Why Endpoint Power Is Not Necessarily Trial Power
A multi-endpoint trial can have a reported power value for each endpoint and still leave an important question unanswered: what is the probability that the trial will meet its stated success criterion? If success requires rejecting both co-primary hypotheses, the relevant probability is the probability of both rejections. If success requires at least one endpoint to succeed, it is the probability of at least one rejection. Neither quantity is generally the same as either endpoint’s power.
Repeated monitoring adds another layer. An endpoint may meet its boundary at an interim look, at the final look, or at neither. The trial-level rule combines these possibilities across endpoints and looks. Endpoint-wise power summarizes one endpoint’s chance of crossing its boundary; composite trial power summarizes the chance of the specified overall decision. The latter depends on the dependence among endpoint statistics, their dependence across looks, and the exact success rule.
Modeling Correlated Endpoints and Repeated Looks
A useful planning model treats the endpoint test statistics across all looks as a jointly normal vector under the alternative. Write \(Z_{e,k}\) for the standardized test statistic for endpoint \(e\) at look \(k\), and \(b_{e,k}\) for its upper rejection boundary in a one-sided test. Then endpoint \(e\) rejects if \(Z_{e,k}\ge b_{e,k}\) at any permitted look, subject to any stopping or multiplicity rules in the design.
For a common group-sequential model with information fractions \(t_k\), the within-endpoint correlation is canonical:
This expression assumes nested data, increasing information, and standardized statistics formed from the same accumulating endpoint data. Across endpoints, a planning model also specifies correlations such as \(\rho_{ef,k\ell}=\operatorname{Corr}(Z_{e,k},Z_{f,\ell})\). In a simple separable model with common information fractions, these may be written as the endpoint correlation times the canonical look correlation. More complex designs may require a directly specified covariance matrix. The covariance structure should reflect the endpoint definitions, data collection, and analysis plan; it should not be chosen merely because it makes a calculation convenient.
Under a specified alternative, the mean of each statistic also matters. For example, with a standardized effect \(\delta_e\) and information \(I_{e,k}\), a common large-sample approximation is \(E(Z_{e,k})=\delta_e\sqrt{I_{e,k}}\). The joint distribution, its means, and the boundaries determine the probability of each combination of rejections. A marginal power calculation uses only the distribution of one endpoint’s statistics. A composite calculation uses the joint distribution.
The multiplicity procedure determines the boundaries or rejection rules, including any applicable alpha allocation, gatekeeping, or recycling. A nominally significant statistic is not automatically a rejection: the statistic must satisfy the boundary and the current multiplicity rule. Power is evaluated under an alternative; type I error control is evaluated under the null configuration or configurations relevant to the procedure. A high joint power calculation does not itself establish valid type I error control.
Worked Example: Two Correlated Co-Primary Endpoints at One Look
Consider a hypothetical one-look design with two co-primary endpoints. Success requires rejecting both null hypotheses. Suppose the standardized statistics under the planning alternative have means 1.96, variances 1, and correlation 0.50. Both are compared with the one-sided boundary 1.96. This is a simple bivariate normal planning model, not a report of an actual study.
For either endpoint, \(Z_e\sim N(1.96,1)\), so its power is
The composite success event is \(Z_1\ge1.96\) and \(Z_2\ge1.96\). After centering and standardizing, this is the event that two standard normal variables with correlation 0.50 are both nonnegative. For equal zero thresholds, the bivariate normal quadrant probability is \(1/4+\arcsin(\rho)/(2\pi)\). Therefore,
Thus each endpoint has 50.0% power, but the probability of meeting the co-primary success criterion is about 33.3%. The endpoint powers cannot be multiplied here: their product, \(0.500\times0.500=0.250\), would be correct only if the two endpoint rejection events were independent. Positive correlation raises the probability of simultaneous success relative to that independence calculation, while it still leaves joint power below either marginal power.
What Repeated Looks Change
With repeated monitoring, define \(A_e=\bigcup_k\{Z_{e,k}\ge b_{e,k}\}\) when the endpoint’s rejection rule is simply crossing any of its assigned boundaries. More complicated rules can be represented by defining \(A_e\) to include the full prespecified decision logic. For a co-primary rule, the overall event is \(\bigcap_e A_e\): each endpoint must reject at some permitted look. The endpoints need not reject at the same look unless the protocol explicitly requires that.
The distinction between “each endpoint succeeds at some look” and “both succeed at the same look” is consequential. The first is \((\bigcup_k R_{1,k})\cap(\bigcup_k R_{2,k})\), where \(R_{e,k}\) denotes rejection for endpoint \(e\) at look \(k\). The second is \(\bigcup_k(R_{1,k}\cap R_{2,k})\). The second event is contained in the first, so its probability cannot be larger. The actual trial criterion, not a convenient shorthand, determines which event to calculate.
Worked Example: Two Looks and Independent Endpoint Processes
Suppose, hypothetically, that each of two endpoints is monitored at information fractions \(t_1=0.50\) and \(t_2=1.00\). For each endpoint, use upper boundaries 2.80 and 2.00, respectively. Under the planning alternative, let the final-look mean be 2.50 and assume the canonical mean relationship. Thus the interim mean is \(2.50\sqrt{0.50}=1.768\), and the correlation between that endpoint’s interim and final statistics is \(\sqrt{0.50}=0.707\). Assume the two endpoints’ entire monitoring processes are independent of one another. The boundaries are stipulated for this illustration; no particular alpha-control claim is being made about them.
For one endpoint, the rejection event is \(Z_1\ge2.80\) or \(Z_2\ge2.00\). The marginal power is the probability of this union. Equivalently, subtract from one the probability of not crossing either boundary:
The probability 0.304 is obtained from the bivariate normal distribution with means \(1.768\) and \(2.50\), unit variances, and correlation 0.707. It accounts for the overlap between the two look-specific rejection events; simply adding their separate tail probabilities would double-count paths that cross both boundaries.
If trial success requires both endpoints to reject at some look, the assumed independence between endpoint processes gives
So the endpoint-wise power is about 69.6% for each endpoint, while the probability of co-primary success is about 48.4%. The multiplication is justified here by the explicit independence assumption between endpoint processes, not by the fact that there are two endpoints. If the endpoint statistics were correlated, the joint rejection probability would need the cross-endpoint covariance as well as the within-endpoint look correlation.
Same Marginal Power, Different Composite Power
Marginal powers alone do not determine joint power. This can be seen without choosing a particular normal model. Suppose the endpoint rejection events each have probability 0.70. If the events are independent, their intersection has probability 0.49. If their intersection instead has probability 0.55, each endpoint can still have marginal probability 0.70: the probabilities of endpoint 1 only and endpoint 2 only are then 0.15 each, and the probability of neither is 0.15. If the intersection is 0.42, the corresponding probabilities of endpoint 1 only and endpoint 2 only are 0.28 each, and neither is 0.02. Both joint distributions have the same 70% endpoint-wise powers but different probabilities of co-primary success.
Worked Example: Dependence Changes the Probability of Any Success
Use the two-event distribution with \(P(A_1)=P(A_2)=0.70\) and \(P(A_1\cap A_2)=0.55\). If the trial succeeds when at least one endpoint rejects, then
If instead \(P(A_1\cap A_2)=0.42\), while keeping both marginal powers at 0.70, then the probability of at least one success is \(0.70+0.70-0.42=0.98\). Greater overlap makes co-primary success more likely but makes “at least one” success less likely, given the same marginal powers. Therefore, even the direction in which correlation affects composite power depends on the success criterion.
Calculation, Diagnostics, and Limitations
A practical calculation starts by writing the trial success event in terms of endpoint-and-look rejection events. The joint probability can then be obtained from a multivariate normal probability calculation, numerical integration, or simulation from the specified joint model. Simulation is particularly useful when the decision rule includes complex stopping logic or multiplicity updates, but it must reproduce the planned boundaries, information times, endpoint dependence, and stopping rules. Monte Carlo error should be reported or made small enough that it does not obscure the design comparison.
Check that the covariance matrix is valid (symmetric and positive semidefinite), that the marginal distributions reproduce the intended endpoint powers, and that the look correlations match the accumulating-data design. Also check that simulated or integrated rejection frequencies agree with simple special cases, such as a single endpoint or a one-look design. Sensitivity analyses are appropriate when endpoint correlation, event rates, missingness, or endpoint maturity are uncertain. In particular, a correlation estimate from limited data may be imprecise, and a single assumed value can give a misleadingly definite composite-power estimate.
Endpoint powers are useful for describing the operating characteristics of individual tests and for comparing endpoint alpha allocations. They do not by themselves answer whether the trial is likely to satisfy a joint success rule. Conversely, a composite power value can conceal an endpoint with weak marginal power or a design that is sensitive to assumptions about dependence. Reporting both marginal and composite probabilities, with the alternative, boundaries, dependence assumptions, and success rule stated, makes the interpretation clearer.
- Do not multiply marginal powers automatically. Multiplication requires independence of the relevant rejection events.
- Do not confuse endpoint correlation with look correlation. Statistics may be correlated across looks even for one endpoint, and different endpoints may add further dependence.
- Do not silently change the success criterion. “Both endpoints at some look” differs from “both at the same look.”
- Do not interpret power under the alternative as an error rate under the null. These are probabilities under different parameter configurations.
The central limitation is that joint power is conditional on a model for effects, covariance, information timing, and the decision rule. If those assumptions are poor approximations, a precise numerical result may still be a poor forecast. The introductory material on endpoint alpha allocation helps explain how boundary choices affect marginal power; the additional step here is to evaluate the probability of the trial’s actual composite decision.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Two co-primary endpoints each have 60% power. Is their joint power necessarily 36%? Explain.
- For a rule requiring at least one of two endpoints to reject, write the event in terms of \(A_1\) and \(A_2\).
- In a canonical two-look design with information fractions 0.40 and 1.00, what is the correlation between the same endpoint’s standardized statistics?
- Why can’t a nominally significant interim statistic automatically be counted as an endpoint rejection in a multiplicity-controlled sequential design?
Solutions. (1) No. It is 36% only if the endpoint rejection events are independent, because then \(P(A_1\cap A_2)=P(A_1)P(A_2)=0.60^2\). Dependence changes the intersection probability. (2) The event is \(A_1\cup A_2\), with probability \(P(A_1)+P(A_2)-P(A_1\cap A_2)\). (3) The canonical correlation is \(\sqrt{0.40/1.00}=0.632\). (4) Rejection must satisfy the prespecified sequential boundary and the applicable multiplicity rule; nominal significance alone does not establish either condition.