POWER ANALYSIS OF INDEPENDENCE TESTING FOR CONTINGENCY TABLES

Transcription

1 ZESZYTY NAUKOWE AKADEMII MARYNARKI WOJENNEJ SCIENTIFIC JOURNAL OF POLISH NAVAL ACADEMY 05 (LVI) (00) Piotr Sulewki, Ryzard Motyka DOI: / X.660 POWER ANALYSIS OF INDEPENDENCE TESTING FOR CONTINGENCY TABLES ABSTRACT Six tet for idepedece i a two way cotigecy table amely chi quared tet, log likelihood ratio tet, Neyma modified chi quared tet, Kullback Leibler tet, Freema Tukey tet, Creie Read tet, were examied. It wa accomplihed with the Mote Carlo method. The Goodma Krukal τ idex wa ued to fix depedece i two way cotigecy table i Mote Carlo experimet. The examiatio coited i determiig power fuctio of the tet. Next, the power fuctio were compared to each other. It wa revealed that differece i power are egligible. Key word: two way cotigecy table, idepedece tet, Mote Carlo method. BASIC OF CONTINGENCY TABLE Let X ad Y be two feature of the ame object havig level X, X,..., X w ad Y, Y,..., Yk. Tetig for idepedecy of thee two feature with appropriately arraged cotigecy table ad χ tatitic applied i probably oe of the mot commo tatiticia tak. Table recall a patter of two dimeioal cotigecy table. There are item claified with repect to X ad Y. Thi produce a table of a cheme how below a table. Here are the cout, beig the umber of object claified to belog to the cell ( X i, Y j ). The ymbol,,..., w ad,,..., k are margial cout. The Pomeraia Academy, Ititute of Mathematic, Arcizewkiego Str., Słupk, Polad; e mail: {piotr.ulewki; ryzard.motyka}@apl.edu.pl 37

2 Piotr Sulewki, Ryzard Motyka Table. A cheme of w k cotigecy table Criterio X Criterio Y Y Y Yj... Yk Margial cout X j... k X... k Xi i i ik Xw w w wj... wk Margial cout j.... k i w The table of a form a above i a bai to tet hypothei commoly called the mai ad deoted Ho that X ad Y are idepedet. Pearo tet tatitic i of the form []: where: w k ( e ) χ =, () e i= j = e i j = (A) are cout expected whe ditributio o ( )( k ) H o i true. Statitic () have aymptotically a chi quare w degree of freedom, whe H o i true. Goodma ad Krukal [4] put forward the followig meaure of depedece i two dimeioal cotigecy table: w k p / p j pi i= j= i= * τ =, p w * =. (B) p i= w i 38 Zezyty Naukowe AMW Scietific Joural of PNA

3 Power aalyi of idepedece tetig for cotigecy table ALTERNATIVE TESTS OF INDEPENDENCE The opportuity to make omethig better that it curretly i occur everywhere if omeoe chooe to act. A the reult tatitical ciece ha bee eriched with five other tatitic iteded to tet idepedecy. The followig competitive tet tatitic were developed after Pearo: The G tatitic of Sokal ad Rohlf [] The N tatitic of Neyma [0] G r c = l ; () i= j= e N = r c ( e ) i= j= ; (3) The KL tatitic of Kullback ad Leibler [6] r c e KL = e l (4) i= j= The FT tatitic of Freema ad Tukey [3] r k ( e ) FT = 4 ; (5) i= j= The CR tatitic of Creie ad Read [] 9 CR = 5 w k i= j= e / 3. (6) Both Pearo ad other origiator claim that their tatitic follow the chi quare ditributio o ( w )( k ) degree of freedom provider H o i true. Thi claim eem irrelevat to N ad KL tatitic ice they have i deomiator. (00) 05 39

4 Piotr Sulewki, Ryzard Motyka The cout are radom variable. Radom variable that iclude reciprocal or ratio of radom variable do ot have expected value ad frequetly alo ome higher momet. I cotrat chi quare ditributio ha all the momet. From the defiitio D λ = λ ( ) λ + λ w k = = j e i (7) it ca eaily be ee that tatitic () ( λ = ), the log likelihood ratio tatitic () ( λ = 0), the Neyma tatitic (3) ( λ = ), the modified log likelihood ratio tatitic (4) ( λ = ), the Freema Tukey tatitic (5) ( λ = 0,5) ad Creie Read tatitic (6) ( λ = / 3), are all pecial cae. 0 D ad Statitic (6) emerge a a excellet ad compromiig alterative to the D. Variou propertie ad compario of thee o called chi quared tet ca be foud i [, 5, 7, 9, 3]. ANALYSIS OF THE POWER OF TESTS Fillig i cotigecy table Phae. Preparatio tep Step P. Settig parameter of imulatio: dimeio of cotigecy table w k, for itace 3 3; cocrete value of τ that hold i virtual geeral populatio we ample from, for itace τ = 0, 5 ; ample ize, for itace = 00. Step P. Labelig cell. To cell of cotigecy table aig iteger deoted v that take value from v = to v max = w k tartig from the leftmot upper corer. Thee are cell label. For itace p i label otatio refer to 5 p i matrix otatio. 3 Step P3. Determiig a patter of cotigecy table i.e. uch a et of ubtituted ito (B) yield τ = 0, 5. p value that 40 Zezyty Naukowe AMW Scietific Joural of PNA

5 Power aalyi of idepedece tetig for cotigecy table Phae. The mai tep Repeat the followig tep from M to M4 time: Step M. Geerate radom umber r k uiformly ditributed withi < 0, >. Step M. Fid uch cell label v that atifie the followig two ided iequality v v= p v < r k v v= p v. (8) Step M3. Havig cell label v determie cell idex. Step M4. Icreae cell cout i.e. +. = Etimatig power of tet fuctio Let τ be a appropriately defied meaure of XY depedecy for itace of (B) form. A tadard coure of actio i that two competitive hypothee are formed, amely: the ull hypothei, H 0 that ay: X ad Y are idepedet; alterative hypothei, H that ay: X ad Y are depedet. The power of the tet fuctio ha the meaure of depedecy τ a it argumet ad retur the probability of rejectig H 0 a depedecy i growig. Sice there i o way to determie the fuctio i aalytical way we employ the Mote Carlo method we ca rely o i uch ituatio. Preparatio phae et dimeio of the cotigecy table i.e. umber of row w ad umber of colum k ; et cocrete value of τ = τ for which the tet ha to be carried out; et ample ize = ; et the cofidece level α = α ; calculate the umber of degree of freedom d = ( w )( k ) ; determie α quatile of the chi quare ditributio with d degree of freedom beig the tet critical value χ crit = q ; α, d et umber of repetitio m ; rep create a vector CoR of 6 row that are couter of rejectio of H 0 ; row relate to particular tet tatitic from () to (6); et iitial value to 0; determie the patter of cotigecy table a it wa decribed i ectio 3. (00) 05 4

6 Piotr Sulewki, Ryzard Motyka The mai phae Repeat time what pecified below: mrep fill i the cotigecy table i accordace with the patter a it wa decribed i ectio 4; calculate value of tet tatitic from () to (6); compare each of above value to critical value; icreae by the couter tied with thi tet tatitic that exceed critical value. Calculate power of tet uig tatitic () (6) ( ) = CoR m ; =,..., 6 PoT τ. (9) i rep The reult Let u chooe Pot ( τ) a the referece power fuctio that i well grouded becaue the tatitic of Pearo i both widely kow ad ued. Power of chi quared tet i how i figure. To compare other tet fuctio defie the relative differece a: ( τ ) ( τ ) Pot Rd ( τ ) = ; =,...,6. (0) Pot Thee relative differece are how i figure from to 5.,00 0,90 Power of chi quared tet 0,98 0,94 0,87 0,99 0,96,00 0,99 0,90 0,80 0,79 0,8 0,70 0,70 0,64 Power of χ tet 0,60 0,50 0,40 0,30 0,0 0,49 0,44 0,43 0,9 0,53 x, =00 3x3, =00 4x4, =00 4x, =00 0,0 0,06 0,05 0,00 0 0,0 0,04 0,06 0,08 0, The Goodma Krukal τ coefficiet Fig.. Power of chi quared tet 4 Zezyty Naukowe AMW Scietific Joural of PNA

7 Power aalyi of idepedece tetig for cotigecy table 5% Table x, =00 G 4% N Relative differece 3% % KL FT CR % 0% 0,0 0,04 0,06 0,08 0, The Goodma Krukal τ coefficiet Fig.. Relative differece: table x, = 00 6% Table 3x3, =00 Relative differece 5% 4% 3% % G N KL FT CR % 0% % 0,0 0,04 0,06 0,08 0, The Goodma Krukal τ coefficiet Fig. 3. Relative differece: table 3 x 3, = 00 (00) 05 43

8 Piotr Sulewki, Ryzard Motyka 30% Table 4x4, =00 Relative differece 5% 0% 5% 0% G N KL FT CR 5% 0% 0,0 0,04 0,06 0,08 0, The Goodma Krukal τ coefficiet Fig. 4. Relative differece: table 4 x 4, = 00 5% Table 4x, =00 G Relative differece 0% 5% 0% N KL FT CR 5% 0% 0,0 0,04 0,06 0,08 0, Goodma Krukal τ coefficiet Fig. 5. Relative differece: table 4 x, = Zezyty Naukowe AMW Scietific Joural of PNA

9 Power aalyi of idepedece tetig for cotigecy table CONCLUSIONS Differece (0) vary withi iterval of few percet. Some of them are eve egative. Oly tet procedure that employ N, KL tatitic appear a outtadig oe. A explaatio i rather of thi ort that N, KL tatitic imply, a it wa metioed i ectio, do ot follow chi quare ditributio. Cocludig we may ay that o oticeable progre wa achieved i the domai of cotigecy table by puttig forward tatitic () (6). REFERENCES [] Creie N., Read T., Multiomial Goode of Fit Tet, Joural of the Royal Statitical Society, Serie B (Methodological), 984, Vol. 46, pp [] Fieberg S. E., The ue of chi quared tatitic for categorical data problem, Joural of the Royal Statitical Society, Serie B, 979, Vol. 4, pp [3] Freema M. F., Tukey J. W., Traformatio related to the agular ad the quare root, Aal of Mathematical Statitic, 950, Vol., pp [4] Goodma L., Krukal W., Meaure of Aociatio for Cro claificatio, Joural of the America Statitical Aociatio, 954, Vol. 49, pp [5] Hor S. D., Goode of fit tet for dicrete data. A review ad a applicatiob to a health impairmet cale, Biometric, 977, Vol. 33, pp [6] Kullback S., Iformatio Theory ad Statitic, Wiley, New York 959. [7] Lacater H. O., The chi quared ditributio, Wiley, New York 969. [8] Light R. J., Margoli B. H., A Aalyi of Variace for Categorical Data, Joural of the America Statitical Aociatio, 97, Vol. 66, No. 335, pp [9] Moore D. S., Recet developmet i chi quare tet for goode of fit, Mimeograph erie 459, Departmet of Statitic, Purdue Uiverity, 976. [0] Neyma J., Cotributio to the Theory of the x Tet, Proc. (Firt) Berkeley Symp. o Math. Statit. ad Prob. (Uiv. of Calif. Pre), 949, pp [] Pearo K., O the criterio that a give ytem of deviatio from the probable i the cae of a correlated ytem of variable i uch that it ca be reaoably uppoed to have arie from radom amplig, Philoophy Magazie Serie, 900, Vol. 50, pp [] Sokal R. R., Rohlf F. J., Biometry: the priciple ad practice of tatitic i biological reearch, Freema, New York 0. [3] Wato G. S., Some recet reult i chi quare goode of fit tet, Biometric, 959, Vol. 5, pp (00) 05 45

10 Piotr Sulewki, Ryzard Motyka ANALIZA PORÓWNAWCZA TESTU NIEZALEŻ NOŚ CI DLA TABLIC DWUDZIELCZYCH STRESZCZENIE W artykule zbadao ześć tetów iezależości dla tablic dwudzielczych, do których ależą: tet chi kwadrat Pearoa, tet ajwiękzej wiarygodości, tet Neymaa, tet Kullbacka Leiblera, tet Freemaa Tukeya, tet Creiego Reada. Dokoao tego metodą Mote Carlo. Wyiki ą do iebie podobe. Na wyróżieie załugują tet chi kwadrat Pearoa oraz tet Creiego Reada. Moża uzać, że jakość tych dwóch tetów (wyrażoa w fukcji mocy) jet porówywala, ale lepza od pozotałych. Idex τ Goodmaa Krukala wykorzytao do badaia zależości w tablicach dwudzielczych za pomocą ekperymetów Mote Carlo. Słowa kluczowe: tablice dwudzielcze, tet iezależości, metoda Mote Carlo. 46 Zezyty Naukowe AMW Scietific Joural of PNA