A comparative study between ridit and modified ridit analysis

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1 Ameca Joual o Theoetcal ad Appled Statstcs 3; (6): Publshed ole Decembe, 3 ( do:.648/j.ajtas.36.3 A compaatve stud betwee dt ad moded dt aalss Ebuh Godda Uwawuoe, Oea Iewelugo Cpa Aaee Depatmet o Statstcs, Facult o Phscal Sceces, Namd Azwe Uvest, Ngea Emal addess: ablegod7@ahoo.com (Ebuh G.U.) To cte ths atcle: Ebuh Godda Uwawuoe, Oea Iewelugo Cpa Aaee. A Compaatve Stud betwee Rdt ad Moded Rdt Aalss. Ameca Joual o Theoetcal ad Appled Statstcs. Vol., No. 6, 3, pp do:.648/j.ajtas.36.3 Abstact: Ths pape compaes dt aalss wth moded dt aalss. The compaso was the llustated wth a eample. It was obseved om the eample at least, that whe the sample szes o the two samples beg compaed ae too dspaate, a moe elable cocluso usg the Boss dt aalss s lel to be eached ol whe the goup wth the lage sample sze s used as the eeece goup. Othewse Boss dt aalss would lead to colctg coclusos, depedg o whch goup s used as the eeece goup. Moded dt aalss teats the goups beg studed as samples daw om some lage populatos whch the vaaces o stadad devatos as well as the esults obtaed ae the same o matte whch sample s used as the eeece goup. The moded pocedue s theeoe peeable to dt aalss especall cases whee the goups beg compaed ae samples om some populatos. Kewods: Samples Estmates, Populatos, Boss Mea Rdt, Ch-Squae, Sgcace. Itoducto Suppose that we have sample data daw om a umbe o populatos each o whch s assumed to have -bult qualtatvel odeed categoes o classes. Fo eample suppose we have adom samples o patets b age sa o a ceta dsease whose codto s odeed om ctcal to sevee, poo, mpoved, most mpoved, etc. I a automoble accdet volvg some passeges, a passege s level o ju ma age om oe though mld, sevee to atal. Although these gaduatos ma be coase, dscete ad stll te, the ae eve-the-less moe descptve ad ehaustve tha meel usg some dchotomous classcatos such as oe o all, es o o, peset o abset, etc, whch ae al cude ad ot ull descptve. To compae these samples ad each clea cocluso s ote dcult. Howeve the above ad smla cases, the gadg o the degee o seousess s subjectve ad ma ot be elable. Futhemoe, t s dcult to d a eadl tepetable summa de o such a data-set ad to mae compasos amog deet samples a tellget wa. The covetoal ch-squae aalss ma be peomed, but mpotat omato o the atual odeg o the categoes would be lost. A equetl emploed pocedue s to umbe the categoes om sa, o the least seous to some hghest umbe o the most seous, calculate meas ad stadad devatos, ad the appl the t test o aalss o vaace. Thee s howeve also a poblem wth ths appoach. The assgmet o odeed umecal codes wth equal spacg to the vaous categoes o the vaable ude stud s ote abta. It s a devce that dees a metc o the categoes o a qualtatve vaable whch ma o ma ot epeset the tue patte o elatoshps amog these categoes. A techque that does ot attempt to quat the categoes but athe wos wth the atual odeg s the dt aalss developed b Boss(958). The tem dt s a acom o elatve to a deted dstbuto o the popotos o equeces ove the vaous odeed categoes o some chose stadad o eeece populato, elatve, the sese that the popotos o equeces o occuece o obsevatos the vaous odeed categoes o a populato o teest, ae compaed wth the popotos o equeces the coespodg odeed categoes o the eeece populato o goup. Vtuall the ol assumpto made dt aalss s that the dscete categoes epeset tevals o a udelg but uobsevable cotuous dstbuto. No assumpto s made about omalt o a othe om o the dstbuto. I ths pape, we bel dscuss dt aalss, peset a moded pocedue, ad use data to llustate ad compae the two techques.

2 Ameca Joual o Theoetcal ad Appled Statstcs 3; (6): Rdt Aalss Rdt aalss begs wth the selecto o oe o the goups o data wth the speced odeed categoes to seve as a stadad o eeece populato o the othe goups, ote eeed to as compaso goups. Havg selected a eeece goup o populato, oe the calculates a dt o scoe o each o ts categoes. The scoe o dt o a gve catego s calculated as the cumulatve equec o all the categoes lowe degee o seousess tha the catego o teest plus oe-hal o the equec o that catego, all dvded b the total equec o the populato sze o the eeece goup. Thus usg the data the om o equeces show table, the dt o a catego o the eeece populato s the popoto o all subjects o obsevatos om the eeece goup allg the lowe ag categoes plus hal the populato allg the gve catego.. Methodolog GROUPS Odeed catego o cteo vaable (C) Table : Data Fomat o Rdt Aalss Y (Reeece, ) X (Compaso, ) Total (t ) C t ( ) C t ( ) C t ( ) Total Oce the dts o all the categoes o the eeece populato ae detemed, the ae tae as values o a depedet vaable o the othe goups. Gve the dstbuto o a othe goup ove the same categoes the mea dt o that goup ma be calculated. It s smpl the sum o the poducts o obseved equeces, tmes the dts obtaed om the eeece populato o the coespodg categoes dvded b the total equec o that goup. Thus usg the equeces Table, the dt o the th catego o the eeece populato Y s j j j p j p () Whee p j ad p ae espectvel the popotos o the total obsevatos the j th ad th categoes o the eeece populato Y o j, ,,. The mea dts o a othe goup X s the calculated as () p whee s the popoto o cases, o the elatve P equec o the th catego o goup o populato X, o,.... The mea dt o the eeece populato Y s b the detos equatos - alwas equal to.5. Ths meas that a two subjects ae selected at adom om the eeece populato Y, the oe o them would be epected to epeece a moe seous codto o the cteo vaable hal o the tme, ad a less seous codto also hal o the tme tha the othes subject the eeece populato. The mea dt o a othe goup X, s tepeted as ollows; Gve the eeece goup Y ad a othe goup X, the the mea dt o the compaso goup X s a estmate o P(X Y), that s o the pobablt that a adoml selected subject om goup X, the compaso goup, has a codto that s at least as seous as that o a adoml selected subject om goup Y the eeece goup o the cteo vaable. Thus the mea dt o a gve compaso goup X s moe tha.5, the moe tha hal o the tme o adoml selected subject om t wll have a moe seous codto tha a adoml selected subject om the eeece goup Y. I o the othe had the dt o the goup s less tha.5, we would coclude that a adoml selected subject om t would be epected to epeece a less seous codto tha a adoml selected subject om the eeece goup Y. A mea dt o.5 o a goup would mpl that subjects om that goup would ted to epeece ethe moe o less seous codto tha subjects om the eeece goup. Theeoe R s the mea dt o a compaso populato X om whch a adom sample o sze has bee daw to obta, the a ull hpothess that eeds to be tested s H : R.5, vesus H: R.5 (3) Whee.5 s the mea dt o the stadad o eeece populato Y. It has bee show b Boss(958) that o sucetl lage sample sze, s appomatel omall dstbuted wth mea R ad vaace Va ( ) (4) Hece the ull hpothess o Eq 3 ma be tested usg the test statstc. ( ).5 Va( ) (.5) χ (5)

3 5 Ebuh G. U. ad Oea I. C. A.: A Compaatve Stud betwee Rdt ad Moded Rdt Aalss Whch has appomatel a ch-squae dstbuto wth degee o eedom o sucetl lage. Ho s ejected at the α level o sgcace χ χ (6) α; Othewse H s accepted... Illustatve Eample The data o Table shows the degee o the eects o the cocetato o some posoous chemcal o the blood steam o thee goups o emploees b wo place. Table : Dstbuto o Emploees b wo place ad level o eacto o some posoous chemcal Wo place Reacto to posog (sevet o codto) A(Y) B(X) C(Z) Total T T z T z Noe Modeate Sevee Ctcal Fatal Total 5( ) 65( ) 4( z) It ca be see om Table that goups Y ad X have a sample sze ato o appomatel :6, goups Y ad Z, a sample sze ato o appomatel : ad goups X ad Z a sample sze ato o appomatel 3:. To d out the eect deet sample szes have o dt aalss, we have use each o the goups o populatos X, Y ad Z alteatvel as the compaso ad eeece populatos. Fst usg Y(A) as the eeece goup we calculate the dts o the categoes o Y om Table usg Equato as Noe :.67; Modeate:.48; Sevee:.7; 3 Ctcal:.89; Fatal:.933; 5 Fom the dt scoes we calculate the mea dt o X(B) Equato as ( 58)(.67 ) ( 39 )(.48 ) ( 96)(.7 ) ( )(.89 ) ( 48)(.933 ) Thus the dt aalss estmates that the pobablt s.534 that a adoml selected emploee wo place B(X) has as seous o moe seous eacto to the chemcal posog tha a adoml selected emploee wo place A(Y), the eeece goup. The coespodg vaace s om Equato 4 Va ( ). ( 65) 738 Hece the test statstc o the ull hpothess o Equato 3 s χ (.534.5).56. whch wth degee o eedom s hghl statstcall sgcat. A smla calculato ma be made o wo place C(Z) as a compaso goup aga usg wo place A(Y) as the eeece goup eldg a estmated mea dt. Fom Equato as z 4 wth a vaace o Va ( z ).3 ( 4) Hece the coespodg ch-squae test statstcs s (.573.5).53 χ whch s aga hghl statstcall sgcat. I stead wo place B(X) has bee used as the eeece goup o populato the dt scoes o X would be Mld:.47; Modeate: ; Sevee: ; Ctcal: ; Fatal: 96 Usg these values Equato we calculate the mea dt o wo place A(Y), ow teated as a compaso goup as. 466

4 Ameca Joual o Theoetcal ad Appled Statstcs 3; (6): Ths value s smpl equal to whch s meagull electg the act that a complemeta pobablt s beg estmated. The vaace o ths mea dt om Equato 4 s Va ( ).8 5 ( ) The coespodg ch-squae test statstcs s χ (.466.5) whch s ot sgcat at the 5% sgcace level. Thus the chage choce o the eeece goup o populato has esulted a o sgcat eect. I othe wods, we had used wo place B(X) as the eeece goup o populato stead o wo place A(Y) we would coclude that emploees wo place A(Y) ow teated as the compaso goup ae as seousl aected b the chemcal posog as the emploees wo place B(X). But we had used woplace A(Y) as the eeece goup we would coclude that emploees woplace B(X) ae moe seousl aected b the chemcal posog tha the emploees woplace A(Y). Hece coclusos eached usg dt aalss ote deped o whch goup s used as a eeece goup o populato, ad whch s used as a compaso goup. We ow peset a moded method o estmatg dts that ae depedet o whch populatos ae used as eeece ad whch as compaso goups. 3. Moded Rdt Aalss Implct the Boss dt pocedue s the assumpto that the eeece goup s a populato. Although the autho dd meto the dcult o selectg a appopate eeece goup, he aled to eplctl suggest a appopate pocedue whe ethe o the two goups to be compaed mght seve as a eeece goup. The ma cause o the poblem wth the Boss pocedue s the dcult o detemg a appopate stadad devato to use the deomato o the test statstc. Itechagg the eeece ad compaso goups meel techages the oles o these goups, whle the mea dts estmated ae stll meagul ad useul pobabltes. Howeve, the szes o the two goups that ae beg compaed ae ve deet usg oe o the goups athe tha the othe as a eeece goup aects the stadad devato ad hece esult o the test. Futhemoe, all avalable goups ae egaded as samples om the espectve populatos, a addtoal souce o vaablt s also toduced, sce the dt scoes ae the subject to vaatos themselves. The esults o the ollowg pocedue ae smla to those o dt aalss tepetato but the pocedue maes the eplct assumpto that all the goups ae to be egaded as samples om the espectve populatos. The method s based o Ma ad Whte (947). Aea wos b Coove (973) ad Oea (99) povde theoetcal bases. Othe eseaches clude Mee etal (9), Pouplad etal (997), ad Rao ad Calgu (993). 3.. Methodolog Let be a set o obsevatos made o goup j ( j,,..., ) o populato Y (Boss eeece goup), occug wth equeces,, acoss c, c... c the categoes o a cteo vaable such that. Smlal let be a set o obsevatos made o (,,..., ) a othe goup o populato X eeed to as the compaso goup occug wth equeces,.... espectvel acoss the categoes o the same cteo vaable, such that The data omat s as Table Now o the eeece goup Y ad a compaso goup X dee the ucto u as o,.... ; j,... Let P u, > j, j, < ( ) : u P( u ) : P( u ) Also dee Now E Whee W u ( u ) Va( u ) Also OR Note that j j (7) (8) (9) () ; () ( ) ( ) E E W E u j ( W ), () ad ae espectvel the

5 5 Ebuh G. U. ad Oea I. C. A.: A Compaatve Stud betwee Rdt ad Moded Rdt Aalss pobabltes that a adoml selected subject om the compaso populato X s a moe seous, as seous as o less seous codto o the cteo vaable tha a adoml selected subject om the eeece populato Y. I tems o Boss mea dt _ ; (3) These pobabltes ae estmated as uctos o, ad whee, ad ae espectvel the umbe, o equeces o occuece, o s s ad - s the equec dstbuto o the values o these umbes u,, ; j,. The sample estmates o betwee ad s om equato, the deece W (4) Also the sample estmate o the vaace o w s obtaed usg equato 3 as ˆ Va w ˆ ( ) Hece the sample estmate o s w (5) ˆ (6) Also t s eas to show usg equatos 9 ad 4 that the estmated values o ad ae espectvel ( ) ˆ ˆ W (7) Ad ( ) ˆ ˆ W (8) It has bee show b Haje (969) ad Coove (973) that the vaace o W s estmated as 3 ( ) ( ) t t 3 ( ) ( ) Va( W ) 3 (9) that a adoml selected subject om the compaso populato X s a moe seous codto o the cteo vaable tha a adoml selected subject om the eeece populato Y ad the pobablt that the adoml selected subject s a less seous codto, povdes a measue o the elatve seousess o the codto the populatos. Now populato X ad populato Y ae act the same populato, the the elatve deece betwee the pobabltes o seousess o the codto the two populatos would be Zeo, so that Hece a moe geeal ull hpothess that eeds to be tested hee s H: δ vesus H: < δ, δ < ( < ) () Now o sucetl lage ad, W has appomatel the omal dstbuto wth mea E(W ) o equato ad vaace, Va(W ) o equato 9. Hece the ull hpothess o equato ma be tested usg the test statstc OR ( W E( W )) Va( W ) χ () ( W ) δ χ () Va ( W ) Whch has appomatel a ch-squae dstbuto wth degee o eedom o sucetl lage, whee Va(W ) s gve equato 9. H s ejected at the α level o sgcace Equato 6 s satsed. Othewse H s accepted. 3.. Illustatve Eample Let us ow use the moded method to e-aalze the data Table eale used to llustate dt aalss eample. We st use wo place A(Y) as the eeece populato ad wo place B(X) as the compaso populato Now om equato 3 ad Table, we have that ( 39 )( 35 ) ( 96 )( 35 3 ) ( )( ) ( 48 )( ) 667 Also ( 58 )( ) ( 39 )( ) ( 6 )( 8 4 ) ( )( 4 ) 754 Ad ( 58)( 35) ( 39 )( 3) ( 96)( 7) ( )( 8) ( 48)( 4) 5954 Hece om equato 4 we have that

6 Ameca Joual o Theoetcal ad Appled Statstcs 3; (6): ˆ ˆ ˆ Also om equato 3.3 Fom equato 7 ˆ ˆ 5954 ( 65)( 5) (.4.68). 333 Ad om equato 8 ˆ (.4.68) Theeoe the estmated pobabltes ae ˆ.333;.4 ad.65 Hece the moded appoach estmates that the pobablt s.333 that a adoml selected emploee wo place B(X) (the compaso populato) s moe seousl aected b the chemcal posog tha a adoml selected emploee wo place A(Y) (the eeece populato).4 that the emploee s as seousl aected ad.65 that the emploee s less seousl aected. Notce that as eale obtaed usg Goss mea dt ˆ ( ) 534 The act that the pobablt o epeecg equal sevet o codto has ow bee estmated sepaatel s a mpotat ad useul addtoal omato ad advatage o the moded method ove Boss dt aalss. The vaace o W s estmated om equato 9 as ( 65)( 5)( 7) Va ( W ), 68, The ull hpothess o Equato (wth δ ) ma ow be tested usg the test statstc ( 443) 9, χ.587,68,44,6844 whch wth degee o eedom s ot statstcall sgcat. I we ow techage the oles o Y ad X, that s X ow becomes the eeece goup ad Y, becomes the compaso goup, we would have that j j 358 ( ) 7( ) 8( ) 4( ) 667 Ad j j j 35( ) 3( 96 48) 7( 48) 8( 48) 754 Also 5954 as beoe Theeoe, W ˆ.68 ˆ Ad ˆ ( )( 5) ( 65 )( 5 ) ( 65 )( 5 ).4 Also (.4.68). 65 ˆ Ad (.4.68). 333 ˆ Hece we have that.65;. 4 ad.333 These ae the same values obtaed whe wo place A(Y) s used as the eeece populato ad wo place B(X) s used as the compaso populato. Hece the value o the estmated vaace o W emas the same as that o W ad the value o the test statstc ad the attaed sgcace level ema uchaged. Hece the same coclusos ae eached wth the moded dt aalss espectve o whch o the populatos s used as the eeece goup ad whch as the compaso goup. Thus the ol chages that esult whe the oles o the two goups ae techaged ae that the sg o W chages; s techaged wth ˆ ad s techaged wth, ule s the case wth Boss dt aalss. Table 3 shows the esults o the aalss o the data Table b both the dt aalss ad the moded dt aalss usg each o the wo places tu as the eeece goup o populato 4. Results ad Dscussos The ch-squae test statstcs o Equato 5 ad ma be epessed tems o ut-omal z-scoes ad ae hece so epoted Table 3 as ca be see om ths table, techagg the eeece ad compaso goups has a maed eect o the Boss aalss, ote chagg the attaed sgcace level dastcall. I moded dt aalss o such eect s obseved. Hee the vaace (stadad devatos) ad the attaed sgcace levels ema the same whe the eeece ad compaso goups ae techaged. Futhemoe the sgcace levels attaed b the Boss dt aalss appea to be geeall lowe tha those b the moded dt aalss. I act t s ol whe the same sze o the eeece goup s ve lage, as goup X, that the sgcace levels attaed b the two pocedues ae appomatel the same. Thus, t would seem, om the above eample at least, that whe the sample szes o the two samples beg compaed ae two dspaate, a moe elable cocluso usg the Boss dt aalss s lel to be eached ol whe the goup wth the lage sample sze s used as the eeece goup. Othewse Boss dt aalss would lead to colctg coclusos, depedg o whch goup s

7 54 Ebuh G. U. ad Oea I. C. A.: A Compaatve Stud betwee Rdt ad Moded Rdt Aalss used as the eeece goup. Wth the moded dt aalss o the othe had, o such poblem s ecouteed. Futhemoe whe the deece betwee the sample szes o the goups beg compaed, s ot too lage, as the Table 3: Compaso o two Rdts methods Data Rdt Aalss Moded Rdt Aalss ˆ Set S Z p-value Woplace A(Y) as eeece goup case o goups X ad Z, the attaed levels o sgcace seem to suggest that the moded pocedue would stll be the peeed method tems o cosstec o coclusos. ˆ ˆ S Z P-value Woplace B(X) Woplace C(Z) Woplace B(X) as eeece goup Woplace A(Y) Woplace C(Z) Woplace C(Z) as eeece goup Woplace A(Y) Woplace B(X) Cocluso Moded dt aalss teats the goups beg studed as samples daw om some lage populatos whch the vaaces o stadad devatos ad the esults obtaed ae the same o matte whch sample s used as the eeece goup. The moded pocedue s theeoe peeable to dt aalss especall cases whee the goups beg compaed ae samples om some populatos. Reeeces [] Boss, DJ. (958) How to Use Rdt Aalss. Bometcs 4 Pgs8-38. [] Ma, HB. ad Whte, DR. (947) O a Test o whethe oe o two adom vaables s stochastcall lage tha the othe. The Aals o Mathematcal Statstcs 8():5-6 [3] Coove, WJ. (973) Ra Tests o Oe Sample, Two Samples ad K Samples Wthout the Assumptos o a Cotuous Dstbuto Fucto. Aals o Statstcs. Pgs 5-5 [4] Oea, ICA. (99) A Moded Rdt Aalss. Joual o Ngea Statstcal Assocato (JNSA) Vol8, No Pgs [5] Mee, PW. J; Log, MA.; Be, KJ.; ad Johso, JE.(9) g-teatmet dt aalses:resamplg Pemutato methods Statstcal Methodolog, Volume 6, Issue3, Ma 9, Pgs 3-9. [6] Pouplad, N; Quaa, EM.; ad Smo, SC.(997)Use o dts to aalse Categocal data peeece studes.food Qualt ad Peeece Volume 8,Issues 5-6,Septembe-Novembe 997, Pgs 49-4 Thd Sesometcs Meetg. Do:.6/S95-393(97)-7 [7] Rao, CR. ad Calgu, MP. (993) Aalss o odeed Categocal Data though appopate Scalg. Hadboo o Statstcs Volume 9, 993, Pgs Computatoal Statstcs. Do:.6/S69-76(5)839-7 [8] Haje, J. (969) Nopaametc Statstcs. Holde-Da, Sa Facsco.