Practical tests for randomized complete block designs

Transcription

1 Journal of Multivariate Analysis 96 (2005) Practical tests for randomized complete block designs Ziyad R. Mahfoud a,, Ronald H. Randles b a American University of Beirut, Beirut , Lebanon b University of Florida, Gainesville, FL 32611, USA Received 9 June 2003 Available online 11 November 2004 Abstract A class of affine-invariant test statistics, including a sign test and a related family of signed-rank tests, is proposed for randomized complete block designs with one observation per treatment. This class is obtained by using the transformation retransformation approach of Chakraborty, Chaudhuri and Oja along with a directional transformation due to Tyler. Under the minimal assumption of directional symmetry of the underlying distribution, the null asymptotic distribution of the sign test statistic is shown to be chi-square with p 1 degrees of freedom. The same null distribution is also proved for the family of signed-rank statistics under the assumption of symmetry of the underlying distribution. The Pitman asymptotic relative efficiencies of the tests, relative to Hotelling Hsu s T 2 are established. Several score functions are discussed including a simple linear score function and the optimal normal score function. The test based on the linear score function is compared to the other members of this family and other statistics in the literature through efficiency calculations and Monte Carlo simulations. This statistic has an excellent performance over a wide range of distributions and for small as well as large dimensions Elsevier Inc. All rights reserved. Keywords: Affine-invariance; Complete block designs; Pitman efficiencies 1. Introduction Randomized complete block designs with one observation per treatment are settings in which each of the p treatments is observed once on every experimental unit. Let Corresponding author. addresses: zm15@aub.edu.lb (Z.R. Mahfoud), rrandles@stat.ufl.edu (R.H. Randles) X/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi: /j.jmva

2 74 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) Y i = (Y i1,...,y ip ) denote the p-dimensional vector observed for the ith experimental unit where Y ij denotes the observed response of the ith experimental unit to treatment j.a possible model for the data is Y i = β i 1 p + τ + i, i = 1,...,n, (1) where 1 p is the p 1 vector of 1 s, τ = (τ 1,...,τ p ) is the vector of fixed treatment effects, β i represents the random effect of the ith subject, and i is the vector of random error of the ith subject. It is assumed that the β i s are iid with mean 0 and variance σ 2 β and the i s are iid continuous random p-vectors with median 0 and having a general dispersion structure. The error distribution may or may not have finite first (second) moments. The random variables β i, i for i = 1,...,nare all mutually independent. The test of equal treatment effects, H 0 : τ 1 = =τ p versus H a : τ i = τ j for some i = j, (2) is of interest. Two classical parametric approaches are often recommended. The first procedure, due to Hsu [16], is performed by applying Hotelling s T 2 to the (p 1)-variate random vectors, X i s, obtained from the Y i s via X i = (X 1,...,X p 1 ) = (Y 1 Y p,...,y p 1 Y p ) i = 1,...,n. This procedure is affine-invariant, and if the Y s come from the p-variate normal distribution, T 2 will have a null distribution that is a multiple of the F-distribution with p 1 and n (p 1) numerator and denominator degrees of freedom, respectively. The affineinvariance property is important because it ensures that the performance of the test is not affected by the shape of the population variance covariance matrix, as is the case with the test we describe next. The second procedure is the classical analysis of variance (ANOVA) F test. The original observations are used to form the test statistic F = 1 n 1 n p j=1 (Ȳ j Ȳ ) 2 ni=1 p j=1 (Y ij Ȳ i Ȳ j + Ȳ ), 2 where Y ij denotes the jth element of the vector Y i, Ȳ j = (1/n) n i=1 Y ij, Ȳ i = (1/p) p j=1 Y ij, and Ȳ = (1/np) n p i=1 j=1 Y ij. When the Y i s are (multivariate)-normally distributed, the null distribution of the ANOVA test is an exact F-distribution with p 1 and (n 1)(p 1) degrees of freedom if and only if the population variance covariance matrix of Y, say Σ = (σ tt ), satisfies the Huynh and Feldt condition. Define ε = (p 1) ( p t=1 p 2 ( σ d σ ) 2 p t =1 σ2 tt 2p p t=1 σ2 t + p2 σ 2 where σ d = (1/p) p t=1 σ tt, σ t = (1/p) p t =1 σ tt, and σ = (1/p) p t=1 σ t, then Σ satisfies the Huynh and Feldt condition if ε = 1, in which case the matrix is referred to as a matrix of type H. Note that ε is the multiplicative adjustment factor for the degrees of freedom proposed by Greenhouse and Geisser [8]. Two disadvantages of using the ANOVA F test is the dependence of its performance on the structure of the variance covariance matrix, and, unlike Hotelling s T 2, its lack of the affine-invariance property. However, if ),

3 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) the underlying distribution is p-variate normal with H-type variance covariance matrix, the F test will have a greater power than the Hotelling Hsu T 2 test for the same alternatives (see [29]). Morrison [29] developed several statistics to test (2) under various assumptions on the variance covariance matrix Σ. Morrison demonstrated that his tests have higher efficacies than that of Hotelling Hsu T 2. Yet, Morrison only considered the case where the observations come from a multivariate normal distribution. Nonparametric competitors to Hotelling s T 2 appeared as early as 1937 and 1939 when Friedman developed his well known method of ranks procedure and Kendall and Babington proposed an equivalent test, respectively. Friedman s procedure only uses within-subject information and ignores between-subjects information. Under H 0, Friedman s test has an asymptotic chi-square distribution with (p 1) degrees of freedom. An alternative approximation of Friedman s test uses the statistic (n 1)T F R = n(p 1) T, where T represents the Friedman statistic. F R is then compared with the usual F distribution with (p 1) and (n 1)(p 1) numerator and denominator degrees of freedom, respectively. Iman and Davenport [18] showed that the chi-square large sample approximation can be inaccurate and recommended using the F approximation. In their paper, Iman and Davenport also proposed two new statistics related to Friedman s test. The first statistic is a linear combination of the T and F R, while the second is obtained by using the ranks to estimate the degrees of freedom for F R. In order to reduce the effect of subject-to-subject variation, Koch [26] proposed using a test statistic based on the ranks of the aligned observations. These observations are obtained by subtracting the subject average from each component of that subject s response vector. After this centering, the entire set of the np observations is ranked. Koch s aligned-ranks test statistic has an asymptotic chi-square distribution with (p 1) degrees of freedom under the null hypothesis of the exchangeability of the components of Y i. In another attempt to retrieve some of the between-subjects information, Quade [32] presented a class of procedures of weighted within-subject rankings. The class contains, as a special case, a p-sample extension of Wilcoxon signed-rank test. Quade s method gives larger weights to observation vectors Y i that exhibit more variability among the components of Y i. Under the null hypothesis of the exchangeability of the components of the Y i, Quade s test has an asymptotic chi-square distribution with (p 1) degrees of freedom. In an attempt to find natural nonparametric analogs for Hotelling s T 2 and the ANOVA F test, Iman et al. [19] and Agresti and Pendergast [1] used the rank transformation approach. This approach consists of ranking all np observations from 1 to np, replacing each original observation with its corresponding rank and then applying a parametric method to the ranks instead of the original observations. Iman et al. [19] proposed computing the ANOVA F test using these ranks, say F RT. Iman, Hora and Conover considered a null hypothesis of the form H 0 : F ij = F i for i = 1,...,nand j = 1,...,p, where F ij is the continuous distribution function of the random variable Y ij. Under the assumption that the Y ij s are mutually independent and for some regularity conditions on

4 76 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) F i, Iman, Hora, and Conover showed that the null asymptotic distribution of (p 1)F RT is χ 2 p 1. Agresti and Pendergast [1] suggested replacing each of the np observations by its rank among all np values and then computing the Hotelling Hsu statistic based on the ranks instead of the original values. They considered the case where the null hypothesis of no treatment effects is expressed as F 1 F 2 F p, where F 1,...,F p are the one dimensional marginals distributions of Y = (Y 1,...,Y p ). Under H 0, Agresti and Pendergast compare their test statistic to a χ 2 p 1 distribution or to an F distribution with (p 1) and n (p 1) degrees of freedom. Kepner and Robinson [25] developed the asymptotic properties, including efficiencies, for this test. See [5] for other performance characteristics of this test, and [9] for performance comparisons of other tests in this problem setting. Agresti and Pendergast [1] proposed another statistic that made use of this joint ranking. The null hypothesis was considered as the exchangeability of the components of the Y i. Their statistic included Koch s statistic and the ANOVA F rank transformation statistic as special cases. Jan and Randles [20] proposed applying the interdirection sign test of Randles [33] and the interdirection signed-rank test of Peters and Randles [30] to this design. Their tests are shown to have a limiting χ 2 p 1 when the underlying distribution is elliptically symmetric. In this paper, the sign test of Randles [34] and a related family of signed-rank tests (see [27,12]) are applied to this problem setting. In Section 2, the sign test is introduced and its null asymptotic distribution is established under the minimal assumption of directional symmetry of the underlying distribution. The family of signed-rank tests along with its null limiting distribution, under the assumption of symmetry of the underlying distribution, is presented in Section 3. In Section 4, the Pitman asymptotic relative efficiencies of the tests, relative to T 2, are established. In Section 5, we discuss several score functions to be used and we study the score function s effect on the efficiencies of the corresponding test by comparing four members from this family with different score functions, including a test based on an optimal score function as well as a test based on a simple linear score function. Monte Carlo studies are presented along with an example in Section 6. These Monte Carlo studies compare the performance of many of the tests described in the paper. 2. The sign test Consider the (p 1)-variate vectors, X 1,...,X n obtained from Y 1,...,Y n via: X i = (X i1,...,x ip 1 ) = (Y i1 Y ip,...,y ip 1 Y ip ) where C is the (p 1) p matrix, C = = CY i,i= 1,...,n, (3)

5 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) Under this transformation, the location parameter of the X s is θ = Cτ =[τ 1 τ p,...,τ p 1 τ p ]. Moreover, using the p-variate Y s to test (2) is equivalent to using the (p 1)-variate X s to test H 0 : θ = 0 versus H a : θ = 0. (4) Randles [34] suggests using the transformation retransformation approach where each observation X i is transformed by a data-determined nonsingular matrix   x, to form ÂX i. Unit vectors, V i = ÂX i / ÂX i, where is the Euclidean norm, are created. The test statistic is then a quadratic form in the mean unit vector, i.e., S n = nv (n 1 n V i V i ) 1 V. (5) i=1 In his paper, Randles chooses the matrix  =  d that was proposed by Tyler [37], which satisfies: ( )( ) 1 n Âd X i Âd X i = 1 n  d X i  d X i p 1 I. i=1 Beside being nonsingular,  d is also invariant under sign changes among the x i s. Randles [34] showed that if n>(p 1)(p 2), then S n is affine-invariant, i.e. that its value is the same when computed on the X i, i = 1,...,nas it is when based on DX i, i = 1,...,n, for any nonsingular (p 1) (p 1) matrix D. An iterative scheme for computing  d can be found in Randles [34] paper. With this choice of Â, (5) becomes S n = n(p 1)V V (6) and H 0 is rejected for large values of S n. The test statistic S n is affine-invariant and distribution-free under H 0 for the class of distributions with elliptical directions (see Randles [34], for more details). An absolutely continuous p-vector Y is said to be directionally symmetric around 0 if Y Y d = Y Y. Lemma 2.1. If Y 1,...,Y n are iid from an absolutely continuous directionally symmetric distribution, then for any fixed matrix C, the vectors X i = CY i, i = 1,...,n, are also iid from a directionally symmetric distribution, provided CY i is absolutely continuous. Proof. Using the transformation H(u) = Cu, results in CY i Y i d = CY i Y i. Applying the transformation ψ(v) = v/ v, v = 0gives X i X i = CY i CY i = CY ( ) ( i Y i Y i CY i = ψ CYi d = ψ CY ) i Y i Y i

6 78 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) = CY i Y i = CY i CY i = X i X i. Thus, the X i s have a distribution that is directional symmetric. Y i CY i Theorem 2.1. If Y 1,...,Y n are iid from a directionally symmetric distribution, then, under H 0, the quadratic form, S n, has a limiting chi-square distribution with p 1 degrees of freedom. Proof. The proof follows readily from Theorem 1 of [34]. 3. A family of signed-rank tests We now propose a class of signed-rank statistics related to (5) and (6). These statistics use the ranks of the distances of the transformed observations from the origin. Specifically, ( ) let Q i be the rank of  d X i among  d X 1,...,  d X n and define V si = Qi n+1 V i, where ( ) is a nonnegative, nondecreasing, uniformly bounded continuous function over [0, 1] that may depend on the dimension p. The test statistic is then the quadratic form W np 1 = n V s (n 1 n V si V si ) 1 V s (7) i=1 and H 0 is rejected for large values of W np 1. The test statistic W np 1 is affine-invariant as long as n>(p 1)(p 2). Moreover, if under H 0 the Y i s are iid from a symmetric distribution, then W np 1 is conditionally distribution-free (see [27] for more details). This family of test statistics was developed by Mahfoud [27] and Hallin and Paindaveine [12]. In order to obtain the null distribution of W np 1, the following lemma, the proof of which is easily established, is needed. Lemma 3.1. If Y 1,...,Y n are iid from a symmetric distribution around τ, then for any fixed matrix C, the vectors X i = CY i, i = 1,...,n, are iid symmetric around θ = Cτ. Theorem 3.1. If Y 1,...,Y n are iid from a symmetric distribution, then, under H 0, the quadratic form, W np 1, defined in (7) has limiting chi-square distribution with p 1 degrees of freedom. Proof. The proof follows readily from Theorem of [27]. The test statistics S n and W np 1 are both affine-invariant. Thus their performance does not depend on the variance covariance matrix of the i s and just like with Hotelling s T 2 computed on the X i s, the particular contrast scheme in (3) is used without loss of generality.

7 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) For example, many authors use the differencing scheme: X i = (X i1,...,x ip 1 ) = (Y i1 Y i2,y i2 Y i3,...,y ip 1 Y ip ) = C Y. (8) The matrix C in (8) can be written as DC, where D is the (p 1) (p 1) nonsingular matrix , thus, using the contrast scheme in (3) or the one in (8) will result in the same value of these test statistics. 4. Asymptotic relative efficiencies Pitman asymptotic relative efficiencies (ARE) relative to Hotelling Hsu T 2 will be used to compare several test statistics. The ARE s are developed under the assumption that the Y s come from the class of elliptically symmetric distributions. Thus, the Y s come from a distribution of the form f τ,σ (y) = K p Σ 1 2 h{(y τ) Σ 1 (y τ)}, (9) where h(t) is a nonnegative function defined over the positive reals, K p > 0, and Σ is proportional to the covariance matrix of Y, provided it exists. Note that when Y has a p-variate elliptically symmetric distribution with parameters τ and Σ and a density given by (9), then X = CY, defined in (3), has a (p 1)-variate elliptically symmetric distribution with parameters Cτ and CΣC and a density given by where f Cτ,CΣC (X) = K p CΣC 1 2 h {(X Cτ) (CΣC ) 1 (X Cτ)}, h (t) = + h(t + s 2 ) ds. (10) Therefore, if Y has a density function given by (9), and CΣC = I p 1, then, under H 0, R = X X has a density function of the form g R (r) = p 1 2π 2 Γ( p 1 2 )K p r p 2 h (r 2 ), 0 r, where h ( ) is as defined in (10). The ARE s of W np 1 and S n are given in the next theorem and corollary.

8 80 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) Theorem 4.1. If Y 1,...,Y n are iid from an elliptically symmetric distribution then the Pitman asymptotic relative efficiency of W np 1 with respect to Hotelling Hsu T 2 is [ ] ( 2 ARE(W np 1,T 2 p 1 )2 EH 2 0 R h (R 2 ) h (R 2 ), (G(R)) [ E H0 R 2 ] ) = E H0 [ 2. (11) (G(R))] where R 2 = X X, h (R 2 ) = h (t) t t=r 2, and G(t) = P H0 (R 2 t), t 0. Proof. Without loss of generality, let CΣC = I p 1. Thus, under the null hypothesis, the transformed observations are iid from an elliptically symmetric distribution with a density function of the form f X (x) = K p h (x x). Therefore, the proof follows from applying Proposition (5) of [12] to the transformed sample or see [27] for more details. Corollary 4.1. If Y 1,...,Y n are iid from an elliptically symmetric distribution then the Pitman asymptotic relative efficiency of S n with respect to Hotelling Hsu T 2 is [ ] ARE(S n,t 2 2 ) = ( p 1 )2 EH 2 0 R h (R 2 ) h (R 2 E H0 [R 2]. (12) ) Proof. The proof follows from the fact that S n can be obtained from W np 1 by setting (u) 1. Note that the assumption CΣC = I p 1 is used without loss of generality since the test statistics involved are affine-invariant. 5. The score function In this section we compare the ARE (Pitman Asymptotic Relative Efficiency) values for four specific score functions. If (u) 1, then W np 1 reduces to S n. This test statistic, under elliptically symmetric distributions, has AREs equal to those of the test obtained by applying Randles s [33] interdirection sign test to the complete block design setting (see, [20]). These Sign test statistics have good efficiencies for heavy-tailed distributions but do not perform well for light-tailed distributions as well as the normal distribution. If (u) u, then, for elliptically symmetric distributions, W np 1 will have the same AREs as the test statistic obtained by applying Peters and Randles [30] signed-rank interdirection test to the complete block design setting (see, [20]). These test statistics generally perform well over a wide range of distributions, yet their performance deteriorates as the dimension increases, especially in the light-tailed cases and the normal case. The test will be denoted by PR. Optimal score functions have been discussed in the literature (see [12,14]). For example, if the underlying distribution is the multivariate normal then the van der Waerden score

9 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) function, (u) = (χ 2 ) 1 (u), is optimal where ( χ 2) 1 (u) is the inverse of the distribution function of a χ 2 p 1. With such a score function and for elliptically symmetric distributions, W np 1 AREs with respect to Hotelling Hsu T 2 will always be greater than 1 with equality when the distribution is the multivariate normal. Note that with this score function the test will be denoted by W wn p 1. While tests based on optimal scores have nice properties, they are seldom used in practice. The score function that we now recommend combines excellence in performance with simplicity in implementation. It is defined as (u) = λu + (1 λ)1, (13) where λ ={ln(p + e 2)} 1 depends on p, the dimension of the data. This score function was introduced in [28]. With this linear score function, W np 1 has an excellent performance as good as and in some cases better than W wn p 1 but yet it is still easy to compute as S n and PR. W ln p 1 will be used to denote W np 1 with the score function defined in 13. We now show the effect of the score function on the efficiencies of the corresponding test by computing and comparing the AREs, with respect to Hotelling Hsu s T 2, of the four test statistics, W ln p 1, W wn p 1, PR, and S n. Tables 1 and 2 display the AREs of these four statistics, for the case when the Y s come from the power family of distributions and the family of t-distributions, respectively. These two families of distributions belong to the class of elliptically symmetric distributions. The power family is such that h(t) = exp { ( t c 0 ) ν }, ν > 0, where c 0 = pγ(p/2ν) Γ((p + 2)/2ν), and K νγ(p/2) p = Γ(p/2ν)[πc 0 ] p/2. It includes the multivariate normal density (ν = 1), as well as heavier-tailed distributions (0 < ν < 1) and lighter-tailed distributions (ν > 1). For the multivariate t-distribution, h( ) and K p in (9) are defined as h(t) = [1 + 1ν ] (p+ν)/2 (t) and K p = Γ((p + ν)/2) Γ(ν/2)(πν) p/2. It is clear from Tables 1 and 2 that W ln p 1 has very competitive ARE values in a wide variety of distributions, ranging from very heavy-tailed distribution to very light-tailed ones. These efficiencies do not deteriorate as the dimension increases (as is the case for PR) or as the underlying distribution shifts from being heavy-tailed to being light-tailed (as is the case with S n ). For normal alternatives, W ln p 1 not only performs virtually as well as Hotelling Hsu s T 2 and W wn p 1 (recall this test is optimal for the normal distribution) but also has higher efficiencies than S n and PR n. For the heavy-tailed distributions of the power

10 82 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) Table 1 Asymptotic relative efficiencies under the power family of distributions Dimension Test ν W ln p W wn p PR n S n W ln p W wn p PR n S n W ln p W wn p PR n S n W ln p W wn p PR n S n W ln p W wn p PR n S n Entries: Pitman asymptotic relative efficiencies with respect to Hotelling Hsu s T 2 : Distribution = Power family; W ln p 1 = W np 1 with the linear ( ); W wn p 1 = W np 1 with the van der Waerden score function; PR n =Peters and Randles procedure; S n = Randles [34] procedure. family, W ln p 1 has higher efficiencies than W wn p 1 and PR but slightly lower efficiencies than those of S n. For the light-tailed distributions of this family, the efficiencies of W ln p 1 exceed those of S n but are slightly less than those of PR and W wn p 1. However, the slight advantages of some tests over W ln p 1 whether in the light-tailed cases or the heavy-tailed cases disappear slowly as the dimension increases. For the multivariate t-distribution family, W ln p 1 has higher efficiencies than PR for all degrees of freedom. W ln p 1 also has higher efficiencies than W wn p 1, except when p = 2, ν = 20. Moreover, W ln p 1 has better efficiencies than S n when p<5 and virtually equal efficiencies when p 5.

11 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) Table 2 Asymptotic relative efficiencies under the multivariate t-distributions Dimension Test ν W ln p W wn p PR n S n W ln p W wn p PR n S n W ln p W wn p PR n S n W ln p W wn p PR n S n W ln p W wn p PR S n W ln p W wn p PR n S n Entries: Pitman asymptotic relative efficiencies with respect to Hotelling Hsu s T 2 : Distribution= t-distribution; W ln p 1 = New procedure, with the linear ( ), applied to the X s; W wn p 1 = W np 1 with the van der Waerden score function; PR= Peters and Randles procedure applied to the X s; S n = Randles [34] procedure applied to the X s. 6. Monte Carlo results and an example To compare the performances of S n and W ln p 1 with respect to other procedures in the literature, a Monte Carlo study for dimension p = 3, sample size n = 40, and significance

12 84 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) level α =.05 was conducted. Besides S n and W ln p 1, the study included W wn p 1, PR, the ANOVA F test, Hotelling Hsu T 2, Friedman s test (F R ), Iman et al. [19] rank transformation version of the ANOVA F test (F RT ), the rank transformation test applied to centered within block data (F CRT ) and Quade s test (W Q ). Note that Quade s test was computed using the range as the location-free statistic that measures the variability within the components of each observed vector. Five of the 10 procedures considered are computed on the transformed observations, that is the X s. These are T 2, S n, W wn p 1, PR, and W ln p 1, while the rest of the procedures use the original data, the Y s. The test statistics F, F RT, F CRT, and W Q are compared to upper αth quantile of an F distribution with (p 1) and (n 1)(p 1), numerator and denominator degrees of freedom, respectively. No adjustment for the degrees of freedom was made since we noticed that doing so will only correct the α level of the test but will not have any significant effect on the power (see [20]). On the other hand, the tests F R, S n, W wn p 1, PR, and W ln p 1 are compared to the upper αth quantile of a chi-square distribution with p 1 degrees of freedom. The Hotelling Hsu T 2 is compared to (n 1)(p 1)/(n p + 1) times the upper αth quantile of an F distribution with (p 1) and (n p + 1), numerator and denominator degrees of freedom, respectively. The test statistics were compared under elliptically symmetric distributions with densities from the power family and the family of multivariate t-distributions. From the power family, the heavy-tailed case (ν =.1), the multivariate normal case (ν = 1), and the light-tailed case (ν = 5) were used. From the family of multivariate t-distributions, distributions with degrees of freedom of 1, 3, and 10 were considered. For test statistics that use the original observations, the distributions were located at τ = (kδ,kδ, 0), k = 0, 1, 2, 3, and for the rest of the tests, the distributions were located at θ = (kδ,kδ), k = 0, 1, 2, 3. The value δ was adjusted for different distributions to obtain somewhat similar points on the power curve. Two different types of the population variance covariance matrix were considered, the first is Σ = I 3, the identity matrix and the second is Σ = EE = (14) The first choice satisfies the Huynh and Feldt condition, but Σ in (14) does not. This was done in order to show the dependence of the performances of F, F R, F RT, F CRT, and W Q on the type of the population variance covariance matrix. Tables 3 and 4 contain the results from the Monte Carlo study for distributions from the power family, while Tables 5 and 6 contain those for the distributions from the family of t-distributions. Entries in each table are the proportion of times out of 3000 iterations in which each test statistic exceeded the upper α-percentile of the corresponding null distribution. For the affine-invariant test statistics considered, that is, T 2, W wn p 1, W ln p 1, PR, and S n, the results from the Monte Carlo studies agree with those obtained from the efficiency comparisons. Moreover, these performances did not change when the structure of the population s variance covariance matrix changed from I 3 to EE. This can be seen by comparing Table 3 with Table 4 and Table 5 with Table 6. Comparing all 10 tests, the following observations are made. When the underlying distribution is multivariate normal with Σ = I 3, all 10 tests showed a good performance, with

13 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) Table 3 Monte Carlo results for the power family, with Σ = I 3 Test statistic ν Amount of shift.1 0 1Δ 2Δ 3Δ T W wn p W ln p PR S n F F RT F CRT F R W Q T W wn p W ln p PR S n F F RT F CRT F R W Q T W wn p W ln p PR S n F F RT F CRT F R W Q Entries: the proportion of times out of 3000 iterations in which each test statistic exceeded the upper α-percentile of its asymptotic null distribution: Distribution= power family; α=.05; Σ=I 3 ; Sample size=40; Dimension=3; T 2 = Hotelling Hsu procedure; W wn p 1 = signed-rank procedure withvan der Waerden scores; W ln p 1 = signed-rank procedure with the linear ( ) function; PR= Peters and Randles signed-rank test; S n = Randles [34] procedure; F= ANOVA F test; F RT = Rank transformation ANOVA test; F CRT = Rank transformation ANOVA test using centered within block data; F R =Friedman s test; W Q = Quade s test.

14 86 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) Table 4 Monte Carlo results for the power family, with Σ = EE Test statistic ν Amount of shift.1 0 1Δ 2Δ 3Δ T W wn p W ln p PR S n F F RT F CRT F R W Q T W wn p W ln p PR S n F F RT F CRT F R W Q T W wn p W ln p PR S n F F RT F CRT F R W Q Entries: the proportion of times out of 3000 iterations in which each test statistic exceeded the upper α-percentile of its asymptotic null distribution: Distribution = power family; α=.05; Σ=EE ; Sample size=40; Dimension=3; T 2 = Hotelling Hsu procedure; W wn p 1 = signed-rank procedure withvan der Waerden scores; W ln p 1 = signedrank procedure with the linear ( ) function; PR= Peters and Randles signed-rank test; S n = Randles [34] procedure; F= ANOVA F test; F RT = Rank transformation ANOVA test; F CRT = Rank transformation ANOVA test using centered within block data; F R =Friedman s test; W Q = Quade s test.

15 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) Table 5 Monte Carlo results for the family of t-distributions, with Σ=I 3 Test statistic df Amount of shift 1 0 1Δ 2Δ 3Δ T W wn p W ln p PR S n F F RT F CRT F R W Q T W wn p W ln p PR S n F F RT F CRT F R W Q T W wn p W ln p PR S n F F RT F CRT F R W Q Entries: the proportion of times out of 3000 iterations in which each test statistic exceeded the upper α-percentile of its asymptotic null distribution: Distribution= multivariate t-distribution; α=.05; Σ=I 3 ; Sample size=40; Dimension=3; T 2 = Hotelling Hsu procedure; W wn p 1 = signed-rank procedure withvan der Waerden scores; W ln p 1 = signed-rank procedure with the linear ( ) function; PR= Peters and Randles signed-rank test; S n = Randles [34] procedure; F= ANOVA F test; F RT = Rank transformation ANOVA test; F CRT = Rank transformation ANOVA test using centered within block data; F R =Friedman s test; W Q = Quade s test.

16 88 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) Table 6 Monte Carlo results for the family of t-distributions, with Σ = EE Test statistic df Amount of shift 1 0 1Δ 2Δ 3Δ T W wn p W ln p PR S n F F RT F CRT F R W Q T W wn p W ln p PR S n F F RT F CRT F R W Q T W wn p W ln p PR S n F F RT F CRT F R W Q Entries: the proportion of times out of 3000 iterations in which each test statistic exceeded the upper α-percentile of its asymptotic null distribution: Distribution = multivariate t-distribution; α=.05; Σ=EE ; Sample size =40; Dimension =3; T 2 = Hotelling Hsu procedure; W wn p 1 = signed-rank procedure with Van der Waerden scores; W ln p 1 = signed-rank procedure with the linear ( ) function; PR = Peters and Randles signed-rank test; S n = Randles [34] procedure; F = ANOVA F test; F RT = rank transformation ANOVA test; F CRT = rank transformation ANOVA test using centered within block data; F R =Friedman s test; W Q =Quade s test.

17 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) F being the best. However, when Σ = EE, the affine-invariant tests maintained a strong performance, while all the other five tests, showed a significant decrease in power. For the heavy-tailed distribution from the power family, with Σ = I 3, S n and F R showed the highest powers, followed by a good performance with F RT, F CRT, and W ln p 1, while T 2, W wn p 1, PR, F, and W Q did not perform well. On the other hand, when Σ = EE, only S n kept a strong performance, followed by F R and W ln p 1, while the rest of the tests performed poorly. For the light-tailed distribution from the power family, with Σ = I 3, all tests had similar performances with the exception of S n and F R, which performed poorly. Again, the affine invariant tests maintained their performances when the variance covariance matrix became Σ = EE, but not the other tests. For the multivariate t-distribution with one degree of freedom and Σ = I 3, F R, and S n performed the best, followed by good performances from F RT, F CRT, and W ln p 1, while the rest of the tests performed poorly, especially F, and T 2. On the other hand, when Σ = EE, the performance of F RT dropped to become comparable to that of W ln p 1 with S n still being the best and all the other tests performing poorly. For the multivariate t-distribution with three degrees of freedom and Σ = I 3, all nine tests performed well. However, when Σ = EE, only S n, W ln p 1, PR, W wn p 1, and T 2 maintained a strong performance, with a slight advantage of S n and W ln p 1 over the others. Finally, for the multivariate t-distribution with 10 degrees of freedom and Σ = I 3, all nine tests have virtually the same performance. Again, the affine invariant tests maintained their performances when the variance covariance matrix became Σ = EE, but not the other tests. We also note that centering the data (within block) had a slight improvement on the performance of F RT only in the family of t-distributions when Σ = EE. In summary, the dependence of the tests F, F RT, F CRT, F R, and W Q on the structure of the variance covariance matrix of the underlying population was clear under all distributional assumptions. The effect was obvious as the powers of these tests deteriorated and, in some cases, the sizes of the tests shifted grossly from the.05 α-level. For heavy-tailed distributions, S n, was the best overall performance, with W ln p 1 and F R having good performances. While for the light-tailed distributions W ln p 1, W wn p 1, PR, and T 2 were always good choices followed by S n. Finally, we present the following example. The data is taken from [2, p. 214]. It consists of the fasting blood glucose test (Table 7) for 52 females observed in three pregnancies. The data were analyzed using eight tests, F, F R, W Q, T 2, S n, F RT, F CRT, and W ln p 1. Results are summarized in the Table 8. All tests, except W Q, showed significant differences (p <.05), moreover the test of sphericity (Mauchly criterion in SAS GLM) was not significant (p =.73) indicating that the population may be H-type. The data was then altered to include extreme observation. Indeed, the last observation was changed from (52, 70, 76) to (527, 70, 76) and the data were re-analyzed. The sphericity test was significant (p <.0001) and therefore the adjusted H F p-value for the F test was reported. It can be seen that the presence of an outlier had a large effect on the tests, F, T 2. In conclusion, this paper presents some affine-invariant test statistics for data from a randomized complete block setting with one observation per treatment and random block effects. They are designed to perform well regardless of the variance covariance structure within block errors. The test statistics are easy to compute and resistant to the influence of extreme observations.

18 90 Z.R. Mahfoud, R.H. Randles / Journal of Multivariate Analysis 96 (2005) Table 7 Fasting blood glucose test Table 8 Analyses of blood glucose data Test Original data Modified data Test statistic p-value Test statistic p-value T W lnp S n F F RT F CRT F R W Q

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