\operatorname{Cov}(\theta_1,\theta_2)=(-0.40)(0.10)(0.30)=-0.012$, in the product of the two effect units. (2) $\arcsin(0.50)=\pi/6$, so the probability is $1/4+(\pi/6)/(2\pi)=1/3$, or approximately $0.3333$. (3) Under a bivariate normal prior, zero correlation implies independence; for a general joint distribution it does not. (4) The correlation changes sign because the covariance changes sign while the standard deviations remain the same.