Association for the Chi-square Test

Transcription

1 Assocaton for the Ch-square Test Davd J Olve Southern Illnos Unversty February 8, 2012 Abstract A problem wth measures of assocaton for the ch-square test s that the measures depend on the number of observatons and on the dmenson of the r c contngency table Hence C = 05 for one contngency table and C = 02 for another contngency table does not necessarly mean that the assocaton s hgher n the frst table than the second There are two measures of assocaton that tend to be small when the ch-square test statstc X 2 s not sgnfcant provded > 10(r 1)(c 1) KEY WORDS: categorcal data, contngency coeffcent C, Cramer s V Davd J Olve s Assocate Professor, Department of Mathematcs, Southern Illnos Unversty, Malcode 4408, Carbondale, IL , USA E-mal address: dolve@mathsuedu 1

2 1 THE MAXIMUM VALUE OF X 2 The ch-square test s used to test whether there s an assocaton between two categorcal varables: the row varable wth r categores and the column varable wth c categores The ch square test statstc = X 2 = r c (O j E j ) 2 j E j where O j s the observed count of the jth cell and the expected cell count under ndependence s E j = (th row total)(jth column total)/ where = number of observatons j O j s the total ote that X 2 = 0 f all of the rc observed counts equal the expected counts: O j = E j Let q = mn(r, c) Cramér (1946, p 443) showed that the maxmum value of X 2 s X 2 M = (q 1), and that the maxmum occurs when all of the cell counts are zeros except q nonempty cells such that there s at most one nonempty cell n each row and each column Thus X 2 s smallest under ndependence or no assocaton, and X 2 s largest f the categorcal varables are functons of each other n that f q = r, then the th level of the row varable was observed only wth the j()th level of the column varable and vce verca If q = c, then the jth level of the column varable was observed only wth the (j)th level of the column varable and vce verca For example, n the followng table, let a be the count of the nonempty cell n the th row Then categores r1 and c2, r2 and c5, and r3 and c3 occur together 2

3 Row/Column c1 c2 c3 c4 c5 row total r1 0 a a 1 r a 2 a 2 r3 0 0 a a 3 column total 0 a 1 a 3 0 a 2 To see that such a confguraton of q nonempty cells has X 2 = (q 1), defne (0 0) 2 /0 = 0 n the sum for X 2 Snce the varables are categorcal, the categores of each varable can be arranged so that the nonempty cells counts are O 11 = a 1,, O qq = a q and the r q rows or c q columns where all of the counts are zeros can be omtted, resultng n the followng computatonal table wth X 2 = q q (O j E j ) 2 j and = q a ote that 2 = ( q a )( q j a j ) = j a a j = a 2 + j a a j E j Row/Column c1 c2 c3 cq row total r1 a a 1 r2 0 a a 2 r3 0 0 a 3 0 a 3 rq a q a q column total a 1 a 2 a 3 a q Hence X 2 = (a a2 )2 + a 2 j (0 a a j )2 a a j = ( a a 2 ) 2 a 2 + j a a j = 3

4 [a ( a )] 2 a [ j a a j + a 2 ] 1 a 2 = ( a ) a 2 = + [ ( a ) 2 a 2 ] = + [ 2 2a + a 2 ] a2 = + [ 2 ] 2a = + [ 2a ] = + q 2 a = + q 2 = (q 1) 2 MEASURES OF ASSOCIATIO Gbbons (1985), Goodman and Kruskal (1954) and Wkpeda (2012) revew measures of assocaton for the ch-square test The Cramér (1946, p 443) contngency coeffcent V 2 = X 2 (q 1) and Cramer s V = V 2 The coeffcent of mean square contngency or contngency coeffcent C = X 2 + X 2 The Sakoda (1977) adjusted contngency coeffcent q q X C = q 1 C = 2 q 1 + X 2 Let A be V 2, V, C or C 2 Then assocaton measure A satsfes 0 A 1 wth A = 0 f all of the O j = E j and A = 1 f X 2 = (q 1) Hence C > C and A s near 0 f X 2 s small and A s near 1 f X 2 s near ts maxmum so that the assocaton s large 4

5 Measures of assocaton need to be used wth care For multple lnear regresson, the coeffcent of multple determnaton R 2 s a measure of lnear assocaton If the populaton coeffcent s δ 0, then for large enough sample sze n, the Anova F statstc wll be sgnfcant and R 2 close to δ Cramér (1946, pp ) suggests that R 2 should be consderably larger than p/n f the p predctors are useful, and for d normal errors and 0 slopes, notes that E(R 2 ) = (p 1)/(n 1) If n 1 >> p 1 and n 2 >> p 2 then R 2 1 = 07 suggests stronger lnear assocaton than R 2 2 = 06, but f n 1 = k 1 p 1 and n 2 = k 2 p 2 where k 1 and k 2 are small, then no such comparson can be made In fact, R 2 = 1 f k 1 = k 2 = 1 These types of problems are compounded for assocaton measures for contngency tables Goodman and Kruskal (1954, p 740) note that such assocaton measures depend on r and c Hence A = 06 for one contngency table and A = 02 for another contngency table does not necessarly mean that the assocaton s hgher n the frst table than the second Conover (1971, p 177) notes that V 2 depends on r and c for ts nterpretaton snce X 2 tends to be larger the larger (r 1)(c 1) s, and dvdng by q 1 only partally offsets ths tendency Smth and Albaum (2004, p 631) suggests that C can be larger f r c than f q = r = c To further examne these problems, frst note that the 98th percentle of the χ 2 d dstrbuton s approxmately d+3 d d d Let d = (r 1)(c 1) Assocaton measures A seem comparable for tables of the same dmenson and As, A 0 f X 2 s close to d + 3 d If X 2 = d, and = kd, then V 2 = 1/[k(q 1)] and C = q q 1 1 k + 1 Hence > 10d suggests C 2 and V 2 wll be small (< 02) when X 2 s not sgnfcant 5

6 REFERECES Conover, JW (1971), Practcal onparametrc Statstcs, Wley, ew York, Y Cramér, H (1946), Mathematcal Methods of Statstcs, Prnceton Unversty Press, Prnceton, J Gbbons, JA (1985), Shrnkage Formulas for Two omnal Level Measures of Assocaton, Educatonal and Psychologcal Measurement, 45, Goodman, LA, and Kruskal, WH (1954), Measures for Assocaton for Cross-Classfcaton, Journal of the Amercan Statstcal Assocaton, 49: Sakoda, JM (1977), Measures of Assocaton for Multvarate Contngency Tables, Proceedngs of the Socal Statstcs Secton of the Amercan Statstcal Assocaton (Part III), Smth, SC, and Albaum, GS (2004), Fundamentals of Marketng Research, Sage Publcatons, Thousand Oaks, CA Wkpeda (2012), Contngency Table, onlne at ( Contngency table) 6