Multilevel Logistic Regression for Polytomous Data and Rankings

Transcription

1 Outlne Multlevel Logstc Regresson for Polytomous Data and Rankngs 1. Introducton to Applcaton: Brtsh Electon Panel 2. Logstc Models as Random Utlty Models 3. Independence from Irrelevant Alternatves (IIA) 4. Logstc Models wth Observed Heterogenety 5. Logstc Models wth Shared Unobserved Heterogenety 6. Multlevel Logstc Regresson 7. Applyng Multlevel Logstc Regresson: Brtsh Electon Panel ANDERS SKRONDAL Dvson of Epdemology Norwegan Insttute of Publc Health 8. Generalzed Lnear Latent and Mxed Models (GLLAMM): Framework, Estmaton and Predcton Jont work wth SOPHIA RABE-HESKETH EFRON-SEMINAR September 3, 2002 Slde 1 Slde 2 Brtsh Electon Panel: Outcome Brtsh Electon Panel: Covarates Alternatves: Conservatve, Labour, Lberal (exclude mnor partes) Polytomous / Frst choce: Party voted for Rankngs: Partes not explctly ranked rankngs derved A. Party voted for gven rank 1 B. Partes not voted for ranked nto second and thrd place usng ratng scales: 1 5 strongly aganst strongly n favour Electon-specfc covarates: [1987] [1992] Voter-specfc covarates: [Male] [Age] Age n 10 year unts [Manual] Father a manual worker Electon and voter-specfc covarate: [Inflaton] Snce last electon, prces fallen a lot ncreased a lot Electon, voter and alternatve-specfc covarate: [LRdst] Absolute dstance between voters s poston on left-rght poltcal contnuum and party s mean poston Slde 3 Slde 4

2 Brtsh Electon Panel: Three-Level Data Requrements of Methodology 1608 people partcpated n survey of votng n 1987 and 1992 electons. Excluded votng occasons wth votng on mnor partes and mssng covarates voter 1 consttuency voter 2 voter 3 Level 3: 249 consttuences Level 2: 1344 voters Level 1: 2548 votng occasons Methodology should handle: Rankngs as well as frst choces (rankngs benefcal for effcency & dentfcaton) Multlevel data (dependence nduced at several levels) Dfferent types of covarates (ncludng alternatve specfc) Varyng alternatve sets Tes Responses Mssng at Random (MAR) and be Implemented n publcly avalable software Slde 5 Slde 6 Random Utlty Models Frst Choce as Utlty Maxmzaton Utlty formulaton useful: Insght nto logstc regresson models (e.g. specfcaton, dentfcaton) Facltates extenson of conventonal logstc regresson for frst choce and rankngs to MULTILEVEL desgns Unobserved utlty U a assocated wth each alternatve a=1,..., A for unt =1,..., N Random utlty models composed as U a = V a + ɛ a V a s fxed lnear predctor representng observed heterogenety ɛ a s random term representng unobserved heterogenety (ndependent over and a) Alternatve f s chosen f U f > U g for all g f ɛ a ndependent (over and a) Gumbel or extreme value dstrbuted of type I: g(ɛ a = exp { ɛ a exp( ɛ a } McFadden (1973), Yellott (1977): ɛ a ndependent Gumbel Pr(f = exp(v f Aa=1 exp(v a [Conventonal multnomal logt] Slde 7 Slde 8

3 Rankng as Utlty Orderng Identfcaton r s s alternatve wth rank s for unt. Rankng defned as R = (r 1, r 2,, r A, e.g. (2, 1, 3) R s obtaned f U r1 > U r2 > > U ra Luce & Suppes (1965); Beggs, Cardell & Hausman (1981): Pr(R = ɛ a exp(v r1 As=1 exp(v rs ndependent Gumbel r 2 r A exp(v As=2 exp(v rs exp(v As=A 1 exp(v rs [Exploded logt] Probablty of choosng alternatve 1 among alternatves 1, 2 and 3 can be expressed n terms of utlty dfferences Pr(U 1 U 2 > 0 U 1 U 3 > 0) Probablty of rankng alternatves as smlarly becomes Pr(U 1 U 2 > 0 U 2 U 3 > 0) Frst choce and rankng probabltes (and lkelhoods) depend only on utlty dfferences Locaton of V a s arbtrary Scale of V a not arbtrary snce varance of ɛ a fxed (at π 2 /6), exp(v 1 ) a exp(v a ) = exp(v 1 + c a exp(v a + c exp(s V 1 ) a exp(s V a At each stage, a frst choce s made among the remanng alternatves. Dualty wth partal lkelhood contrbuton of stratum n Cox regresson ( survvng alternatves as rsk sets and choces as falures) = Survval software applcable No exploson for normally dstrbuted utltes! Slde 9 Slde 10 Independence from Irrelevant Alternatves (IIA) The Party Merger Problem Multnomal logt: Odds of alternatve a versus b becomes Pr(a) Pr(b) = exp(v a V b Odds ndependent of propertes of other alternatves Luce (1959) calls ths Independence from Irrelevant Alternatves 1. Intally three poltcal partes: Lab1, Lab2 and Cons. Lab partes are ndstngushable and have the same lnear predctor V Lab, whereas Cons party has lnear predctor V Cons Pr(Lab1 or Lab2 Cons, Lab1, Lab2) = 2. Lab1 and Lab2 merge to form a sngle Lab party 3. Follows that Pr(Lab Cons, Lab) = 2 exp(v Lab 2 exp(v Lab )+exp(v Cons exp(v Lab exp(v Lab )+exp(v Cons Pr(Lab Cons, Lab) < Pr(Lab1 or Lab2 Cons, Lab1, Lab2) Merger reduces the probablty of votng Lab and ncreases the probablty of votng Cons whch s contrantutve! Would expect no change n probablty of votng Lab. Slde 11 Slde 12

4 Heterogenety and IIA Observed Heterogenety Lnear predctor for unt and alternatve a: Numercal Examples of Probablty Votng Lab Margnal Probablty Before merger After merger Heterogenety (Lab1 or Lab2) (Lab) none V Lab V Cons = observed men: V Lab V Cons = 1.2 women: V Lab V Cons = observed & shared unobserved men: V Lab V Cons = 0.8+4δ women: V Lab V Cons = 3.2+4δ δ N(0, 1) V a Covarates and Parameters: = m a + g a x + b x a m a alternatve specfc constants x vares over unts (but not alternatves) and has fxed effects g a varyng over alternatves Examples: [Age] and [Male] for voter x a vares over alternatves (and possbly unts) and has fxed effects b not varyng over alternatves Example: [LRdst] between dfferent partes and voter Identfcaton: Alternatve 1 reference alternatve, set m 1 =0 and g 1 k =0 for all k. Common specal cases: V a = m a + g a x [statstcs/bostatstcs] V a = m a + b x a [econometrcs/psychometrcs] Slde 13 Slde 14 Shared Unobserved Heterogenety Unt-specfc unobserved heterogenety shared between alternatves Utltes for alternatves dependent wthn unts Use latent varables δ a(1) to obtan flexble yet parsmonous covarance structure for utltes: I. Random Coeffcent Models II. Factor Models Covarance Structure I: Random Coeffcent Models For alternatve-specfc covarates, z a, we consder random coeffcents β representng unt-specfc effects of the covarates where U a = V a + δ a(1) + ɛ a δ a(1) = β z a β N(0, Ψ β ) Example: β l s the voter-specfc effect of poltcal dstance when z a l =[LRdst] Slde 15 Slde 16

5 Covarance Structure II: Factor Models Multlevel Desgns and Latent Varables One-factor model: where U a = V a + δ a(1) + ɛ a δ a(1) = λ a η η N(0, ψ η ) η s a common factor, λ a are factor loadngs and ɛ a are unque factors (ndependent Gumbel as before) Two nterpretatons of factor models: 1. λ a alternatve-specfc effect of unobserved unt-specfc varable η 2. λ a an unobserved attrbute of alternatve a and η random effect Identfcaton: Lkelhood depends only on utlty dfferences, V a V b + (λ a λ b )η + ɛ a ɛ b one loadng must be fxed, e.g. λ 1 =0, and scale of factor must also be fxed, e.g. λ 2 =1 Fragle dentfcaton for frst choces unless alternatve-specfc covarates ncluded Can be extended to multdmensonal factors Three-level applcaton: Consttuences (level 3) ndexed k Voters (level 2) ndexed j Electons (level 1) ndexed Latent varables ntroduced at each level to represent unobserved heterogenety at that level (nduces dependence at all lower levels): Latent varables at electon level = Cross-sectonal dependence between utltes wthn voter j at gven electon Latent varables at voter level = Longtudnal dependence between utltes wthn voter j over electons Latent varables at consttuency level = Dependence between utltes between voters j wthn consttuency k Slde 17 Slde 18 Multlevel Logstc Regresson Brtsh Electon Panel: Retaned Model and Estmates The general three-level model U a jk = V a jk + δ (1) jk + δ(2) jk + δ(3) jk + ɛa jk Electon level latent varables δ (1) jk are composed as δ (1) jk = β(1) jk za(1) jk + λa(1) η (1) jk Voter level latent varables δ (2) jk are composed as δ (2) jk = β(2) jk za(2) jk + γa(2) jk z jk + λ a(2) η (2) jk Random Coeffcents I: β (2) jk are voter level random coeffcents for alternatve-specfc covarates z a(2) jk (EX: effect of [LRdst] on party preference vares between voters) Random Coeffcents II: γ a(2) jk are voter level alternatve specfc random coeffcents for electon-specfc covarates z jk (EX: effects of [1987] and [1992] on party preference vary between voters) Factors: λ a(2) η (2) k for a voter nduces dependence between dfferent electons Consttuency level latent varables δ (3) jk are composed as δ (3) jk = β(3) k z a(3) jk + γa(3) k z jk + λ a(3) η (3) k Latent varables at voter and consttuency levels Correlated alternatve specfc random ntercepts FIXED PART: Corr. Random Intercepts Independence Lab vs. Cons Lb vs. Cons Lab vs. Cons Lb vs. Cons Est. (SE) Est. (SE) Est. (SE) Est. (SE) g a 1 [1987] 0.70 (0.51) 0.71 (0.35) 0.38 (0.20) 0.12 (0.17) g a 2 [1992] 1.24 (0.53) 0.75 (0.37) 0.51 (0.20) 0.13 (0.18) g a 3 [Male] (0.31) (0.20) (0.11) (0.09) g a 4 [Age] (0.10) (0.04) (0.04) (0.03) g a 5 [Manual] 1.63 (0.35) 0.12 (0.21) 0.65 (0.11) (0.10) g a 6 [Inflaton] 1.27 (0.18) 0.72 (0.13) 0.87 (0.09) 0.18 (0.03) b [LRdst] (0.04) (0.02) RANDOM PART: Voter Level ψ (2) γa (2.02) 5.73 (0.85) ψ (2) 8.20 (1.09) γ 2,γ3 Const. Level ψ (3) γa 5.15 (1.07) 0.76 (0.28) ψ (3) 1.39 (0.47) γ 2,γ3 logl Slde 19 Slde 20

6 The GLLAMM Framework Generalzed Lnear Latent and Mxed Models (GLLAMM): gllamm: Stata program for estmaton and predcton 1. RESPONSE MODEL: Generalsed lnear model condtonal on latent varables Lnear predctor: observed covarates multlevel latent varables (factors and/or random coeffcents) Lnks and dstrbutons: as for GLM s plus ordnal and polytomous responses and rankngs 2. STRUCTURAL MODEL: Equatons for the latent varables Regressons of latent varables on observed covarates Regressons of latent varables on other latent varables (possbly at hgher levels) 3. DISTRIBUTION OF LATENT VARIABLES (DISTURBANCES) Multvarate normal Dscrete wth unspecfed dstrbuton To obtan the lkelhood of GLLAMM s, the latent varables must be ntegrated out Sequentally ntegrate over latent varables, startng wth the lowest level usng a recursve algorthm Use Gauss-Hermte quadrature to replace ntegrals by sums Scale and translate quadrature locatons to match the peak of the ntegrand usng adaptve quadrature Maxmum lkelhood estmates obtaned usng Newton-Raphson Emprcal Bayes (EB) predctons of latent varables and EB standard errors obtaned usng adaptve quadrature Slde 21 Slde 22 Some lnks and references Skrondal, A. & Rabe-Hesketh, S. (2002). Multlevel logstc regresson for polytomous data and rankngs. Psychometrka, n press. GLLAMM framework: Rabe-Hesketh, S., Skrondal, A. & Pckles, A. (2002a). Generalzed multlevel structural equaton modellng. Psychometrka, n press. Skrondal, A. & Rabe-Hesketh, S. (2003). Generalzed latent varable modelng: Multlevel, longtudnal and structural equaton models. Boca Raton, FL: Chapman & Hall/ CRC. gllamm software: gllamm and manual can be downloaded from Rabe-Hesketh, S., Skrondal, A. & Pckles, A. (2002b). Relable estmaton of generalzed lnear mxed models usng adaptve quadrature. The Stata Journal, 2, Slde 23