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

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. If t parameters are assumed equal to 0, the resultant model (unsaturated) is represented as If either of the tests for carryover is significant, an analysis of the first treatment period is where Xl results from deleting the columns of X recommended. In this case, the GLS estimator of which correspond to the t parameters set equal to the treatment effect (T) under the saturated model O. The vector PI results from deleting the trows corresponds to the OLS estimator. from p. 5. CONCLUSIONS Since the variance-covariance matrix ofy is unknown,. the GiS estimator of PI is The unified framework proposed by Kenward and Jones for the analysis of 2 x 2 crossover trials with ~I = (Xi 5;1 XI)"I (Xi 5;ly) baselines can be easily programmed in SAS. The OLS estimators derived from PROC GLM provide where a simple but flexible analysis. The F statistics produced by PROC GLM assume normality of the observations but standard nonparametric tests can be used if this assumption is in doubt. The use of the GLS estimators derived from PROC IML is equivalent to the use of analysis of covariance. Neither procedure makes any assumptions about the underlying covariance structure of the four measurements.

n + n - 1 2

1104 References TEST2: PERFORM mANGE FROM BASELINE ANAL VSI5. Chi, E.M. (1989). ''Recovery of Inter-Block Information in Cross-Over Trials." Submitted to Biometrics. Dec. 1989. DATA DBL;SET ASTI-lMA; RETAINXt X2DY; IF PERIOD=} mEN Xl=Y; Kenward, MG, Jones, B. (1987). "The Analysis of Data IF PERIOD--2.1HEN DY",Y·Xl; from 2 x 2 Cross-over Trials with Baseline IFPERIOD=3 mEN X2",Y; Measurements." Statistics in Medicine 6:911-926. IF PERIOD=4 mEN DY=Y-X2; RENAME SEQ;-LAMBDA; DROPXIX2; SAS Institute Inc. (1985). SAS User's Guide: Statistics, Version 5 Edition. Cary, NC: SAS Institute Inc. DATA DUL; SET DBL; IF PERIOD=2 OR PERIOD=4; SASlnstitute Inc. (1985). SAS/IML User's Guide. Version TITLE4 'TEST 2: TEST FOR TMT·PERIOD INTERACTION USING 5 Edition. Cary, NC: SAS Institute Inc. CHANGE FROM '; TInES' BASELINE (THAT IS, USE PER2 - PERI AND PER4- PER3)'; FIGURE 1 SAS CODE FOR TIlE OIS ESTIMATORS PROCGLM; cr.ASS LAMBDA SUB TMT PERIOD; TITI.E1 'ANALYSIS OF ASTI-IMA DATA INKENWARD/JONES (1987)'; MOOEL DY=LAMBDA SUB(LAMBDA) 1MT PERIOD; TITI.E2 '-> DERIVA TIrn OF THE OLS ESTIMATORS USIN'G PROC GLM <-'; TFST H=LAMBDA E:SUB(LAMBDA); ESTIMATE'LAMBDA'LAMBDA-ll; OPTIONS NODATE NONUMBER LINESIZE=80; RUN; PROCFORMAT; --_.. _ .... _ ...... - VALUETMTA 1='A' ....TFST3: PERFORM... GRIZZLE ANALYSIS... _-_.; ON 1MT PERIODS. -1 ",'B'; __._ __ ...... DATA 1MT; SET ASTI-IMA; ----_.. __ _._ _. - IF PERlOD=2 OR PERIOD=4; INPUT THE DATA (EACH PATIENT WILL HA VE4__ RECORDS). OEFINE PERIOD AND 1MT...... VARIABLES. _____ .0_ ... _; TITLE4 'TEST 3: PERFORM USUAL ANALYSIS ON TREATMENT IGNORING BASELINE'; TITI.E5' (USEONLY PERIODS 2 AND 4) '. DATA ASfHMA; INPUT SUB SEQ Y@; PROCGLM; PERIOD",1; OlJ'l1'UT; CLASS SEQ SUB TMT PERIOD; INPUT Y@;PERIOD=2;IFSEQ=1 THEN TMT=I; EISE 1MT=-I; OUTPUT; MODEL Y:SEQ SUB(SEQ) TMT PERIOD; INPUTY@;PERIOD=3;OUTPUT; ESTIMATE '1MT EFFECf(2"TAU)' TMT -11; INPUTY; PERIOD=4;IFSBQ=1 THENTMT=-I; ELSE1MT=l; OUTPUT; LSMEANS 1MT/PDIFF srDERR; CARDS; FORMAT 1MT TMTA.; 11 U)91281.24 1.33 II 21 1381.60 1.90221 31 2272.46 2192.43 411.341.411.471.81 F!GUl

1105 FlGURBl (Omt'd) SASOUlPUTroa nIB OLS ESIlMAlOllS FIGURE 3 lliST2: TBST lOR 'IMT'PRlUOO INTIiRACI'ION USJNC CHANGIl FROM BASliLINEmtAT IS.USEiPBlll· PERt ANDPmU.PHR3) SASCODE FOR THE GLS ESTIMATORS

DIirENDENTVARIABLE: m' THIS MACRO WILL CALCULATE THE GIS ESTIMAlORS FOUND If" KENWARD &JONES'(l987) PAPER ON 2X2 CROSSOVER DATA. SOURCE OF SUM OF SQUABES MEANSJUARE 'VALUE DATASET", DATASET TO BE ANALYZED MOOEL 17 U"""'" .,"""" 5EQ:SFl;lUENCE ERROR 1).61613150 '" RESPl ;: BASELINE RESPONSE " 0.""""'" Pb' RHSP2;: PERIOD 1 RESPONSE CORRECTED TOTAL at 0.0144 RFSP3 = WASHOUT RESPONSE RESP4 = PERIOD 2 RESPONSE R"""ARt< c.v. ROOn.","""""" OY """" NOTE: THE4 RESPONSES AND THE SEQUENCE MUST BE FOUND 0.8005J8 7~.2b02 ON TIlE SAME RECORD. '.>Jm51' 0-"250000 ...... __ .. _-_ ...... _.... _ ...... _...... ; SOURCIi OF TYPElSS FVALUE Pb,

LAMBOA 0.0.061250 %MAcrO MACGLSCDATASET$EQ.RESPl,RESP2.RESP3,RESP4); 5UB(LAMBDA) "">57 ""16'""'" nIT ", "'"""'"0.137812.50 , Il£m4SCOI PERlOO "'UO .-""" SOURCE OF ,ypm,. FVALUIi Ph' CALCULATE THE SAMPLE DISPERSION MATRIX FIRST UMBOA 0.040612.50 0.3S30 SUB(t.AMB0A) 2.20198750 ,,,"" O.oll& ...... --...... ; nIT ", _ U37812S0 __ _--_ _ PERIOD 1 """"" .""' """O.1~3 TESTS OF HYPOTHESI1S USING lHETYPE mMS FOR Slf6Cl.AMBDAl AS AN IiR.RO~ TERM - CALCULATE AVERAGES OF 4 MEASURES BY SEQUENCE ...... ; SOURCE OF 1YPEmss FVAUJE PR>' PROCSORT DATA=&DATASET; BY &SEQ; LAMllDA 0D4061250 0.26 0.6193 PROC MEANS NOPRINT; BY &:SEQ; VAR &RESl'1 &RESl'2 &RESP3 &RESP4; TFORHll: PR> m SIDERRORQF OUTPUT OUT=AVG MEAN=XlBAR YIBAR X2BAR Y2BAR; PARAMETER ESTIMATE P Al

ANAL YSISOF ASI"HMA DATA IN KIiNWARDIJONllS (19(17) PROC CORR DATA=SEQI COY NOCORR NOPRINT -> DERlVAnON OFTImOIS ESTIMATORS U5n

MOOEL 17 14.9244lSOO 0.""", ..... 1. FIND THE NUMBER OF OBS FOR EACH SEQUENCB "" - 2. MERGE IT ONTO TIlE VAR-COV MATRIX DATASET ..wR " 0..86017500 Cl.06144107 PR>F ...... 3. CALCULATE TIlE POOLED SAMPLE DISPERSION MATRIX; CORRECtEDrotAL 31 15.78460000 O.£OO} DATA COY; MERGE COVI COV2;

R.sQUARE CV. ROOTMSE YMEAN DATA NS(KEEP=Nl N2) COV(DROP=Nl N2);

13.1672 1.882S0000 SETCOV; """" """"" IF _lYPE_ = 'COY' THEN OUTPUT COY; IF _lYPE_='N' TIlENDO; SOURCE 0' lYPEI" FVAlUE PR>' Nl=Xll;N2=X21; SEQ , 2.02OOOXlO 31-M omo, OUTPUTNS; ,."""'" 14.6Il MOO' END; 1MTSU"""" ", .. ~ 0.0640 PERlOO 1 """""O.D%i'61lSO ... 05135

SOURCE OF lYPEm" FVAWE PR>' DATACOV; SEQ 1 1.0200500O "-" 0"'" IF _N_=1 TIlE\! SET NS; SUBISEQ) ,..,.,.,. I.... 00001 1M' "1 O.l486125D 0_ SETCOV; PERIOD 1 "".~ 0.5135 AKRA Y A XlI YIl X12 YI2; """"" AKRA Y B X21 Y21 X22 Y22; ARRAY C SXl SYI 5X2 SY2; TFORHO: PR>11I STDERROROF PARAMlITER tonMATIl PA~ ESllMAn OOOVERA; C= «(NI~WA + (N2.t)fB) / (NhN2·2); TMTEFFliCf 0.17625000 0.... 0D8763637 END; (l"TAlJ) "" KEEP Nt N2 SXl SYI SX2 SY'l;

1106 ' ______"M" __ ~'~"_"'_'_

CALCULATE THE GIS ESTIJvlATORS USING IML _____""'_"MUM'M"<;' :;''';';'';';Ij",,;;,,~ ____ FIGURE 4 SAS OUTPUT FOR mE GiS ESfIMATORS

-1. READ POOLED DISPERSION MATRrX INTO IML ANALYSIS OF ASTI-IMA DATA IN KENWARD &: JONES (1987) - 2. READ IN YBAR VECTOR -> DERIVATION OF GIS ESTIMATORS USING PROCIML <- -3.SET-UPXMATRlX - 4. CALCULATE GtsFSTIMATOR AND VARIANCE ¥BAR VECTOR IS: ... FOR EACH HYPOTI-IESIS TFST -- YBAR OOLt

PROCIML; ROWI 1.3375 USEOOV; R0W2 1.5725 READ ALL INTOSMAT; ROW3 1.4312 Nl :SMATQUD; ROW4 1.6900 N2 ..sMATOl.2D; ROWS 1.8137 SMAT1=5MATQ, 3:6 DiNt; ROW6 22512 SMAT2=5MATQ, 3:6 J)/N2; R0W7 1.8175 SfAR:: {OOOOOOOO, ROWS 2.0162 00000000, 00000000, 00000000, 1. GIS ESTIMATOR FOR mETA, ASSUMING SEQ=O 00000000, 00000000, FSTIMATOR OF THETA IS! 00000000, Bl COLt 00000000 }; STARQ 1:4, 1:4 ))::SMAT1; ROWI 0.00350 STARQ 5:8,5:8 »::SMAT2; slNV '" INV(STAR); WITH VARIANCE: VBl COLt USE AVGi READ ALL INTO YB; ROWl 0.00746 YBT =YS'; YBAR= I 0,0,0,0, O,O,O,O}; 2. GLS ESTIMATOR FOR LAMBDA, ASSUMING SEQ.. 1HETA::O YBARQ 1:4,1]) '" YBTQ 2:5,1]): YBARO ~8, 1D = YBTO 2,5 ,2]); FSTIMATOR OF LAMBDA IS: PRINT "YBAR VEO'OR IS: ", YBARi B2 COLt x" 1 0 0 -1 0 0 0, ROWl 0.09409 0 1 0 -1 -1 0 0, 0 0 1 -1 0 -1 0, WITH VARIANCE: -1 -1 -1 -1 1 0 -I, VB2 COLt 1 0 0 0 0 0, 0 1 0 1 0 0, ROWl 0.01527 0 0 1 0 1 0, -1 -1 -1 -1 0 1); 3. GIS ESTIMATOR FOR TAU, ASSUMING SEQ.. THETA, LAMBDA=O

ESTIMATOR OF TAU IS: B3 COLt

PRINT, "1. GIS ESTIMATOR FOR 1HETA, ASSUMING SEQ=:() Hi ROWI 0.08222 Xl=X(}1:8,{1234678]]); BIGLS '" lNV (Xl' 'SlNV"Xl)" (Xl" SINV 'YBAR)i WITH VARIANCE: VBIGIS '" INV (Xt'"SINV' Xl); VB3 COLt Bl= BIGLS(] 6, ]); VBl=VBlGlS(] 6,6]); ROWI 0.00078 PRINT ~ESTIMATOR OF THETA IS: H,Bl(lFORMAT::8.51), "WITH VARIANCE: ",VBIQFORMAT:8.5])"i

PRINT,"2. GLS ESTIMATOR FOR LAMBDA, ASSUMING SEQ, 1HETA=:()~' X2::XQ1:8,{1234'6 alD; B2GLS ""!NY 00'" SINV 'X2)' (X2" SINV 'YBAR)i VB2GLS::!NY 00" S1NV' X2); 82= B2GLS(] 6, ]); VB2=VB2GLSO 6,6»; PRINT "ESTIMATOR OF LAMBDA is: ",B2(]FORMAT=8.5J), "WITH VARIANCE: ",VB2QFORMAT:8.5]),,;

PRINT ,"3. GLS ESTIMATOR FOR TAU,ASSUMING SEQ, 1HETA, LAMBDA=:()"· X3=X(Jl:8,ll234 6' J]); B3GLS:: INV

%MEND MACGLS;

1107