Naresuan University Journal: Science and Technology 2018; (26)1

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Narsuan Unvrsty Journal: Scnc and Tchnology 018; (6)1 Th Dvlopmnt o a Corrcton Mthod or Ensurng a Contnuty Valu o Th Ch-squar Tst wth a Small Expctd Cll Frquncy Kajta Matchma 1 *, Jumlong Vongprasrt

1 Narsuan Unvrsty Journal: Scnc and Tchnology 018; (6)1 Th Dvlopmnt o a Corrcton Mthod or Ensurng a Contnuty Valu o Th Ch-squar Tst wth a Small Expctd Cll Frquncy Kajta Matchma 1 *, Jumlong Vongprasrt and Npaporn Chutman 3 1 Dpartmnt o Educatonal Rsarch and Evaluaton, Faculty o Educaton, Ubon Ratchathan Rajabhat Unvrsty, Ubon Ratchathan, Dpartmnt o Statstcs, Faculty o Scnc, Ubon Ratchathan Rajabhat Unvrsty, Ubon Ratchathan, Dpartmnt o Mathmatcs, Faculty o Scnc, Mahasaraham Unvrsty, Mahasaraham, * Corrspondng Author. E-mal addrss: ajta.m@ubru.ac.th Rcvd: 13 March 017; Accptd: 6 Jun 017 Abstract Usng th ch-squar tst wth a small xpctd cll rquncy s an mportant problm n gnrally survy and xprmntal rsarch bcaus t cannot control typ-i rror ld to amss conclud th rsult n our wor. Th purposs o ths wor wr rst to dvlop a corrcton mthod or nsurng a contnuty valu o th ch-squar tst and scondly to compar ts cncy wth othr mthods, namly; Yat s corrcton and Wllam s corrcton by usng smulaton data. Th comparsons wr mad wth th ollowng condton; two sgncant lvls o 0.01 and 0.05, sx contngncy tabl szs (x, x3, x4, 3x3, 3x4 and 4x4), a small xpctd cll rquncy up to 30% o th total cll and a sampl sz btwn 5 to 10 tms that o th total cll. W ound that typ I rror n ch-squar tst wth dvlopd corrcton and sgncant lvl s smlar valus (can control typ I rror). Th smlarty valus ar hghr than ch-squar tst wthout corrcton, Yat s corrcton and Wllam s corrcton. Largr sampl szs rsultd s bttr control typ I rror at both lvls o sgncanc. For th contngncy tabl sz x to 4x4, chsquar tst wth dvlopd corrcton can control typ I rror bttr than ch-squar tst wthout corrcton and Wllam s corrcton at both 0.01 and 0.05 sgncant lvls. Th corrcton mthod usd to control th typ-i rror was obtand usng a dvlopd corrcton n vry condton. Kywords: Tst o ndpndnt, Ch-Squar Tst, Corrcton mthod, Typ-I rror Introducton In socal and bhavoural scnc rsarch, survys and xprmnts us qualtatv or catgorcal masurmnt mthods to dtrmn th rsults rathr than quanttatv mthods; that s, a qualty or charactrstc s masurd or ach xprmntal unt. W can summarz ths typ o data by cratng a lst o th catgors or charactrstcs and rport a count o th numbr o masurmnts that all nto ach catgory. Ths ar som o th many stuatons n whch th data st has charactrstcs approprat or th multnomal xprmnt. Th statstcs tst cratd or th multnomal xprmnt was drvd by a Brtsh statstcan namd Karl Parson n 1900 and s calld th Ch-squar statstc tst. (Mndnhall, Bavr, & Bavr, 013) Th Ch-squar tst s only an approxmat largsampl tst and t s rcommndd that t not usd whn on (or mor) o th xpctd rquncs s lss than v (Frund, 004). Whn th sampl szs ar too small, you should not us ch squar tst or G tst. Howvr, how small s "too small"? Th convntonal rul o thumb s that all o th xpctd numbrs ar gratr than 5, t's accptabl to us th ch squar or G tst; an xpctd numbr s lss than 5, you should us an altrnatv (McDonald, 014). At larg sampl szs, many asymptotc proprts o tst statstcs drvd or ndpndnt sampl comparson ar stll applcabl n adaptv randomzaton provdd 98

2 Narsuan Unvrsty Journal: Scnc and Tchnology 018; (6)1 that th patnt allocaton rato convrgs to an approprat targt asymptotcally. Howvr, th small sampl proprts o commonly usd tst statstcs n rspons-adaptv randomzaton ar not ully studd (Gu & L, 010). Th rsarchrs rcommndd that w should not us ch-squar tst whn th xpctd numbrs ar small bcaus t wll lad to rronous rsults o a rsarch study, manng th concluson o th rsarch cannot b rghtly ntrprtd. Yat s corrcton (Yats, 1934) and Wllam s corrcton mthods (McDonald, 014) ar usd to tst ndpndnc o vnts n a cross tabl. It s don by rducng th drnc btwn ach obsrvd valu and ts xpctd valu. Ths tsts ar commonly usd whn xpctd rquncs ar lss than tn. In ths artcl, w prsnt our own dvlopd corrcton mthod to mantan a contnuty valu to b usd whn small xpctd cll rquncs on chsquar tst or ndpndnc xst n th rsarch data. Th objctvs o ths study ar to compar our dvlopd corrcton mthod s cncy o control o a typ-i rror wth Yat s corrcton and Wllam s corrcton mthods. Th smulaton data usd th Mont Carlo tchnqu wth R programmng languag n drnt stuatons; sgncanc lvls, contngncy tabl szs, sampl szs and th numbr o small clls. Our powrul corrcton mthod wll control a typ-i rror mor than th othr corrcton mthods. Matrals and Mthods Ch-Squar Tst Many xprmnts rsult n masurmnts that ar qualtatv or catgorcal rathr than quanttatv. In ths nstancs, a qualty or charactrstc s dntd or ach xprmntal unt. Data assocatd wth such masurmnt can b summarzd by provdng th count o th numbr o masurmnts that all nto ach o th dstnct catgors assocatd wth th varabl. In 1900 Karl Parson proposd th ollowng tst statstc whch s a uncton o th squars o th dvatons o th obsrvd counts rom thr xpctd valus wghtd by th rcprocals o thr xpctd valus: 1 n E En n 1 n np np (1) Whr thr ar catgors wth probablts p and n s sampl sz n ach catgors. Yat s Corrcton For th tst o ndpndnt cratng a x contngncy tabl that usd ch-squar tst, th Yat s corrcton s usually rcommndd spcally mor than 0% o th xpctd cll rquncs ar blow 5. Th ch-squar ormula quaton s blow (), Whr O s obsrvd rquncs, s xpctd rquncs. Yats O 0.5 () Wllam s Corrcton For th ndpndnt tst n contngncy tabl wth R (row) and C (column), th Wllam s corrcton or contngncy to comput q to dvd th ch-squar tst ar as ollows (3 and 4). 99

3 Narsuan Unvrsty Journal: Scnc and Tchnology 018; (6)1 Whr O s obsrvd rquncs and s xpctd rquncs. O Wllam q (3) Whr q s dnd as (4) n 1/ raw1total... 1/ rawrtotal 1n 1/ column1total... 1/ columnctot al 1 q 1 (4) 6R 1C1 Whr n s sampls sz, contngncy tabl wth R rows and C columns. Dvlopd Corrcton A major lmtaton o th tst o ndpndnt wth ch-squar tst s ts nablty to control a typ-i rror whn an xpctd rquncy s small. An accurat tst o ndpndnt was ndd whn th typ-i rror and sgncant lvl had smlar valus. Atr tst o ndpndnt by classcal ch-squar (wthout corrcton or contnuty) w consdr typ-i rror (numbr rjcton o null hypothss dvdd by 10,000) and sgncant lvl. Whr th typ-i rror s gratr than th sgncant lvl, th ch-squar tst quaton to b usd s as ollows (5) O C (5) Whr th typ-i rror s lss than th sgncant lvl, th ch-squar tst s (6) O C (6) Whr th ch-squar tsts n both stuatons ar dnd n (5) and (6), w can dn th dvlopd corrcton o ch-squar tst by th ollowng (5) Whr C s dvlopd corrcton valu. It was computd n two cas as ollows; th typ-i rror s gratr than th sgncant lvl w try to rplac th valu C nto th quaton (5) start rom 0.01, 0.0, 0.03,.. I th typ-i rror s lss than th sgncant lvl w try to rplac th valu C nto th quaton (5) start rom -0.01, -0.0, -0.03,.. Atr w rplac valu C and computd typ-i rror thn to compard wth sgncant lvl. Dvlopd corrcton valu (C) s th valu whch gt typ-i rror and sgncant lvl ar vry smlar valus. 100

4 Narsuan Unvrsty Journal: Scnc and Tchnology 018; (6)1 Smulaton Study Th prormancs o th thr corrctons or contnuty (Yat s Corrcton, Wllam s Corrcton and our Dvlopd Corrcton) wr valuatd usng a smulaton study wth a pattrn o data st at a sgncant lvl o 0.05 and Contngncy Tabls wr gnratd btwn x to 4x4 clls. Th numbrs o small xpctd cll rquncs up to 30% o total cll wr usd. Sampl szs wr gnratd at 5 to 10 tms th total cll sz. Th data was smulatd usng R programmng languag on th Mont Carlo tchnqu. Th data was smulatd 10,000 tms or ach pattrn. For comparson, th accuracy o th thr corrcton mthods was valuatd. For ach pattrn, th smulaton was usd to nd th corrcton valus that bst controlld th typ-i rror. Th rsults wr tabulatd to dsplay th rlatonshps btwn th contngncy tabl s pattrn, th sgncant lvls and th corrcton valus. Rsults Th corrctons or contnuty nclud Yat s corrcton (Y), Wllam s corrcton (W), our Dvlopd corrcton (D) and th ch-squar wthout corrctons or contnuty (N). Thr accuracy was compard usng a smulaton study classd by contngncy tabl sz, th numbr o small xpctd cll rquncs (No. Sc.), th sampl szs () and th sgncant lvl. Tabl 1-1 ndcats th typ-i rror and dvlopd corrcton valu (C) n ach stuaton. From tabl 1-1 ound that whn w usd our dvlopd corrcton valu (C), typ-i rror was narr th sgncant lvl than th othr corrcton mthods rvwd n vry stuaton and ts was control typ I rror bttr whn sampl szs was ncrasd at both lvls o sgncanc. Th rsult showd that dvlopd corrcton can control typ-i rror bttr than othr mthods. Tabl 1 Typ-I rror o x contngncy tabl n 0.01 sgncant lvl. N Y W D C Tabl Typ-I rror o x contngncy tabl n 0.05 sgncant lvl. N Y W D C Tabl 3 Typ-I rror o x3 contngncy tabl n 0.01 sgncant lvl

5 Narsuan Unvrsty Journal: Scnc and Tchnology 018; (6)1 Tabl 4 Typ-I rror o x3 contngncy tabl n 0.05 sgncant lvl Tabl 5 Typ-I rror o x4 contngncy tabl n 0.01 sgncant lvl Tabl 6 Typ-I rror o x4 contngncy tabl n 0.05 sgncant lvl Tabl 7 Typ-I rror o 3x3 contngncy tabl n 0.01 sgncant lvl

6 Narsuan Unvrsty Journal: Scnc and Tchnology 018; (6)1 Tabl 8 Typ-I rror o 3x3 contngncy tabl n 0.05 sgncant lvl Tabl 9 Typ-I rror o 3x4 contngncy tabl n 0.01 sgncant lvl Tabl 10 Typ-I rror o 3x4 contngncy tabl n 0.05 sgncant lvl

7 Narsuan Unvrsty Journal: Scnc and Tchnology 018; (6)1 Tabl 11 Typ-I rror o 4x4 contngncy tabl n 0.01 sgncant lvl Tabl 1 Typ-I rror o 4x4 contngncy tabl n 0.05 sgncant lvl Concluson and Dscusson W appld thr mputaton mthods to trat th problm o small xpctd cll rquncy whn usng th ch-squar tst. W rvwd and provdd tchncal dtals o th drnt mthods usd, ncludng Yat s corrcton, Wllam s corrcton and our Dvlopd corrcton. As dpctd n Tabl 1-1, all mthods ld to an mprovmnt n accuracy, as masurd by typ-i rror or ach stuaton. Th mthod whch outprormd th control o typ-i rror was th dvlopd corrcton mthod n all condton. 104

8 Narsuan Unvrsty Journal: Scnc and Tchnology 018; (6)1 Assocaton n tabls tradtonally has bn tstd usng th ch-squar tst or largr sampls. Thr ar Yat s corrcton or amng to mprov th small xpctd cll rquncy. For Wllam s corrcton usd to contnuty ch-squar tst or ndpndnc whn contngncy tabl lagr than. In ths study to dvlopd contnuty corrctons or contngncy tabl btwn to 4 4, w ound that typ I rror n ch-squar tst wth dvlopd corrcton and sgncant lvl s smlar valus. And t s smlarly valus mor than ch-squar tst wthout corrcton, Yat s corrcton and Wllam s corrcton. Whn sampl szs was ncrasd th rsultd s bttr control typ I rror at both lvls o sgncanc. For th sz o contngncy tabl x to 4x4, chsquar tst wth dvlopd corrcton can control typ I rror bttr than ch-squar tst wthout corrcton and Wllam s corrcton at both 0.01 and 0.05 sgncant lvls. It outprormd to control typ I rror wth C valu bttr than othr corrcton n all condton. Espcally 4 4 tabl, thr s smlar valus o typ I rror and sgncant lvl. A corrcton valu or ch-squar tst dpnds on th pattrn o a contngncy tabl. Th approprat corrcton valu s not ncssarly qual to Yats corrcton valu o 0.5. Whn a contngncy tabl wth small xpctd rquncs s usd as an nput, th tst procdur s to dnty th tabl s pattrn and us th approprat corrcton valu or that pattrn. Th approprat corrcton valus or th pattrns assocatd wth x to 4x4 contngncy tabls ar tabulatd n ths artcl. Rrncs Frund, J. E. (004). Modrn Elmntary Statstcs. Nw Jrsy: Parson Educaton, Inc. Mcdonald, J. H. (014). Handboo o Bologcal Statstcs. Maryland: Spary Hous Publshng. Yats, F. (1934). Contngncy tabls nvolvng small numbrs and th χ tst. Supplmnt to th Journal o th Royal Statstcal Socty, 1(), Mndnhall, W., Bavr, R. J., & Bavr, B. M. (013). Introducton to Probablty and Statstcs. Australa: Boos/Col, Cngag Larnng. Gu, X., & L, J. J. (010). A smulaton study or comparng tstng statstcs n rspons-adaptv randomzaton. BMC mdcal rsarch mthodology, 10,

Chapter 6 Student Lecture Notes 6-1

Chapter 6 Student Lecture Notes 6-1 Chaptr 6 Studnt Lctur Nots 6-1 Chaptr Goals QM353: Busnss Statstcs Chaptr 6 Goodnss-of-Ft Tsts and Contngncy Analyss Aftr compltng ths chaptr, you should b abl to: Us th ch-squar goodnss-of-ft tst to dtrmn