SAS Macros for Multivariate Two-Sample Tests in the Case of Incomplete Data Thomas Bregenzer, Walter Lehmacher Institut ftir Biometrie und Epidemiologie, Tierarztliche Hochschule Hannover and Olaf Gefeller Abteilung Medizinische Statistik, Universitiit Gottingen Abstract Multivariate two-sample tests offer the opportunity to condense the statistical comparison between two samples containing data on several variables into a single p-value. This approach is preferable in decision-oriented situations, for example, in clinical trials when the evaluation of therapeutic superiority of one treatment over the other is based on more than one endpoint describing treatment efficacy. In such situations, Hotelling's T2 statistic detects any departure from the global null hypothesis of no difference between the samples with respect to all variables, whereas O'Brien's OLS- and GLS-test looks for equidirected alternatives. In the paper, a SAS macro collection is introduced to address the computational realization of those multivariate two-sample tests within the SAS software. The statistical methods implemented in this collection incorporate the case of incomplete data using all available information from the observed variables. The statistical background of the test procedures is briefly summarized, the syntactical requirements for the usage of the macros are delineated, and a worked example of their application to a data set is presented. 1 Introduction The evaluation of treatment efficacy in clinical trials often leads to a situation in which several endpoints are of (equal) importance for the clinical judgement, for example, \ \.. 760 r-""'''''''''''·==''''''''''.... ~-'''''''-~>-~~~·~~~~"--~--~~~-"-_C-C--' I i I i if repeated measurements are obtained to assess longitudinal differences between two groups of subjects, one group receiving the treatment, the other a pl~cebo. The separate statistical analysis of such multiple endpoints in a comparison between two treatments results, however, in a multitude of p-values, whereas in many cases a single p-value would be preferable. When assuming normality of the observed data and homoscedasticity between groups, the classic Hotellings T2 test or a multivariate analysis of variance (MANOVA) can be performed to test the equality of the multivariate distributions of two (or more) groups. An important drawback of these procedures refers, however, to the ordered (software-) applicability only to complete observations with respect to the information of the endpoints. When dealing with incomplete data the statistical software packages (including SAS) fall back upon complete case analysis (meaning that in practice all observation vectors containing missing values will be excluded from the analysis), which is an inefficient "solution" to the problem of incomplete measurements loosing a lot of information. In the SAS software, even the standard form of Hotelling's T2 is not directly implemented in any of the procedures of the SAS/STAT component. Moreover, one is often concerned with equidirected treatment differences, where one is only interested in whether a treatment reveals some improvement upon placebo with respect to all response variables. Multivariate two-sample tests provide a solution to the last problem by summarizing the differences between two treatments with respect to all endpoints in a global test statistic leading to a single p-value. This approach, originally proposed by O'Brien [5], turned out to be a considerably more powerful test for detecting restricted alternatives than Hotelling's T2, which, on the other hand, is optimal for detecting any departure from the null hypothesis "no treatment effect in any of the endpoints". SAS/IML macros for O'Brien's test procedures, handling only balanced data sets (i.e. without missing observations), have been introduced at SEUGI '93 [1]. In this paper, we present an extension to the case of incomplete data, taking into account each available observation. Furthermore, an asymptotic version of Hotelling's T2 for incomplete data is proposed. The macros are programmed in SAS/IML, using the conventional SAS procedure syntax as far as possible, and can be used as a profitable supplement to existing SAS procedures in order to efficiently analyse unbalanced multivariate data. 2 Notation and Theoretical Considerations Let X ijk , i = 1, 2jj = 1, ... , nij k = 1, ... , K the k-th measurement of subject j in group i, where some of the Xijk may be missing at random. The vectors Xii' = (XijI , ••. ,XijK ) are assumed to satisfy the following conditions: 1. The independent Xij have cumulative distribution functions (c.d.f.) Fi(x) with marginal c.d.f. F ik ( Z ) 2. E(Xij) = Pi = (Pi}, ... , PiK)' 761 3. COV(Xij) = Ei (not singular), El = E2 =: E = (Ukl)k.l=l •...• K, Without loss of generality we can assume Ukk := 1 for k = 1, ... , K, i.e., the data are expressed in common units by subtracting the overall mean from each observation and dividing by the pooled within-group sample standard deviation. Define J := (1, ... ,1)1< and N := nl + n2, let nik be the number of nonmissing observarions in group i for the k-th measurement, and finally let 5 denote the indicator function 5( z) = l{x observed}. Consequently, nikl := Ej 5ijk5ijl is the number of observations being observed for the k-th and l-th variable together. Define X i .k := Ej X ijk , 1 Xi.k:= n'" E j 5ijkXijk and Xi:= (Xi.t, ... ,Xi.K)'. In the sequel the 5 are omitted to simplify the notation. The following conditions N --+ 00, ndN --+ consti > 0, nik/ni --+ constik > 0 are assumed to hold whenever the limiting behaviour of test statistics and distributions is regarded. Consistent estimators of P.i and Ei can now be obtained by using all available data. By applying multivariate central limit theory, the limiting distributions of Hotelling's T2 and O'Brien's statistics can be derived. ,- ;.: .~ ,'- ~ ~ '" .;,' ·i· ;:," " !:; . .~ .f, >, j: " '~ ;~ 2.1 Estimators of Jti and Ei The covariance matrix Ei can be estimated by Ei := (Uikl)k.l with ~; .... Uikl = Ej~l(Xijk - Xi.k)(Xijl - Xi.,) nikl - 2 + l{k=l} :~. ~.;: ;', ~~ Then the pooled estimator of E ist ~~; (U.kl)k.l with I)nikl - i) 2 )Uikl i=l ~ i~ U.kl ~; ~- = Lnikl- 4 2 L(nik - I)Uik .~ j: i=l ~ ~ • , k= I Let {li denote the covariance matrix of the mean vector Xi: cov(X·) = (COV(X'k X'I») with WOkl = ..!!ib.LUokl = ni"ni/ n~", cOV*(Xok XOI) 1 I., I. k.1 =: (W'kl)kl I. I ni"ni/ 1 I., •• where cov*(.) denotes the conditional covariance based on the nikl observations which ~, ~ W: % ~. ;~ ~ 2 • have nonmissing data for both variables. The factor n~1tl01 can be regarded as a missing correction factor with the property of adjusting the conditional covariances in order to obtain a positive semidefinite covariance matrix. A similar approach can be found in [7]. Note that the missing correction factor can take values between 0 and 1, where the value 1 is attained if the data are complete, and 0 if each nonmissing observation in one variable has a missing counterpart in another variable. In the latter case, the variables are regarded as conditionally independent. The main result of this section is that the Xi and Ei constructed in this way are consistent estimators of P.i and E i , respectively, using all available data. ~ R I ~. '.' , , I . 2 ,C ~' (, Ii' E= , \ t_ \.. ~ ~ ~ 762 l ~ h - " -_._--,-----~. "-'- - "-'-'-'-"', :: . 2.2 Distribution of the Mean Treatment Difference Vector The difference vector between the two treatment groups Y := (Yk)k=l, ...,K, Yk := X u X 2.k , is a consistent estimator of the treatment difference 1'1 - 1'2. Applying a version of the multivariate central limit theorem ([6], p. 127) yields the asymptotic distribution of Y: VNY with r = ('Ykl )k,l ~ NK(O, Nr) and 'Ykk 'Ykl Finally, 3 3.1 r can be consistently estimated by E. Multivariate Test Statistics for Incomplete Data Hotelling's T2 By means of the preceding section, we extend the classic T2 statistic to incomplete data by where r is estimated by using all nonmissing data as described above:· fkk - tT.kk ~ (1 fk, - tT.kl ~ (nUl nu 1) , + -n2k nUnll k = 1, ... ,K n2kl ) + n2k , n 21 k =1= I. Therefore, T2 is asymptotically X2 distributed with K degrees of freedom. In order to detect any departure from the null hypothesis Ho: 1'1 = 1'2 Hotelling's T2 is uniformly most powerful (when assuming normally distributed data), but, on the other hand, is inefficient for investigating directional treatment differences [2,4]. 3.2 Tests for Directional Alternatives O'Brien [5] proposed two parametric multivariate statistics for evaluating Ho : 1'1 = 1'2 vs. HI : 1'1 - 1'2 = cD, where D is a (k x I)-vector describing the direction of the 763 !?"t.......""""'"_ _ ?0~~x""--:,r-~";;:;;.::;,.:v-~~0-:.;::.;::,:-=,,:>~<::.---'-' ~ I ~ y treatment difference, and c denotes a real-valued scalar. The OLS- and the GLS-test statistics - their names reHect their relation to ordinary least sq~ares and generalized least squares techniques - are J'(X1 - X2 ) TOLS (J'l'J)1/2 J':r-l~Xl TGLS - X2 ) (J'r- 1J)1/2 . Both statistics are asymptotically normally distributed under Ho. 4 A Data Example A data set named HEP is assumed to contain data for three variables GOT, GPT, and GGT corresponding to three measurements of GOT, GPT, and -y-GT taken repeatedly in a clinical trial. The efficacy of two hepatitis treatments has to be investigated, whereby one is especially interested in comparing treatment difference effects which occur in all three relevant variables simultaneously and with equal direction (the printed example only shows the pre-post-treatment differences in the two groups). Some of the observations are missing at random. \ Treatment Patient A A A A A A A A A A A A A A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 endpoints GOT GPT -y-GT 4.0 1.6 4.1 3.0 . 4.4 4.5 2.6 3.8 5.2 0.4 3.1 2.6 3.0 \, 764 8.2 4.6 5.1 5.6 4.3 7.4 6.4 7.6 3.2 5.7 4.6 7.1 2.4 -0.6 2.6 0.1 4.1 0.3 4.8 3.3 1.7 7.1 1.3 3.3 1.6 2.2 . . Treatment Patient B B B B B B B B B B B B B B 15 16 17 18 19 20 21 22 23 24 25 26 27 28 endpoints GOT GPT I'-GT 9.1 4.8 2.3 6.0 . 2.1 5.2 4.5 . 6.7 4.5 3.4 2.9 5.2 0.1 4.8 2.0 3.7 3.9 2.4 1.7 2.2 3.1 4.2 3.9 1.2 1.7 2.1 2.7 8.3 4.4 4.2 3.6 4.7 4.8 -2.7 -1.5 2.4 7.4 1.6 0.6 3.8 The printed output of our SAS macros for multivariate two-sample tests contains, among other details, the estimated correlation matrix CORRELATION (GOT GPT GGT) 1 0.1250937 -0.142397 0.1250937 1 0.3709465 -0.142397 0.3709465 1 and the values ot the test statistics for Hotelling's T2 (T_HOT) and O'Brien's OLS- and GLS-test (T_OLS, T_GLS) as well as the accompanying p-values (p _HOT, P_OLS, P_GLS) O'BRIEN'S STATISTICS T_OLS 0.815113 T_GLS -0.277358 P_OLS 0.2075038 P_GLS 0.3907526 765 '~'- ----------. - :./... - '~'- '-' ~ - -.'-' HOTELLING'S STATISTICS 20.599375 0.0001275 As can be shown, for example, by PROC GLM or PROC TTEST, considerable differences between the two treatments only occur in variable GPT, whereas GOT shows a small treatment difference (but into the other direction!) and GGT indicates no distinction between the two treatments: TTEST PROCEDURE Variable: GOT CL Mean N Std Dev StdError -------------------------------------------------------------~-- A B 13 12 3.25384615 4.72500000 T DF Prob>ITI -2.1802 -2.2170 18.7 23.0 0.0422 0.0368 Variances Unequal Equal 1.29461409 1.97903787 0.35906134 0.57129902 **************************************************************** Variable: GPT N Mean Std Dev Std Error 13 14 6.17692308 2.64285714 1.95369472 1.32690421 0.64185742 0.35436281 CL A B T DF Prob>ITI 3.9140 3.9701 20.9 26.0 0.0008 0.0006 Variances Unequal Equal **************************************************************** Variable: GGT CL A \ N Mean Std Dev Std Error 13 2.82307692 2.43623817 0.67669089 t. \. 766 3.16428571 14 B T DF Prob>ITI -0.3242 -0.3216 24.6 26.0 0.7485 0.7506 Variances Unequal Equal 3.01932421 0.80694834 This behaviour is reflected in the p-values of the various multivariate two-sample tests, too: Hotellings T2 detects the differences between the two treatments which occur in two variables, but with contrary directions, i.e., treatment A reveals an improvement upon B with respect to GOT while GPT allows to suppose a superiority of treatment B. Application of Hotelling's T2 adresses thus the wrong question: in spite of the small p-value no treatment can be regarded as being actually the best. O'Brien's tests give no indication of equidirected treatment differences. 5 ..; ,. Macro Call Usually a data set in SAS is arranged in a form similar to the hepatitis example above: one treatment variable, one variable enumerating the subjects receiving the treatment and one variable containing the values measured for each endpoint. For making such a data set usable for our macros, it must be reorganized by creating a single variable containing the names of the endpoints and another variable with the corresponding endpoint values (Note that PROC GLM needs exactly the same structure to perform a repeated measurement analysis). Thus, the hepatitis data set has to be organized as follows (for example by means of PROC TRANSPOSE): TREATH PATIENT VARS A A A 1 1 1 GOT GPT GGT B B B 13 13 13 GOT GPT GGT POST_PRE 4.0 8.2 1.1 9.1 0.1 2.1 The general form of a macro call corresponding to the multiple endpoint test problem IS % procedure ( <DATA = dataset ,> SUBJ = subjects variable, 767 I ~ ~ .~ .~ lJ .}: ~, ~; ENDP = endpoints variable, VAL UES = endpoint values, CLASS = class variable <,BY = by variable> ~: ¥. :,.: ~ ~' ~ ; }; ;.," where i~ " ~: ":~ t procedure names the procedure to be used: OBRIEN - for O'Brien's test procedures (OLS and GLS) - or HOTELL - for Hotelling's T2 - respectively) ,~ :s~ :1', " t dataset names the SAS data set containing the data to be analysed, default is _LAST_ (as in SAS) f,. subj denotes the variable witch enumerates the subjects receiving the treatments f endp names the variable containing the names of the endpoint variables .:" ,. .' . " ~~' 'j; ,-r, .0, .< .r: values specifies the variable containing the values of the endpoints &: ~ .'t :~ class variable names the class variable defining the two samples (treatments) to be compared 'f; '). { ...)-!, by variable leads to seperate analyses on observations in groups defined by the specified by variable (as in SAS) to' " '.t" P \' ~ " For example, the macro call for analysing the hepatitis data set by means of O'Brien's OLS- and GLS-tests as described in the previous section is: :,:;- ~~ 1~ J:, "t: f % obrien( DATA = hep, SUBl = patient, ENDP = vars, VALUES = posLpre, CLASS = treatm r i· l: 1. .~ f~ ~ t, t "r,~: ); ~~. ii'-'!; ,j: f; ~.rI· 6 Closing Remarks ~. X 1~ While Hotelling's T2 is the method of choice for detecting any nonspecific departure from the null hypothesis, O'Brien's procedures are optimal for detecting equidirectional ' alternatives [3]. Even the simpler but slightly less powerful OLS-test has been shown to perform quite well, especially in small samples where the GLS-procedure turned out to be a little anti conservative. More details concerning the performance in simulation studies and comparisons between the tests can be found in [4]. To our knowledge the computational realization of multivariate two-sample tests for incomplete data with respect to the exploitation of all available data (i.e., without ~ it;.... ~ I ~ h i t .~ \ ~ \. 768 deleting observations) has not been approached in any statistical sQftware package. To overcome this unsatisfactory situation, we have developed a SAS macros collection which makes use of the SAS /IML matrix language. It can be implemented on every platform with SAS version 6.04 or higher. Since under DOS working with larger data sets in PROC IML often exeeds the memory capacity, the SAS/WINDOWS version should be preferred. A copy of our macro collection is available on request from the authors. Please direct the correspondance to the following address: Thomas Bregenzer Institut fiir Biometrie und Epidemiologie Tierarztliche Hochschule Hannover Bischofsholer Damm 15 D-30173 Hannover Tel. +49 511 856 7549 FAX +49511 856 7695 email [email protected] References [1] Th. Bregenzer and W. Lehmacher. SAS macros for comparisons with multiple endpoints. In Proceedings of the SAS European Users Group International Conference, pages 784-790. SAS Institute Inc., Heidelberg, 1993. [2] D. Follmann. Multivariate tests for multiple endpoints. Report, National Heart, Lung, and Blood Institute, Bethesda, 1993. [3] H. Frick and V.W. Rahlfs. Comparison of multivariate and univariate tests for directional alternatives. Biometrics, to appear, 50, 1994. [4] W. Lehmacher, G. Wassmer, and P. Reitmeier. Comment on: On the design and analysis of randomized clinical trials with multiple endpoints. Biometrics, to appear, 50, 1994. [5] P.C. O'Brien. Procedures for comparing samples with multiple endpoints. Biometrics, 40:1079-1087, 1984. [6] C.R. Rao. Linear Statistical Inference and its Applications. John Wiley, New York, 1972. [7] W.M. Stanish, D.B. Gillings, and G.G. Koch. An application of multivariate ratio methods for the analysis of a longitudinal clinical trial with missing data. Biometrics, 34:305-317, 1978. SAS, SAS/STAT, and SAS/IML are registered trademarks of SAS Institute Inc., Cary, NC, USA. 769