will have three arms when matched sets include both versions of control, which might complicate inference. However, we only use the matching with sets including both versions of control to conduct inference under the assumption that the versions are irrelevant, and thus the three arm randomization distribution collapses to the simpler two arm design. Consider the two null hypotheses about additive effects for the two versions of control, H a τ$_{0}$ : δ a ij = τ$_{0}$ , ∀ i, j and H b τ$_{0}$ : δ b ij = τ$_{0}$ , ∀ i, j . Here, H a τ$_{0}$ might be true when H b τ$_{0}$ is false, or conversely. Define H$_{τ}$$_{0}$ to be the hypothesis that both H a τ$_{0}$ and H b τ$_{0}$ are true, that is, H$_{τ}$$_{0}$ : δ a ij = δ b ij = τ$_{0}$ , ∀ i, j , so the two versions of control yield the same effect τ$_{0}$ and need not be distinguished. By the definition of H$_{τ}$$_{0}$ , if either H a τ$_{0}$ or H b τ$_{0}$ is false, then H$_{τ}$$_{0}$ is false; that is, if there are two versions of treatment or control with different effects, then there is not a constant effect. Suppose there are two versions of a constant additive treatment effect, δ a ij = τ a and δ b ij = τ b for every i, j . Let τ$_{min}$ = min $^{(}$τ $^{a}$, τ b ) and τ$_{max}$ = max $^{(}$τ $^{a}$, τ b $^{)}$. If τ a = τ b = τ , then τ$_{min}$ = τ and τ$_{max}$ = τ , so the versions do not matter. Our approach in § 5.2 is to build two confidence intervals, one interval for τ and another interval designed to contain [ τ$_{min}$, τ$_{max}$ ]. If there is no need to consider versions of treatment or control because Y$_{ij}$ (0 , a ) = Y$_{ij}$ (0 , b ), implying that τ a = τ b = τ , then with probability at least 1 − α , both It is straightforward to test H a τ$_{0}$ or H b τ$_{0}$ using the methods in § 4 simply by restricting attention to controls of one type or the other. These tests will be based on a smaller sample size than the test in § 4 because not all of the controls are used. Moreover, if H$_{τ}$$_{0}$ , H a τ$_{0}$ and H b τ$_{0}$ are each tested at level α , then the chance of at least one false rejection would typically exceed α unless something is done to control the family-wise error rate. Understandably, an investigator would like to avoid weakening the inference about H$_{τ}$$_{0}$ by virtue of considering H a τ$_{0}$ and H b $_{τ$_{0}$}$, and the question is how to achieve the investigator’s goals. 12