Posterior Decisions With Correlated Outcomes

Why Joint Posterior Decisions Matter

An arm can appear promising on efficacy and acceptable on safety when each outcome is considered alone, yet the probability that it satisfies both requirements can be quite different. If efficacy and toxicity are related, the posterior probability of joint success depends on how their effects move together, not just on their separate posterior probabilities.

This tutorial focuses on decisions that require benefit without unacceptable toxicity. The central quantity is a joint posterior probability for one treatment arm, conditional on the data and the model. It is not the probability that the arm is “truly effective” in isolation, and it is not a frequentist error rate.

Definition: Let $\delta_E$ denote an arm’s efficacy effect, with larger values representing greater benefit, and let $\delta_T$ denote its toxicity effect, with larger values representing more harm. For clinically specified thresholds $d_E$ and $d_T$, define the joint eligibility probability as $$ q=P(\delta_E>d_E,\ \delta_T<d_T\mid \text{data}). $$ The arm meets a posterior decision rule if $q\geq\gamma$, where $\gamma$ is a prespecified posterior probability threshold. The thresholds and $\gamma$ are decision inputs, not quantities estimated by this probability calculation.

The direction of each inequality matters: efficacy must exceed its minimum acceptable target, while toxicity must remain below its maximum acceptable limit. The thresholds need a clinical interpretation and units. For example, if efficacy is an improvement in response probability, $d_E$ may be a minimum increase in response probability; if toxicity is a risk difference, $d_T$ may be the maximum tolerated increase in toxicity risk.

Joint Posterior Model and Assumptions

A convenient model for calculation is a bivariate normal posterior for the two arm effects:

\begin{pmatrix}\delta_E\\ \delta_T\end{pmatrix}\mid\text{data} \sim N_2\left[ \begin{pmatrix}\mu_E\\ \mu_T\end{pmatrix}, \begin{pmatrix} s_E^2 & \rho s_Es_T\\ \rho s_Es_T & s_T^2 \end{pmatrix} \right].

Here $\mu_E,\mu_T$ are posterior means, $s_E,s_T$ are posterior standard deviations, and $\rho$ is the posterior correlation between the two effects. The covariance matrix must be valid: $s_E>0$, $s_T>0$, and $-1<\rho<1$ for a non-degenerate bivariate normal distribution. In applications, a different joint posterior model may be more appropriate; the decision event itself does not require normality.

The efficacy and toxicity effects, their scales, and the favorable directions are defined consistently with the thresholds.
The joint posterior distribution represents uncertainty about both effects after conditioning on the observed data and specified prior model.
The bivariate-normal form is an approximation or modeling assumption, not a consequence of having two outcomes.
The decision threshold $\gamma$ and clinical limits are set before applying the rule; choosing them after seeing the result changes the meaning of the decision.

Posterior correlation is not automatically the same as the correlation between recorded efficacy and toxicity measurements in participants. It describes the joint posterior uncertainty about the arm-level effects. The data structure and likelihood contribute to that uncertainty, together with the prior and model. In particular, an observed association alone does not tell us the posterior correlation of the treatment effects.

Work With Favorable Directions

For calculation, it is helpful to express both coordinates so that larger values are favorable. Define

U=\frac{\delta_E-d_E}{s_E}, \qquad V=\frac{d_T-\delta_T}{s_T}.

The event of interest is $U>0,V>0$. Both variables have unit posterior standard deviation, but their means are generally not zero. Their correlation is $-\rho$, because the toxicity coordinate has been reversed. Thus, positive posterior correlation between efficacy and toxicity effects can induce negative correlation between the favorable efficacy and favorable safety coordinates. This sign change is an important check when setting up the calculation.

Worked Example: Correlation Reverses an Eligibility Decision

Suppose a hypothetical arm has posterior means exactly at both clinical boundaries: $\mu_E=d_E$ and $\mu_T=d_T$. Assume a bivariate-normal posterior with marginal standard deviations $s_E=0.10$ efficacy units and $s_T=0.04$ toxicity-risk units, and posterior correlation $\rho=0.80$. The decision threshold is $\gamma=0.20$.

Each marginal probability of being on the favorable side of its boundary is $0.5$. The favorable coordinates $U$ and $V$ therefore have zero means, unit variances, and correlation $-0.80$. For centered standard bivariate normal variables with correlation $r$, $$ P(U>0,V>0)=\frac14+\frac{\arcsin(r)}{2\pi}. $$

Substitute $r=-0.80$: $$ P(U>0,V>0)=\frac14+\frac{\arcsin(-0.80)}{2\pi} =0.25-\frac{0.927295}{6.283185} \approx 0.1024. $$

The joint posterior probability is about $0.1024$, below $\gamma=0.20$, so the arm does not meet the rule. If one incorrectly assumed posterior independence, the product of the two marginal probabilities would be $0.50\times0.50=0.25$, which would exceed $0.20$. In this hypothetical case, ignoring correlation reverses the decision. The positive correlation of efficacy and toxicity effects means that more efficacy tends to accompany more toxicity; this makes simultaneous benefit and safety less likely.

A Short Derivation for the Centered Case

The centered formula used above makes the effect of correlation transparent. Let $g(r)=P(U>0,V>0)$ for a centered standard bivariate normal pair with correlation $r$. At $r=0$, the coordinates are independent and $g(0)=1/4$. For a bivariate normal density $\phi_2(u,v;r)$, differentiation with respect to $r$ gives

\frac{\partial}{\partial r}\phi_2(u,v;r) = \frac{\partial^2}{\partial u\,\partial v}\phi_2(u,v;r).

Integrating this identity over $u>0,v>0$, the boundary terms at infinity vanish, leaving the density at the corner $(0,0)$. Hence

g'(r)=\phi_2(0,0;r)=\frac{1}{2\pi\sqrt{1-r^2}}.

Integrating from $0$ to $r$ gives

g(r)=g(0)+\int_0^r \frac{1}{2\pi\sqrt{1-t^2}}\,dt =\frac14+\frac{\arcsin(r)}{2\pi}.

This derivation applies when both posterior means are exactly at their thresholds after standardization. In the general case, the means shift the boundaries relative to the posterior distribution, so the arcsine expression alone is not enough.

Non-Centered Posterior Probabilities

For nonzero favorable-coordinate means, standardize the posterior draws and evaluate a bivariate normal probability. Equivalently, if $Z_1,Z_2$ are centered standard normal variables with correlation $r=-\rho$, and the favorable-coordinate means are $a=(\mu_E-d_E)/s_E$ and $b=(d_T-\mu_T)/s_T$, then

q=P(Z_1>-a,\ Z_2>-b).

This probability can be evaluated using a bivariate normal distribution function or numerical integration. A product of marginal probabilities is valid only when the posterior coordinates are independent. A standard bivariate normal distribution with zero correlation has independent coordinates; for other posterior models, zero correlation does not in general imply independence.

Worked Example: Joint Eligibility With Favorable Posterior Means

Consider a hypothetical arm with $a=b=0.50$: its efficacy posterior mean is $0.50s_E$ above the efficacy threshold, and its toxicity posterior mean is $0.50s_T$ below the toxicity limit. Suppose the raw effects have posterior correlation $\rho=-0.60$, so the two favorable coordinates have correlation $r=0.60$. The rule uses $\gamma=0.55$.

Each marginal probability is $\Phi(0.50)\approx0.691462$. To obtain the joint probability, use the identity $$ \Phi_2(h,k;r)=\Phi(h)\Phi(k)+\int_0^r \phi_2(h,k;t)\,dt, $$ where $\Phi_2$ and $\phi_2$ are the centered bivariate standard normal distribution function and density. Here $h=k=0.50$. The integrand is $$ \phi_2(0.50,0.50;t) =\frac{\exp[-0.25/(1+t)]}{2\pi\sqrt{1-t^2}}. $$

Numerical integration from $0$ to $0.60$ gives approximately $0.0844$. Therefore $$ q=\Phi(0.50)^2+0.0844 \approx 0.4781203+0.0844 \approx 0.5625. $$

Because $0.5625\geq0.55$, the arm meets the stated posterior rule. Under an independence assumption, the calculation would instead be $$ \Phi(0.50)^2=(0.691462)^2\approx0.4781, $$ which is below $0.55$. In this example, negative correlation between efficacy and toxicity effects corresponds to positive correlation between favorable efficacy and favorable safety, increasing the probability of meeting both conditions. The product is calculated from the unrounded marginal probability; squaring a rounded value can introduce avoidable numerical error.

Worked Example: Independence as a Special Case

Suppose a hypothetical arm has posterior probability $0.90$ of exceeding its efficacy target and posterior probability $0.80$ of staying below its toxicity limit. Assume the joint posterior is bivariate normal with posterior correlation $\rho=0$. For this bivariate normal model, the two effects are independent, so $$ q=P(\delta_E>d_E,\delta_T<d_T\mid\text{data}) =0.90\times0.80=0.72. $$

With a prespecified threshold $\gamma=0.70$, the arm meets the rule because $0.72\geq0.70$. The calculation is justified here by the stated bivariate-normal model and zero posterior correlation, not merely by having separate marginal probabilities available. If the joint posterior were not normal, zero correlation alone would not establish independence, and the product could be wrong.

Interpreting and Checking the Decision

The joint probability answers a specific question: conditional on the observed data, prior, and model, how much posterior probability lies in the region that satisfies both clinical criteria? It does not tell us that the arm will benefit every treated patient, nor does it establish that treatment causes a particular outcome beyond the causal interpretation supported by the study design and model.

Before applying the rule, check the following:

Directions and units: verify that larger $\delta_E$ is beneficial, larger $\delta_T$ is harmful, and each threshold uses the same scale as its effect.
Correlation sign: if toxicity is transformed to make larger values favorable, reverse the sign of its correlation with efficacy.
Joint model: confirm that the covariance matrix is valid and that the assumed joint distribution is plausible for the posterior tails near the decision boundaries.
Numerical evaluation: compare the computed probability with marginal bounds. Necessarily, $$ \max\{0,\ P(E\mid\text{data})+P(S\mid\text{data})-1\} \leq q \leq \min\{P(E\mid\text{data}),P(S\mid\text{data})\}, $$ where $E$ is the efficacy event and $S$ the safety event. A result outside these bounds signals an error.
Decision sensitivity: examine whether reasonable changes in the joint model or clinical thresholds change the decision, especially when $q$ is close to $\gamma$.

A common mistake is to calculate $P(E\mid\text{data})P(S\mid\text{data})$ automatically. That product is the joint probability only under posterior independence. The correction is to calculate the probability under the joint posterior or to justify independence from the model. Marginal probabilities can still be useful summaries, but they cannot generally determine the probability of satisfying both requirements.

How This Rule Relates to Other Decisions

Approach	What it answers	Main caution
Separate marginal rules	Whether efficacy and safety each have sufficient posterior probability individually	Both marginal criteria passing does not guarantee that a joint-probability threshold passes.
Product of marginals	The joint probability under posterior independence	Incorrect if the outcomes are dependent; the direction of the error depends on the relevant correlation and thresholds.
Joint posterior eligibility rule	Whether the posterior probability of meeting both clinical criteria reaches $\gamma$	Depends on a defensible joint model and on clinically meaningful thresholds.
Posterior expected utility	Expected value of consequences across possible efficacy, toxicity, and other outcomes	Requires utilities or losses; it is not equivalent to a hard eligibility boundary.

A joint eligibility rule is particularly clear when the decision is explicitly “benefit above this minimum and toxicity below this limit.” It can be conservative if either threshold is uncertain or if clinically important dimensions are omitted. Expected utility may be preferable when trade-offs are gradual rather than governed by hard boundaries. Neither approach removes the need to justify the model and clinical inputs.

Key takeaway: A decision requiring efficacy and safety simultaneously is based on a joint posterior probability. Preserve the dependence in the posterior calculation, keep the favorable directions straight, and compare the resulting probability with a threshold chosen for the decision context.

Check Your Understanding

Answer each question, then compare with the solutions below.

If the posterior correlation between efficacy and toxicity effects is positive, what is the correlation between efficacy and the transformed safety coordinate $V=(d_T-\delta_T)/s_T$?
For centered standard bivariate normal favorable coordinates with correlation $r=-0.50$, calculate the joint probability that both coordinates are positive.
An arm has marginal probabilities $0.85$ for efficacy and $0.75$ for safety. Is their product automatically the joint probability? State the condition that would justify using it.
Why is a joint probability of $0.64$ not sufficient by itself to decide whether an arm is eligible?

Solutions. (1) It is negative: reversing the toxicity coordinate changes the sign, so the correlation is the negative of the raw efficacy-toxicity posterior correlation. (2) Using the centered formula, $q=1/4+\arcsin(-0.50)/(2\pi)=0.25-(\pi/6)/(2\pi)=1/6\approx0.1667$. (3) No. The product $0.85\times0.75=0.6375$ is the joint probability only if the events are posterior independent; for a bivariate normal posterior, zero correlation gives independence. (4) Eligibility also depends on the prespecified threshold $\gamma$, and the probability must have been calculated under an appropriate joint posterior model.