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Sequential and Multiple Testing for Dose-Response Analysis

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Sequential and Multiple Testing for Dose-Response Analysis Drug Information Journal, Vol. 34, pp. 591±597, 2000 0092-8615/2000 Printed in the USA. All rights reserved. Copyright 2000 Drug Information Association Inc. SEQUENTIAL AND MULTIPLE TESTING FOR DOSE-RESPONSE ANALYSIS WALTER LEHMACHER,PHD Professor, Institute for Medical Statistics, Informatics and Epidemiology, University of Cologne, Germany MEINHARD KIESER,PHD Head, Department of Biometry, Dr. Willmar Schwabe Pharmaceuticals, Karlsruhe, Germany LUDWIG HOTHORN,PHD Professor, Department of Bioinformatics, University of Hannover, Germany For the analysis of dose-response relationship under the assumption of ordered alterna- tives several global trend tests are available. Furthermore, there are multiple test proce- dures which can identify doses as effective or even as minimally effective. In this paper it is shown that the principles of multiple comparisons and interim analyses can be combined in flexible and adaptive strategies for dose-response analyses; these procedures control the experimentwise error rate. Key Words: Dose-response; Interim analysis; Multiple testing INTRODUCTION rate in a strong sense) are necessary. The global and multiplicity aspects are also re- THE FIRST QUESTION in dose-response flected by the actual International Confer- analysis is whether a monotone trend across ence for Harmonization Guidelines (1) for dose groups exists. This leads to global trend dose-response analysis. Appropriate order tests, where the global level (familywise er- restricted tests can be confirmatory for dem- ror rate in a weak sense) is controlled. In onstrating a global trend and suitable multi- clinical research, one is additionally inter- ple procedures can identify a recommended ested in which doses are effective, which dose. minimal dose is effective, or which dose For the global hypothesis, there are sev- steps are effective, in order to substantiate eral suitable trend tests. For multiple compar- the choice of a dose. Dose differences are isons, there are several approaches for multi- considered effective if they are clinically rel- ple test procedures, some of which are evant and statistically significant. Therefore, described in the following. It is easy to per- multiple comparisons which control the ex- form global tests in group sequential or adap- perimentwise level (familywise type-I error tive interim versions, but it seems that there are only a few methods which combine mul- tiple test procedures with interim analyses. Reprint address: Prof. Dr. Walter Lehmacher, Institut In this paper, some new approaches are pro- fuÈr Medizinische Statistik, Informatik und Epidemiolo- gie der UniversitaÈtzuKoÈln, Joseph-Stelzmann-Str. 9, D- posed for the combination of these multiple 50931, KoÈln, Germany. E-mail: Walter.Lehmacher@ and sequential dose-response analyses, medizin.uni-koeln.de. which can link proof of global trend with 591 592 Walter Lehmacher, Meinhard Kieser, and Ludwig Hothorn proof of efficacy of a chosen dose. Sequential K = 3:H0 = H123 = H13 : µ1 =µ2 =µ3 procedures can be especially useful in this H12 : µ1 =µ2 area because the multiplicity of this problem K = 4:H = H = H : µ =µ =µ =µ makes a reasonable sample size calculation 0 1234 14 1 2 3 4 = difficult. H123 H13 H12 K = 5:H = H = H : TESTS OF TREND 0 12345 15 µ1 =µ2 =µ3 =µ4 =µ5 K groups with increasing dose levels are con- H1234 = H14 sidered, where the smallest dose is often the H = H null-dose (placebo) as a control. A one-way 123 13 H12 layout with nk experimental units tested at the k-th dose level, k = 1,...,K, is consid- As usual, the definition H1...k : µ1 =µ2 = ... ered. All x ki observations are assumed to be 2 =µk is used; the monotonicity implies H1...k mutually independent with xki , N(µk, σ ), = H1k. For each K, these families of K − 1 k = 1,..., K,andi = 1,...,nk. In the fol- lowing, a monotone dose-response relation- hypotheses are closed under intersection. Un- der the assumption of a monotone dose-re- ship is assumed: µ1 ≤ ...≤µK. The global sponse relationship all relevant hypotheses hypothesis is H0: µ1 = ...=µK, and the trend which are necessary to identify an (mini- alternative is *H*: µ1 ≤ ...≤µK with at least mally) effective dose are included. In practi- µ1 <µK. Related trend tests are reviewed in Tamhane et al. (2). Most are based on special cal applications, the recommended dose will contrasts (eg, Helmert contrasts or simple be chosen among the effective doses by care- pairwise contrasts between the maximal and fully considering the magnitude of the ob- minimal dose). Phillips (3) described meth- served benefits and side effects. ods for sample size estimation in the random- For this family of hypotheses the closed ized k-sample design including single-step test procedure is identical to the method of dose-response studies. Recently, a review of a-priori ordered hypotheses, where the the performance of several tests of trend for hypotheses are tested step by step, beginning = dose-response studies was given by Phillips with the global hypothesis H0 H1K , fol- (4). Here a detailed discussion of the perfor- lowed by H1,K−1, H1,K−2,..., H13 and ending mance of these trend tests is not repeated, with the ªsmallestº hypothesis H12. All tests α because in the framework of this paper all are performed at local level , and all hypoth- such test statistics can be used providing one eses are rejected as long as significance at α considers their well-known pros and cons. level is reached; if a test for a hypothesis H1k is not significant at level α, H1k is not rejected, and no further test (for H1,k−1, H1,k−2, MULTIPLE TEST PROCEDURES etc.) is necessary because no further rejection is possible. For a more detailed description Three multiple test procedures which can be of this a-priori method see Hothorn and Leh- extended to designs with interim analyses are macher (5), Kieser and Lehmacher (6), or mentioned here. Maurer et al. (7). For local tests of these elementary hypotheses, all (local) trend tests A-priori Ordered Hypotheses can be chosen (eg, linear contrasts or com- parisons of highest vs. lowest doses), as men- The family of the K − 1 elementary hypothe- tioned earlier. In view of the monitonicity, ses which contain the smallest dose group is these tests should be 1±sided. The assump- considered, that is, H1k: µ1 = ...=µk,k= 2, tion of monotonicity can be omitted, because ...,K. Then for K = 3, 4, 5 one obtains the the a-priori method works generally with following families of hypotheses: each predefined order of hypotheses. In this Sequential and Multiple Dose-Response Analysis 593 case, however, pairwise t-tests should be pre- application of the closed test principle was ferred as local test statistics to trend tests. proposed by Marcus et al. (9). Some other multiple testing methods fail to Each hypothesis H of this closed family control the experimentwise error rate if mon- is tested at local level α; a hypothesis H is otonicity is not given; see Bauer (8). rejected if its local test and all local tests of hypotheses H′ inducing H (ie, H′ , H)are significant at level α; if a local test of H is The Closed Test Procedure not significant at level α, H is not rejected, ″ , The K − 1 elementary hypotheses Hk,k+1 stat- and all tests for H induced by H (ie, H ing the identity of two neighboured dose H″) are not necessary because rejections of ″ groups are considered, that is, H12 : µ1 =µ2, H are not possible. H23 : µ2 =µ3,...,HK−1,K : µK−1 =µK. The fam- Alternatively to the original approach of ily of hypotheses is chosen, which is gener- Marcus et al. (9), a partition hypothesis = > ated by all intersections of these elementary H12/34 H12 H34 can be tested very sim- hypotheses. As usual, the abbreviation Hk k ply by combining the two 1-sided p-values 1 2 with the Bonferroni method or with the ...k : µk = ...µk is used. For example, for m 1 m Fisher combination test; or the test statistics K = 3, 4, 5 the closed systems of hypotheses can be combined to (z12 + z34)/√2 or another are given by the families: weighted sum of these statistics. Rom et al. (10) proposed a modified closed test proce- = = = K 3:H0 H123 H13 dure, where the partition hypotheses are H12 H23 tested with a test for the ªsmallestº partial hypothesis, for example, H is tested by a K = 4:H0 = H1234 = H14 12/34 level-α-test for H12, which is evidently a H123 = H13 H234 = H24 level-α-test for H12/34 as well. As a conse- H := H > H 12/34 12 34 quence, H34 can be tested only after the rejec- H12 H23 H34 tion of H12, and according to the general closed test principle after the rejection of all K = 5:H0 = H12345 = H15 other inducing hypotheses H , H , and so H = H H = H 0 234 1234 14 2345 25 forth. As tests for the simple pairwise com- = > = > H123/45 : H123 H45 H13 H45 parison hypotheses all (local) trend tests (see H12/345: = H12 > H345 = H12 > H35 above) can be chosen. The above mentioned H = H H = H H = H H Rom procedure is ªuniformly betterº than 123 12 234 24 345 35 12/34 the a-priori method, because after the rejec- H12/45 H23/45 tion of a simple pairwise comparison hypoth- H12 H23 H34 H45 esis H1k of the family described in the above a-priori ordered hypotheses section, only ad- For each K, these families of hypotheses ditional partial hypotheses induced by H1k are closed under intersection by definition. can be tested without loss of power concern- They consist not only of simple pairwise ing these K − 1 tests.
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