CHAPTER FOUR REPEATED MEASURES IN TOXICITY TESTING

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1 CHAPTER FOUR REPEATED MEASURES IN TOXICITY TESTING 4. Inroducon The repeaed measures sudy s a very commonly used expermenal desgn n oxcy esng because no only allows one o nvesgae he effecs of he oxcans, bu also enables one o look a how he effec changes over a me perod. The repeaed measures sudy n oxcology shares smlares wh he spl-plo desgn, whch s very popular n agrculural research. I s carred ou wh a leas wo facors of reamens n a nesed fashon where a whole plo s dvded no several sub-plos. The frs facor of reamens s randomly assgned o he whole-plos, hen he second facor of reamens s randomly assgned o he sub-plos whn each whole plo. When he subjecs are measured repeaedly hrough me, each me slo can be hough of as a sub-plo whn each subjec, and each subjec can be hough of as a whole plo n he expermen. Some auhors call he whole-plos or subjecs n he expermen clusers (Longford, 993). However, he measuremens whn each subjec are very lkely o be correlaed and s mpossble o randomzaon me. Hence, he analyss requres a model ha ncludes an specal covarance srucure for he observaons. Ths ype of repeaed measures sudy s ofen referred o as a longudnal sudy. Longudnal sudes have some specfc advanages over classcal cross-seconal sudes (Dggle e al. 994). The Cerodaphna duba es menoned n Chaper Three could be consdered as a longudnal sudy snce he number of offsprng s obaned on a 76

2 daly bass. The praccal problem agan s he exsence of moraly and he mxng of zeroes n he daa. Moreover, he repeaed measuremens whn he same subjec would lkely be correlaed. The model proposed n he Chaper hree only deals wh he daly egg couns or he oal egg couns. I does no address he repeaed measure effec or he me effec of he expermen. Wh repeaed measuremens hrough me, here are dfferen approaches ha can be used. The purpose of hs chaper s o examne he exsng approaches and modfy her applcaon n longudnal oxcy sudes o handle he problem of mxng zeroes due o morales. 4.2 Common approaches o Longudnal Daa In general, here are hree approaches o handle longudnal daa. The frs one s he me-wse approach whch s also he eases way o handle longudnal daa. Ths approach reas each repeaed measuremen as an ndependen expermen and analyzes he daa on a me by me bass. Ths s a raher sraghforward approach, snce any sandard sascal mehod can be appled; for example, one can use a me by me ANOVA or a me by me regresson analyss o examne he longudnal daa. The second approach s he derved varable approach whch summarzes he repeaed values no one number and hen analyzes he summary varable as a funcon of he reamens. Ths mehod s somemes referred as he wo-sage mehod whch can be daed back o Wshar (938). I works well when he covaraes are ndependen of me. Bu, s no very useful when he covaraes change over me. The las approach s based on he generalzed esmang equaons (GEE) by Lang and Zeger (986). Ths mehod combnes esmang funcon heory (Godambe 960, 976; Godambe and Heyde, 987; Godambe and Thompson, 989; Godambe and Kale, 99; Schabenberger and Gregore, 995) wh a mulvarae exenson of quaslkelhood prncple (Wedderburn, 974; McCullagh, 983; McCullagh and Nelder, 989; 77

3 Nelder and Lee, 992). By allowng a block dagonal covarance marx of he response, esmaes boh he regresson coeffcens and he covarance srucure a he same me. As menoned earler n hs Chaper, he moraly problem n chronc oxcy esng furher complcaes he analyses n hese approaches. The secons below wll dscuss he followng: he ssue of ncorporang moraly zeroes, he ssue of handlng longudnal daa, and how o use varous approaches (he me, derved varable and GEE approach) o ncorporae moraly problems, he applcaons n he Cerodaphna duba es example. 4.3 Tme-wse approach The me-wse approach s a raher smple and sraghforward approach ha reas he me-wse daa as f hey are ndependen. I apples sandard mehods, such as - ess, ANOVA, or regresson analyses a each me slo. The advanage of hs approach s s smplcy, bu here are wo drawbacks n hs approach. The frs one s he nably of he me wse approach o handle he longudnal aspec of he daa,.e. he me effec and he change n he reamen effec over me. The second problem concerns wh he dependence of he analyses a dfferen me slos, here s no well-defned way o combne hese analyses. Hence, by assumng ndependence of he observaons of he same subjec a dfferen me slos, he p-values of he ess would be napproprae. In he Cerodaphna duba es example, one can apply regresson analyss, snce he man neress are n he regresson coeffcens (β) and he nhbon concenraons (IC x ). Wh he dea of mxure dsrbuon and he ZIP model developed n Chaper Three, he daa obaned from he Cerodaphna duba es example can be fed o he ZIP model on a daly bass. As dscussed n Chaper Three, he ZIP model wll also adjus for he exra zeroes ha are caused by moraly. Comparng he esmaes of he 78

4 daly regresson coeffcens and nhbon concenraons can gve us only a rough dea of he longudnal effec of he daa whou any sascal concluson. The dea of he ZIP approach s based on he ndependen assumpon of he repeaed measures whch s no very relable n mos longudnal sudes. Neverheless, snce he ndvdual analyses can be carred ou usng raher sandard mehods, hs approach s smple and easy o use. 4.4 Derved varable approach The derved varable approach, whch s lke he me-wse approach, aemps o smplfy he daa and apples sandard analyss mehod. There are wo seps nvolve n hs approach. I sars wh summarzng he repeaed values no one number and hen analyzng he summary varable as a funcon of he covaraes. Ths mehod s somemes referred as he wo-sage mehod and can be daed back o Wshar (938). I works well when he covaraes reman consan over me. In Cerodaphna duba reproducon oxcy ess, esmaon and nferences are radonally done based on he mean number of he offsprng produced per adul. In hs case, he derved varable can smply be he mean (average) number of he eggs produced by each reamen whn a seven-day perod. The regresson analyss can hen be carred ou n he second sage o nvesgae he effluen oxcan effec. However, when moraly exss, he calculaon of mean s complcaed. In order o adjus for he moraly effec, Hamlon (986) dscusses wo dfferen ways o defne he mean when moraly occurs. They are he mean overall (MOA) and he mean gnorng moraly (MIM). The reasonng behnd Hamlon s mean gnorng moraly s based on he mxure model dea whch separaes moraly effec from fecundy. Ths s smlar o he dea behnd he zero nflaed Posson model dscussed n Chaper Three. 79

5 4.4. The Mean Overall (MOA) and Mean Ignorng Moraly (MIM) A sample daa s shown n Table 4. whch s obaned from a 7 days Cerodaphna duba es. Table 4. also demonsraes and compares how he esmaes of MOA and MIM can be found usng smple calculaon. The mean overall s calculaed by averagng he number of offsprngs produced by all he anmals ha ake par n he es. If an anmal des before reproducng he nex generaon, hen zero reproducon s recorded for ha anmal. If an anmal des afer reproducng he nex generaon, hen he oal number of offsprng produced before deah s used. Ths MOA s jus our usual mean f moraly does no exs n he es. For MIM, only he survvng anmals are used n mean. Snce he anmals do no maure and reproduce on he frs hree days, only four days of daa are recorded. Table 4. shows only wo reamens (A and B), and each reamen s replcaed 0 mes. The mean reproducon for each replcaon s calculaed on a daly bass. The daly means of each of he four days are hen summed ogeher o oban MOA and MIM. Table 4.. Sample daa of Cerodaphna es Treamen Replcaon number Mean Per 0 anmals Per survvng anmals A Day Day Day Day Toal (MOA) 8.20 (MIM) B Day Day Day Day Toal (MOA) 2.26 (MIM) A negave sgn - ndcaes he deah of he anmal. 80

6 Noce ha MOA and MIM are he same under reamen A because moraly does no exs. On he oher hand, when moraly occurs under reamen B, he wo means are dfferen. In order o llusrae he mxure concep, a MIM hypohecal model s examned n he followng secon. Mahemacal dervaon of MOA and MIM Le he varable X be he number of offsprngs reproduced by a randomly chosen anmal a me ( =,2,, ) and Z be he survval ndcaor varable, where Z = 0, f he anmal des a me ; Z =, f he anmal survves unl me. A me, he observed number of offsprng can be denoed as Y and Y = X *Z. If he anmal des a me, hen X, X +,, X wll be unobservable. Hence, he oal observed number of offsprng produced by he anmal s denoed as: Y = = Y = = X Z and he oal number of offsprng produced by he anmal (assumng he anmal survves ll me ) s denoed as: X = = X. MIM and MOA are he average of X and he average of Y over all anmals respecvely. Wh he assumpon ha X and Z I are ndependen, he expeced value of Y I, E Y ) = E( X ) E( Z ). Ths s exacly he same mxure dea as for he ZIP where: ( 8

7 E(X I ) s he λˆ modeled by a Posson regresson, and E(Z I ) s he ( pˆ ) modeled by a logsc regresson. Furhermore, he expeced value of X I can be wren as E( Y ) E ( X ) =, and E( Z ) E( Y MIM can be calculaed as = ) E( X). E( Z ) = = By usng MIM n he second sage of he analyss, one s able o nvesgae he effluen oxcan effec wh he adjusmen of moraly. However, n order o calculae MIM, moraly nformaon s requred. In addon o he requremen of he moraly nformaon, MIM also has he drawback of no addressng he longudnal feaure (me effec) of he daa. Bu hs may sll be a sensble mehod o use f he me effec s no he neres of he expermen. 4.5 Generalzed esmang equaons approach Zeger and Lang (986) propose usng generalzed esmang equaons (GEE) o analyze longudnal daa. Ths mehod combnes he esmang funcon heory (Godambe, 960 and 976; Godambe and Heyde, 987; Godambe and Thompson, 989; Godambe and Kale, 99; Schabenberger and Gregore, 995) wh an exenson of he quas-lkelhood prncple (Wedderburn, 974; McCullagh, 983; McCullagh and Nelder, 989; Nelder and Lee, 992). By allowng a block dagonal covarance marx of he response, one s able o esmae boh he regresson coeffcens and he covarance srucure a he same me. Dggle e al. (994) dscuss how GEE apples o hree dfferen modelng approaches. They are he margnal model, he random effec model and he ranson model. The complee deals of he generalzed esmang equaon model and s heory can be found n Zeger and Lang (986), Lang and Zeger (986), Zeger e al. 82

8 (988), and Zeger and Lang (992). In he example of oxcy esng, he focus s on he populaon-average, and we wll concenrae on he margnal model because can be used o address quesons such as: How does he oxcan affec he reproducon of he waer flea? How does he fecundy of he Cerodaphna change over me? The margnal GEE model can be vewed as a mulvarae exenson of he generalzed lnear model (GLM) wh correlaed observaons among ndependen subjecs. For example, suppose observaons are obaned from n subjecs and le he response be Y j (he j measuremen of he subjec) wh =,,n, j=,,, and covarae x j. By usng he GLM noaon, we have E(Y j ) = µ j = β x j, where β s he regresson coeffcen, Var(Y j ) =ϕv(µ j ), and he lnk funcon g(µ j ) = η j whch lnk he lnear predcor (β x j ) o he mean. The covarance srucure beween he repeaed measuremen s modeled as Cov ( Y, Y ) = c(, µ ; ) where j j, for some. Under mld regulary j j µ j j condons, Lang and Zeger (986) prove ha he GEE esmaor of β s gven by solvng he followng esmang equaon: where n U = D Var( Y ) ( Y µ ) = 0 ; = µ D =, µ = ( µ, K, µ n ), and β / 2 / 2 Var( Y ) = A R ( ) A, where A s a n n dagonal marx wh a = Var(Y j ), j =, K,n, and R () a n n workng correlaon marx. The esmaes of he regresson coeffcens and he correlaon can hen be obaned eravely. The erm workng mples ha he esmae of he correlaon s updaed a each eraon. Boh βˆ and he esmae of Var( β ˆ) are 83

9 conssen even f R () s no correcly specfed (Zeger and Lang, 986). The GEE s acually a mulvarae verson of quas-lkelhood score equaon (Wedderburn, 974): n = µ β j var( y ) ( y µ ) where j =,,p. Wh hs GEE seup, dfferen correlaon srucures (R ()) can be specfed o reflec dfferen me dependence whn each subjec. Some common correlaon srucures are shown n Table 4.2. In margnal models, he subjecs are assumed o be ndependen, he man neres s n he populaon-average nformaon, and hence he focus of he model s on he regresson coeffcens. The regular regresson models have a smlar focus on he regresson coeffcens. Snce he mehod of GEE s sll n s early sage, he goodness of f crera s no ye well developed (Schabenberger, 995). In he followng secon, a GEE based approach usng he mxure dea s presened o ncorporae he exra zeroes due o moraly, and he focus of he dscusson wll be on he nference of he regresson coeffcens. 84

10 85 Table 4.2. Common Correlaon Srucures (Assumng = 4) Varance-Covarance srucure R () (symmerc) Independen Auo-regressve Compound Symmerc (Exchangeable) Unsrucured The GEE model dscussed above s able o model longudnal daa wh a preassumed correlaon srucure. Usng he zero nflaed Posson dea n Chaper Three, a GEE based approach s developed o handle longudnal oxcy daa. Ths approach furher exends he zero nflaed Posson model n a mulvarae sense and makes use of he GEE o esmae he coeffcens. The model can be nerpreed as a regresson model,

11 and he esmaes can be obaned by usng he eraed re-weghed leas squares procedure. Furhermore, many sandard programs have been developed o f he GEE model by smply specfyng he dsrbuon and he lnk funcon. For example, SAS, S- plus, Sudaan, and Saa all have sandard procedures o handle GEE. Based on he dsrbuon of he ZIP model, and he assumpon of he ndependence of he subjecs over me, a lkelhood funcon smlar o (3.2) can be obaned. Even hough he assumpon of ndependence s no realsc n hs case, s made o smplfy he followng developmen. Here, he marx G conans he covaraes of he reproducon par of he GEE model, and he marx B conans he covaraes of he moraly par of he GEE model. The daa s measured from n subjecs wh repeaed measures on each subjec. There are a oal of n observaons. By assumng all he observaons are ndependen (boh over me and subjecs,.e. n ndependen observaons), he lkelhood funcons can be wren as follows: L (γ, β; y, z) = n n Gγ ( zgγ log( + e )) + = = ( z )( y B β e B β ) n = ( z )log( y!) (4.) where G and B are he h row of he covarae marx G and B, Z = 0 f Y s from a survvng anmal and Z = f Y s from a dead anmal. Agan he wo componens whch nclude β and γ n he lkelhood can be re-wren as: n L(γ; y, z) = = ( z G γ log( + e )) (4.2) G γ 86

12 n = B β L(β; y, z) = ( z )( y B β e ) (4.3) The esmaon of β and γ would agan nvolve an erave E-M ype procedure by alernang beween Z, β and γ. The (k+) h eraon of he procedure s as follows:. Esmae Z I As n Chaper 3, gven he mos recen esmaes of β and γ, β (k) and γ (k), Z can be found by usng he Bayes rule. Here Z (k) = Prob(moraly y, γ (k), β (k) ) = Prob( y Prob( y moraly) Prob( moraly) moraly) Prob( moraly) + Prob( y survval) Prob( survval) = ( + e G γ ( k ) exp(b β ( k ) ) ) f y = 0 = 0 f y > 0 (4.4) 2. Esmae γ Use he funcon (4.2) o oban he esmae of γ (k+). As n Chaper 3, snce Z (k) = 0 whenever y > 0, (4.2) can be expressed as: L(γ; y, z) = y = 0 y = 0 G γ n Z G γ Z log( + e ) ( Z )log( + e ) (4.5) = G γ Now suppose here are n 0 anmals who recorded zero a leas on one ou of he four days. Le her responses be y y, K, y ), hen we defne he followng: l = ( l ln0 4 87

13 y G * = ( n 4 l ln0 4 = ( G y, K, y, y, K, y ), n 4 l, K, G, G, K, G ), * l n0 4 * P p, K, p, p, K, p ). = ( n 4 l ln0 4 Then a dagonal marx W (k) wh dagonal elemen s defned as: w ( k ) ( k ) ( k ) n 4 ( k) l = ( Z, K, Z, V, K, V ( k ) ln0 4 ) ( k) ( k) l l where V = Z when y = 0, and V = 0 when y > 0. l ( k) l l W (k) has a smlar weghng scheme as ha n Chaper 3, excep: when here s a leas a zero observaon from an anmal, hen all he responses from ha anmal wll be added o he y *, and he nonzero observaons of hese anmals are forced o become zero wh he weghs V = 0. Now he funcon (4.2) becomes: n + n0 n + n0 ( k) ( k) * * = = L(γ; y, Z (k) * ) = y w G γ w log( + e ) (4.6) G γ Followng he dscusson n Chaper hree, consder he score funcon of weghed logsc regresson. The funcon (4.6) wll agan gve rse o he score funcon for (4.2) whch can also be wren as: * * P G W (k) ( y *) = 0 88

14 as n Chaper hree, and he negave nformaon marx s: G (k) * W Q* G* where Q * s he dagonal marx wh P * (-P * ) on he dagonal. The score funcon s he same as for he unvarae case for he logsc GEE G* γ wh D Var( y* ) = G* ( Var( y* )), E(Y ) = µ p, and excep wh an exra γ wegh called W (k). The varances n he equaons cancel each oher. Ths can be exended o a mulvarae score funcon usng he underlyng developmen n he GEE model wh Var(y *I ) as a marx. Hence, hs leads o a weghed GEE equaon (k) D * W Var ( Y * ) ( Y* µ * ) = 0 wh responses Y * and weghs W (k). The esmae of γ for he curren eraon can be obaned. 3. Esmae β Use he funcon L(β; y, Z (k) ) o oban he esmae of β (k+). In Chaper hree, hs s acheved by usng a weghed log-lnear Posson regresson wh weghs - Z (k) (McCullagh and Nelder, 989). When he ndependen assumpon s relaxed, hs can be exended o a weghed GEE model usng he Posson dsrbuon wh weghs (-Z (k) ). Agan, by pre-assumng a correlaon srucure (R ()) among he repeaed measures, he esmaes of he regresson coeffcens can be obaned. The esmaes of β and γ can be obaned by hs E-M ype procedure. The weghed GEE model program s avalable hrough sandard sofware whch apples an 89

15 erave re-weghed leas squares procedure ogeher wh a pre-assumed correlaon srucure (R ()) among he repeaed measures. Some possble correlaon srucures are shown n Table 4.2. In he followng example, hree dfferen correlaon srucure wll be used. They are he ndependen, auo-regressve, and compound symmerc correlaon srucure. 4.6 Examples for GEE Approach In he Cerodaphna duba es example, here are 660 anmals. Each anmal s observed for a seven-day perod. Snce he anmals do no reproduce n he frs hree days, only four days of daa are used n he analyss (n equaon 4.6, n = 660 and = = 4). The approach s based on he zero nflaed Posson densy. There are wo pars n he model, one s modelng reproducon (Posson daa), and he oher s modelng moraly (bnary daa). I can be vewed as a reproducon model wh an adjusmen due o moraly. In order o examne me and oxcan effecs, boh are used as he covaraes n he model, however, only one oxcan s used o keep he dscusson smple. An S-plus compuer program s wren o f he GEE-ZIP model (see Appendx C). The model has sx regresson coeffcens and he resuls are shown n Table 4.3 o Table

16 Table 4.3. GEE-ZIP model for he cerodaphna daa; S.E. n () (Independen correlaon srucure) Chromum Copper Mercury Znc β 0 (n.).093 (0.0606).084 (0.5695).5 (0.0543).2 (0.0625) β (me) 0.28 (0.028) (0.028) (0.27) 0.28 (0.029) β 2 (oxc) (0.0270) (0.0083) (0.0087) (0.006) γ 0 (n.) 0.69 (0.0984) (0.092) 0.72 (0.0932).045 (0.035) γ (me) (0.0320) (0.0339) (0.033) (0.032) γ 2 (oxc) (0.047) 0.28 (0.025) 0.57 (0.032) (0.0028) Sum squared Resdual Table 4.4. GEE-ZIP model for he cerodaphna daa; S.E. n () (Compound Symmerc correlaon srucure) Chromum Copper Mercury Znc β 0 (n.).086 (0.0487).078 (0.0467).7 (0.0453).096 (0.0499) β (me) (0.03) (0.03) 0.29 (0.03) (0.04) β 2 (oxc) (0.098) (0.0058) (0.0064) (0.002) γ 0 (n.) (0.0958) (0.0897) (0.0906) 0.82 (0.00) γ (me) (0.0294) (0.03) (0.0303) (0.0288) γ 2 (oxc) (0.0394) 0.09 (0.007) 0.3 (0.024) (0.0027) Sum squared Resdual

17 Table 4.5. GEE-ZIP model for he cerodaphna daa; S.E. n () (Auo-regressve correlaon srucure) Chromum Copper Mercury Znc β 0 (n.).4 (0.0546).09 (0.059).35 (0.0500).25 (0.0558) β (me) (0.03) (0.030) (0.028) (0.032) β 2 (oxc) (0.024) (0.0067) (0.0070) (0.003) γ 0 (n.) (0.0995) (0.0928) (0.0939) 0.96 (0.047) γ (me) (0.039) (0.0337) (0.0329) (0.03) γ 2 (oxc) (0.040) 0.25 (0.022) 0.53 (0.030) (0.0028) Sum squared Resdual Dfferen correlaon srucures can be embedded n he wo pars o model he wo dfferen mechansms. However, n hs example, he same correlaon srucure s used n boh he reproducon and moraly par. The esmaes of he correlaon srucures for dfferen models are shown n Table 4.6. Based on he sum of squared resduals, he mercury and copper model are found o be he beer model among he four sngle oxcan models. The day effec s conssenly sgnfcan n all he models, and also all four models have very smlar me effec n boh pars of he model (0.282 o 0.28 n reproducon, o 0.48 n moraly par). The znc model has he lowes oxcan effecs among he four models whle he chromum model has he hghes. In addon, he models wh auo-regressve correlaon srucure have he lowes sum of squared resduals (25895 for Mercury and for Copper). The auo-regressve correlaon srucure mples ha he furher apar he daa pons are, he lesser he correlaon s beween hem. In he moraly pars of he copper and mercury model, he off-dagonal erms are very close o zero. I mples he use of he assumpon of ndependen correlaon srucure n he model would be accepable. 92

18 The advanage of GEE-ZIP approach s he flexbly of allowng dfferen correlaon srucures for he models. Even hough he ZIP model n Chaper Two s n fac a unvarae case of he GEE-ZIP model, and me can be reaed as a covarae n he ZIP model, he esmae of he ZIP model would sll be napproprae or based. Ths s because he ZIP model does no consder he possble correlaon beween he repeaed measures. Unlke he derved varable approach, he GEE-ZIP model s able o handle covaraes ha change over me. By nvesgang he correlaon srucure ogeher wh he me effecs, a beer undersandng of he underlyng mechansm can be acheved. 93

19 94 Table 4.6. The correlaon esmaes for dfferen models (symmerc) Chromum Copper Mercury Znc Compound Symmerc Reproducon Moraly Auo-regressve Reproducon Moraly

20 4.6. Inhbon Concenraon In order o nerpre he resuls for he GEE approaches, an overall predcon and he nhbon concenraon (IC) can be defned as hose n Chaper hree. By usng he same defnon n Chaper hree, he overall mean can be expressed as λ ˆ( pˆ ), where λˆ s he predced mean of he Posson GEE and pˆ s he predced mean of he logsc GEE. Ths can be vewed as he reproducon weghed by moraly probably. Snce here are wo ndependen varables (me and oxcan concenraon), he mean response s a surface and he IC levels are a se of conour lnes. Snce he auo-regressve correlaon models have he lowes sum of squared resduals, her conours of he IC levels are ploed n Fgure 4.. The IC levels of he chromum model are he lowes, whch means ha chromum has a relavely sronger oxc effec han he oher effluens. However, he slopes of he IC lnes are relavely fla when compared wh oher models. Ths means he oxc effec of chromum does no change as fas as he oher effluens. Fgure 4.. IC levels of he four sngle oxcan models wh me effec (PDF, 4K, Fg4-.pdf) 4.7. MIM revsed Wh he GEE-ZIP model esablshed, we now can compare he GEE-ZIP esmae wh he MIM esmae. The mxure dea suggess he observaon Y s equal o he produc X*Z, where X presens fecundy and Z ndcaes moraly. The GEE-ZIP model s hen based on he wo pars model: log(λ) = Bβ and log(p) = log(p/( p)) = Gγ, 95

21 where he Posson par models fecundy (X), and he logsc par models moraly (Z). The predcon of he GEE-ZIP can be wren as µ ˆ = λˆ( pˆ ), where µˆ s jus he expeced value of he observaon Y, λˆ s he expeced value of X, and ( pˆ ) s he expeced value of Z. Ths s exacly he dea n he MIM esmaon E( Y ) E( X ) =, E( Z ) = = where he fecundy s esmaed wh adjusmen for he moraly effec. In he GEE-ZIP, he fecundy (X) s esmaed (or predced) by he Posson par of he model and he adjusmen for moraly s esmaed by he logsc par (model for Z). However, he MIM esmaon does no nvolve any covarae. The MIM esmaon of he mean s hen equvalen o he predcon of he nercep only GEE-ZIP model. An esmae for he varance of he X can hen be developed based on he GEE-ZIP esmae. The quany = E( ) = λ ˆ can be expressed as a lnear combnaon c λˆ where c s a vecor X = of s, and λˆ s a vecor of λˆ ( =, K, ). The varance of = E( ) becomes he X varance of c λˆ, whch s equal o c Σˆ c where Σˆ s he esmaed varance-covarance marx of λˆ. The value of Σˆ can be obaned from he esmaed varance-covarance marx of he GEE-ZIP model. Snce he vecor c s jus a column of ones, c Σˆ c s jus he sum of all he erms n Σˆ. For example, f he subjec s measured four mes (=4), hen c s a 4 vecor, and Σˆ s a 4 4 marx wh elemens σ where, j =,2,3,4. Then c Σˆ c s jus 2 c σ jc j. Ths esmae of he varance of mean can hen be compared wh = j= 2 j 96

22 Hamlon s (986) boosrap mehod (Table. 4.7) usng Hamlon s se hree daa (Hamlon, 986). Table 4.7. MIM and GEE-ZIP mean esmaes and her varance MIM GEE (Auo-regressve) GEE (Exchangeable) mean varance of mean Two correlaon srucures are used n he GEE-ZIP model, and hey are compared wh he MIM mehod. The esmaed mean reproducon for he exchangeable correlaon model has he lowes varance among he hree. Hamlon uses a boosrap mehod o oban he varance whou explcly assumng any correlaon srucure n he daa. The esmae obaned by he Hamlon mehod has a relavely hgh varance compared o ha obaned by he GEE-ZIP model. The varance of he GEE-ZIP s obaned by usng he Posson par of he model and s esmaed covarance marx as dscussed n he prevous secon. Snce he calculaon ncludes he covarance erms, he resuls of he GEE-ZIP model depend on he correlaon assumpon Concluson There are wo major problems wh explorng longudnal oxcy daa. One s he possble correlaon srucure among he repeaed measures whn each subjec. The oher s he problem of exra zeroes due o moraly. 97

23 In hs chaper, hree dfferen approaches ha handle longudnal oxcy daa are dscussed. The frs wo approaches: he me-wse approach and he derved varables approach, are no able o smulaneously resolve he wo problems ha are lsed above n one model. Bu hey analyze he daa usng very smple sraghforward mehods. The hrd approach s based on he generalzed esmang equaons (GEE) echnque and he zero nflaed Posson (ZIP) model whch s he man focus of hs chaper. Ths approach s able o model fecundy wh he adjusmen of exra zeroes due o moraly, and a he same me, esmae he correlaon srucure whn each subjec. Snce he GEE modelng echnque s sll n s early sage, here s no welldefned goodness-of-f creron. The example n hs Chaper uses he sum of squared resduals each model as a goodness of f creron o decde whch one s he bes model among he four sngle oxcan GEE-ZIP models n Table 4.3 Table 4.5. Copper model and mercury model are found o be he wo beer models among he four sngle oxcan models. The models also can be used o examne he me effec and look a he changes of he oxcan effec over me. The same defnon of nhbon concenraon (IC) can also be obaned as ha n Chaper Three. The GEE model dscussed n hs Chaper s a margnal model wh he assumpon ha he subjecs are ndependen of each anoher. The dscusson s lmed o sngle oxcan models. A sraghforward exenson o he GEE model s o nclude he me oxcan neracon n he model. In addon, by relaxng he assumpon of ndependen subjecs, more complcaed models can be developed. For example, may be possble o apply oher ypes of GEE models lke he random effec model or he ransonal model o longudnal oxcy daa. However, more research s needed for models ha are able o ncorporae exra zeroes. 98