Comparing Disjunctive and Conjunctive Power

Two Different Meanings of “The Trial Succeeds”

In a multi-arm trial, success can mean that at least one treatment comparison meets its specified criterion, or that every comparison in a specified set meets its criterion. These are different objectives, even when they concern the same arms and the same pairwise tests. Their probabilities are called disjunctive power and conjunctive power, respectively.

The distinction matters when a trial has several candidate treatments. A decision-maker looking for evidence that any treatment may be useful may care most about disjunctive power. A decision-maker who needs every treatment in a specified group to demonstrate benefit may care about conjunctive power. Neither criterion is inherently preferable; the scientific objective determines which one is relevant.

Definition: For a fixed treatment-effect configuration, let \(S_i\) be the event that the prespecified success criterion is met for comparison \(i\), for \(i=1,\ldots,K\). The disjunctive power for these comparisons is \(P(\bigcup_{i=1}^{K}S_i)\), the probability of at least one success. The conjunctive power is \(P(\bigcap_{i=1}^{K}S_i)\), the probability that all \(K\) comparisons succeed. The marginal pairwise power for comparison \(i\) is \(p_i=P(S_i)\), evaluated under the specified treatment-effect configuration.

The events \(S_i\) must represent the actual trial success rules: for example, rejection of the relevant null hypothesis using the prespecified test and threshold. The probabilities are calculated under a stated configuration of true effects, not under a configuration chosen after seeing the results. If a multiplicity procedure sets the rejection thresholds, those thresholds are part of the success definition.

The General Comparison

For any configuration and any dependence pattern among the success events, success on every comparison implies success on at least one comparison. In set notation, \(\bigcap_i S_i\) is a subset of \(\bigcup_i S_i\). Taking probabilities therefore gives the basic ordering.

P\left(\bigcap_{i=1}^{K} S_i\right) \leq P\left(\bigcup_{i=1}^{K} S_i\right).

This inequality does not require independent tests, equal pairwise powers, or equal treatment effects. If \(K=1\), the two criteria coincide. With multiple comparisons they can still coincide in special cases, such as when all success events are the same event, but they commonly differ: an outcome with some successful comparisons and some unsuccessful ones counts toward disjunctive power but not conjunctive power.

There is also a useful comparison with each marginal power. Since the intersection is contained in each event, and each event is contained in the union,

P\left(\bigcap_{i=1}^{K}S_i\right) \leq p_i \leq P\left(\bigcup_{i=1}^{K}S_i\right), \qquad i=1,\ldots,K.

Thus, the all-success probability cannot exceed the power of any one required comparison, while the any-success probability cannot be less than the power of any one comparison. These comparisons do not, by themselves, provide exact numerical values for either joint probability.

Calculating Both Criteria Under Independence

A convenient special case occurs when the success events are mutually independent under the effect configuration being considered. This is a substantive assumption about the joint distribution of the tests, not a consequence of having separately calculated each pairwise power. Under independence, the probability that no comparison succeeds is the product of the probabilities of individual failures. The probability that all comparisons succeed is the product of their individual success probabilities.

P\left(\bigcup_{i=1}^{K}S_i\right) =1-\prod_{i=1}^{K}(1-p_i), \qquad P\left(\bigcap_{i=1}^{K}S_i\right) =\prod_{i=1}^{K}p_i.

These equations are useful for understanding how the criteria respond to the same power profile. The first aggregates opportunities for at least one success; the second requires every success. In particular, a low marginal power has a direct and often substantial effect on the product for conjunctive power.

Worked Example: Equal Pairwise Powers

Suppose a hypothetical trial has three specified treatment-control comparisons, each with marginal power \(p_i=0.80\) under the effect configuration of interest. For this calculation, assume the three success events are mutually independent. Calculate disjunctive and conjunctive power.

For disjunctive power, each comparison has failure probability \(1-0.80=0.20\). Independence gives

P(\text{at least one success}) =1-(0.20)^3 =1-0.008 =0.992.

For conjunctive power, all three comparisons must succeed, so

P(\text{all three succeed}) =(0.80)^3 =0.512.

Under these assumptions, the probability of at least one success is 0.992, or 99.2%, while the probability all three succeed is 0.512, or 51.2%. The same pairwise power of 80% leads to very different trial-level probabilities because the two criteria count different outcomes.

Unequal Powers and Mixed Effect Configurations

Pairwise powers need not be equal. Different treatment effects, sample sizes, variances, or comparison-specific thresholds can produce a power profile in which some arms have high power and others have modest power. The effect configuration is therefore essential to interpretation: “conjunctive power” is always the probability that all comparisons in the stated set succeed under the stated configuration.

Worked Example: Strong, Moderate, and Weaker Effects

Consider a hypothetical three-arm trial in which the marginal powers for comparisons A, B, and C are 0.90, 0.70, and 0.40, respectively. Suppose these values correspond to a strong, a moderate, and a weaker treatment effect, and assume independent success events for this calculation.

The probability that none succeeds is the product of the failure probabilities:

P(\text{none}) =(1-0.90)(1-0.70)(1-0.40) =(0.10)(0.30)(0.60) =0.018.

Therefore, disjunctive power is

P(\text{at least one}) =1-0.018 =0.982.

Conjunctive power is the product of the three success probabilities:

P(\text{all three}) =(0.90)(0.70)(0.40) =0.252.

Thus the any-success probability is 98.2%, but the all-success probability is 25.2%. The weaker comparison does not prevent the trial from having high disjunctive power, because either of the other comparisons can succeed. It does limit conjunctive power, because that comparison must succeed along with both others.

A null or nearly null effect makes the contrast sharper. Under a null effect, the probability of a false positive is controlled by the comparison’s test threshold, rather than being the power associated with a beneficial effect. If the all-success criterion includes that comparison, the trial is unlikely to meet the criterion unless the false-positive probability is large. This is not a defect in the definition: it reflects that the criterion requires evidence of success even for the arm with no effect.

Worked Example: One Null Comparison in the Required Set

Suppose a hypothetical trial requires success on all three specified comparisons. At the fixed test thresholds, assume the marginal powers are 0.80 and 0.70 for two comparisons with beneficial effects, while the third comparison has a null effect and a 0.025 probability of rejection. Assume independent success events for this illustration.

The probability of no success is

$$ (1-0.80)(1-0.70)(1-0.025) =(0.20)(0.30)(0.975) =0.0585. $$

So the disjunctive power is

P(\text{at least one}) =1-0.0585 =0.9415.

The probability of all three succeeding is

P(\text{all three}) =(0.80)(0.70)(0.025) =0.014.

The resulting disjunctive power is 94.15%, while conjunctive power is 1.40%. The latter is low because the required set includes a comparison whose null rejection probability is only 2.5%. This example is hypothetical; the 0.025 null rejection probability is an assumed threshold-specific value, not a universal value for every multi-arm procedure.

Why the Difference Matters in Design

Choosing a power criterion is part of translating a scientific objective into a design target. If a trial is intended to identify whether any candidate treatment merits further study, disjunctive power may describe the relevant chance of obtaining at least one qualifying result. If the decision requires evidence for every treatment in a specified set, conjunctive power is the relevant criterion. Reporting one as though it were the other can make a design appear much more or less capable than it is for the decision at hand.

When the objective is neither “at least one” nor “all,” an at-least-\(k\) criterion provides an intermediate target: success occurs when at least \(k\) of the \(K\) comparisons succeed. Disjunctive power is the at-least-one case, and conjunctive power is the at-least-\(K\) case. A different summary, the expected number of successful comparisons, describes an average count rather than the probability of meeting either endpoint criterion; a large expected count does not alone establish a high probability of all comparisons succeeding.

The independence formulas are not generally automatic in multi-arm trials. For example, two comparisons may use a shared control group, creating dependence between their test statistics and success events. Under dependence, the ordering of conjunctive and disjunctive power remains valid, but the product formulas need not give the correct probabilities. A joint model for the success events, or a design-specific joint probability calculation, is needed for numerical evaluation. The individual marginal powers alone are not enough to justify multiplying probabilities.

Diagnostics and Common Pitfalls

Using the wrong objective. A high probability of at least one positive comparison does not show that every arm is likely to succeed. State explicitly whether the requirement is any, all, or at least \(k\) successes.
Using the wrong set of comparisons. Conjunctive power can change sharply if the required set includes an arm with little chance of success. Define which comparisons count before computing the probability.
Multiplying marginal powers without checking independence. The product formula for all-success power requires independent success events. The complement-product formula for disjunctive power has the same requirement. Shared-control or other dependence must not be ignored without justification.
Confusing power with a result after observing data. Power is a probability under a specified effect configuration and design. It is not the probability that a particular observed trial will have a given conclusion after its data are known.
Interpreting a null arm’s low contribution as a calculation failure. If the criterion requires every specified arm to succeed, a null arm should make success unlikely. That behavior can reveal a mismatch between the scientific question and the chosen criterion.

Key takeaway: For the same specified comparisons, conjunctive power is the probability all succeed and disjunctive power is the probability at least one succeeds. The former cannot exceed the latter. Their numerical values depend on the marginal powers, their joint dependence, and the exact success criterion.

Check Your Understanding

For each problem, identify the criterion and show the calculation. Assume independence only where the question states it.

Two comparisons have marginal powers 0.75 and 0.60, and their success events are independent. Find disjunctive and conjunctive power.
Three independent comparisons each have marginal power 0.70. Find the probability of at least one success and the probability all three succeed.
For any two success events, explain why the probability of both succeeding cannot exceed the probability of at least one succeeding, even if the events are dependent.
Three independent comparisons have marginal powers 0.90, 0.80, and 0.50. Calculate the probability that at least one succeeds and the probability all three succeed.

Solutions. (1) The probability of at least one success is \(1-(1-0.75)(1-0.60)=1-(0.25)(0.40)=0.900\). The probability both succeed is \((0.75)(0.60)=0.450\). (2) Disjunctive power is \(1-(0.30)^3=0.973\); conjunctive power is \((0.70)^3=0.343\). (3) The event “both succeed” is a subset of the event “at least one succeeds,” so its probability cannot be larger; independence is not needed. (4) Disjunctive power is \(1-(0.10)(0.20)(0.50)=0.990\). Conjunctive power is \((0.90)(0.80)(0.50)=0.360\).