Joint Success Criteria for Co-Primary Endpoints

Why joint success needs a joint calculation

In a trial with co-primary endpoints, the decision rule may require convincing evidence of benefit on each endpoint. A treatment that succeeds on one endpoint but not the other does not satisfy this rule. Consequently, the design quantity of interest is the probability of the intersection of two success events, not either endpoint’s power considered alone.

That intersection depends on endpoint dependence. If both endpoint estimates tend to be favorable together, the probability of passing both criteria can be much greater than the product of their marginal probabilities. If favorable evidence on one endpoint tends to accompany unfavorable evidence on the other, the joint probability can be smaller. This matters for both conditional power—calculated at specified true effects—and assurance, which averages over uncertainty in the true effects.

Definition: Let \(S_j\) be the event that endpoint \(j\) satisfies its prespecified posterior success criterion, for \(j=1,2\). With a co-primary “both must succeed” rule, the trial succeeds on \(S=S_1\cap S_2\). Conditional power at a specified true effect vector is \(P(S\mid \theta)\), or \(P(S\mid \theta,\text{interim data})\) when an interim result is given. Assurance is the prior-predictive probability of success, \(P(S)=\int P(S\mid\theta)\,\pi(\theta)\,d\theta\), where \(\pi(\theta)\) is the design prior.

From posterior criteria to a joint decision region

Suppose the treatment effects are \(\theta_1\) and \(\theta_2\), with positive values representing benefit. For each endpoint, let \(\delta_j\) be the minimum effect that counts as benefit for the decision, and let \(q_j\) be the required posterior probability. A common rule is to declare endpoint \(j\) successful when \(P(\theta_j>\delta_j\mid Y)\ge q_j\), where \(Y\) denotes the trial data.

For a normal posterior \(\theta_j\mid Y\sim N(m_j,V_{jj})\), this criterion is equivalent to a one-sided boundary on the posterior mean. Writing \(z_{q_j}\) for the \(q_j\) quantile of the standard normal distribution, the posterior probability is at least \(q_j\) exactly when:

P(\theta_j>\delta_j\mid Y)\ge q_j \quad\Longleftrightarrow\quad m_j-z_{q_j}\sqrt{V_{jj}}\ge\delta_j.

Thus the final data must put both posterior means beyond their respective boundaries. If the posterior covariance is fixed by the design, the two marginal posterior criteria define a region in the space of final data. In a simple model that region is a pair of coordinate-wise thresholds. In a more general model, correlated outcomes and priors can make each posterior mean depend on data from both endpoints; the success region is then still the intersection of the two posterior criteria, but may not be a pair of separate thresholds on the endpoint estimates.

A useful working model for calculation is a bivariate normal final estimator \(Y=(Y_1,Y_2)\) with \(Y\mid\theta\sim N_2(\theta,\Omega)\), where \(\Omega\) is the sampling covariance matrix. Under a normal prior, the posterior is normal as well. Once the posterior rule has been applied, conditional power is the probability that \(Y\) falls in the resulting success region under the assumed \(\theta\). Assurance is the same probability under the prior-predictive distribution of \(Y\).

Important distinction: The correlation relevant to the joint probability is the correlation of the final-data quantities that determine the two success decisions. It may arise from correlated outcomes, correlated prior uncertainty, or both. Marginal posterior probabilities alone do not determine the probability of passing both criteria.

Worked calculations

Worked Example: Translating Two Posterior Criteria Into Cutoffs

Suppose each endpoint requires \(P(\theta_j>0\mid Y)\ge0.975\). In a prespecified normal model, the final posterior standard deviation for each endpoint is \(0.50\), and each posterior mean equals \(0.80Y_j\), where \(Y_j\) is the final effect estimate. The prior mean is zero. Find the corresponding cutoff on each \(Y_j\).

For a standard normal distribution, \(z_{0.975}=1.96\). The posterior criterion is \(m_j\ge1.96(0.50)=0.980\). Since \(m_j=0.80Y_j\), endpoint \(j\) succeeds exactly when:

Y_j\ge\frac{0.980}{0.80}=1.225.

Both endpoints must therefore satisfy \(Y_1\ge1.225\) and \(Y_2\ge1.225\). These are effect-estimate cutoffs in the units of their respective endpoints. This calculation also shows why the posterior probability threshold does not, by itself, specify a data cutoff: the posterior standard deviation and the prior-to-posterior mapping are needed.

Worked Example: Conditional Power With Independent Endpoints

Use the two cutoffs \(c_1=c_2=1.225\) from the preceding example. Assume that, conditional on the true effects, \(Y_1\) and \(Y_2\) are independent normal estimators, each with standard error \(0.50\). At the design alternative, let \(\theta_1=1.725\) and \(\theta_2=1.225\). Calculate the probability that both endpoints succeed.

For endpoint 1, the standardized distance from the true mean to the cutoff is \((1.725-1.225)/0.50=1.00\). Its success probability is \(\Phi(1.00)=0.8413\). For endpoint 2, the true mean is exactly at the cutoff, so the success probability is \(\Phi(0)=0.5000\). Independence allows multiplication:

P(S\mid\theta)=0.8413(0.5000)=0.4207.

The conditional power for the co-primary rule is therefore \(0.4207\), or about 42.1%. Neither the larger marginal probability of 84.1% nor the smaller marginal probability of 50.0% is the trial’s joint power. Multiplication is justified here only because conditional independence was assumed.

Worked Example: How Positive Dependence Changes Joint Power

Now suppose both true effects equal their cutoffs, so each standardized final estimate has mean zero relative to its success boundary. Let their standard errors be as above, but let their conditional correlation be \(\rho=0.60\). Each endpoint still has marginal success probability \(0.50\). Calculate the joint conditional power.

Let \(Z_1,Z_2\) be the standardized estimates relative to their cutoffs. They have a bivariate standard normal distribution with correlation \(0.60\), and success requires \(Z_1\ge0\) and \(Z_2\ge0\). For a centered bivariate normal pair, \(P(Z_1\ge0,Z_2\ge0)=\frac14+\frac{\arcsin(\rho)}{2\pi}\). Hence:

P(S\mid\theta) =\frac14+\frac{\arcsin(0.60)}{2\pi} =\frac14+\frac{0.6435}{6.2832} =0.3524.

If one incorrectly multiplied the marginal probabilities, the result would be \(0.50(0.50)=0.2500\). Here positive dependence raises the probability that both estimates are on the favorable side of their cutoffs, giving joint power of about 35.2%. This does not mean dependence always increases joint success: with negative correlation, the corresponding probability at these centered boundaries is lower.

Conditional power and assurance with dependence

In general, suppose the final-data decision quantities \(W_1,W_2\), after standardization, are bivariate normal conditional on \(\theta\), with means \(\mu_1(\theta),\mu_2(\theta)\), unit variances, and correlation \(\rho\). For separate cutoff rules \(W_j\ge a_j\), conditional power is:

P(S\mid\theta) = P\{W_1\ge a_1,\ W_2\ge a_2\mid\theta\} = \int_{a_1}^{\infty}\int_{a_2}^{\infty} \phi_2(w_1,w_2;\mu_1(\theta),\mu_2(\theta),\rho)\,dw_2\,dw_1.

Here \(\phi_2\) is the bivariate normal density. When \(\rho=0\), the integral factors into two marginal tail probabilities. Otherwise it is a joint tail probability and can be evaluated with a bivariate normal distribution function or numerical integration. If the success region is not a pair of coordinate-wise cutoffs, integrate the joint predictive density over the actual region defined by the two posterior criteria.

Assurance averages this conditional probability over the design prior. If \(\theta\sim\pi(\theta)\), then:

\text{Assurance} = \int P(S\mid\theta)\,\pi(\theta)\,d\theta.

Equivalently, draw \(\theta\) from the design prior, generate final data from the sampling model at that \(\theta\), apply the posterior success rule, and record whether both endpoints pass. The proportion of successes estimates assurance. This simulation must retain the dependence specified by the joint prior and joint outcome model; simulating each endpoint separately and multiplying two average powers generally gives the wrong answer.

Worked Example: Assurance at Two Boundary-Centered Endpoints

Suppose that under the design prior predictive distribution, the standardized final-data quantities for the two endpoint decisions are jointly normal with unit marginal variances, correlation \(0.60\), and means zero relative to their respective success cutoffs. Each endpoint has prior-predictive probability \(0.50\) of passing. Find the assurance when both must pass.

The assurance is the probability of the upper-right quadrant of this bivariate normal distribution. Applying the centered bivariate normal formula:

\text{Assurance} =\frac14+\frac{\arcsin(0.60)}{2\pi} =0.3524.

The product of marginal prior-predictive success probabilities would be \(0.25\), which would be correct only if the two decision quantities were independent under the prior predictive distribution. This example uses a joint distribution directly; in a full design calculation, that distribution combines uncertainty in the true effects with sampling variability.

Conditional power given interim data

When interim results are available, conditional power is often evaluated at a specified assumed true effect vector while conditioning on the observed interim data. For a simple equal-information normal model, let \(Y_I\) be the interim estimator based on \(n_I\) observations and \(Y_F\) the final estimator based on \(N\) observations, with \(n_I<N\). If observations after the interim are independent of those already observed, then:

Y_F=\frac{n_I}{N}Y_I+\frac{N-n_I}{N}Y_{\mathrm{new}}, \qquad Y_F\mid Y_I,\theta \sim N_2\left( \frac{n_I}{N}Y_I+\frac{N-n_I}{N}\theta,\, \frac{N-n_I}{N^2}\Sigma \right),

where \(Y_{\mathrm{new}}\) is the estimator from the remaining observations and \(\Sigma\) is the per-observation covariance matrix. The conditional power is the probability that this conditional final-data distribution enters the joint success region. This calculation uses the actual interim estimates, not the prior distribution for \(\theta\). A conditional assurance variant would additionally average over a specified distribution for the unknown effects given the interim information; the conditioning distribution must be stated explicitly.

Calculation choices, diagnostics, and limitations

For a bivariate normal model, direct evaluation of the joint normal tail is often efficient and accurate. For more complex posterior rules, nuisance parameters, or nonnormal outcomes, simulation can be more practical. At each simulated draw, generate the two endpoints jointly, fit or update the planned posterior, and record whether both criteria pass. Simulation methods for assurance and Monte Carlo error are useful here, but the event being counted must match the co-primary decision rule.

A practical diagnostic is to compare the estimated joint probability with its marginal bounds. For any two events, \(P(S_1\cap S_2)\le\min\{P(S_1),P(S_2)\}\); it also satisfies \(P(S_1\cap S_2)\ge\max\{0,P(S_1)+P(S_2)-1\}\). A result outside these bounds indicates a calculation or coding error. Check that the simulated correlation matches the model, that each endpoint uses its own units and threshold, and that posterior success is assessed separately for both endpoints in every replicate.

A frequent mistake is to calculate each endpoint’s power and multiply the two values without checking independence. That multiplication is valid only if the two success indicators are independent under the probability distribution being used—for conditional power, conditional on the specified effects; for assurance, under the prior predictive distribution. Another mistake is to treat a rule requiring both endpoints as if it required either endpoint. The “either” rule uses a union of success events and answers a different clinical decision question.

The joint calculation is only as credible as the assumed endpoint model, prior, and posterior decision rule. A normal approximation may be poor for sparse binary outcomes, strong skewness, or boundary effects. Dependence may also be uncertain; a design should assess plausible correlations rather than rely on a single convenient value. A simulated assurance estimate can have small Monte Carlo error while still being misleading if the model omits important sources of dependence or the success rule is implemented incorrectly.

Key takeaway: For co-primary endpoints with a both-must-pass rule, calculate the probability of the intersection of endpoint success events. Preserve the relevant dependence in conditional power and assurance calculations; marginal probabilities alone are not enough.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

Two independent endpoint success events have conditional probabilities 0.70 and 0.60. What is joint conditional power under a both-must-pass rule?
Why is multiplying two marginal endpoint powers generally invalid when endpoint estimates are correlated?
For a centered bivariate standard normal pair with correlation zero, what is the probability that both variables exceed zero?
State the difference between conditional power at a fixed effect vector and assurance.

Solutions. (1) Independence permits multiplication: \(0.70(0.60)=0.420\), so joint conditional power is 42.0%. (2) The product equals a joint probability only when the success events are independent under the relevant distribution; correlated endpoint estimates can make the probability of passing both different from the product. (3) With zero correlation, the variables are independent and each has probability \(0.50\) of exceeding zero, so the joint probability is \(0.50(0.50)=0.25\). (4) Conditional power is the probability of satisfying the final joint rule at specified true effects, optionally conditional on interim data. Assurance averages that probability over a stated design prior for the unknown effects.