Parameter Estimation and Determination of Sample Size in Logistic Regression

Size: px

Start display at page:

Download "Parameter Estimation and Determination of Sample Size in Logistic Regression"

Jacob Hudson
6 years ago
Views:

1 Jourl of Mhmic d Siic 8 (4): , ISSN Scic Publicio doi:.3844/mp Publihd Oli 8 (4) (hp:// Prmr Eimio d Drmiio of Smpl Siz i Logiic Rgrio Adlk Kzm Addyo d Dwud Adbyo Agubid Dprm of Mhmic, Fculy of Scic, Obfmi Awolowo Uivriy Il-If, Ou S, Nigri Dprm of Mhmicl Scic, Fculy of Scic, Olbii Obo Uivriy, Ago-Iwoy, Nigri Rcivd -6-6, Rvid --; Accpd -- ABSTRACT Th drmiio of Smpl-Siz i of impor p i plig iicl udy d i i uully difficul k. Amog h impor hurdl o b urpd, o mu obi im of o or mor rror vric d pcify ffciv mpl iz of imporc. Th udy w crrid ou o chck for h imio of prmr d mpl iz i logiic rgrio, lhough, hr i h mpio o k om horcu. W o lookd wo mhod of obiig mpl iz hvig obi h prmr im by vryig h rpo probbiliy. Th rul from h rl lif d howd h wh h rpo probbilii r mll, pproximio of corrcd rm Equio prform br h h pproximio Equio 8, bu i highly ovr im wh h rpo probbilii r lrg. Kyword: Logiic Rgrio, Smpl Siz, Powr, Proporiol Odd Modl. INTRODUCTION Logiic rgrio i yp of rgrio ud wh h dpd vribl r cgoricl Adlk d Adpou (). Th dpd vribl my hv wo cgori (.g., liv/dd; ml/fml; Rpublic/Dmocr) or mor h wo cgori. If i h mor h wo cgori hy my b ordrd or uordrd. Howvr, lo of iic i cocrd wih prdicig h vlu of coiuou vribl lik Blood prur, illigc, oxyg lvl, wlh d o o. Bu hi kid of iic domi wh your rpo vribl i biry. I i highly robu d h idpd vribl do o hv o b ormlly diribud, or hv qul vric i ch group. Logiic rgrio i uful i om iuio wh umpio of lir rgrio fil. I rquir diffr yp of d d i coffici hv diffr irprio. Lik lir rgrio, logiic rgrio llow rul o b grphd wih rgrio li d prdicio o b md giv of codiio. I hi udy, our ir focu o h drmiio d prmr imio of mpl iz uig logiic rgrio lyi. Lirur rviw hv how my udi imd drmiig whhr priculr vribl h ffc o biry rpo. Agri (7) rgud h h udy dig hould drmi h mpl iz dd o provid good chc of dcig ffc of giv iz. H ud impl logiic rgrio c udy. Hi udy did o provid much rul for h mulipl logiic rgrio. Thi udy hrfor coidr horough lyi o h mulipl c h hc br pproch o mpl iz drmiio. W bgi by giv bckgroud iformio o h rld rm lik powr lyi. Powr lyi c opimiz h rourc ug d dig of udy, improvig chc of cocluiv rul wih mximum fficicy. Powr lyi i h mo ffciv wh prformd h udy plig g d uch i courg rly collborio bw rrchr d iici. Mullr d Bigu (99); O Bri d Mullr (993) d Rull (), provid cog dicuio of h d rld cocp. Powr lyi i of problmic i prcic, big prformd ifrquly or improprly. Thr r vrl ro for hi: i i chiclly complicd, uully udr-rprd i iicl curricul d of o prform rly ough o b ffciv. Good ofwr ool for powr lyi c llvi h difficuli d hlp you o bfi from h chiqu. Corrpodig Auhor: Adlk Kzm Addyo, Dprm of Mhmic, Fculy of Scic, Obfmi Awolowo Uivriy Il-If, Ou S, Nigri Scic Publicio 48

2 Adlk Kzm Addyo d Dwud Adbyo Agubid /Jourl of Mhmic d Siic 8 (4): , W propo o dvlop mpl iz clculio mhod wihi h proporiol odd modl rucur. Such mpl iz i dd o coruc of hypohi i Ordil Logiic Rgrio (OLR) hvig dird powr. Th u of logiic rgrio h widly b ccpd i ciific fild (bioiic, pidmiology, girig). Thi i bcu i i impl d ffciv mhod o dcrib h ffc of om xplory vribl o cgoricl rpo vribl. Sudi o prmr imio i logiic rgrio rvld h h powr d mpl iz imio of diffr iicl pproch wihi logiic rgrio modl. Whimor (989) coidrd for igl prmr wih ohr prmr rd uic prmr. Much lirur xi o pproximio o h powr d mpl iz of diffr iicl wihi logiic rgrio modl (Mh d Tii, 984; Hilo d Mh 993; Lui, 993). Whimor (989) coidrd mpl iz pproximio i h c of drd logiic rgrio wih mll rpo probbiliy. A pr, mpl iz iu i ordil logiic rgrio ig do o ppr o hv b udid i dph i h lirur. Smpl iz drmiio i mulilvl dig rquir io o h fc h iicl powr dpd o h ol mpl iz for ch lvl. I i uully dirbl o hv my ui poibl h op lvl of h mulilvl hirrchy (Sidr, 5). Rull () offr om uggio for uccful d migful mpl-iz drmiio d lo dicud i h poibiliy h mpl iz my o b h mi iu; h h rl gol i o dig highquliy udy. Li l. () dicud om crucil iu i h problm formulio, prmr pcificio d pproch h r commoly propod for mpl iz imio i microrry xprim. Roy l. (7) coidr h problm of mpl iz drmiio for hr-lvl mixd-ffc lir rgrio modl for h lyi of clurd logiudil d. Thr-lvl dig r ud i my r, bu i priculr, mulicr rdomizd logiudil cliicl ril i mdicl or hlh-rld rrch. Powr lyi mo ffciv wh prformd h udy plig g d uch i courg rly collborio bw rrchr d iici. I lo focu io o ffc iz d vribiliy i h udrlyig ciific proc, cocp h boh rrchr d iici hould coidr crfully hi g. Mullr d Bigu (99) d O Bri d Mullr (993) provid cog dicuio of h d rld cocp. Th rfrc lo provid good grl iroducio o powr lyi. Our focu i hi udy i hrfor o fi uibl modl d chck h rlibiliy of h modl uig logiic rgrio d o ugg mpl iz d powr clculio mhod for ordil logiic rgrio o iicl hypohi. Scic Publicio 48. MATERIALS AND METHODS W dl wih udi i which rdom mpl i drw from h oi diribuio of (Y, X) whr Y i ordil rpo d X= (x, x, x 3,, x p ) i vcor of covri Equio : 3 k () L (x'), (x'), (x'),..., (x') ohprdicor X' Sic our rpo cgori hv url ordrig, w u h proporiol odd modl h i Equio : logi[p (Y / X)] = +X' =,,...k () r whr, i vcor of h ircp prmr d γ = (γ, γ,, γ p ) i h lop prmr vcor wihou ircp rm. If < + hold hi modl fi commo lop cumuliv modl bd o cumuliv probbilii of h rpo cgori Equio 3 d 4: (3) L ϕ (X') = (X') + (x') (x ') k ϕ (X ') = (X ') (X ') (X ') (X ') ϕ = + M M ϕ (X ') = (X ') + (X ') + (X ') (x ') = 3 k Th OLR follow h Equio 5 d 6: ϕ i Logi( ϕ i ) = log( ) ϕi = +γ X +γ X γ X p p Logi( ϕ ) = log ϕ ϕ = +γ X +γ X γ X p p ϕ Logi( ϕ ) = log ϕ = +γ X +γ X γpxp : : :... ϕ k Logi( ϕ k ) = log ϕk = k +γ X +γ X γpx p (4) (5)

3 Adlk Kzm Addyo d Dwud Adbyo Agubid /Jourl of Mhmic d Siic 8 (4): , Whr: ϕ (X') = (X') + (X') + Scic Publicio +γ 'X' (X') (X') = + X' 3 k + (6) Thi modl i kow Proporiol odd Modl bcu, h odd rio of h v (Y ) i idpd of cgory idicio... Mximum Liklihood Fucio Wh mor obrvio o Y occur fixd X vlu, i i uffici o rcord h umbr of obrvio d h umbr of oucom, for =,, k. Thu w l Y, =,,, b idpd muliomil rdom (rpo) vribl, h Y i ~ muliomil,..., wih E(Y ) = ϕ (X ) Whr Equio 7: W dfi: k S = S = + : : : Sk = k = = k (7) Sic w r dlig wih cumuliv probbilii, i rm of h prmr of h cumuliv rformio, h liklihood c b wri h produc of k- quii. Th oi probbiliy m fucio of (Y, Y ) i proporiol o h produc of muliomil fucio. For giv mpl iz, h liklihood of h obrvio y, x, =,,, i: L( ', ) = f (x )f (y / x ) = ϕ ϕ ϕ = ϕ ϕ 3 ϕ ϕ3 ϕ *...* ϕ3 ϕ3 ϕ ϕ k k k ϕk ϕk k ϕk * 483 whr, F(x)i oi p.d. of x. i i umd h f (x) γ. do o dpd o ukow prmr ( ', ') Th vlidiy of hi modl how h MLE ) ( ˆ ', ) ify pproximly (% ', γˆ ) ~ N[( ˆ ', γ% '),I ( ', )]... Smpl Siz Eimio O of h mi obciv of hi wri up i imio of mpl iz d hi i chivd by obi mpl iz h i u ufficily lrg ough o b cofidc of big bl o chiv ifrc wih rquird prciio. I i dircly rld o h co d im ivolvd i urvy or d collcio. L u h ull hypohi: H : γ= V H : γ=γ% A lvl wih powr -β wh h diribuio of ˆγ i rd orml wih m γ d vric σ, h criicl rgio i: σ σ γ< γ< γ> ˆ Z, ifγ> % ˆ Z,if % whr, Z i (-)% of h drd orml diribuio. Th mpl iz will b foud o h h h pcifid powr (-β) h lriv H :γ=γ%, h mpl iz i hu cho o h: σ z Pr γ ˆ > =γ> ˆ, σ=σ = ( β) γ σ z or Pr γ< ˆ =γ< ˆ, σ=σ = ( β) γ Thi c b wri σ σ ϕ Z γ % / = ( β) if γ< % : σ σ σ ϕ γ % = = β γ> % Z / ( ) if σ

4 Adlk Kzm Addyo d Dwud Adbyo Agubid /Jourl of Mhmic d Siic 8 (4): , If γ> % d ppropri, oluio of h bov formul ifi: Hc: Z Ohrwi if γ< % : z σ + z σ = γ% o β σo γ% = Zβ σ σ σ γ% Z = σ σ Z β (8) For boh c γ> % d γ< % For modl of h form i Equio (), (5) d (6) wih o prdicor i..: logi( π ) = +γ X Hih (989) u pproxim mpl iz formul o obi h mpl iz dd for ig H : γ =. Hr w d o gu h probbiliy of ucc π h m of x. h iz of hi ffc i h odd rio θ comprig π o h probbiliy of ucc o drd dviio bov h m of x. L k = log(θ) A pproxim mpl iz i Equio 9: Z + Z (+ πδ) / ( πk ) k 4 β Whr Equio : k 4 ( k /4) δ= + (+ k ) / [+ ] (9) () I h c of Proporiol Odd Modl (POM), imio of mpl iz wih grl rpo probbilii whr w hv mor h wo cgori which c ihr mll or lrg, h: i+ i ( ) z, if z z < < i= f (z) = = + z i+ i ( ) z, if < z< i= Ad i imply pproximd i Equio 6 by: +γ 'X' f (z) =ϕ (X') = = +γ + 'X' +X ' + +γ O( ),if 'X' ( + x ') + O( ),if +γ'x' whr, i mll wh + γ X (or - i mll wh +γ X ). W ow prov for rpo vribl wih hr cgori wih ordrd probbilii. i..: Th: Ad: + + x' X' <.5 d > X' X' +γ + x' + ( X') + 'x' (+ x ') +γ ' + X' O( ) = + ( +γ ' X ') O( ) = + ( +X ') +γ { } + I E( E( 'X') ( ) { } + = * m( γ') m( γ') ( ) I I,I3 =,I I + m( γ') I = m ( )d I = m ( γ') 3 I 33 II Thrfor vric i Equio : Vr( γ') m( γ) {m( γ)m II( γ') m I ( γ)} () γ If X ~N (, ) h Vr ( ) = d Equio : γ zσ o + zβ 4 γ% () Equio dicovrd mhod of obiig σ ud i (3.) which i Vr( ) obid i Equio bov. Hc, w c grliz i o mulipl prmr, whr w h hypohi of: Scic Publicio 484

5 Adlk Kzm Addyo d Dwud Adbyo Agubid /Jourl of Mhmic d Siic 8 (4): , Ad l: Whr: Ad: Scic Publicio H : γ = V H : γ = γ% = ( γ, γ,..., γ ) = ( γ, γ ) ' ' γ = ( γ, γ,..., γ ) ' p γ = ( γ, γ,..., γ ) ' p+ p+ 3. RESULTS W illur by uig h d o dibic pi from Uivriy Collg Hopil Ibd. Th d covr 678 rpord c of pi wih dib. 3.. Tbl. Logi Dib vru Smokig Logiic rgrio: Numbr of ob = 678 LR chi () =.5 Prob > chi =.6 Log liklihood = Pudo R =. Eimio of mpl iz uig h mhod propod by Hih (989) i Equio 9. Aum π=.873 if w go by h hypohi h H : γ = gi h lriv H : γ from Tbl h: =.5 d β=., Z =.96 d Z =.64 k.7 β k = log (odd rio) = log ( ) =.79 d = δ =.9 d 56,946 If w ow coidr h ffc of mokig d drikig of lcohol o iducd dib pi i.., prdicor, h h bov c b i oupu of Tbl b bov whr boh h coffici hvig giv ffc o iducd dibic pi. Alhough h Logliklihood rio for modl lcio uppor h full modl of wo (full modl) prdicor wih.54 >.83 vlu of chi wih df. Sic h pudo R i. which impli h hr i hrdly mulipl corrlio bw h 485 prdicor d h rpo vribl, h odd rio i Tbl ( d b) how h for mokr, hr i pproxim vlu of 6% l im of hvig dibi wh comprd wih ho who r o mokig, giv h ll ohr vribl rmi co. Th odd of hvig dib for idividul ddicd o lcohol i u.6% l im ho who r o drikig lcohol. Alhough hi rul look omhow, bu h p-vlu for mokr (.3) d idividul dicd o lcohol (.87) r o igific mig h boh fcor coidrd r o rlly coribuig o dib problm. Th urpor h rul obid i R. w compu: = R whr, i h obid wh w hv o prdicor Hc: 56,946 Thrfor, w rquir lmo 57 mpl for ig H : γ =. Uig h bov iformio w hv h followig rul from our imulio of mpl iz for boh Equio 8 d rpcivly. Mo Crlo mhod for lcd vlu of α =.5, β =. d α =.5 wll h vlu of γ> & k = wh h xplory vribl h h drd orml diribuio. Th rul i Tbl blow how u h h pproximio (3.) i uibl wh h rpo probbilii r mll bu i lwy udr im. 4. DISCUSSION Acordig o h rul of hi udy, h im of h prmr d mpl iz r obid from boh rl lif d of dib d imulio udy, Tbl d. Smpl iz obid wh h prdicor i o i pproximly h m wh h h prdicor r wo uig rl lif d. Th pproximio wih corrcd rm (3.4) prform br h h pproximio (3.) wh h rpo probbilii r mll, bu i highly ovr im wh h rpo probbilii r lrg. Alo, h grphicl rprio of h mpl iz for h imulio i giv i Fig. -3. Sic h mpl iz dpd o h wo prmr, γ d α, imulouly, w fixd o prmr o obi h ohr. If w chg h wo prmr imulouly, h imd mpl iz flucud oo much.

6 Adlk Kzm Addyo d Dwud Adbyo Agubid /Jourl of Mhmic d Siic 8 (4): , Fig.. Th gph of mpl iz fixd for k =, xp (α) =.5 Tbl. Logiic rgrio: Dib v Smokig Dib Cof. Odd rio Sd. Err. z P> z Smokig Co LR =.5, Prob> (chi) =.83 Tbl b. Logiic rgrio: Dib vru Smokig d Alcohol Dib Cof Odd Rio Sd. Error Z P</z/ Smokig Alcohol Co LR=.54, Prob>(chi) =.83 Tbl. (Eim of mpl iz for boh quio (3.) d (3.4)) (k =, α =.5) (k =, α =.5) (k =, α =.5) γ N (3.4) N (3.) γ N (3.4) N (3.) γ N (3.4) N (3.) Scic Publicio 486

7 Adlk Kzm Addyo d Dwud Adbyo Agubid /Jourl of Mhmic d Siic 8 (4): , Fig.. Th gph of mpl iz fixd for k =, xp (α) =.5 Fig. 3. Th gph of mpl iz fixd for k =, xp (α) =.5 Scic Publicio 5. CONCLUSION Thi udy h dvlopd mhodologicl frmwork o im h prmr of logiic 487 rgrio d obi mpl iz diffr lvl of d β. W hv lo propod mpl iz clculio mhod for logiic rgrio o for iicl hypoh. W hv lo coidrd ig h mulipl

8 Adlk Kzm Addyo d Dwud Adbyo Agubid /Jourl of Mhmic d Siic 8 (4): , prmr. W gv impl clod-form formul for pproximd mpl iz wh h probbilii of h rpo cgori r mll. Th rul howd h pproximio of corrcd rm Equio prform br h h pproximio Equio 8 wh h rpo probbilii r mll, bu i highly ovr im wh h rpo probbilii r lrg. 4. REFERENCES Adlk, K.A. d A.A. Adpou,. Ordil logiic rgrio modl: A pplicio o prgcy oucom. Am. J. Mh. S., 6: Agri, 7. A Iroducio o Cgoricl D Alyi. Ed., Wily, Nw York, ISBN-: , pp: Hih, F.Y., 989. Smpl iz bl for logiic rgrio. S. Md., 8: DOI:./im Hilo, J.F. d C.R. Mh, 993. Powr d mpl iz clculio for xc codiiol wih ordrd cgoricl d. Biomric, 49: DOI:.37/53573 Li, W.J. H.M. Huh d J.J. Ch,. Powr d mpl iz imio i microrry udi. BMC Bioiform., : DOI:.86/ Lui, 993. Smpl iz drmiio for cohor udi udr xpoil covri modl wih groupd d. I. Biomric Soc., 49: Mh, P. d Tii, 984. Exc Sigificc ig o blih rm quivlc wih ordrd cgoricl d. Biomric, 4: DOI:.37/5397 Mullr, K.E. d V.A. Bigu, 99. Icrig ciific powr wih iicl powr. Nurooxicol. Trol., 4: -9. DOI:.6/89-36(9)99-7 O Bri, R.G. d K.E. Mullr, 993. Uifid Powr Alyi for -T Through Mulivri Hypoh. I: Applid Alyi of Vric i Bhviorl Scic, Edwrd, L.K. (Ed.), Nw York. Roy, A., D.K. Bhumik, S. Aryl d R.D. Gibbo, 7. Smpl iz drmiio for hirrchicl logiudil dig wih diffril riio r. J. I. Biomric Soc., 63: DOI:./ x Rull, V.L.,. Som prcicl guidli for ffciv mpl iz drmiio. Am. S., 55: DOI:.37/ Sidr, T.A.B., 5. Powr d mpl iz i mulilvl lir modl. Ecyclopdi S. Bhv. Sci., 3: DOI:./4739.b49 Whimor, A.S., 989. Smpl iz for logiic rgrio wih mll rpo probbiliy. J. Am. S. Aoc., 76: 7-3. DOI:.8/ Scic Publicio 488

Advanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S.

Advanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S. Rfrc: (i) (ii) (iii) Advcd Egirig Mhmic, K.A. Sroud, Dxr J. Booh Egirig Mhmic, H.K. D Highr Egirig Mhmic, Dr. B.S. Grwl Th mhod of m Thi coi of h followig xm wih h giv coribuio o h ol. () Mid-rm xm : 3%