Testing the homogeneity of variances in a two-way classification

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1 Biomelrika (1982), 69, 2, pp Printed in Ortal Britain Testing the homogeneity of variances in a two-way classification BY G. K. SHUKLA Department of Mathematics, Indian Institute of Technology, Kanpur, India SUMMARY Approximations have been obtained to the null distribution of Bartlett's type of statistic for testing the homogeneity of one-way variances in a two-way classification. Simulation studies show that the approximations give satisfactory results even for a very small number of rows and columns. Some key words: Bartlett's test statistic; Dirichlet distribution; Heterogeneity of correlated variances. 1. INTRODUCTION In the analysis of variance one of the important assumptions is that the variances of all the observations are equal. In two-way classifications there are situations variances may differ between columns or rows but may remain the same within the same column or row. In such cases it may be advisable to make a preliminary test of their homogeneity before proceeding to the test of the equality of row means. In some cases testing the homogeneity of such variances may itself be of primary interest. For example, it may be of interest to test the homogeneity of variances of different measuring devices or judges in assessing the performance of certain subjects, it may not be desirable to take repeated observations on the same subject by the same device. Let y i} (i = 1,...,n; j = 1,...,p) be a random variable representing the observation in the ith row and the jth column. The usual two-way additive model is y l} = \i -V a, + /?, + e l}, n, a, and fij are the general mean and the effects of the tth row and the^'th column, respectively, e tj represents the error associated with the (i,j)th cell, and we assume that the e tj are independent and normally distributed, e y ~ N(0, oj). Russell & Bradley (1958) showed that the maximum likelihood method applied to y^'a does not give consistent estimators of the aj's. They, however, obtained such estimators based on contrasts of the y,/s for p ^ 3. For p > 3, no explicit solution exists. They suggested the likelihood ratio test for testing the homogeneity of <jj's for p = 3. Han (1969) suggested a test based on the multiple correlation coefficient. This test is, however, applicable only when the a/s are assumed to be random effects. Shukla (1972) modified this test to the case the a/s may be random or fixed effects. Han (1969) also suggested a short cut test based on the maximum F ratio. A related problem was also dealt with by Han (1968). Johnson (1962), while considering the distribution of estimators suggested by Ehrenberg (1950), also suggested a test of homogeneity of the aj'a based on Bartlett's (1937) test of the homogeneity of variances. In 2 a test statistic of Bartlett type is proposed. In 3 we consider three approximations to the null distribution of the test statistic for moderately large n. In 4 the result of a simulation study on these approximations is reported.

2 412 G. K. SHUKLA 2. A SIMPLE TEST STATISTIC Consider a new variable z ( j defined as We have say; *ij = yij-yt. ( i= l,,n;j = l,...,p). E(Zij) = Pj-P, var (z tj ) = {a 2 + (p - 2) a 2 }/p = X }, cov (z ijt z u.) = (d 2 -aj-aj,)/p = c jr, say for j =#/; cov (z, v,z, T ) = 0 for i * i'. Here /F = Z/J)/p and a 2 = E<x 2 /p. Testing the hypothesis a 2 = a 2 for all j +/ is equivalent to testing a slightly different hypothesis X } = k r for all j =)=/, when p > 2. Now consider a Bartlett type statistic T = p log S 2 -Z, \ogs], (1) * 2 = I, (z y - 2 _,) 2 /(n- 1); 8 2 = -Lj sj/p. For testing the null hypothesis H Q :Xj = X y for all j=t=/, one could find the null distribution of T, which we have done in the next section. 3. APPROXIMATION TO NULL DISTRIBUTION OF TEST STATISTIC 31. A generalization of Dirichlet 's distribution For simplicity here we assume that Xj = 1 for all j. We define a new variable Uj = ^vsj (j= l,...,p; v = n 1). The joint distribution of u l,...,u p, will be a multivariate gamma distribution on m = ^v degrees of freedom. The form of the distribution given by Krishnamoorthy & Parthasarathy (1951) can be written in terms of Laguerre polynomials L: after substituting (=1 * = 0 IC- X (E Ci}L(u i,m)l(u J,m)m~ C l2 p L(u 1,m)...L(u p,m)m~ p } k (2) L a (Ui, m) L"(Uj, m)/(m' m?) = L a (u,, m) L p (u p m)/{m (a) m 0 "}, C 12 = P 2, C 123 = 2p\...,C 12,,, p = (p-l)p'', p=-(p-])~\ ), L k (u,m) = (-d/du) k {u k <f>(u)}/4>(u), Using the transformation z 0 = EM,,ZJ = ujz o (j = 1,...,p), the joint distribution of Zj,...,z p _! can be obtained, after integrating out z 0, as f 2 lz 1,...,z p - 1 ) = D{m,...,m)+ 2 G, (3) t= 1 K! say. The infinite sum is obtained from (2) on replacing terms corresponding to L'liii, m) L fi (uj, m)l{m w m^},

3 Homogeneity of variances in a two-way classification 413 by (-l) a+ " t t r)( P )(-l) i+j D(m,m,...,m + i,m,...,m+j,...,m), i = oj=o\ij\jj D(n lt...,n p ) is a Dirichlet's probability density function defined as =i Thus the expression in (3) gives a generalization of Dirichlet's distribution for variates which are not independent as the sum of Dirichlet's distributions. From this one can find the hth moment of x = ft z, as expressions of the type in (3) are replaced by and n h is given by i=0jo i=0j=o i=oj=o H h {n lt...,n p ) = i= 1 w \ (4) 3-2. Moments of the test statistic By noting that T 1 = p~ p e~ T, T is the test statistic defined in (1), we can obtain the cumulant generating function of T as = }oge(e-" T ) = hp\ogp + logn h (m,...,m) \ A(2) = (l+h/m) 2 (l+ha 1 /mr l (l+ha 2 /m)- 1, cti 1 = ]+(mp)~\ aj 1 = }+2(mp)~ 1. In (5) we have retained only terms of order p 2 and p 3 as the terms of higher order become very small even for small values of p and m. In (5), \ogh h {m,...,m) = \ogr(pm) p\ogr(m)+p\ogr(m + h) logr(j)m+ph), (6) and this can be expanded in powers of h and \/m by using the generalized Stirling's expansion of the logarithm of a gamma function. One obtains the coefficients of h, h 2 and

4 414 G. K. SHUKLA h 3 from (6) to the appropriate degree of approximation as (7) From the last part in (5) one obtains the coefficients of h, h 2 and h 3 as <t> t, (p 2 <f>l/2 and 4>3 4>i 02 + ^1/3. 0i = {k l ct l + 2k 2 (<x 2 -a 1 )}/(mp), 4>i = -[k l <x k 2 {a 2 2-tx l <x 2 (mp)- 1 -a 2 }]/(m 2 p), 4> 3 = [k l <x 3 Now, by putting r= h, and collecting coefficients of r, r 2 /2!,r 3 /3! and substituting v = 2m, one obtains from (7) and (8) H^vT) = ( P -l) + (p 2 -l)/(3pv)-2( P 4 -l)/(l5 P 3 v 3 )- V (t> 1, 2 * ( 8 ) > 2-4> 2 /2), (9) 33. Approximation of null distribution of T Three approximations to the null distribution of T, corresponding to equating (i) the first moment of x 2, (») the first two moments of % 2, and (iii) the first three moments of F, as suggested by Box (1949), have been considered. (i) For the first approximation, a correction constant C t such that ^(vtcf 1 )- (p-1) is given by C\ = ^/(p-1). (ii) For the second approximation, the first two moments of vtc 2 1 are equated to that of x 2 on p* degrees of freedom, that is EiyTCl 1 ) = p*, V(vTCl l ) = 2p*, which gives C 2 = n 2 /(2n Y ),p* = 2fi\/n 2. (iii) The distribution vt could be approximated by a Pearson type I distribution, if we note that /ij^/^/if.) < 1. If X has a beta (qi,q 2 ) distribution then by equating the three moments of vt to that of C 3 X one obtains A =^111 B = fi 3 /nl D = A(B + 2A)/(2A 2 -B). Hence q 2 vt/{q l (C 3 vt)} will have approximately an F distribution on (2<7J,2</ 2 ) degrees of freedom. (iv) Johnson (1962) has also considered an approximation of the null distribution of T as XJ = v i GjT having a x 2 distribution on (p 1) degrees of freedom,

5 Homogeneity of variances in a tioo-vxiy classification RESULTS OF A SIMULATION STUDY One thousand samples were generated from a normal population for p = 3,5,10, n = 3,8 and 20. Table 1 gives the empirical probabilities of rejecting the null hypothesis corresponding to a = 005. The results for a = 001 follow a similar pattern. In the table Xi,Xu,F represent the statistics given in (i), (ii) and (iii) of 33, respectively; XJ represents Johnson's statistic, given in (iv); Xv denotes the statistic in (i) obtained by taking p = 0, that is as if the sj's were independent. The empirical probabilities are satisfactory for Xi, Xiu F ar >d XJ f r n = V = 3. The results for Xi an d Xn do not differ much and give a satisfactory test statistic for p ^ 3 and n ^ 3. The statistic F is not satisfactory for p = 3, showing that higher order terms should be included while considering the moments. However, it gives satisfactory results for p ^ 5. Johnson's approximation Xj gives satisfactory results for n larger than p, but may not be used when n and p are of the same order. When p is as large as 10 then Xu> which does not take into account p, gives a satisfactory result, as one would expect because p becomes quite small. Table 1. Result of Monte Carlo study giving empirical probabilities for a = V Xi O O Xn O r O (K) XJ O Xu o-on OO40 Thus the simulation study shows that Xi ar >d Xu can be use d safely even for n and p as small as 3. Tests of goodness of fit with the corresponding null distributions show that the fit is good for n > 8 and p > 3, except for the cases of Xj, Xv ar >d F, for which larger n and p are required for giving a satisfactory fit. I am very grateful to the referee for valuable comments regarding the presentation of the results. REFERENCES BABTLETT, M. S. (1937). Properties of sufficiency and statistical tests. Proc. R. Soc. A 160, Box, G. E. P. (1949). A general distribution theory for a class of likelihood criteria. Biometrika 36, EHRENBEKO, A. S. C. (1950). The unbiased estimation of heterogeneous error variances. Biometrika 37, HAN, C. P. (1968). Testing the homogeneity of a set of correlated variances. Biometrilca 55, HAN, C. P. (1969). Testing the homogeneity of variances in a two-way classification. Biometrics 25, JOHNSON, N. L. (1962). Some notes on the investigation of heterogeneity in interactions. Trab. Eslad. 13, KRISHNAMOOHTHY, A. S. & PABTHASARATHY, M. (1951). A multivariate gamma-type distribution. Ann. Math. Statist. 22, Correction (1960) 31, 220.

6 416 G. K. SHUKLA RUSSELL, T. S. & BRADLEY, R. A. (1958). One-way variances in a two-way classification. Biomelrika45, SHTJKLA, G. K. (1972). An invariant test for the homogeneity of variances in a two-way classification. Biometrics 28, [Received December Revised October 1981]