intervals simultaneously cover the true τ . If τ a = τ $^{b}$, then H$_{τ}$$_{0}$ is false for every τ$_{0}$ , but with probability at least 1 − α the second interval covers the interval [ τ$_{min}$, τ$_{max}$ ]. Moreover, the first interval for τ is the interval reported in § 4 ignoring versions of treatment, so the investigator has ensured that under either the assumption of a constant treatment effect or the assumption of versions, the two intervals he reports control the family-wise error rate at α , while paying no additional price in power to detect a constant effect for consideration of versions of treatment. The inference is simultaneous in that the two intervals provide the correct coverage for τ under the assumption of a constant effect and also the correct coverage for τ a and τ b under the presence of versions. The theory in this section is derived for one-sided intervals but, as we will see shortly, is easily extended to the two-sided 1 − α intervals described in the previous section. There is a valid, one-sided P -value, say P a $_{τ$_{0}$}$, testing H a τ$_{0}$ against τ a > τ$_{0}$ , so that Pr $^{(}$P a τ$_{0}$ ≤ α ) ≤ α if H a τ$_{0}$ is true. In parallel, there is a valid one-sided P -value P b $_{τ$_{0}$}$, testing H b τ$_{0}$ against τ b > τ$_{0}$ , and a valid one-sided P -value, P$_{τ}$$_{0}$ , testing H$_{τ}$$_{0}$ against τ > τ$_{0}$ . With a slight abuse of notation, write the probability that a random interval I contains a fixed real number τ as Pr ( I ⊇ τ ). Under the assumption, perhaps incorrect, that the there is a single version of the treatment, τ a = τ b = τ , let I − c be the usual one-sided 1 − α confidence interval for τ formed by inverting the test of H$_{τ}$$_{0}$ , so I − c is the smallest set of the form [$_{˜}$ τ , ∞ ) containing { τ$_{0}$ : P$_{τ}$$_{0}$ > α } . If there are no versions of treatment, so τ a = τ b = τ for some τ , then Pr ( I − c ⊇ τ ) ≥ α by the familiar duality of tests and confidence intervals; see Lehmann and Romano (2005, Chapter 3). The investigator would like to report this standard interval I − c for a constant effect, without lengthening it for multiple testing, yet would like to also say something about the possibility that there are versions of treatment with τ a = τ $^{b}$. Of 5.2 Inference when there may or may not be two versions of treatment 13