Conjunctive Power as All-Success Probability

Why “Every Effect” Is a Different Success Criterion

A multi-arm trial can produce evidence for some treatment comparisons but not others. The power question therefore depends on what counts as trial success. If a trial is intended to detect every effect in a prespecified set, its success criterion is stricter than simply detecting at least one. Conjunctive power quantifies the first objective: the probability that all the required treatment effects are detected.

The event must be defined before calculating its probability. In particular, the set of comparisons, the statistical decision rule for each comparison, and the parameter configuration under which power is evaluated must be specified. A collection of marginal powers alone does not describe the probability of joint success; the joint outcomes of the tests matter.

Here, “detects an effect” has an operational meaning: the test for that comparison meets its prespecified rejection criterion. For example, \(R_j\) may mean rejecting \(H_{0j}\) under a family-wise error rate procedure. If success also requires a particular direction or a minimum estimated effect, that requirement must be included in the definition of \(R_j\). The probability is evaluated under the specified alternative configuration, not under the global null.

The assumptions are part of the definition’s use, not optional details. The comparisons in \(S\) must be fixed rather than selected after seeing results; the rejection rules must be specified; and the assumed sampling distribution must represent the planned design and parameter configuration. If these conditions change, the probability being calculated may describe a different success criterion or a different trial.

All Successes Versus at Least One Success

For the same set \(S\), disjunctive success occurs if at least one comparison is significant. Its event is the union of the individual success events, \(\bigcup_{j\in S}R_j\). Conjunctive success is their intersection, \(\bigcap_{j\in S}R_j\). These criteria answer different design questions: “Will every required effect be detected?” versus “Will any effect be detected?”

For two or more comparisons, the all-success event is contained in the at-least-one event. Therefore, under the same design, analysis, and parameter configuration, conjunctive power cannot exceed disjunctive power. It is also no greater than the marginal power for any one required comparison: if every comparison succeeds, then that particular comparison succeeds. These are checks on a reported result, not methods for calculating the joint probability.

A useful justification follows directly from event containment. If \(A\subseteq B\), then \(B\) is the disjoint union of \(A\) and \(B\setminus A\). Hence \(P(B)=P(A)+P(B\setminus A)\geq P(A)\). Taking \(A=\bigcap_{j\in S}R_j\) and \(B=R_k\) for any \(k\in S\), or \(B=\bigcup_{j\in S}R_j\), gives the stated bounds. This argument does not require the tests to be independent.

The complement form provides another way to express all-success power:

The complement event is that at least one required comparison fails. This identity is exact regardless of dependence. It does not, by itself, simplify the calculation: determining the probability of at least one failure still requires the joint behavior of the decisions or an appropriate joint probability model.

Using the Joint Decision Distribution

In practice, the joint distribution of the test decisions can be obtained from a model for the trial data, a suitable multivariate calculation, or simulation of the complete planned analysis. Regardless of the computational approach, the target remains the same: evaluate how often the entire prespecified set of success events occurs under the chosen alternative configuration.

The decision procedure matters. If each comparison is tested using a multiplicity-adjusted rule, the events \(R_j\) must represent decisions from that rule, not from a different set of unadjusted tests. Family-wise error control concerns the probability of one or more false rejections under a null configuration; conjunctive power concerns the probability of all required rejections under a specified alternative configuration. Controlling the former does not make the latter equal to any particular value.

The main alternatives to conjunctive power correspond to different questions. Pairwise power is the probability of success for one comparison. Disjunctive power is the probability of at least one success in the family. A summary such as the expected number of significant comparisons also differs: it averages the count of successes and does not tell us directly how often every comparison succeeds. Choose the measure that matches the scientific and operational objective.

Estimating Conjunctive Power by Simulation

When the joint probability is difficult to calculate analytically, a common approach is to simulate complete trial data under the specified alternative, apply the planned analysis to each simulated trial, and count how often every required decision is successful. This estimates the probability in the definition. The simulation must reproduce the planned allocation, outcome model, analysis procedure, and any multiplicity adjustment relevant to the success events.

If \(B\) simulated trials are generated and \(A\) of them satisfy every required success event, the estimated conjunctive power is \(\hat{p}=A/B\). Since this is a Monte Carlo estimate based on a finite number of simulated trials, it has simulation error. Its approximate Monte Carlo standard error is \(\sqrt{\hat{p}(1-\hat{p})/B}\), provided the simulated trials are independent draws from the specified design model. This uncertainty measures simulation precision; it is not uncertainty about the true treatment effects in a real trial.

Diagnostics, Limitations, and a Common Mistake

Before interpreting a conjunctive power calculation, check that the event being counted matches the design objective. Verify that the comparison set was prespecified, that each success event corresponds to the intended test and rejection rule, and that the alternative configuration is stated. For simulation, confirm that every replicate applies the same complete analysis procedure and that the reported denominator includes all simulated trials or explains any exclusions.

A basic numerical diagnostic is event containment: conjunctive power should not exceed any required marginal power or the probability of at least one success. If it does, the events or calculations have likely been defined inconsistently. For simulation, report the number of replicates and consider whether the Monte Carlo standard error is small enough for the precision claimed. More replicates reduce simulation error but cannot repair a misspecified model or an incorrectly coded success rule.

Common mistake: A researcher reports marginal powers of 80% for each of two comparisons and concludes that conjunctive power is 64% by multiplying \(0.80\times0.80\). That multiplication is justified only under an independence assumption for the two success events. In multi-arm trials, comparisons may be dependent, so the product need not equal the probability of both succeeding. The correction is to use the joint distribution implied by the design and analysis, or simulate the complete procedure under the specified model. Do not infer an all-success probability from marginal powers alone.

Conjunctive power is specific to the defined family, success rule, design, and parameter configuration. It does not guarantee that every effect will be detected in a particular trial, nor does it establish that all effects are clinically important. It is a design probability: under the stated assumptions, it describes how often the planned procedure would meet every prespecified statistical success criterion across repeated trials.