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Recent Developments in Multiple Comparison Procedures

Institute of Mathematical Statistics

Lecture Notes - Monograph Series

Vol. 47 (2004) 76-88

c ?Institute of Mathematical Statistics, 2004

On O"Brien"s OLS and GLS tests for

multiple endpoints

Brent R. Logan

1 and Ajit C. Tamhane 2 Medical College of Wisconsin and Northwestern University Abstract:In this article we obtain some new results and extensions of the OLS and GLS tests proposed by O"Brien (1984) for the one-sided multivariate testing problem. In particular, we empirically obtain an accurate small sample approximation to the critical point of the OLS test. Next we give a power com- parison between the OLS test and a competing test proposed by L¨auter(1996). Lastly, we extend the OLS and GLS tests to the heteroscedastic setup where the control and treatment populations have different covariance matrices.1. Introduction Most clinical trials are conducted to compare a treatment group with a control group on multiple endpoints. Often, the treatment is expected to have a positive effect on all endpoints. O"Brien (1984) proposed two global tests, known as the ordinary least squares (OLS) and generalized least squares (GLS) tests, to demon- strate such an overall treatment effect. In this article we obtain some new results and extensions of these tests. The following is an outline of the paper. Section 2 gives the notation, the problem formulation and the assumptions. Section 3 deals with the homoscedastic case. First it gives a review of the OLS and GLS tests, including an improved approximation to the small sample critical value of the OLS test. Next it gives a power comparison between the OLS test and a test proposed by L¨auter. Section 4 derives extensions of the OLS and GLS tests to the heteroscedastic case. Section 5 gives some concluding remarks. The appendix gives derivations of asymptotic power expressions of the OLS and L¨auter"s tests required for thepowercomparisoninSection3.

2. Notation and preliminaries

Suppose that there are two independent treatment groups with n1 andn 2 subjects on each of whomm≥2 endpoints are measured. Treatment 1 is the test treatment and treatment 2 is the control. Letx ijk denote the measurement on thekth endpoint for thejth subject in theith treatment group. For treatment groupi, assume thatx ij =(x ij1 ,x ij2,...,x ijm ,j=1,2,...,n i , are independent and identically distributed (i.i.d.) random vectors from a multivariate normal (MVN) distribution with mean vectorμ i i1 i2 im and covariance matrixΣ i (i=1,2). In the homoscedastic case, we assumeΣ1 2 =Σ(say). The elements ofΣare kk =Var(x ijk )andσ k? =Cov(x ijk ,x ij? 1 Division of Biostatistics, Medical College of Wisconsin, 8701 Watertown Plank Rd., Milwau- kee, WI 53226, USA. e-mail: blogan@mcw.edu2 Department of Statistics, Northwestern University, 2006 Sheridan Rd., Evanston, IL 60208,

USA. e-mail:ajit@iems.northwestern.edu

Keywords and phrases:clinical trials, one-sided multivariate test, homoscedastic, het- eroscedastic. AMS 2000 subject classifications:primary 62J15; secondary 62P10. 76
On O"Brien"s OLS and GLS tests for multiple endpoints77 The corresponding correlation matrix will be denoted byRwith elements k? =Corr(x ijk ,x ij? k? kk

In the heteroscedastic case, the elements ofΣ

i will be denoted byσ i,k? m) and the corresponding correlation matrices will be denoted byR i i,k? }(i= 1,2).

Letδ=μ

1 2 1 2 m denote the vector of mean differences. To establish an overall treatment effect, a global null hypothesis of no difference is tested against a one-sided alternative H 0 :δ=0vs.H 1 :δ?O ,(1) where0is the null vector and O ={δ|δ≥0,δ?=0} is the positive orthant. Let x i· =(x i·1 ,x i·2 ,...,x i·m denote the vector of sample means of then i subjects from theith group and let?Σ i denote the sample covariance matrix from the ith group withν i =n i -1 degrees of freedom (d.f.) (i=1,2). In the homoscedastic case, we use the pooled estimate ofΣgiven by?Σ={(n 1 -1)?Σ 1 +(n 2 -1)?Σ 2 }/(n 1 n 2 -2) withn 1 +n 2 -2 d.f. Denote the elements of?Σby?σ k?

3. Homoscedastic case

3.1. OLS and GLS Tests

O"Brien (1984) considered a simplified version of the hypothesis testing problem (1) obtained by restrictingthe mean difference vectorδ=μ 1 2 to a ray 11 mm whereλ≥0. In other words, ifδ k kk k denotes the standardized treatment effect for thekth endpoint then O"Brien assumed that k =λ≥0 for allk. In that case the hypothesis testing problem (1) simplifies to H 0 :λ=0 vs.H 1 :λ>0.(2) O"Brien solved this problem by using a univariate regression framework that models the standardized responses as y ijk =x ijk kk k kk 2I ijk ijk i whereμ k 1k 2k )/2,I ijk =+1ifi=1and-1ifi=2,and? ijk ≂N(0,1) r.v."s with correlations

Corr(?

ijk i j k? ifi=i andj=j ,Corr(? ijk i j )=0 otherwise.

Note that the vectorsy

ij =(y ij1 ,y ij2 ,...,y ijm are independent, each with corre- lation matrixR={ρ k? Assuming thatRis known, O"Brien showed that the OLS estimate ofλand its standard deviation (SD) equal OLS =j (y 1· -y 2· m= y

1··

-y

2··

and SD(?λ OLS )=1 m? n 1 +n 2 n 1 n 2 ??j Rj?,

78Brent R. Logan and Ajit C. Tamhane

wherejis a vector of all 1"s of an appropriate dimension. Therefore the OLS statistic withRreplaced by the sample correlation matrix?Requals t OLS ?SD(?λ)=? n 1 n 2 n 1 +n 2 ?j (y 1· -y 2· j ?Rj? =j t j ?Rj,(4) wheretis a vector of thet-statistics, t k n 1 n 2 n 1 +n 2 x

1·k

-x

2·k

kk for comparing the treatment and control groups on the individual endpoints. Each t k is marginallyt-distributed underH 0k withn 1 +n 2 -2d.f.

Since the errors?

ijk in the regression model (3) are correlated, one may prefer the generalized least squares (GLS) estimate ofλ(which is also its MLE) to the OLS estimate. Assuming thatRis known, O"Brien showed that GLS =j R -1 (y 1· -y 2· j R -1 jand SD(?λ GLS n 1 +n 2 n 1 n 2 ??1 j R -1 j? The test statistic using this GLS estimate with the estimated correlation matrix ?R substituted in place ofRequals t GLS ?SD(?λ)=? n 1 n 2 n 1 +n 2 ?j ?R -1 (y 1· -y 2· j ?R -1 j? =j ?R -1 t j ?R -1 j.(6) Both the OLS and GLS statistics are standardized weighted sums of the individ- ualt-statistics for themendpoints. The OLS statistic uses equal weights, while the GLS statistic uses unequal weights determined by the sample correlation matrix?R. If some endpoint is highly correlated with the others then the GLS statistic gives a correspondingly lower weight to itst-statistic. The convergence oft GLS to the standard normal distribution is slower than that oft OLS because of the use of the estimated correlation matrix?Rboth in the calculation of?λ GLS and?SD(?λ GLS ). Also, the simulation study by Reitmeir and Wassmer (1996) has shown that the powers of the OLS and GLS tests are comparable when used to test subset hypotheses in closed testing procedures. Finally, the linear combination of thet k -statistics used in the GLS test can have some negative weights, which can lead to anomalous results; this problem does not occur with the OLS test. For all these reasons, the OLS test is preferred.

The exact small sample null distribution oft

OLS is intractable. O"Brien (1984) proposed to approximate it by thet-distribution withn 1 +n 2 -2md.f. For large sample sizes, the standard normal (z) distribution may be used as an approximation; however, this approximation is liberal for small sample sizes. Thet-approximation is exact form= 1 and conservative form>1 if the d.f. is small. For example, ifquotesdbs_dbs24.pdfusesText_30

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