Simultaneous Estimation of Adjusted Rate of Two Factors Using Method of Direct Standardization

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1 Biometics & Biosttistics Itetiol Joul Simulteous Estimtio of Adjusted Rte of Two Fctos Usig Method of Diect Stddiztio Astct This ppe pesets the use of stddiztio o djustmet of tes d tios i compig two popultios usig sigle idices the th seies of specific tes o tios. Hee the ovell djusted cude te o the udjusted cude te fo two popultios will hve sme estimte iespective of the tue of the stdd popultio distiutio. These esults e otied i ll cses wheeve the two stdd distiutios e of the totl smple. I these cses the ovell djusted cude tes sed o the two sets of diectly djusted tes would e equl to ech othe, lthough ot ecessily lwys equl to the ovell udjusted cude te s is foud to e the cse hee. Howeve, if the stdd popultio distiutio chose fo give popultio is diffeet fom tht chose fo othe, the the two esultig estimted djusted o stddized cude tes would most likely ot e equl to ech othe. Volume 2 Issue Resech Aticle Deptmet of Sttistics, mdi Azikiwe Uivesity, igei 2 Deptmet of Idustil Mthemtics d Applied Sttistics, Eoyi Stte Uivesity, igei *Coespodig utho: Oyek ICA, Deptmet of Sttistics, mdi Azikiwe Uivesity, Awk, igei, Emil: Received: August 24, 205 Pulished: Octoe 27, 205 Keywods: Stddiztio; Adjusted specific; Udjusted cude te; Adjusted cude te; Rtios Itoductio Stddiztio o djustmet of tes d tios is ofte ecessy ecuse it is usully esie i compig two popultios, sy, to mke the compiso usig sigle summy idices the th seies of specific tes o tios. This ppoch lso helps void the polem of smll impecise d sometime o-existece of specific tes d tios [-3]. Stddiztio of tes d tios my e doe fo oly oe fcto o sevel fctos of clssifictio of citeio vile of iteest. I pticul if citeio vile o coditio is ssocited with ech of two fctos of clssifictio which my y themselves lso e ssocited with ech othe, the stddiztio of tes o tios my sometimes e ecessy fo clee lysis d ite-pesettio of esults to simulteously stddize o djust the tes fo the two fctos of clssifictio, fist specific to the levels o ctegoies of oe of the fctos coss the levels of the othe fcto, d the lso specific to the levels of the secod fcto sy holdig costt the levels, tht is fo ll levels o ctegoies of the fist fcto [4,5]. Resech iteest i this cse would e to idetify d mesue the septe effects of the two fctos of clssifictio o the citeio vile o coditio. This ppe poposes, develops d pesets fomtted systemtic sttisticl method fo this pupose. The poposed method Resech iteest hee is usig the diect method of stddiztio of tes to mesue o estimte the septe effects of two fctos of clssifictio which my e ssocited o the vile eig studied d to oti smple estimte of udjusted d djusted cude tes specific to the levels of ech of the fctos holdig the levels of the othe fcto of clssifictio costt. ow to develop the method of estimtio of diect stddized o djusted te, suppose A d B e two viles of clssifictio with d goups o levels espectively. Fctos A d B my e ssocited o elted. Resech iteest is to estimte the tes of occuece of citeio vile o coditio specific to ech of the levels of fcto A coss, tht is fo ll levels of fcto B d lso the tes of occuece of the specified coditio specific to ech of the levels of fcto B fo ll levels of fcto A s well s the coespodig mgil tes d ovell te. Suppose totl dom smple of size of suject e domly dw fom tecedet o pedisposig popultio C fo ll levels of fctos A d B of which ij is the ume of sujects t the i th level of fcto A d j th level of fcto B, fo i,2, j d j,2,,. Also suppose thee e totl of outcomes o cses i coditio o set D of cses fo ll levels of fctos A d B of which ij cses e t the i th level of fcto A d j th level of fcto B, fo i,2, j d j,2,, whee popultio D is possily suset of popultio C. ow the te of occuece of cses i popultio D s fuctio of cses i popultio C specific to the i th level of fcto A d j th level fcto B is Fo i,2, j ; j,2,. Let ij ij ij ; i. ij. j ij j i Sumit Muscipt Biom Biostt It J 205, 2(7): 0005

2 Pis i.( Simulteous Estimtio of Adjusted Rte of Two Fctos Usig Method of Diect Stddiztio Copyight: 205 Oyek et l. 2/6 e espectively the totl o mgil ume of sujects o osevtios i popultio C t the i th level of fcto A d j th level of fcto B. Similly let e espectively the totl o mgil ume of cses o outcomes i popultio D t the i th level of fcto A d j th level of fcto B. The the estimted udjusted cude tes of occuece of cses o outcomes i popultio D s fuctio of outcomes i popultio C specific to the i th level of fcto A fo ll levels of fcto B fo ll levels of fcto A e espectively the tios Ad Fo i,2,.; j,2,. ote tht ;.3 i. ij. j ij j i ;. j 4 i. i.; udj. j; udj Theefoe the ovell udjusted cude te of occuece of evet D s fuctio of evet C fo ll levels of fctos A d B is As oted ove esech iteest is to oti stddized o djusted cude te specific to ech level of fcto A fo ll levels of fcto B d lso specific to ech level of fcto B fo ll levels of fcto A s well s the ovell djusted o stddized cude i.. j.(5) i.. j ij i j j i (6) i.. j ij i j j i udj. (7) te. To oti estimtes of djusted o stddized cude tes specific to ech level of fcto B fo ll levels of fcto A we use the popotiote distiutio of totl ume of osevtios. coss the levels o goups of fcto A, mely P is the witig fcto, fo i,2,,. Thus P is i. Similly to oti estimtes of djusted o stddized cude te specific to ech level of fcto A fo ll levels of fcto B we use the popotiote distiutios coss the levels o goups of fcto B, mely P sj the witig fcto, fo j,2,,. Thus P sj. j Hece the djusted o stddized cude te of coditio D s fuctio of popultio C specific to the j th level of fcto B fo ll levels of fcto A is Similly the djusted o stddized cude te of coditio D s fuctio of popultio C specific to the i th level of fcto A fo ll levels of fcto B is i.; dj. j; dj j We the oti the smple estimte of the ovell djusted cude te of coditio D s fuctio of popultio C fo ll levels of fctos A d B s.(8) (9) p is ij i p. sj ij.(0).() p p. s; dj dj is. i. sj. j i j 2 Cittio: Oyek ICA, Okeh UM (205) Simulteous Estimtio of Adjusted Rte of Two Fctos Usig Method of Diect Stddiztio. Biom Biostt It J 2(7): DOI: /ij

3 Simulteous Estimtio of Adjusted Rte of Two Fctos Usig Method of Diect Stddiztio Copyight: 205 Oyek et l. 3/6 These esults e summized i Tle. Tle : Dt fomt fo Estimtio of Udjusted d Adjusted Rtes i two Fcto Stddiztio y Diect method. FACTOR B FACTOR A 2 Totl Popotio U djust djust ( ) 2 2. ( ( )) ( ( )).... ( p sj ).;dj.;dj.;dj.. 2 ( ) 2 2 ( ) 2 2 ( ) ;dj ;dj ( ) 2 2 ps. p s ( ) ( ). ;.. j. ; j dj. j ; dj.. Totl ( ( ). j. j ) ( ).. ( ) ( )..... popotio ( sj ) p p p s 2 s 2 p.. s U djust... j ; dj.udj.. Adjust (. j ; udj ).; dj. ;dj. ; dj. udj ;dj Cittio: Oyek ICA, Okeh UM (205) Simulteous Estimtio of Adjusted Rte of Two Fctos Usig Method of Diect Stddiztio. Biom Biostt It J 2(7): DOI: /ij

4 Simulteous Estimtio of Adjusted Rte of Two Fctos Usig Method of Diect Stddiztio Copyight: 205 Oyek et l. 4/6 I tle the eties i ech of the cells e the ume of cses i coditio D the ume of osevtios i popultio D d the tios of these umes. Illusttive Exmple We ow illustte the poposed method with the smple dt of Tle 2 o pemtue d live iths y ith ode d ge of mothe i ceti popultio. Tle 2: Smple Dt o Pemtue d Live iths y Bith ode d Mtel ge i popultio. Bith Ode Mtel Age Totl ( ( )) i.. Popotio of totl iths ( p is ) Ude 20 (23) 3(72) 3(32) (43) 0(33) 8(203) (329) 5(327) 7(76) 3(69) 8(67) 47(968) (5) (209) (207) 6(32) 6(23) 40(786) (78) 8(83) 0(7) 9(98) 2(50) 43(526) (42) 8(56) (90) 4(56) 3(04) 40(348) d ove 3(34) 4(457) 8(72) 0(48) 4(68) 29(267) Totl ( ). j. j 42(62) 49(792) 47(694) 45(446) 33(545) 27(3098) Popotio of totl iths ( p sj ) The dt of Tle 2 is used to oti estimtes of the udjusted d djusted cude te specific to ech of the levels o goups of the two fctos of clssifictio. Specificlly to estimte djusted o stddized cude tes specific to ith ode, we pply the popotiote distiutio of the totl life iths coss mtel ge s the stdd popultio, mely p is i the lst colum of Tle 2 to ech of the colums of tes, p s of the Tle, fo j,2,3,4,5.similly to estimte djusted o stddize cude te specific to Mtel ge we pply the popotiote distiutio of totl life iths coss ith ode p s the stdd popultio, mely sj i the lst ow of Tle 2 to ech of the ows of tes, ps of the Tle, fo i,2,3,4,5,6.the esults e peseted i Tle 3. Cittio: Oyek ICA, Okeh UM (205) Simulteous Estimtio of Adjusted Rte of Two Fctos Usig Method of Diect Stddiztio. Biom Biostt It J 2(7): DOI: /ij

5 Simulteous Estimtio of Adjusted Rte of Two Fctos Usig Method of Diect Stddiztio Copyight: 205 Oyek et l. 5/6 Tle 3: Simulteous Estimtes of Udjusted Adjusted Pemtue Bith tes y mtel ge d Bith ode: Diect Stddiztio Bith Ode Mtel Age Popotio of totl ith ( p is ) i i 2 2 i3 3 i4 4 (.; ) d 5+ Udjusted cude te ( ).; dj Adjusted cude te ( ).; dj less th d ove Popotio of totl ith ( p ) Udjusted cudete ( ). j ; udj Adjusted cude te ( ). j ; dj j Cittio: Oyek ICA, Okeh UM (205) Simulteous Estimtio of Adjusted Rte of Two Fctos Usig Method of Diect Stddiztio. Biom Biostt It J 2(7): DOI: /ij

6 Simulteous Estimtio of Adjusted Rte of Two Fctos Usig Method of Diect Stddiztio Copyight: 205 Oyek et l. 6/6 Summy d Coclusio The djusted cude te of pemtue iths specific to ith ode fo ll ge goups show i the lst ow of Tle 3 e estimted usig equtio 0,while the coespodig djusted cude te specific to mtel ge fo ll ith odes show i the lst colum of Tle 3 e estimted usig equtio. Thus the lst two ows of Tle 3 show tes specific fo ith ode d diectly djusted fo mtel ge, with the stdd mtel ge distiutio of iths eig tht of the totl smple of iths. The lst two colums of the Tle show tes specific fo mtel ge d diectly djusted fo ith ode, with the stdd ith ode distiutio of ith eig tht of totl smple of iths. The estimted djusted specific pemtue ith te of Tle 3 seem to idicte tht icidece of pemtue iths my ot e stogly ssocited with ith ode, ut my poly e some how ssocited with icesig mtel ge, especilly fom ge 25 yes. The ovell djusted cude pemtue ith te is estimted to e sevelly 70 pe 000 live iths whethe the stdd popultio distiutio is eithe the popotiote distiutio of totl ith y ith ode o y mtel ge. The udjusted cude te is lso hee estimted to e 70 pe 000 live iths. These esults e usully the cse wheeve the two stdd distiutios e those of the totl smple. I these cses the ovell djusted cude tes sed o the two sets of diectly djusted tes would e equl to ech othe, lthough ot ecessily lwys equl to the ovell udjusted cude te s is foud to e the cse hee. Howeve, if the stdd popultio distiutio chose fo popultio A (hee mtel ge)is diffeet fom tht chose fo fcto B(hee ith ode),the the two esultig estimted djusted o stddized cude tes would most lively ot e equl to ech othe. Ackowledgemet oe. Coflict of Iteest oe. Refeeces. Fleiss JL (98) Sttisticl Method fo Rtes d popotios (2d ed), ew Yok:Joh Wiley & Sos. 2. Pepe MS (2003) The Sttisticl Evlutio of Medicl Tests fo Clssifictio d Pedictio. Oxfod sttisticl seies 28, Oxfod: Uivesity Pess, U.K.; 3. Geeeg RS, Diels SR, Fldes WD, Eley JW d Boig JR (200) Medicl Epidemiology, Lodo: Lge-McGw-Hill. 4. Coch WG (950) The compiso of pecetges i mtched smples, Biometik 37: Gio JD (97) o Pmetic Sttisticl Ifeece McGw Hill Ic. Cittio: Oyek ICA, Okeh UM (205) Simulteous Estimtio of Adjusted Rte of Two Fctos Usig Method of Diect Stddiztio. Biom Biostt It J 2(7): DOI: /ij