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THE ANALYSIS of DATA from 2 X 2 CROSSOVER TRIALS with Baseline MEASUREMENTS

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THE ANALYSIS of DATA from 2 X 2 CROSSOVER TRIALS with Baseline MEASUREMENTS THE ANALYSIS OF DATA FROM 2 X 2 CROSSOVER TRIALS WITH BASElINE MEASUREMENTS Joe DiGennaro, Merrell Dow Research Institute William C. Huster, Merrell Dow Research Institute One remedy for this problem is to measure the outcome during both a 'run-in' and a 'washout' Kenward and Jones (1987, Statistics in Medicine. 6:911~ 926) have proposed a unified framework for the analysis period. The run-in period precedes and the of data from 2 x 2 crossover trials which include baseline washout follows the first treatment period. These measurement:». Und_er this framework, a simple analysis baseline measurements as they are called can based on ordInary least-squares (OLS) estimators was provide a within-subject test of the treatment-by­ developed and shown to be applicable whether the data period interaction. were normal or not. Generalized least-squares (GLS) estimators were also derived and shown to be related to In an expository paper, Kenward and Jones (1987) covariance analysis. The OLS and GIS estimators do not have proposed a unified framework for the rely on any assumptions about the covariance structure of analysis of crossover studies with baseline the repeated measurements. measurements. This framework is based on a linear model for the expectations of the The OLS estimators can be obtained using the SAS observations and uses either ordinary least-squares procedure GLM (1985, SAS User's Guide: Statistics). The (OLS) or generalized least-squares (GLS) GLS estimators are a liHle more difficult to obtain using estimators. The GLS estimators are similar to the SAS, requ~ing utilization of PROC IML (1985, SAS/IML analysis of covariance. The OLS and the GLS Users GUide). The SAS code and output for these 2 estimators do not rely on any assumptions about procedures are presented using as an example analyzed the covariance structure of the repeated measures. the dataset in Kenward and Jones. A brief review of the linear model is presented in 1. INTRODUCTION Section 2. The OLS estimators are derived using the SAS procedure GLM (SAS Institute Inc., 1985) A crossover study is a repeated measures design in in Section 3. The GLS estimators require which each subject receives a different treatment in manipulation of the data using SAS PROe IML each time period. A major component of variation (SAS Institute Inc., 1985) and are presented in in clinical trials is the difference in response Section 4. The methods of analysiS will be between subjects. Because crossover studies illustrated by using the set of data analyzed by provide. a comparison of treatments using Kenward and Jones. These data are from a 2 x 2 observatIons on the same subject, the between­ crossover study of the effect of single doses of two subject comp?nent is eliminated. Therefore, active treatments (A and B) for asthma on resting crossover tnal.!! make efficient use of the forced expiratory volume in one second (FEV 1)' information from a subject and are particularly Sixteen subjects with mild to moderate asthma important when experimental subjects are costly or were randomized to either sequence AB or BA. scarce. The study had a run-in as well as a washout period. The observed data are given in Table 1. A particular order of treatment administration in a crossover study is called a sequence. Usually, subjects are randomized to a particular sequence. For the simplest crossover study, a 2 x 2 design, there are 2 treatments (call them A and B) given in 2 periods. There are 2 different sequences: A followed by Band B followed by A. The treatment effect is estimated by taking the differences of the pairs of observations for each subject. This estimate is unbiased if there is no treatment-by­ period interaction. Treatment-by-period interaction can have many causes. For example, it may be caused by residual drug present in the second time period because of a too-short washout period or a priming effect of the treatment on the experimental subjects causing a rebound to a worse (or better) disease state. The test for this interaction is not very powerful because it utilizes the between-subject component of variation. For this reason, crossover trials are reconunended only when the probability of this interaction is very low. 1102 Table 1. 2 x 2 Crossover Trial of Single Oral Doses of Two The parameters are as follows: IA is the overall Active Drugs (A and B) in Patients with Bronchial Asthma: mean, 'l'j the jth period effect (where '1'1 + '1'2 + '1'3 + Forced Expiratory Volume in One Second (FEV!, liters) '1'4 = 0), 7 is the sequence effect, f' is the treatment effect, 8 is the first-order carryover effect onto the Sequence 1 (AB) washout and A is the second-order carryover effect onto the second treatment period. The parameter ). can also be interpreted as the treatment-by­ period interaction. Subject Run·in Perind 1 Washout Period 2 3. OLS ESTIMATORS 1 1.09 1.28 1.24 1.33 As a simple analysis of crossover data with 2 1.38 1.60 1.90 2.21 baselines, Kenward and Jones suggest using the 3 2.27 2.46 2.19 2.43 OLS estimators of the carryover and treatment 4 1.34 1.41 1.47 1.81 effects in a sequential procedure consisting of three 5 131 1.40 0.85 0.85 tests. The OLS estimators can be derived easily 6 0.% 1.12 1.12 1.20 and are based. on contrasts among the four means 7 0.66 0.90 0.78 0.90 from each sequence. Test 1 is a test for presence of 8 1.69 2.41 1.90 2.79 the first order carry--over effect onto the washout, 8, under the full model. The estimator of 8 is: Sequence 2 (AB) Subject Run-in Period 1 Washout Period 2 Use of this estimator corresponds to testing the equality of the two sequences with respect to the 1 2.41 2.68 2.13 2.10 change in baseline measurements. Test 2 is a test 2 3.05 2.60 2.18 2.32 for presence of the treatment-by-period interaction, 3 1.20 1.48 1.41 1.30 A, under the assumption that 8=0. The estimator 4 1.70 2.08 2.21 2.34 of A is: 5 1.89 2.72 2.05 2.48 6 0.89 1.94 0.72 1.11 7 2.41 335 2.83 3.23 ! (>;11. -Yll. +x12. -YI2. -x21. -tJ21. -<22. +Y22.)· 8 0.96 1.16 1.01 1.25 This is the standard estimator of the treatment-by­ period interaction except that the treatment minus 2. TIlE UNEAR MODEL baseline differences are used instead of the treatment measurements. Test 3 is a test of the Let the four observations for subject k of sequence i direct treatment effect, 'T, under the assumption be xilk, Yilk' xi2k' Yi2k, in order of collection, that A, 8=0. The estimator of 'T is: where k=l,. .. ,ni' i=l,2. That is, the baseline measurements are represented by Xijk and the treatment measurements by~. The expectations of these observations can be 'tten as in Table 2. This is the usual estimator of the direct treatment effect used in a 2 x 2 crossover trial without baselines. Table 2. Expectations of the Responses in Each Sequence Period Sequence 1 Sequence 2 Run-in E(xllk) = P-7+<1 E(x21k) = P+7+<1 First Treatment E(Yllk) = P-7+"2-< E(Y21k) = P+7+"2+< Wash-out E(xI2k) = P-7+<3- 8 E(x22k) = 1'+7+'3+8 Second Treatment E(YI2k) = P-7+"4+<-}. E(Y22k) = P+7+"4-<+}. 1103 and H the test for either 8 or). is significant (p < .10" for 1 n. example), then the estimator of Tis: ~1 ( -) ( - )' S i~ --- "" Y -Y Y -Y . ik i ik i n - 1 k=l Also, the variance<ovariance matrix of ~1' is Use of this estimator corresponds to analyzing the treatment minus baseline differences in the first (Xi 5;! XI)"!' period only. Chi (1989) has demonstrated that by combining Figure 1 presents the SAS code for the 3-step intra- and inter-block estimators, as is done in an sequential procedure and Figure 2 presents the incomplete block design, we will obtain the output from SAS using the asthma dataset as an generalized least squares estimators. example. The sequence of hypothesis tests that was applied 4. GLS ESTIMATORS to the asthma dataset using the OLS estimators is now applied using the GLS estimators. The SAS The following description of the derivation of the code is presented in Figure 3 and the SAS output GLS estimators is abstracted from the Appendix of produced by this code is in Figure 4. By Kenward and Jones' 1987 paper. Let comparison with the results from the OLS procedure, there is some improvement in the Yik = [xili<- Yilk' "i2k' Yi2kl' precision of the estimate of the treatment effect: be the four measurements for subject k of sequence i. Define Estimaters and 1heir Standard Errors from the 01.5 and GI.5 Models Y= [9'1',9'2'1', 01.5 GI.5 and p = [. 'I '2 '3 7 r 8 ~I', where the parameters in fJ were defined in Section TEST!: ~ -.045 .004 2. Then. when all the parameters are in the model Se(~) .092 .086 (Kenward and Jones call this a saturated model) TEST 2: ~ .071 .094 5e(~) .140 .124 A where the elements of XfJ are just the expectations TEST 3: 7 A .088 .082 in Table 2 with X containing the appropriate Se(7) .044 .028 coefficients.
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