Why Specify the Target Set?
A multi-arm trial may enroll several treatments, but its design objective need not require success for every one of them. Investigators might care about whether at least one of two named treatments succeeds, or whether both of those treatments succeed, even when other treatments are also included in the trial. The power calculation must match that objective: it must use the success events for the named treatments, not automatically those for every enrolled arm.
The earlier material on disjunctive and conjunctive power defines the general any-success and all-success criteria. Here the essential extra step is to state exactly which arms count. That set should be specified as part of the design objective, rather than selected after looking at the results.
The events \(E_j\) are defined by the trial’s comparison and success rule. For example, \(E_j\) might mean that the comparison of treatment \(j\) with control meets a specified significance criterion. The rule, including any testing threshold, is held fixed when calculating power. The probabilities are evaluated under a stated alternative configuration; power is not a single number independent of assumed treatment effects and the trial design.
The set \(S\) describes the objective, while \(T\) describes the enrolled treatments. An arm in \(T\setminus S\) does not contribute an event to either subset criterion. It may still affect the design or the joint distribution of the test results, but its success is not required for the stated subset objective.
Calculating Power From Joint Success Probabilities
When success events are dependent, subset power is determined by their joint distribution. Marginal power \(P(E_j)\) gives the probability of success for one comparison, but it does not in general determine the probability of at least one success or of success on every target arm. The earlier material on joint power establishes this distinction; the subset calculation applies it only to events indexed by \(S\).
For any dependence structure, the complement of “at least one target success” is “no target success.” This gives a useful identity. If the target events are independent, the probability of no success is the product of their individual failure probabilities. Similarly, under independence, the probability that all target events succeed is the product of their marginal powers.
The independence condition matters: the product formulas are not general formulas for multi-arm trials. If dependence is present, use the joint success distribution or a valid joint model. Before calculating, check that the event definitions correspond to the intended comparison rules, that \(S\) is the specified target set, and that the dependence assumptions match the design and analysis.
Worked Example: Any Success Among Two Named Treatments
Consider a hypothetical trial with four treatment arms, \(A,B,C,D\), each compared with control. The prespecified target set is \(S=\{A,C\}\); treatments \(B\) and \(D\) are enrolled but are not part of this objective. Under a specified alternative, suppose the joint probabilities of success on the two target comparisons are as follows.
| Success on A | Success on C | Probability |
|---|---|---|
| Yes | Yes | 0.54 |
| Yes | No | 0.14 |
| No | Yes | 0.12 |
| No | No | 0.20 |
The entries sum to \(0.54+0.14+0.12+0.20=1.00\), so they form a probability distribution for the target success outcomes. The subset disjunctive event occurs in the first three rows. Therefore,
The disjunctive power for the stated target set is \(0.80\), or 80%. The marginal powers are \(P(E_A)=0.54+0.14=0.68\) and \(P(E_C)=0.54+0.12=0.66\). Adding those marginal powers would give \(1.34\), which is not a probability and double-counts the \(0.54\) probability that both succeed. Equivalently, the complement calculation gives \(1-P(E_A^c\cap E_C^c)=1-0.20=0.80\).
For comparison, suppose that in the \(0.20\) probability of no target success, treatments \(B\) or \(D\) succeed with probability \(0.08\), while no treatment succeeds with probability \(0.12\). Then the probability of at least one success among all four arms is \(0.80+0.08=0.88\). That is a different objective from the \(0.80\) power for \(S=\{A,C\}\).
Conjunctive Power and the Role of Dependence
For a conjunctive objective, every target event must occur. The probability is therefore the probability of the joint-success outcome, not an average of the marginal powers and not the probability that a typical target arm succeeds. A joint outcome table makes the target event explicit and avoids making an unsupported independence assumption.
Worked Example: All Three Treatments in the Target Set Must Succeed
Suppose a hypothetical trial enrolls treatments \(A,B,C,D\), but its conjunctive objective concerns only \(S=\{A,B,C\}\). Under one specified alternative, assume the following joint distribution for the three target success events. The final column identifies each outcome by the arms that succeed; “none” means none of \(A,B,C\) succeeds.
| Target arms succeeding | Probability |
|---|---|
| A, B, and C | 0.36 |
| A and B only | 0.12 |
| A and C only | 0.08 |
| B and C only | 0.10 |
| A only | 0.10 |
| B only | 0.08 |
| C only | 0.06 |
| None | 0.10 |
The probabilities sum to \(1.00\). The conjunctive event is the single outcome in which all three target arms succeed, so the subset conjunctive power is \(0.36\), or 36%. For context, the marginal powers are
These marginal values alone do not establish that the all-three power is \(0.36\); that value came from the joint distribution. The probability that at least one of the three succeeds is \(1-0.10=0.90\), so the disjunctive and conjunctive powers for the same target set differ substantially. Treatment \(D\) is not included in either calculation.
What Independence Would—and Would Not—Allow
If target success events are independent under the specified alternative, the marginal powers are enough to calculate the two subset criteria. The formulas are convenient, but independence must be justified for the success indicators, not merely assumed because the treatment arms have different names. In practice, shared trial components or analysis rules can induce dependence. If independence is not established, the product calculation may be wrong even when every marginal power is correct.
Worked Example: Comparing a Subset Criterion With an All-Arm Criterion
Suppose a hypothetical four-treatment trial has independent success events, with marginal powers \(P(E_A)=0.70\), \(P(E_B)=0.60\), \(P(E_C)=0.50\), and \(P(E_D)=0.40\). The design objective names \(S=\{A,C\}\). The assumptions for the following product calculations are that these success events are independent under the alternative and that the stated marginal powers apply to the same fixed success rules.
For the subset disjunctive objective, at least one of \(A\) and \(C\) must succeed. By the complement rule and independence,
For the subset conjunctive objective, both must succeed:
If instead success were required for at least one of all four enrolled treatments, the corresponding probability would be
If success were required for all four, the probability would be
Thus \(0.85\) is the power for the stated “at least one of \(A,C\)” objective; \(0.964\) answers a different question about any success among all four treatments. Likewise, \(0.35\) and \(0.084\) correspond to different conjunctive targets. None of these values is the probability that each individual comparison succeeds.
Checks, Alternatives, and Limitations
A reliable calculation begins with the objective, rather than with a formula. State \(T\), state the fixed target set \(S\), name the event \(E_j\) for every target comparison, and write the success event as a union or intersection. Then obtain the probability using the joint distribution or a justified assumption such as independence. As a diagnostic, verify that the reported probability lies between 0 and 1 and that its event includes exactly the target arms named in the objective.
The main alternative is to define the criterion over all enrolled arms, \(T\), rather than over a strict subset \(S\). That is appropriate when the scientific or decision objective genuinely concerns all treatments. Other success criteria, such as requiring at least a specified number of successes, answer different questions; they should not be described as subset disjunctive or conjunctive power unless their event definition actually matches those criteria.
Subset power is specific to both the target set and the assumed alternative configuration. A high disjunctive power does not imply high power for every target arm, and a high conjunctive power for one subset does not imply the same power for a larger set. Also, excluding an arm from the success criterion does not guarantee that it has no effect on the design: enrollment, allocation, or testing rules may affect the joint probabilities for the target events.
Check Your Understanding
Answer each question by identifying the target set and success event before calculating.
- A trial enrolls \(A,B,C\), and the objective is at least one success among \(A\) and \(C\). Write the event. Does success on \(B\) alone meet it?
- For \(S=\{A,B\}\), suppose \(P(E_A)=0.60\), \(P(E_B)=0.50\), and \(P(E_A\cap E_B)=0.35\). Calculate subset disjunctive and conjunctive power.
- For \(S=\{A,C\}\), marginal powers are \(0.80\) and \(0.70\). Can subset conjunctive power be calculated by multiplying them without further information? Explain.
- Four independent target events have marginal powers \(0.80, 0.70, 0.60, 0.50\). Calculate the disjunctive power for the subset containing only the first and fourth events.
Solutions. (1) The event is \(E_A\cup E_C\); success on \(B\) alone does not meet it. (2) Disjunctive power is \(P(E_A)+P(E_B)-P(E_A\cap E_B)=0.60+0.50-0.35=0.75\); conjunctive power is \(0.35\). (3) No. The product \(0.80(0.70)\) is valid only if the two success events are independent; otherwise their joint probability is needed. (4) By independence and the complement rule, \(1-(1-0.80)(1-0.50)=1-(0.20)(0.50)=0.90\).