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Communications in Statistics - Simulation andComputationPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lssp20

Exact Percentage Points for Testing MultisampleSphericity in the Bivariate CaseDaya K. Nagar a b & Carmen Cecilia Sánchez aa Departamento de Matemáticas , Universidad de Antioquia , Medellín, Colombiab Departamento de Matemáticas , Universidad de Antioquia , Calle 67 53-108, Medellín,A. A. 1226, ColombiaPublished online: 20 Aug 2007.

To cite this article: Daya K. Nagar & Carmen Cecilia Sánchez (2004) Exact Percentage Points for Testing MultisampleSphericity in the Bivariate Case, Communications in Statistics - Simulation and Computation, 33:2, 447-457, DOI: 10.1081/SAC-120037246

To link to this article: http://dx.doi.org/10.1081/SAC-120037246

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COMMUNICATIONS IN STATISTICSSimulation and Computation®

Vol. 33, No. 2, pp. 447–457, 2004

Exact Percentage Points for Testing MultisampleSphericity in the Bivariate Case

Daya K. Nagar* and Carmen Cecilia Sánchez

Departamento de Matemáticas, Universidad deAntioquia, Medellín, Colombia

ABSTRACT

This article deals with the distribution and percentage points ofthe likelihood ratio statistic for testing q-sample sphericity whensamples are drawn from bivariate Gaussian populations. The exactdistribution is obtained using the inverse Mellin transform andresidue theorem. Tables of the percentage points are given forq = 2�1�6.

Key Words: Distribution; Inverse Mellin transform; Multisamplesphericity; Repeated measures designs; Residue theorem.

AMS 2000 Subject Classification: Primary 62H15; Secondary 62H10.

∗Correspondence: Daya K. Nagar, Departamento de Matemáticas, Universidadde Antioquia, Calle 67 53-108, Medellín, A. A. 1226, Colombia; Fax: 57- 4 -2330120; E-mail: nagar@matematicas.udea.edu.co.

447

DOI: 10.1081/SAC-120037246 0361-0918 (Print); 1532-4141 (Online)Copyright © 2004 by Marcel Dekker, Inc. www.dekker.com

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1. INTRODUCTION

In repeated measures designs involving q between-subjects factors,under normality conditions, a necessary and sufficient condition for thevalidity of the usual tests based on the F -ratio is that

C′�1C = · · · = C

′�qC = �2I

where �g is the p×p covariance matrix of the subjects under the gthbetween subjects factors, C is an orthogonal contrast matrix representingthe particular comparison of interest and I is the identity matrix of orderp. The variance �2 is the same for all q groups.

The above condition has been described as multisample sphericity.Hunyh and Feldt (1970) suggested that this assumption be tested in twosteps. In the first step one tests

Ha � C′�1C = · · · = C

′�qC

and then if Ha is not rejected one tests

Hb � C′�C = �2Ip�

If Ha and Hb are not rejected then multisample sphericity is assumed.Mendoza (1980) suggested a maximum likelihood ratio test which testsfor multisample sphericity hypothesis in a single step. Without any lossof generality the hypothesis of multisample sphericity can be stated asfollows:

H0 � �1 = · · · = �q = �2Ip�

When q = 1, the hypothesis of multisample sphericity reduces to theMauchly sphericity hypothesis H0 � �1 = �2Ip and the likelihood ratiotest criterion in this case is called the sphericity test criterion. Variousdistributional results and properties of the sphericity test criterion areavailable in the literature (see Gupta, 1977; Khatri and Srivastava, 1971,1974; Kulp and Nagarsenker, 1983; Muirhead, 1976; Nagarsenker andPillai, 1973; Sugiura, 1969, 1995).

Let xgj� j = 1� � � � � Ng be a random sample from Np��g� �g�, where �g

and �g are unknown, g = 1� � � � � q. Also, let Ag =∑Ng

j=1 �xgj − xg·�× �xgj − xg·�′ where Ngxg· =

∑Ng

j=1 xgj . Then Ag∼Wp�ng� �g�, ng = Ng − 1.For testing H0, Pillai and Young (1973) considered the statisticR2 = max1≤i≤q Ti/min1≤j≤q Tj , where Tg = trAg/ng, and derived certaindistributional results by noting that the distribution of R2 is same as

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Testing Multisample Sphericity 449

that of Hartley’s Fmax statistic. Mendoza (1980) derived the modifiedlikelihood ratio statistic �∗ for testing H0, its null moments and gavea one term approximation to its null distribution using Box’s method.Gupta and Nagar (1987) derived the hth nonnull moment and obtainedthe null density in a series involving psi and generalized Riemann zetafunctions. Gupta et al. (1992) have also derived the distribution of �∗

in a series involving incomplete beta functions. The modified likelihoodratio test statistic and its moments are given by

�∗ = �pn0�n0p/2∏q

g=1 nngp/2g

∏qg=1 det�Ag�

ng/2

tr�∑q

g=1 Ag�n0p/2

and

E��∗h� = �pn0�n0ph/2∏q

g=1 nngph/2g

��n0p/2���n0p�1+ h�/2

×q∏

g=1

p∏j=1

��ng�1+ h�/2− �j − 1�/2

���ng − j + 1�/2 (1.1)

respectively, where n0 =∑q

g=1 ng and Re�ng�1+ h�� > p− 1, g =1� � � � � q. The asymptotic nonnull distribution of a constant multiple of−2 ln�∗ is available in Gupta and Nagar (1988). The exact distributionof �∗ for n1 = · · · = nq = n is derived in Gupta and Nagar (1987).

In this article, we will compute the exact percentage points fortesting multisample sphericity in the bivariate case. For simplicity, theexact distribution of V1 = ��∗�1/n, for n1 = · · · = nq = n, will be derivedusing the residue theorem. Since the technique is well known (see Guptaand Nagar, 1987), we will outline the main steps of the derivation andgive the final result omitting all the details of the derivation. The exactpercentage points of V1 for q = 2�1�6 are computed using the exactdistribution given in this article.

2. DENSITY OF V1 IN THE BIVARIATE CASE

Substituting n1 = · · · = nq = n, p = 2 in (1.1) and usingGauss-Legendre multiplication formula for gamma function, the hthmoment of V1 = ��∗�1/n is simplified as

E�Vh1 � =

�q�n− 1+ h�

�q�n− 1�

q−1∏k=0

��n+ k/q�

��n+ h+ k/q��

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Now, using the inverse Mellin transform and the above momentexpression, the density of V1 is obtained as

f�v1� = K�n� q��2���−1∫C

�q�n− 1+ h�∏q−1k=0 ��n+ h+ k/q�

v−1−h1 dh� 0 < v1< 1�

(2.1)

where � = √−1, C is a suitable contour and

K�n� q� =∏q−1

k=0 ��n+ k/q�

�q�n− 1��

Substituting n− 1+ h = t and simplifying, the density (2.1) is restated as

f�v1� = K�n� q��2���−1vn−21

∫C1

�q−1�t�

t∏q−1

k=1 ��t + 1+ k/q�v−t1 dt�

0 < v1< 1� (2.2)

where C1 is the changed contour. The poles of the integrand are given byt = −j, j = 0� 1� 2� � � � , and that each pole is of order q − 1 except t = 0which is of order q. Hence by the residue theorem

f�v1� = K�n� q�vn−21

�∑j=0

Rj� 0 < v1 < 1� (2.3)

where Rj is the residue at t = −j. From the calculus of residues, theresidue Rj at t = −j for j≥ 1 is derived as

Rj =vj1

�q − 2�!q−2∑u=0

(q − 2u

)A

�u�j0 �−ln v1�

q−2−u (2.4)

where

A�u�j0 =

u−1∑m=0

(u− 1m

)A

�u−1−m�j0 B

�m�j0

with

A�0�j0 = �−1��q−1�j+1

j�j!�q−1∏q−1

k=1 ��1− j + k/q��

B�0�j0 = �q − 1���1�+ 1

j− �q − 1�

j−1∑�=0

1�− j

−q−1∑k=1

�

(1− j + k

q

)

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Testing Multisample Sphericity 451

and

B�m�j0 = �−1�m+1m!

[�q − 1���m+ 1� 1�+ 1

�−j�m+1

+ �q − 1�j−1∑�=0

1��− j�m+1

−q−1∑k=1

�

(m+ 1� 1− j + k

q

)]�

In the expressions for B�0�j0 and B

�m�j0 , ��·� and ��·� ·� are well known psi

and generalized Riemann zeta functions respectively (see Abramowitz

Table 1. Percentage points of V1 for q = 2.

n � = 0�01 � = 0�025 � = 0�05 � = 0�1

3 0�0276701 0�0475534 0�072533 0�1124314 0�088825 0�127788 0�169741 0�2280035 0�160494 0�211137 0�26157 0�3268376 0�229695 0�286293 0�340054 0�4067197 0�292211 0�351278 0�405628 0�4711228 0�347361 0�406885 0�460419 0�5236169 0�395671 0�454515 0�506535 0�567002

10 0�437995 0�495534 0�545721 0�60335711 0�4752 0�531104 0�579343 0�63420712 0�508058 0�562174 0�60846 0�66068413 0�537227 0�589506 0�633891 0�6836414 0�563258 0�613711 0�656279 0�70372215 0�586606 0�63528 0�676126 0�72143216 0�60765 0�654611 0�693836 0�73716217 0�626704 0�672028 0�709731 0�75122318 0�644028 0�687796 0�724073 0�76386619 0�659844 0�702136 0�737076 0�77529420 0�674336 0�71523 0�748919 0�78567221 0�68766 0�727232 0�759748 0�79513822 0�699951 0�738273 0�769687 0�80380623 0�711321 0�748461 0�778841 0�81177424 0�72187 0�757891 0�787299 0�81912225 0�731682 0�766644 0�795136 0�82591926 0�740831 0�77479 0�802419 0�83222628 0�75739 0�789496 0�815541 0�84356430 0�771972 0�802407 0�827033 0�85347

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452 Nagar and Sánchez

Table 2. Percentage points of V1 for q = 3.

n � = 0�01 � = 0�025 � = 0�05 � = 0�1

3 0�00906789 0�016621 0�0268414 0�04451584 0�0412952 0�0621124 0�085817 0�1207675 0�0894522 0�121765 0�155473 0�2013166 0�14312 0�183406 0�22325 0�2748547 0�196374 0�241655 0�284865 0�3390178 0�246559 0�29471 0�339488 0�3942979 0�292727 0�342308 0�387528 0�441899

10 0�334734 0�384789 0�429752 0�48306711 0�372787 0�422687 0�466967 0�5188912 0�407225 0�456563 0�499909 0�55027113 0�438425 0�486941 0�529208 0�57794214 0�466749 0�51428 0�555397 0�60249815 0�492529 0�538982 0�578921 0�62441916 0�516061 0�561386 0�60015 0�64409617 0�537603 0�581783 0�619392 0�66184918 0�557383 0�60042 0�636906 0�6779419 0�575597 0�617507 0�652908 0�6925920 0�592415 0�633224 0�667582 0�7059821 0�607987 0�647725 0�681084 0�71826422 0�62244 0�661143 0�693546 0�72957323 0�635887 0�673592 0�705082 0�74001624 0�648428 0�685171 0�71579 0�74968925 0�660149 0�695968 0�725755 0�75867326 0�671126 0�706057 0�735052 0�76703928 0�69111 0�724371 0�751888 0�78215230 0�708832 0�740555 0�766725 0�795433

and Stegun, 1965; Luke, 1969; Magnus et al., 1966). The residue at t = 0is obtained as

R0 =1

�q − 1�!q−1∑u=0

(q − 1u

)A

�u�00 �−ln x�q−1−u (2.5)

where

A�0�00 = 1∏q−1

k=1 ��1+ k/q��

B�0�00 = �q − 1���1�−

q−1∑k=1

�

(1+ k

q

)

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Testing Multisample Sphericity 453

Table 3. Percentage points of V1 for q = 4.

n � = 0�01 � = 0�025 � = 0�05 � = 0�1

3 0�00320871 0�00621007 0�0105195 0�01845374 0�020252 0�0316225 0�0451492 0�06603225 0�0519528 0�0727738 0�0952806 0�1270616 0�092221 0�120955 0�150258 0�1894527 0�135763 0�170364 0�204293 0�2480528 0�179358 0�218033 0�254899 0�3012189 0�221307 0�262662 0�301251 0�348786

10 0�260821 0�303825 0�343289 0�39115711 0�297614 0�34152 0�381274 0�428912 0�331669 0�375938 0�415584 0�46259813 0�3631 0�407351 0�446617 0�49278714 0�392084 0�436046 0�474751 0�51993615 0�418822 0�462303 0�500331 0�54444916 0�443511 0�486382 0�523656 0�5666717 0�466344 0�508514 0�544993 0�5868918 0�487496 0�528908 0�56457 0�60535819 0�507128 0�547746 0�582585 0�62228420 0�525384 0�565191 0�59921 0�63784821 0�542394 0�581383 0�614595 0�65220422 0�558273 0�596447 0�628869 0�66548423 0�573125 0�610492 0�642145 0�67780224 0�587042 0�623615 0�65452 0�68925725 0�600105 0�635902 0�666083 0�69993626 0�612388 0�647427 0�676908 0�70991428 0�634869 0�668454 0�696608 0�72802130 0�654933 0�687149 0�71407 0�744019

and

B�m�00 = �−1�m+1m!

[�q − 1���m+ 1� 1�−

q−1∑k=1

�

(m+ 1� 1+ k

q

)]�

Substituting (2.4) and (2.5) in (2.3) we get the density of V1 as

f�v1� = K�n� q�vn−21

[1

�q − 1�!q−1∑u=0

(q − 1u

)A

�u�00 �−ln v1�

q−1−u

+ 1�q − 2�!

�∑j=1

vj1

q−2∑u=0

(q − 2u

)A

�u�j0 �−ln v1�

q−2−u

]�

0 < v1 < 1� (2.6)

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454 Nagar and Sánchez

Table 4. Percentage points of V1 for q = 5.

n � = 0�01 � = 0�025 � = 0�05 � = 0�1

3 0�00118956 0�00241534 0�00426668 0�007861124 0�0102574 0�0165525 0�0243227 0�03678925 0�0309298 0�0444291 0�0594615 0�08135496 0�0606311 0�0811596 0�10263 0�1321177 0�0954651 0�121867 0�148338 0�1832828 0�1324 0�163348 0�193447 0�2320669 0�169487 0�203794 0�236401 0�27735

10 0�205589 0�242284 0�27654 0�31884511 0�240094 0�278419 0�313683 0�3566512 0�272716 0�312088 0�347888 0�39102813 0�303362 0�343337 0�379327 0�42229714 0�332045 0�372289 0�408218 0�45077915 0�358842 0�399104 0�434789 0�47677916 0�383861 0�423953 0�459264 0�50057217 0�407221 0�447002 0�481847 0�52240218 0�429044 0�468412 0�502728 0�54248519 0�449452 0�488331 0�522074 0�56101120 0�468557 0�506894 0�540037 0�57814421 0�486465 0�524222 0�556751 0�59402922 0�503274 0�540427 0�572334 0�60879323 0�519073 0�555608 0�586893 0�62254524 0�533944 0�569853 0�600521 0�63538425 0�547961 0�583242 0�613301 0�64739626 0�56119 0�595846 0�625307 0�65865528 0�585527 0�618954 0�647257 0�67917730 0�607379 0�639617 0�66682 0�697402

3. COMPUTATION

The computation of the exact percentage points of V1 = ��∗�1/n hasbeen carried out by using F�x� = ∫ x

0 f�v1�dv1 where f�v1� is given by(2.6). The computation is carried out by using the series representationgiven in (2.6). First F�x� is computed for various values of x. It ischecked for monotonicity and for the conditions F�x� → 0 as x → 0 andF�x� → 1 as x → 1. Then x is computed for various values of q, n andF�x� = �. A six place accuracy is kept throughout. Tables are given forvalues of q from 2 to 6 (Tables 1–5). The exact percentage points of �∗

can be computed from those of V1 by using the transformation �∗ = Vn1 .

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Testing Multisample Sphericity 455

Table 5. Percentage points of V1 for q = 6.

n � = 0�01 � = 0�025 � = 0�05 � = 0�1

3 0�03455436 0�03965728 0�00177173 0�003411624 0�00531188 0�00883076 0�0133169 0�02076045 0�0187298 0�0275224 0�0375702 0�05260166 0�0404165 0�0551047 0�0708068 0�09286517 0�067914 0�0880464 0�108622 0�1363298 0�0987235 0�123434 0�147884 0�179829 0�130955 0�159319 0�186704 0�221662

10 0�163338 0�194517 0�22405 0�26107611 0�195079 0�228366 0�259413 0�29778212 0�225708 0�260528 0�292597 0�3317613 0�254974 0�290866 0�323575 0�36312614 0�282762 0�319362 0�352419 0�39205615 0�309042 0�346065 0�379246 0�41874716 0�333839 0�371061 0�404197 0�44339817 0�357207 0�394452 0�427416 0�46619918 0�379216 0�41635 0�449044 0�48732419 0�399947 0�436863 0�469216 0�50693320 0�419479 0�456097 0�488056 0�5251721 0�437895 0�474153 0�505679 0�54216422 0�455271 0�491122 0�52219 0�55803123 0�471681 0�507091 0�537682 0�57287224 0�487193 0�522137 0�552242 0�5867825 0�501873 0�536334 0�565945 0�59983626 0�51578 0�549745 0�578862 0�61211328 0�541487 0�574447 0�602582 0�63458630 0�564702 0�596657 0�623836 0�654645

ACKNOWLEDGMENT

This research was supported by the Comité para el Desarrollode la Investigación, Universidad de Antioquia research grant no.IN358CE.

REFERENCES

Abramowitz, M., Stegun, I. A. (1965). Handbook of MathematicalFunctions. New York: Dover Pub.

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Gupta, A. K. (1977). On the distribution of sphericity test criterionin the multivariate Gaussian distribution. Aust. J. Statist. 19(3):202–205.

Gupta, A. K., Nagar, D. K. (1987). Likelihood ratio test for multi-sample sphericity. In: Gupta, A. K., ed. Advances in MultivariateStatistical Analysis. Dordrecht: D. Reidel Publishing Company,pp. 111–139.

Gupta, A. K., Nagar, D. K. (1988). Asymptotic expansion of thenonnull distribution of the likelihood ratio statistic for testingmultisample sphericity. Commun. Statist. Theo. Meth. 17(9):3145–3155.

Gupta, A. K., Nagar, D. K., Jain, K. (1992). Asymptotic distributionof likelihood ratio statistic for testing multisample sphericity.J. Korean Statist. Soc. 21(1):14–26.

Huynh, H., Feldt, L. S. (1970). Conditions under which mean squareratios in repeated measurements designs have exact F -distribution.J. Amer. Statist. Assoc. 65:1582–1589.

Khatri, C. G., Srivastava, M. S. (1971). On exact non-null distributionsof likelihood ratio criteria for sphericity test and equality of twocovariance matrices. Sankhya A33:201–206.

Khatri, C. G., Srivastava, M. S. (1974). Asymptotic expansions of thenon-null distributions of likelihood ratio criteria for covariancematrices. Ann. Statist. 2:109–117.

Kulp, R. W., Nagarsenker, B. N. (1983). Further asymptotic expansionsof the distributions of the sphericity test. Commun. Statist. Theo.Meth. 12(1):119–129.

Luke, Y. L. (1969). The Special Functions and Their Approximations.Vol. I. New York: Academic Press.

Magnus, W., Oberhettinger, F., Soni, R. P. (1966). Formulas andTheorems for the Special Functions of Mathematical Physics.Springer-Verlag, No. 52.

Mendoza, J. L. (1980). A significance test for multisample sphericity.Psychometrika 45(4):495–498.

Muirhead, R. J. (1976). Forms for the distribution of the ellipticitystatistic for bivariate sphericity. Commun. Statist. Theo. Meth.5(5):413–420.

Nagarsenker, B. N., Pillai, K. C. S. (1973). The distribution of thesphericity test criterion. J. Multivariate Anal. 3:226–235.

Pillai, K. C. S., Young, D. L. (1973). The max trace-ratio test ofthe hypothesis H0 � �1 = · · · = �k = ��0. Commun. Statist. 1(1):57–80.

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Testing Multisample Sphericity 457

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Sugiura, N. (1995). Exact nonnull distributions of sphericity tests fortrivariate normal population with power comparison. Amer. J.Math. Manage. Sci. 15(3&4):355–374.

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