[412] A TEST FOR HOMOGENEITY OF THE MARGINAL DISTRIBUTIONS IN A TWO-WAY CLASSIFICATION

Transcription

1 [412] A TEST FOR HOMOGENEITY OF THE MARGINAL DISTRIBUTIONS IN A TWO-WAY CLASSIFICATION BY ALAN STUART Divisio of Research Techiques, Lodo School of Ecoomics 1. INTRODUCTION There are several circumstaces i which we may wish to test the homogeeity of the two sets of margial probabilities i a two-way classificatio. For example, a sample from a bivariate distributio (say height of father, height of so) may be classified ito a two-way table with idetical (height) groupigs i each margi. Or a similar classificatio may be possible for a o-measurable variable (say stregth of right had stregth of left had). Agai, i surveys of the same sample (a 'pael') o two differet occasios, the iterrelatio of the results o the two occasios may be displayed i a two-way table, with oe margi correspodig to each occasio. I all these cases, the questio may arise: are the two sets of margial probabilities idetical? If the variable is measurable, we may test the differece betwee the meas of the two margial distributios by a large-sample stadard-error test. However, we may be iterested i the overall distributios, rather tha oly i their meas. For the more striget hypothesis of homogeeity, a test exists if we have two completely idepedet samples, whe a ordiary %* test of homogeeity may be applied (Cramer, 1946, p. 445). This test does ot meet the essetially bivariate situatios described above, where o-idepedece of the margial distributios is a fudametal feature of the problem. Whe the classificatio is a double dichotomy, the problem of testig margial homogeeity is simple, ad its solutio is a special case of the large-sample solutio of the more geeral 2 K classificatio problem give by Cochra (1950). Bowker (1948) gave a largesample test for complete symmetry i a two-way classificatio, a more restrictive hypothesis which is cocered with the etire set of probabilities i the classificatio, ad ot oly with the margial probabilities as we are here. I the preset paper, a large-sample test for margial homogeeity is derived ad illustrated. 2. VABIANCES AND COVABIANCBS With the usual otatio for atoxto table, we deote the probability of fallig ito the tth row ad jth colum by p lf, ad defie the margial probabilities by while obviously Pi. = 2Pa = Zft = 2P y i i i ' i The correspodig sample umbers are deoted by ijt {.,. t, while the total sample size is.

2 STUABT 413 By stadard multiomial theory, we have for the meas, variaces ad co-variaces of the {j E( ti) = p ip \ C( ik,,,) = - pupt, (t 4= I ad/or k 4= j), J ad also We ow require C( {, ^ = F( jy ) + 2 C( tt, j y ), where t +1 ad/or ifc 4=j. From (1), this is (2) p i p J ), (3) o takig the term i pjy ito the summatio. We ow defie the statistics d { =.-7i l. (i = 1,2,...,ro). Sice the likelihood-ratio priciple yields a itractable result i this case, it seems atural to use the d t as the basis for a test cocerig the differeces betwee the correspodig margial probabilities. For their meas, variaces ad co-variaces, we have, from (2) ad (3), the exact results: Also, for i + «' p. t ), (4) V(d t )= V( i )+V(. t )-2C( {, i ) 3. TESTING THE HYPOTHESIS OF HOMOGENEITY Suppose that we wish to test the hypothesis H i- (Pt.-P.i) = ^i (» = ltom), where, of course, we must have ^ \ = 0. i-l (4), (5) ad (6) the become (!,.j. i p«)j} (7) C(d it d f \H 1 ) = - )

3 414 Homogeeity of margial distributios ad i the special case of particular iterest here, where we have Ho-. A< = 0 (all*), (7) becomes E(d { \ HJ = 0, "j V-VidtlHo) =Mp t.+p. t -2p«), I (8) V ti = C(d it d f \Ho) = - (p {i + Pii ) (» 4= j).j We have deliberately refraied from replacig p < by p t- i T^, as we could have doe, sice if, as will geerally be the case, the true probabilities are ukow, the likelihood estimators of the variaces ad co-variaoes i (8) are It is well kow that the {i have a limitig multiomial distributio with momets give by (1). It follows that the ^. ad. 4 will also be joitly ormally distributed, sice they are liear fuctios of the «y, ad fially, by the same argumet, that the variates d f (t = 1 toto)will be asymptotically multiormally distributed with momets give by (7) i geeral, ad by (8) o the ull hypothesis. To a further degree of approximatio, the same result holds if (9) is used as a estimator of (8). Now it is a stadard result that i the expoet, say \Q, of a multiomial distributio, Q is distributed like x % with degrees of freedom equal i umber to the rak of the distributio. Furthermore, Q i this expoet is simply a quadratic form i the variables. Thus, o HQ, m Q= S a^d, (10) is distributed i the limit like x 2 with (TO 1) degrees of freedom, the rak of the distributio beig (TO 1) sice 2 d 4 = 0, ad i geeral this is the oly costrait o the d t. If the dis- {-i tributio were o-sigular, the coefficiets a it would simply be the elemets of the iverse of the variace-covariace matrix {V ti ). As a result of the sigularity, (V ti ) itself is also sigular, ad so we caot ivert it. However, ay margial distributio of a multiomial distributio is itself ormal, so that we may elimiate the redudat variate (say the last) ad obtai the result that Q = s F%d, (11) U-l is distributed i the limit like x* with (TO 1) degrees of freedom, (TO 1) still beig the rak of the distributio. If we replace the F w i (11) by their rajrimum-likelihood estimators give by (9), the same asymptotic result holds. I (11), (Vy) is ow uderstood to be defied for i,j = 1, toto 1 oly, ad (F 4 ') is its iverse. The fact that (11) takes o explicit accout of d m leds it a appearace of arbitrariess, but although the values of the terms i Q are chaged by elimiatig some other d t istead of d,, their sum Q is uiquely determied. This is i virtue of the fact that sice ay (TO 1) of the d t uiquely determie the other oe, Q must be, apart from costats, the log likelihood of the complete set of d {, irrespective of which of the d { is omitted i Q. I poit of fact, Q could be expressed as a fuctio of all the m values of d i ad the complete matrix

4 ALAN STUAET 415 (V {i ), but this merely complicates the computatioal procedure ad makes o differeoe to the result. Fially, we must determie the appropriate critical regio of the distributio of Q. (7) shows that, whe.flj, is ot true, the variace-co variace matrix of the d { may be writte (I?,) = (w i} ), where («ty) is idepedet of. The iverse matrix is therefore Thus the expected value of Q i (11) is so that, usig (7), (F«) =-(«>«) E(Q \ H x ) = 0(). O the other had, the limitig x 2 distributio of Q o HQ gives us (12) E(Q\H 0 ) = m-l, (13) idepedet of. Thus, with icreasig, the differece betwee (12) ad (13) will exceed ay boud, ad the appropriate critical regio for our large-sample test is the upper tail of the distributio of Q. The same coclusio holds if (Vy) is replaced by its maximumlikelihood estimator 4. COMPUTATIONAL PROCEDURE : AN TT.TAMPT.-R We ow set out the computatio ecessary for the test, ad give a illustratio of its use. (1) Form the matrix (J$ y ) give by (9), for i,j = 1 to (m- 1). (2) Ivert (%) to obtai (F«)- m 1 A (3) Compute Q = 2 Fty^eL, where d t = t {, ad test i the x* distributio with (m 1) degrees of freedom, the critical regio beig the upper tail. Cosider the followig example, which uses data quoted by Stuart (1953). ^\^ ^ Left eye Right eye ^^^^ 7477 toome aged 30-39; uaided distace visio Highest Secod Third Lowest Total Highest Secod Third Lowest Total

5 416 Homogeeity of margial distributios We have d r = = 69, d t = = 34, d 3 = = -51, ad we check that d A = = 52 equals the sum (<2 1 + d 1 + d 3 ), egatively siged. Sice d^ ad d 2 are positive, while ^ ad d 4 are egative, oe may ask whether the sight of the right eye may really be regarded as better tha that of the left eye for the populatio from which this is a sample. The estimated variace matrix of d\, d t, d% is obtaied by use of (9), as j> ^ = ( ) + ( ) = 843, V lt = V = - ( ) = - 500, ( \ / The iverse is obtaied directly as / \ (F<0= lo-mlseo ), \ / ad we have, for our x 2 statistic with 3 degrees of freedom, Q = 10-«{(2482 x 69*) + (1972 x 34 2 ) + (1690 x 51 2 ) + 2(1560 x 69 x 34) - 2(1295 x 69 x 51) - 2(1368 x 34 x 51)} = The 1 % poit for the x* distributio with 3 degrees of freedom is 11*345, so we coclude that the result is sigificat of a differece i the distributio of sight i right ad left eye. If, istead, of elimiatig d t, we had elimiated, say, d t we should have obtaied as above: / \ (V i} ) = , \ / ( \ , ad fially / Q = 10-«{(1332 x 34 2 ) + (1583 x 51 2 ) + (2482 x 52 2 ) + 2(1187 x 51 x 52) - 2(995 x 34 x 51) - 2(921 x 34 x 52)} = as before. I am grateful to Mr J. Durbi for several illumiatig discussios of this problem. REFERENCES BOWKEB, ALBERT H. (1048). A test for symmetry i cotigecy tables. J. Amer. Statist. Ass. 43, COCHBAN, W. Q. (1950). The compariso of percetages i matched samples. Biometrika, 37, CRAMEB, TTAWAT,T> (1946). Mathematical Methods of Statistics. Priceto Uiversity Press. STUABT, A. (1953). The estimatio ad compariso of stregths of associatio i cotigecy tables. Biometrika, 40,