Drug Information Journal - March 2009 - (Page 178) 178 BIOSTATISTICS Lawrence, Li If it is desired to compare the group means of a continuous endpoint, let θ1 be the difference between the means of group 2 and group 1 and let θ2 be the difference between the means of group 3 and group 2. The three alternative hy1 2 potheses are are HA : θ1 > 0, H A : θ2 > 0, and H 3 : A θ1 + θ2 > 0. The procedure also applies in many other situations, such as comparing proportions or comparing slopes or changes from baseline for longitudinal measurements; the procedure applies to any situation where the hypotheses can be written as above. 1 2 3 1 Each of the intersections H0 I H 0 I H 0, H0 I 2, H1 I H 3, and H 2 I H 3 are rejected if Z > z . H0 0 0 0 0 3 α This is an α-level test for each of these intersec3 tions because it is an α-level test for H 0, that is, max P[Z3 > zα ] = α and each intersection is θ1 + θ2 ≤ 0 DESCRIPTION OF THE PROCEDURE Suppose there are three independent random samples from normal distributions with known variance: X1,1, X1,2, . . . , X1,n1 N(μ1,σ2); X2,1, X2,2, . . . , X2,n2 N(μ2,σ2); X3,1, X3,2, . . . , X3,n3 N(μ3,σ2). Define θ1 = μ2 − μ2 and θ2 = μ3 − μ2. 1 Three null hypotheses to be tested are: H0 : 2 3 θ1 ≤ 0, H 0 : θ2 ≤ 0, and H 0 : θ1 + θ2 ≤ 0. The tests are all one-sided and the type I error rates considered are all one-sided. Define the three sample means Xj = 1 nj 3 contained in the parameter space for H 0. The only intersection for which this statement is not 1 2 1 obvious is H0 I H0. For that intersection, if H0 2 and H0 are both true, then θ1 and θ2 are both less than or equal to 0. The sum of two numbers that are each less than or equal to 0 is also less 3 than or equal to 0, that is, H 0 is also necessarily j true. Finally, the individual hypotheses, H0 ( j = 1,2), are rejected as a step in the closed test procedure if Zj > zα. This closed test procedure that rejects each individual hypothesis provided each intersection involving that hypothesis is rejected is logically equivalent to the simpler procedure described in the preceding paragraph. Since it is a closed test procedure, the strong familywise error rate is controlled at level α (8,9). POWER OF THE PROCEDURE The procedure is consistent in the sense that if at least one of θi : i = 1 or 2 is positive and the other is nonnegative or negative and smaller in absolute value, then the probability of rejecting a false null hypothesis goes to 1 as the sample sizes tend to infinity. It dominates the two sequential procedures that start with testing 3 H 0 and then tests the other two hypotheses in either order. Furthermore, the asymptotic relative efficiency (compared to the uniformly most powerful test for the individual hypothesis) is close to 1 under most reasonable circumstances. In other words, this procedure controls the strong familywise error rate, but loses very little in terms of power over the procedure that tests each hypothesis at level α without controlling the familywise error rate. We will again consider the simplest case of comparing normal means with known common variance. Let n be an indexing variable that represents the total sample size; let n1, n2, and n3 be the per-group sample sizes and assume n lim i = pi where p1, p2, and p3 are positive n→∞ n constants that add up to 1. The probability that ∑ X j,i i =1 nj and the test statistics Z1 = X2 − X1 X3 − X2 , Z2 = , 1 1 1 1 σ + σ + n1 n2 n2 n3 , and Z3 = X3 − X1 . 1 1 σ + n1 n3 The proposed procedure is to start by comparing Z3 to the upper α quantile of a standard normal distribution, zα. If Z3 ≤ zα, then stop and accept all three null hypotheses. Otherwise, reject 3 H 0 and continue to test the remaining hypothe1 ses without any adjustment, that is, reject H0 if 2 if Z > z . This is different Z1 > zα and reject H 0 2 α from the ordinary sequential test procedure be1 2 cause both H0 and H 0 are tested without specifying an order for testing them. This procedure is a closed test procedure.
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