Generalized Estimating Equations for Zero-Inflated Spatial Count Data

Transcription

1 Availabl onlin at Procdia Environmntal Scincs 77 ( st Confrnc on Spatial Statistics 0: Mapping Global Chang Gnralizd Estimating Equations for Zro-Inflatd Spatial Count Data Antha Monod * Écol Polytchniqu Fédéral d Lausann, Dpartmnt of Mathmatics, Station 8, CH-05 Lausann, Swzrland Abstract his papr consolidats th zro-inflatd Poisson modl for count data wh css zros proposd by Lambrt (99 and th two-componnt modl approach for srial corrlation among rpatd obsrvations proposd by Dobbi and Wlsh (00 for spatial count data; not only dos this addrss th problm of ovrdisprsion, but addionally provids for gratr flibily whin th zro componnts, allowing for th distinction btwn zros that aris du to random sampling and thos that aris du to an inhrnt charactristic that may induc zro obsrvations. A liklihood and corrsponding scor quations ar drivd for th zro-inflatd Poisson modl; spatial corrlation may b incorporatd via any spatial corrlation structur following Diggl t al. (009. A Matérn (960 corrlation is implmntd as an illustrativ ampl. 0 Publishd by Elsvir Ltd. Opn accss undr CC BY-NC-ND licns. Slction and pr-rviw undr rsponsibily of Spatial Statistics 0 Kywords: Gnralizd stimating quation (GEE; marginal modl; Matérn corrlation; spatial count data; zro-inflatd counts; zro-inflatd Poisson modl.. Introduction Lt y dnot th numbr of occurrncs of an vnt obsrvd at t,,i tim points for ach q subjcti, i,, n, and lt R b a vctor of masurd covariats. Such data is oftn modld in th contt of gnralizd linar modls to allow for gratr flibily, spcifying th pctation to b E Y g ( wh a q vctor of unknown paramtrs, and th link function g ( commonly takn to b th log Var, which in function. For a Poisson probabily spcification, th varianc is st to b Y * Corrspondnc: antha.monod@pfl.ch, tl Publishd by Elsvir Ltd. Opn accss undr CC BY-NC-ND licns. Slction and pr-rviw undr rsponsibily of Spatial Statistics 0 doi:0.06/j.pronv

2 8 Antha Monod / Procdia Environmntal Scincs 7 ( E Y. his is known as ovrdisprsion, which, if lft uncorrctd, may lad to inaccurat infrnc as a rsult of undrstimatd standard rrors. Lambrt (99 prsnts th tchniqu of zro-inflatd Poisson (ZIP rgrssion, giving ris to a nw class of rgrssion modls for count data wh an abundanc of zro obsrvations. In a ZIP modl, th non-ngativ intgr-valud rspons variabl Y is assumd to b distributd as a mitur of a Poisson distribution wh a paramtr and a distribution wh point mass of on at th valu zro, wh miing probabily. In this ZIP modl, th non-zro count rsponss and a portion of th zro count rsponss ar modld by th familiar Poisson spcification. Dobbi and Wlsh (00 adapt th gnralizd stimating quations approach of Liang and Zgr (986 to zro-inflatd spatial count data, and addrss th issu of dpndnc by incorporating a corrlation matri. h abundanc of zros is modld in thir mthod via a two-componnt approach, proposd by Wlsh t al. (996, among othrs: th zro obsrvations ar modld sparatly from th non-zro obsrvations. his mthod considrs first absnc vrsus prsnc (zro vrsus non-zro via a logistic modl, and thn condional on prsnc, modls th non-zro counts by a truncatd discrt modl; Dobbi and Wlsh (00 us a truncatd Poisson distribution. W work in th contt of a Poisson gnralizd linar modl, and consolidat th approachs of Lambrt (99 and Dobbi and Wlsh (00 to construct a gnralizd stimating quation for th zroinflatd Poisson spatial modl, and incorporat spatial dpndnc. Attributing som of th zros to th Poisson distribution avoids condioning on th rsponss, and provids a mor intuiv approach to occurrnc of zros in th data. Dobbi and Wlsh (00 us thir modl to dscrib wkly counts of Noisy Friarbirds (Philmon corniculatus rcordd by obsrvrs for th Canbrra Gardn Bird Survy; attributing a probabily wight of zro obsrvations to a point mass distribution and s complmnt to a Poisson distribution allows for th distinction btwn zro counts arising du to an inhrnt charactristic that may induc zro obsrvations (.g. inadquacy of th rgion whr masurmnts wr takn for th survival or rproduction of Noisy Friarbirds and zro counts arising at random. h rmaindr of this papr is structurd as follows: w provid an ovrviw of th zro-inflatd Poisson tchniqu proposd by Lambrt (99, and outlin th two-componnt approach of Dobbi and Wlsh (00. W thn dvlop th Poisson gnralizd linar modl that borrows idas from th two mthods, w giv s liklihood and comput corrsponding scor quations for th zro-inflatd Poisson modl. W thn follow Diggl t al. (009 to incorporat dpndnc into th scor quations. As an illustrativ ampl, w giv th stimating quations for th non-zro componnt whr th spatial corrlation is dfind by th clbratd Matérn (960 class of spatial corrlations. practic may b too rstrictiv; oftn th data hib Y Var. Ovrviw. h Zro-Inflatd Poisson Modl Dfinion..: Lt Y b a non-ngativ, intgr-valud random variabl dscribing a discrt numbr of occurrncs for a cross-sctional un i at a tim priod t, and lt y b th obsrvd vnt count. Y is said to b Poisson-distributd, or follows a Poisson distribution, wh paramtr ( 0, if th probabily that thr ar actly y occurrncs is givn by th probabily mass function Pr( Y y ; f Y ( y ; y! y.

3 Antha Monod / Procdia Environmntal Scincs 7 ( to dpnd on subjct i and tim priod t ; Hr, as a gnralization, w allow th paramtr s furthrmor, w allow th paramtr to dpnd on som givn information u R (which may vn b providd by th sam masurd covariats, so may b that s q, so that z. As mntiond prviously, th zro-inflatd Poisson modl is a mitur of a Poisson distribution, and a point mass of on at th valu zro, wh a crtain miing probabily. Dfinion..: Lt Y b a random variabl as abov in Dfinion..; is said to follow a zroinflatd Poisson distribution wh paramtr ( 0, and miing probabily (0, if Y 0 wh probabily, Y ~ Poisson( wh probabily (. It follows that. EY ( and VarY ( ( Indd EY Var, sinc, 0 Y. h zro-inflatd Poisson distribution rlas th qual pctation and varianc constraint, and hibs ovrdisprsion.. Modling Corrlatd Zro-Inflatd Count Data: h wo-componnt Approach h two-componnt approach prsntd in Dobbi and Wlsh (00 consists in modling th zro obsrvations sparatly from th non-zro obsrvations; is assumd that givn covariats q r R and z R, Y 0 wh probabily and Y ~ Poissonruncatd (, n0 wh probabily, whr Poisson runcatd (, n0 dnots th zro-truncatd Poisson distribution, truncatd at n 0, wh paramtr. h paramtr and th probabily ar allowd to dpnd rspctivly on auiliary information and z, which nd not b diffrnt. his givs Pr( Y 0 Pr( Y y, z ( y!( y For a two-componnt gnralizd linar modl, th link functions ar g g ( log( z for y ( log( log,, for th first (prsnc vrsus absnc and scond (non-zro, condional on prsnc componnts, rspctivly. Givn this stting that assums indpndnc, and ar orthogonal and th log-liklihood for th two-componnt modl is th sum of th logistic log-liklihood, dnotd by L, and th truncatd Poisson log-liklihood, dnotd by L ; w hav L L L, whr ( L L y 0 y 0 log ( y z z y 0 log log( p( z log y!

4 84 Antha Monod / Procdia Environmntal Scincs 7 ( from which stimating (scor quations can b drivd for ach componnt; dtails may b found in Liang and Zgr (986. o incorporat dpndnc, Dobbi and Wlsh (00 considr drivd rsponss, which compris an indicator variabl for th absnc/prsnc componnt and a count for th non-zro componnt, condional on prsnc. hy procd to modl th dpndnc of th obsrvd y, and dtrmin what this implis about th dpndnc btwn th drivd rsponss. For thir two-componnt modl, thy considr a singl varianc-covarianc matri that taks th structur of a block matri: th varianc bhavior for ach componnt li along th diagonal, whil th off-diagonal blocks dscrib th covarianc bhavior btwn th two componnts: th prsnc/absnc componnt and th posiv count componnt. hy thn show that in assuming a dpndnc structur of an autorgrssiv procss of ordr, AR (, th uncondional varianc dos not dpnd on th covarianc bhavior btwn th two componnts, and thus st th off-diagonal blocks to zro. Furthrmor, th consistncy of th stimators dpnds only on th corrct spcification of th man functions of th drivd rsponss, and not on th corrct choic of corrlation matrics. Nw stimating quations ar thn r-drivd, again following th mthod of Liang and Zgr (986, and th corrsponding robust stimators for th variancs of th paramtr stimators ar givn. 3. Mthod W implmnt th zro-inflatd Poisson modl of Lambrt (99 to dirctly addrss th mattr of ovrdisprsion from css zros, and procd to obtain a liklihood and scor quations, which, following Dobbi and Wlsh (00, turn out to b gnralizd stimating quations in th styl of Liang and Zgr (986, and finally incorporat spatial dpndnc following Diggl t al. (009 into our modl. 3. h Zro-Inflatd Poisson Gnralizd Linar Modl In th contt of gnralizd linar modls outlind in th introduction, w spcify th commonly-usd log-linar link function so that log. In th gnralizd Dfinion.., w allowd th paramtr of th Poisson distribution to dpnd on auiliary information ( z, which dos not ncssarily diffr from th masurd covariats; notic that in using th log-linar link function, w indd hav (. For th zro-inflatd Poisson modl, wh miing probabily, which for simplicy ar also assumd to dpnd on, th obsrvations ar gnratd by p( y Pr( Y y I( y 0 ( ; y! whr I( y 0 is an indicator variabl. h probabily of obsrving a zro is Pr( Y 0 f ( y ( ( p(. 3. Liklihood and Scor Equations h log-liklihood for th zro-inflatd Poisson modl is

5 Antha Monod / Procdia Environmntal Scincs 7 ( (, y 0 log( ( p( y 0 ( y log y!. Applying th tchniqus of Liang and Zgr (986, w notic that th indpndnc stimating quations for th Poisson componnt of th ZIP modl is givn by n i i diag 0, ( ( yi p( whr i ( i,, i. h solution to this quation givs a consistnt stimator of for this i portion of th modl. Procding by classical liklihood analysis, th rsulting scor quation for th ZIP modl wh rgard to is Pr( Y 0 ( y ( y 0. ( y 0 Pr( Y 0 y 0 Modling th miing probabily as any function of anothr paramtr, (, w may also comput th rsulting scor quation for th ZIP modl wh rgard to : Pr( Y 0 0. ( t Pr( Y 0 h ratios of probabilis provid an intuiv odds-ratio intrprtation of th wighting btwn th two probabily componnts of th ZIP modl, howvr s dpndnc on in (, as wll as th dpndnc of ( on through rquirs th two scor quations to b solvd simultanously. 3.3 Introducing Dpndnc Following Dobbi and Wlsh (00 and Diggl t al. (009 in th stting of marginal modls, w may introduc dpndnc into th scor quations ( and (, tnding thm to incorporat a singl i i spatial varianc-covarianc matri dscribing th varianc for both componnts of th modl, which taks th structur of a block matri, as dscribd arlir. Diggl t al. (009 show that for marginal modls undr appropriat paramtrizations, th scor quations assum a form of a gnralizd stimating quation (Liang and Zgr, 986, VarY ( Y 0 (3 whr dnots a man function of th paramtr. As ( assums this form, w may introduc spatial corrlation via th varianc-covarianc matrivar Y. h solution to this quation givs a consistnt stimator of. Corrsponding robust stimators of s varianc ar providd by Liang and Zgr (986, and dtaild in Dobbi and Wlsh ( An Eampl For th scor quation ( concrning th miing probabily of th zro-componnt of th ZIP modl, w follow Dobbi and Wlsh (00 and spcify an AR( dpndnc structur. h clbratd Matérn (960 class of spatial corrlations givs gnral and vrsatil forms for isotropic spatial procsss, ncompassing two othr important modls commonly usd in gostatistical applications. In such an application, th matri Var Y in (3 would tak th form

6 86 Antha Monod / Procdia Environmntal Scincs 7 ( (0 ( s s ( s sn ( s s (0 ( s s (0 whr ( is th spatial smivariogram function, givn by th diffrnc of spatial covariograms ( h C(0 C( h in th isotropic cas, for som sparation vctor h. h Matérn (960 covariogram is givn by h C( h K ( h ( whr dnots th usual Euclidan norm, is th varianc of th spatial procss, th paramtr govrns th smoothnss of th procss, th paramtr is rlatd to th rang of th spatial procss (i.. th lag h byond which th spatial procss valus ar no longr corrlatd, and K ( is th Bssl function of th scond kind of ordr. 4. Conclusion In this papr, w propos a modl for spatial count data comprising css zros, a problm which crats ovrdisprsion lading to undrstimatd standard rrors and thus inaccurat infrnc. W combin mthods proposd by Lambrt (99 and Dobbi and Wlsh (00 to construct a zro-inflatd Poisson modl in which zros ar gnratd by a point-mass probabily at th valu zro as wll as by a usual Poisson distribution, which provids a mor prcptiv insight into th intrprtation of th structural vrsus sampling zros for data whr zros ar abundant. W giv th liklihood for this modl and driv s corrsponding scor quations, thus obtaining gnralizd stimating quations in th styl of Liang and Zgr (986 and Diggl t al. (009 that may incorporat any spatial dpndnc among obsrvations. W illustrat wh an autorgrssiv dpndnc structur of ordr for th componnt corrsponding to th zro obsrvations following Dobbi and Wlsh (00, and a Matérn (960 spatial corrlation modl for th componnt corrsponding to th Poisson distribution in our ZIP modl. Acknowldgmnts I am dply indbtd to Prof. Stphan Morgnthalr for his guidanc, support, and scintific input. Rsarch supportd in part by th Swiss National Scinc Foundation, Grant No. FN Rfrncs [] Lambrt D. Zro-Inflatd Poisson Rgrssion, Wh an Application to Dfcts in Manufacturing. chnomtrics 99;34(: 4. [] Dobbi M, Wlsh AH. Modlling Corrlatd Zro-Inflatd Count Data. Aust. N. Z. Stat. 00;43(4: [3] Wlsh AH, Cunningham RB, Donnlly CF, Lindnmayr DB. Modlling th Abundanc of Rar Spcis: Statistical Modls for Counts wh Etra Zros. Ecological Modlling 996;88: [4] Diggl PJ, Hagrty P, Liang K-Y, Zgr SL. Analysis of Longudinal Data. nd d. Oford Univrsy Prss; 009. [5] Liang K-Y, Zgr SL. Longudinal Data Analysis Using Gnralizd Linar Modls. Biomtrika 986;73(:3. [6] Matérn B. Spatial Variation. nd d. Springr Sris in Statistics No. 36. Springr-Vrlag; 960. n