113. Let R1 be a rejection region with signi cance level a for testingH01: u1 versusHa1: u1, and let R2 be a level a rejection region for testing H02: u 

2 versus Ha2: u 2, where 1 and 2 are two disjoint sets of possible values of u. Now consider testing H0: u  1  2 versus the alternative Ha:

u12. The proposed rejection region for this latter test is R1  R2. That is, H0 is rejected only if both H01 and H02 can be rejected. This procedure is called a union–intersection test (UIT).

a. Show that the UIT is a level a test.

b. As an example, let mT denote the mean value of a particular variable for a generic (test) drug, and mR denote the mean value of this variable for a brand-name (reference) drug. In bioequivalence testing, the relevant hypotheses are H0:

mT /mR dL or mT /mR  dU (not bioequivalent)

versus Ha: dL  mT /mR  dU (bioequivalent). dL and dU are standards set by regulatory agencies;

for certain purposes the FDA uses .80 and 1.25  1/.8, respectively. By taking logarithms and letting hln(m), tln(d), the hypotheses become H0: either hT  hR tL or tU versus Ha: tL  hT  hR  tU. With this setup, a type I error involves saying the drugs are bioequivalent when they are not. The FDA mandates a  .05.

Let D be an estimator of hT  hR with standard error SD such that T  [D  (hT  hR)]/SD has a t distribution with v df. The standard test procedure is referred to as TOST for two onesided tests, and is based on the two test statistics TU  (D  tU)/SD and TL  (D  tL)/SD. If v  20, state the appropriate conclusion in each of the following cases: (1) tL  2.0, tU1.5;

(2) tL1.5, tU2.0; (3) tL2.0, tU2.0.