course, if there are versions of treatment with τ a ̸ = τ b then H$_{τ}$$_{0}$ is false for every τ$_{0}$ and there is no true value of τ for I − c to contain or omit. 

The smallest set of the form [$_{˜}$ τ , ∞ ) containing $^{{}$τ$_{0}$ : P$_{τ}$$_{0}$ > α or P a τ$_{0}$ > α or P b τ$_{0}$ > α } will be denoted I − v . Of course, I − v ⊇ I − c . 

The investigator does not know whether or not there are two versions of treatment, whether or not τ a = τ $^{b}$. The investigator would like to make two inferences appropriate for the two situations, τ a = τ b or τ a ̸ = τ $^{b}$. The investigator would like to make an inference appropriate to this state of ignorance. The investigator says “I do not know whether there are two versions of treatment, whether or not τ a = τ $^{b}$; however, (i) if there are not versions of treatment so that τ a = τ b = τ , then I − c ⊇ τ , and (ii) whether or not there are two versions of treatment, even if τ a ̸ = τ $^{b}$, then I − v ⊇ τ$_{min}$ ; moreover, this method produces two true hypothetical statements with probability at least 1 − α .” Statement (ii) cost nothing, in the sense that I − c is the usual one-sided confidence interval for τ assuming there are not versions of treatment, yet both statements hold jointly without multiplicity correction. This is established in the following proposition. 

Proposition 1 (i) If there is only one version of treatment, τ a = τ b = τ , then Pr ( I − v ⊇ I − c ⊇ τ ) ≥ 1 − α . (ii) In any event, whether there are two versions of treatment, τ a ̸ = τ $^{b}$, or only a single version, τ a = τ b = τ , we have Pr ( I − v ⊇ τ$_{min}$ ) ≥ 1 − α . 

Proof. By the definitions of I − v and I − c , we have I − v ⊇ I − c . Then (i) follows because, if there are not versions of treatment, τ a = τ b = τ , then I − c is a 1 − α confidence interval for τ and Pr ( I − v ⊇ I − c ⊇ τ ) ≥ 1 − α . If there are not versions of treatment, τ a = τ b = τ , then τ$_{min}$ = τ , so Pr ( I − v ⊇ τ$_{min}$ ) ≥ 1 − α , as required for (ii). So suppose there are two versions of treatment. If τ a = τ$_{min}$ < τ$_{max}$ = τ $^{b}$, then τ$_{min}$ / ∈ I − v implies P a τ ≤ α which occurs with probability at most α . If τ b = τ$_{min}$ < τ$_{max}$ = τ $^{a}$, then τ$_{min}$ / ∈ I − v implies P b τ b ≤ α which 

