Doi:10.1016/j.cmpb.2006.06.006

A SAS/IML program for implementing the modiﬁed
Brown–Forsythe procedure in repeated measures designs

Guillermo Vallejo , Joaqu´ın Moris , N´elida M. Conejo
a Methodology Area, Department of Psychology, University of Oviedo, Plaza Feij´oo, s/n, E-33003 Oviedo, Spainb Psychobiology Laboratory, Department of Psychology, University of Oviedo, Plaza Feij´oo, s/n, E-33003 Oviedo, Spain In this paper, we present a computer program written in version 9.1 of SAS’ interactive matrix language in order to implement a new approach for analyzing repeated measures data. Previous studies reported that the new procedure is as powerful as conventional solu- tions and generally more robust (i.e., insensitive) to violations of assumptions that underlie conventional solutions. The program also included a step-wise procedure based on the Bon- ferroni inequality to test comparisons among the repeated measurements. Both univariate and multivariate repeated measures data can be analyzed. Finally, the application of the SAS/IML program is illustrated with a numeric example.
2006 Elsevier Ireland Ltd. All rights reserved.
Modiﬁed Brown–ForsytheapproximationSAS/IMLMultiple post hoc contrasts Introduction
ventional univariate (or mixed-model of Scheff ´e) approachprovides the most powerful tests. If sphericity fails to hold Repeated measures designs are used extensively in epidemi- but the covariance matrices are homogeneous, the resulting ology, psychology, neuropsychology, psychopharmacology, data can be analyzed by either a univariate model with Box’s and other research areas In many repeated measures epsilon (ε) correction for degrees of freedom (d.f.; or a designs, especially those employed in clinical studies, the full multivariate model. The empirical literature indicates, data are collected from N subjects forming J independent however, that both tests are sensitive to departures from groups over K occasions or trials or under different experi- the assumptions of multivariate normality and multisample mental conditions. Several methods have been proposed to sphericity, particularly when group sizes are unequal analyze such designs, many of which can be implemented To counteract the negative impact of the violation of mul- using widely available standard statistical packages such as tisample sphericity on the type I error rates, diverse solu- SAS, S-PLUS, or SPSS. When the variances of all pair-wise tions have been proposed. Algina and Oshima est differences among levels of the repeated measures factor are using the improved general approximation (IGA) test devel- equal (i.e., sphericity) and this constant variance is the same oped initially by Huynh whereas Lix and Keselman for all levels of the between-subjects grouping factor (jointly, proposed a Welch–James (WJ) type test derived by Johansen these two assumptions have been referred to as multisample on the power results presented by Algina and Kesel- sphericity; see details), it is well known that the con- man the WJ test may be preferred over the IGA test, ∗ Corresponding author. Tel.: +34 985 103274; fax: +34 985 104141.
0169-2607/$ – see front matter 2006 Elsevier Ireland Ltd. All rights reserved.
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 3 ( 2 0 0 6 ) 169–177 provided that sample size are sufﬁciently large to obtain a ith subject in the jth group robust WJ test. Nevertheless, when the sample size is small subvectors y , . . . , y
and the data are not extracted from normal distributions, NJ, a general linear model for univariate the WJ approach does not adequately control the type I errorrate for a test of the groups × trials interaction, commonly the most interest test for researchers. For their part, Vallejo andLivacic-Rojas ed the behaviour of a multivariate where Y is the N × K matrix of observed data, B is the J × K
extension of the Brown and Forsythe (BF; rocedure matrix that contains the unknown ﬁxed effects to be esti- with that of a mixed model based on covariance structures mated from the data with known design matrix X and U
selected by means of information criteria such as Akaike’s is the N × K matrix of unknown random errors. The model information criterion Bayesian information criterion assumes that the random vector y
normality and covariance assumptions were vio- dently distributed within each level j, with mean vector ␮ lated. Their simulation studies showed that the BF approach performed as well or better than its competitor, in terms of control of type I error rates, particularly with small sam- are ˆ j = (1/nj − 1)Ej, where Ej = YjYj − ˆ␤ jXjYj are distributed
ple sizes and unstructured covariance matrices. For tests of independently as Wishart WK(nj − 1, j) e also assume the interaction, however, the BF rates were conservative for that nj − 1 ≥ K so that ˆ −1 the negatively paired conditions of group size and covariancematrices—particularly when the sample sizes was small and 2.1.
Modiﬁed BF procedure
the degree of group size inequality was substantial.
However, in a recent article, Vallejo and Ato wed Let us consider the problem of ﬁnding a transformation F that a typically robust test for the interaction effect may be for the common multivariate criteria when homogeneity of obtained by modifying BF’s approach (referred to as modiﬁed covariance matrices is not a tenable assumption, with the BF test hereafter, MBF). In order to obtain better type I error aim of testing hypotheses of the form H0:CBA = 0 using the
control with the BF procedure, Vallejo and Ato, used the d.f.
MBF approach, where C is the (J − 1)×J matrix which deﬁnes
correction proposed by Krishnamoorthy and Yu a set of (J − 1) linearly independent contrasts for the between- of the correction suggested by Nel and van der Merwe groups factor and A is a K × (K − 1) matrix which deﬁnes a set of
which was used in the previous d.f. formulation provided by (K − 1) linearly independent contrasts for the within-subjects Vallejo et al. procedure due to Vallejo and Ato, in addi- factor. The new invariant solution is obtained by modifying tion to being more robust than that provided by Vallejo et al., the approximate d.f. proposed by Vallejo et al. statis- is improved in other two aspects. First, the approximate error tics used to test the hypothesis concerning to the interaction d.f. are invariant to linear transformations of outcome vari- effect using the MBF approach, are functions of the eigenval- ables. Consequently, the p values for testing H01:A1␮ = 0 will
ues of HE*−1, where the hypothesis matrix is
not be different from those for testing H02:A2␮ = 0, by a change
of scale of the elements of A, where A is a known matrix of
H = (C ˆBA) [C(X X)−C ] (C ˆBA),
contrasts with appropriate size and ␮ is the K-variate locationparameter. Second, the new approximate error d.f. always are positive, which is not so clearly the case on applying Nel and Therefore, the purpose of this article is to extend the MBF j A ( 1/2 ˆ
procedure to produce focused tests statistics and to make available a program written in the SAS/IML language order to obtain numerical results. Besides to facilitate the ces E* and H, respectively; c• = 1 − (n
access to this robust method, developing this program within SAS provides an opportunity to utilize other statistical pro- = (c•1 1 + · · · + c•J J), and 1/2 denotes the cedures of this system widely used nowadays. In Sections . Notice that the error matrix in Eq.
e present the modiﬁed BF method and code required equivalent to Eq. Vallejo et al. , since, for the program, along with instructions on its use, for test- is a positive deﬁnite matrix. Then, exist ing omnibus effects and multiple contrast hypotheses related −1/2 such that 1/2 1/2 = and −1/2 −1/2 = −1 to these effects. In Section e use the data from a study −1/2 = IK, where IK is the identity matrix. To
reported by Fitzmaurice et al. illustrate the application ﬁnd the d.f. associated with E*, ﬁrst, the sum
j=1 j A QjA(=
of the computer program with MBF procedure. Finally, some 1A + · · · + c•JA QJA) is approximated as
j A QjA ∼ WK
Deﬁnition of test statistic
Then, proceeding in a fashion similar to Nel Nel ijk, i = 1, . . ., nj; j = 1, . . ., J; k = 1, . . ., K, be the response for the ith subject in the jth group at the kth trial, and let y
and van der Merwe parameter will be found equating ijK) be the random vector of responses associated with the j=1 j A QjA, namely c•jA QjA and
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 3 ( 2 0 0 6 ) 169–177 = det(E*)/det(H + E*), s∗ = [(l2 ∗2 − 4)/(l2 + ∗2 − 5)]
j − 1)−1(c•jA QjA)2, to those of WK(f ∗, A QA). It follows
+ 1)/2 ]s∗ − (l ∗ − 2)/2; here, det(·) denotes the determinant of the matrix.
If the null hypothesis of no interaction between groups and j=1 j A QjA + tr
j=1 j A QjA
trials is rejected, the interpretation of tests of main effects is insufﬁcient to understand the information of the data. When j − 1)[tr2(c•j A QjA) + tr(c•j A QjA)
this occurs, test of possible 2 × 2 interactions (interaction con- where tr(·) stands for the trace of the matrix. In turn, trasts or tetrad contrasts) can be used Under depar- using the so-called multivariate Satterthwaite’s approxima- tures from the assumption of covariance homogeneity, tetrad tion described by Vallejo and Ato the Appendix, the contrasts results are obtained easily using the MBF procedure with appropriate A and C contrast matrices. Speciﬁcally, to test
interaction contrasts using MBF approach, C = cjj and A = akk ,
where cjj is a 1 × J vector of coefﬁcients that contrasts the jth
j=1 j A QjA + tr
j=1 j A QjA
and j th between-subjects means and a
coefﬁcients that contrasts the kth and k th within-subjects j=1 jA ˆ j ˆ −1A +tr
j=1 jA ˆ j ˆ −1A
means. It is important to mention, as a reviewer indicates, that it is not necessary that the omnibus F-test be signiﬁcant prior to testing planned (i.e., the contrasts are determined on V = [tr2(A ˆ j ˆ −1A) + tr(A ˆ j ˆ −1A) ] − 2cj[tr2(A ˆ j ˆ −1
before the data are collected) tetrad contrasts, provided the A) + tr(A ˆ j ˆ −1A) ].
type I error rate is controlled. Contemporary practice favors adopting the family of contrasts as the conceptual unit for ﬁed to tr(I2
K−1) + [tr(IK−1)]2, given that transforming
, A QA → IK−
To control the family-wise error rate (FEW) for all possible 1. Therefore, replacing Qj in Eqs.
2 × 2 interactions, several post hoc procedures may be used.
and y its estimate ˆ −1/2 ˆ j ˆ −1/2 and using the result that For instance, Lix and Keselman that the Hochberg tr(AB) = tr(BA), the approximate d.f. simpliﬁes to p-up Bonferroni, Schaffer sequentiallyrejective Bonferroni, and Studentized maximum modulus crit- ical value ocedures used in combination with Johansen’s j − 1)[tr2(c•jA ˆ j ˆ −1A) + tr(c•jA ˆ j ˆ −1A)
ocedure are largely robust to departures from multi-sample sphericity. Nevertheless, only the Hochberg procedure will be considered in this article. We selected the Hochbergstep-up Bonferroni procedure over the Shaffer and studen- tized maximum modulus approaches because Lix and Kesel- man minimal power differences between them and j=1 jA ˆ j ˆ −1A
j=1 jA ˆ j ˆ −1A
because is very simple to apply. With Hochberg’s the p values corresponding to the r tests statistics for testingthe hypotheses H1, . . ., Hr are rank ordered, where r = J* × K*,J* = J × (J − 1), and K* = K × (K − 1). Then, the largest probability The result in Eqs. considerable theoreti- cal appeal because, as Krishnamoorthy and Yu w, in FEW, where ˛FEW is the family-wise error rate the researcher is willing to tolerate. If p addition to being invariant under any nonsingular transfor- are rejected without further test; otherwise, the next largest mation, lies between min{nj − 1} and N − J for all cjA ˆ jA and
FEW/2. If pr−1 ≤ ˛FEW/2, all hypothe- j − 1 ≥ K − 1. Therefore, the approximate d.f. never could be negative, while this not the case with Nel and van der Merwe’s 1, . . ., Hr−1 are rejected. Continuing in this fashion, at any q where q ≤ q, if pq ≤ ˛FEW/(r − q + 1) for any q = r, r − 1, . . ., 1.
There are several multivariate test statistics for testing the As before, the test used for checking the effect of the tri- null hypothesis of no interaction between groups by trials.
als with unweighted means is given by the determinant of H + ˜E) , where
Hotelling–Lawley trace, and the Pillai–Bartlett trace statistics.
Although their critical values have been widely tabled and charted (see in practice it is usual to obtain the level H = (C ˜BA) [C(X X)−C] (C ˜BA),
of signiﬁcance by deﬁning each of these statistics in termsof an F-variable n the two-group case, all the F-test approximations are interchangeable. For our purpose, the F- multivariate test statistic. According to this transformation, j A ( 1/2 ˆ
the interaction null hypothesis is rejected if where A was deﬁned before, ˜B =
B, C ≡ c is a
(J × 1) vector consisting of all ones, c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 3 ( 2 0 0 6 ) 169–177 and • = 1. Extending the results reported by Krishnamoorthy When there is no interaction and the assumption of mul- and Yu Nel and van der Merwe distribution tisample sphericity is not satisﬁed, the MBF approach also may be applied to obtain robust multiple comparison pro- A QjA can be approximated as a sum of Wishart
cedures for examining all possible pair of groups and trials comparison marginal means. To test between-subjects pair- wise comparison hypotheses using MBF procedure, C = cjj and
A ≡ a. The signiﬁcance of the pair-wise comparisons for the
j A QjA ∼ SWK
within-subjects main effect can be probed in a similar man-
ner, but with C ≡ c (i.e., a 1 × J vector with each element equal
to one) and A = akk .
At present there are numerous simultaneous or sequen- tial multiple comparison procedures that maintain the FEW j=1 j A QjA + tr
j=1 j A QjA
at or below its nominal ˛-level when the validity assumptions of traditional statistics are satisﬁed (see However, when j A QjA) + tr(n−1
normality and covariance homogeneity are not satisﬁed, the Replacing Qj in Eq. by its unbiased estimate
number of procedures that remain relatively unaffected by assumption violations it diminishes considerably. Results of j ˆ −1/2 the approximate d.f. can be written as Keselman Keselman and Lix and Kowalchuk andKeselman est that the Welsch’s p-up range, Schaffer’s y rejective step-down Bonferroni, j A ˆ j ˆ −1A) + tr n−1
j A ˆ j ˆ −1A
and Hochberg’s y rejective step-up Bonferroni procedures performed well in terms of control of type I errorrates and power to detect true pair-wise differences. There-fore, the method used in the preceding paragraphs can be The simpliﬁcation at the Eq. occurs because A QA → IK−1. For the main effect of trials averaged over
the groups, all the F-test approximations are interchangeable.
According to the adaptation of the Rao ansformation,
Program description
the main effect of trials null hypothesis is rejected if To obtain numerical results for the MBF procedure described in the previous section we developed a computational pro- = det( ˜E)/det( ˜H + ˜E), s = [(l2 •2 − 4)/(l2 + •2 − 5)] ,
ming language The program is presented as a set of subroutines or modules and a driver. The subroutines are In turn, using the Mehrotra xtension of the univariate OMNIRESULTS, GROUPTEST, TIMETEST, INTERACTEST, and Brown–Forsythe test applied to the sum of the within-subject DEPVARTEST. They are run sequentially, and each of them variables, the test statistic for testing the effect of the groups checks the conditions of application. The program calcu- is given by determinant of E•(H + E•)−1, where the hypothesis
lates the MBF approximate solution for tests of the main matrix is deﬁned as in Eq. A ≡ a (i.e., a K × 1 vector with
and interaction effects in repeated measures designs. In each element equal to one), the error matrix is addition, contrasts among marginal means or all possibleinteraction contrasts (i.e., tetrad contrasts) can be obtained.
When we have a set of multivariate repeated measures data, the program also can be used to test omnibus effects and cjA ˙jA,
multiple comparison hypotheses related to these effects; bothseparately for each dependent variable and simultaneously.
and •e and • are the approximate d.f. for E• and H, respec-
All of the F tests and the Hochberg adjusted p values are tively. The deﬁnition of the estimators referring to the d.f. as calculate automatically by the program.
it applies to the analysis of repeated measures can be found To implement the program it is assumed that the data in Vallejo and Ato y, the null hypothesis referring to is entered in a SAS data set named DATARECORDED with the equality of the groups, weighted by means of the trials is multivariate format. The hallmark feature of a univariate for- mat is that each subject has multiple rows (or records)—onefor each measurement occasion, whereas the hallmark fea- tures of a multivariate format is that each subject has only one row (or record), regardless of the number of measure- ments made. The program only requires that the user speciﬁes It should be noted that the matrices H and E• are identical
the number of dependent variables (NVD). A run statement to the hypothesis and error sum of squares obtained employ- of the program generates as output F-statistics, along with ing a univariate Brown–Forsythe test with the numerator d.f.
degree of freedom and signiﬁcance levels for hypothesis test- ing. The program also provides as output by default a step-wise Table 1 – Data of CD4 cell count per mm3 for 68 selected subjects from ACTGy 193A
a AIDS clinical trial group.
b Variable that identiﬁes the subject to which the record refers.
c Levels of the between-subjects factor.
d Levels of the within-subjects factor.
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 3 ( 2 0 0 6 ) 169–177 Table 2 – Summary of traditional univariate analysis
a Mean square for the hypothesis.
b Mean square for the error.
c Degrees of freedom.
procedure based on the Bonferroni inequality (i.e., Hochberg for the within-subjects main effect, F = 8.11, with 5 and 320 method) to discover which linear combination of the means d.f. (p < 0.0001). Finally, for the interaction effect, results con- or interaction contrasts have signiﬁcant differences. However, tained in w that F-value is 2.32, with 15 and 320 d.f.
the user program can use several optional scalars (TESTOMNI, (p = 0.0008), so that also is highly signiﬁcant. Consequently, the TESTGROUPS, TESTTIME, TESTINTERAC, TESTDEPVAR), which classical statistic indicates that the shapes of the proﬁles are assume values of 0 or 1, for print the interest information.
not the same across the four groups.
The three univariate tests we have just considered have assumed normality and equal dispersion matrices for the four Example and comparison with
groups under study. However, using Box’s M-test, as given in traditional analysis of variance
Timm (. 134), the hypothesis of equal covariance matri-ces is untenable. The 2-approximation criterion is 180.29 with The application of this procedure is illustrated using data 61 d.f. (p < 0.0001). When multisample sphericity is violated, reported by Fitzmaurice et al. om a study published by the mixed-model of Scheff ´e’s approach suffers from inﬂated Henry et al. the Journal of Acquired Immune Deﬁciency Syn- nominal levels and thus should be used with caution. In order dromes and Human Retrovirology. These authors discuss a ran- to circumvent the problems caused for the lack of homogene- domized, doubly blind, study to determine the relative clinical ity of dispersion matrices, the MBF procedure is a good choice, efﬁcacy of four different reverse transcriptase inhibitor ther- since it becomes more conservative in these cases. A part of apies in AIDS patients with advanced immune suppression the results generated by SAS/IML program appears in (CD4 counts of less than or equal to 50 cells per mm3). Specif- To produce the previous results the following program ically, 1313 HIV-infected patients were randomized to one of four daily regimens containing 600 mg of zidovudine: zidovu-dine alternating monthly with 400 mg didanosine; zidovudine DATA DATARECORDED; INPUT GROUP Y1 Y2 Y3 Y4 Y5 Y6;
plus 2.25 mg of zalcitabine; zidovudine plus 400 mg of didano- sine; or zidovudine plus 400 mg of didanosine plus 400 mg of nevirapine (triple therapy). The time to new HIV disease progression or death, toxicities, the change in CD4 cells, and plasma HIV-1 RNA concentrations in a subset of study sub- TESTOMNI=1; /*‘1’ OMNIBUS TESTS, ‘0’ NO OMNIBUS TESTS*/ jects were evaluated. Measurements of CD4 counts at baseline TESTGROUPS=1; /*‘1’ GROUPS PAIRWISE CONTRASTS, ‘0’ NO (prior to the initiation of treatment) and at 8-week intervals during a 40-week follow-up period for 60 and 8 selected sub- TESTTIME=1; /*‘1’ TRIALS PAIRWISE CONTRASTS, ‘0’ NO TRI- jects are displayed in a multivariate format. Each subject has his or her own row of data containing the values TESTINTERAC=1; /*‘1’ MULTIPLE INTERACTION CONTRASTS, of outcome variable on each of the six levels of the within- subjects factor (0, 8, 16, 24, 32, and 40 weeks, which are denoted TESTDEPVAR=1; /*‘1’ UNIVARIATE TEST OF EACH DEPENDENT by K1, K2, K3, K4, K5, and K6, respectively). Each record also contains two identifying variables: ID, which identiﬁes the subject to which the record refers; J, which identiﬁes the lev-els of the between-subjects factor. The categorical variable According to the results included in t can be appre- treatment is coded: 1 = zidovudine alternating monthly with ciated that the subject’s mean levels are signiﬁcantly different 400 mg didanosine, 2 = zidovudine plus 2.25 mg of zalcitabine, among the four treatments (F = 3.32 with 2.35 and 47.51 d.f., 3 = zidovudine plus 400 mg of didanosine, and 4 = zidovudine p = 0.0373). Also is evident that the within-subjects main effect plus 400 mg of didanosine plus 400 mg of nevirapine.
is highly signiﬁcant (F = 3.95 with 5 and 38.82 d.f., p = 0.0054).
For the data shown in we will carry out a con- However, it is important to note that there is not a signiﬁcant ventional repeated measures analysis of variance, which is difference between the response patterns for the groups over summarized in ding to this analysis, the classical time; in other words, there is some evidence that the groups do F-test statistic gives stronger evidence for effects of treat- not respond differently during the ﬁrst 40 weeks of follow-up ment group, trials, and treatment × trial interaction. A 0.05 (F = 1.72 with 13.16 and 108.31 d.f., p = 0.0666). Consequently, signiﬁcance level is assumed throughout the paper. For the one could no reject the null hypothesis at the 5% level of sig- between-subjects effect, F = 2.89, with 3 and 64 d.f. (p = 0.0327); c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 3 ( 2 0 0 6 ) 169–177 Table 3 – Summary of multivariate MBF analysis
a Degrees of freedom corresponding to Wilks’s b Degrees of freedom corresponding to Rao’s F-approximation.
Table 4 – Hochberg’s adjusted p values for all possible pair-wise differences among the levels of the between-subjects
and within-subjects factors
J, levels of the between-subjects factor; K, levels of the within-subjects factor.
a Hochberg’s adjusted p values.
After the overall null hypotheses referring to the groups Hochberg’s sequentially rejective Bonferroni procedure one and trials are rejected, the next step in the analysis is to decide comparison is signiﬁcant controlling FEW at a level no more which population means differ. As discussed above, both 0.05: J1 versus J4. On the other hand, note that for the within- the pair-wise comparison tests for between-subjects marginal subjects marginal means, four comparisons are declared sig- means and pair-wise comparison tests for measures repeated niﬁcant: K1 versus K2, K1 versus K3, K3 versus K6, and K2 versus marginal means are affected by unequal covariance matrices K6. Because the interaction effect resulted no signiﬁcant at 5%, across the grouping factor. However, it is possible to obtain the tetrad contrasts involving pairs of levels of two factors has robust tests for the pair-wise comparison hypotheses by using not been printed in owever, a signiﬁcant result would the MBF procedure and ﬁtting the p values in step-up fashion not need additional code lines to produce all possible interac- for controlling the FWE. For pair-wise contrast and tetrad con- trast, it can be veriﬁed that the MBF procedure and Johansenas given in Lix and Keselman Lix et al. eequivalent; however, this would not be the case for K > 2.
Conclusion
Given that the null hypothesis of parallel proﬁles for groups is not rejected at 5% level signiﬁcance, the researchers may The basic purpose of this paper was to extend the MBF average over trials and over groups, respectively, to test pair- procedure for testing omnibus effects and multiple contrast wise contrast on group means and repeated measures means.
hypotheses related to these effects, to make available a pro- The program generates automatically all possible pairs among gram written in the SAS/IML language in order to implement the levels of the between-subjects and within-subjects factors.
this procedure, and to illustrate the application of the com- This part of the results has been included in puter program using data for a design grouped measures For all possible contrast in between-subjects marginal repeated. Previous studies had revealed that the MBF proce- means, the results reported in show that applying dure was generally robust (i.e., insensitive) to violations of c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 3 ( 2 0 0 6 ) 169–177 multisample sphericity and to lack of normality of the data [4] H. Huynh, Some approximate tests for repeated in unbalanced designs similar to that employed in the current measurement designs, Psychometrika 43 (1978) 161–165.
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Acknowledgments
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Source: http://causal.uma.es/papers/Vallejo,%20Moris%20&%20Conejo%20(2006%20CMPB).pdf
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