Lecture 7: Overdispersion in Poisson regression

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Ιόλη Λαμέρας
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1 Lecture 7: Overdispersion in Poisson regression Claudia Czado TU München c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

2 Overview Introduction Modeling overdispersion through mixing Score test for detecting overdispersion c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

3 Introduction McCullagh and Nelder (1989) p Overdispersion is present if V ar(y i ) > E(Y i ) Overdispersion can be modeled using a mixing approach: Y i Z i P oisson(z i ) Z i 0 ind. rv s with E(Z i ) = µ i c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

4 Distributional assumption for Z i 1) Z i Gamma with mean µ i and index φµ i Recall X Gamma(µ, ν) f X (x) = 1 Γ(ν) µ mean ν index E(X) = µ V ar(x) = µ 2 /ν ( ) ν ( νx µ exp νx µ ) 1 x Here f Zi (z i ) = 1 Γ(φµ i ) ( ) φµi ( ) φµi z i µ i exp φµ iz i µ i 1 z i E(Z i ) = µ i V ar(z i ) = µ2 i φµ i = µ i /φ c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

5 P (Y i = y i ) = 0 e z i(z i ) y i y i! 1 Γ(φµ i ) ( φµi z i µ i ) φµi exp{ φzi } 1 z i dz i = φ φµ i Γ(φµ i )y i! 0 e zi(1+φ) z (y i+φµ i 1) i dz i s i =z i (1+φ) = φ φµ i Γ(φµ i )y i! 0 e s i 1 s (y i+φµ i 1) 1 (1+φ) (y i +φµ i 1) i φ+1 ds i = = φ φµ i Γ(φµ i )y i!(1+φ) (y i +φµ i ) 0 e s is (y i+φµ i 1) i ds i φ φµ iγ(y i +φµ i ) Γ(φµ i )y i!(1+φ) (y i +φµ i ) = Γ(y i+φµ i ) Γ(φµ i )Γ(y i +1) ( ) φµ φ i ( ) φ 1 + φ }{{}}{{} 1 1/φ 1/φ+1 1/φ+1 y i c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

6 Negative binomial distribution X negbin(a, b) P (X = k) = Γ(a+k) Γ(a)Γ(k+1) ( 1 1+b ) a ( b 1+b) k k = 0, 1,... a, b 0 E(X) = ab V ar(x) = ab(1 + b) c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

7 is given by Marginal distribution of Y i Y i negbin(a i, b i ) with a i = φµ i b i = 1/φ E(Y i ) = φµ i φ = µ i und V ar(y i ) = φµ i φ (1 + 1/φ) = µ i(1 + 1/φ) Remarks: - V ar(y i ) > E(Y i ) if 1/φ > 0, i.e. overdispersion - φ = no overdispersion - To estimate β and φ jointly one needs to maximize the negative binomial likelihood. No standard software available but one can use the Splus library negbin() from http : //lib.stat.cmu.edu/s/ - V ar(y i ) = µ i (1 + 1/φ) is linear in µ i c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

8 2) Z i Gamma with mean µ i and index ν (exercise) Y i negbin(ν, µ i /ν) such that E(Y i ) = ν µ i /ν = µ i V ar(y i ) = µ i + µ 2 i /ν quadratic variance function in µ i c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

9 Multiplicative heterogeneity in Poisson regression Another approach for modeling overdispersion is to use Y i Z i P oisson(µ i Z i ) with E(Z i ) = 1 and V ar(z i ) = σ 2 Z, i.e. Z i i.i.d., Z i is called multiplicative random effect (exercise) E(Y i ) = µ i V ar(y i ) = µ i + σ 2 Z µ2 i If Z i Gamma with expectation 1 and index ν Y i is negbin(a i, b i ) a i = ν, b i = µ i ν c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

10 Summary 1) Y i Z i P oisson(z i ) E(Z i ) = µ i a) b) Z i Gamma(µ i, φµ i ) Y i Negbin(φµ i, 1/φ) E(Y i ) = µ i V ar(y i ) = µ i (1 + 1/φ) linear Z i Gamma(µ i, ν) Y i Negbin(ν, µ i /ν) E(Y i ) = µ i V ar(y i ) = µ i + µ 2 i /ν quadratic 2) Y i Z i P oisson(µ i Z i ) E(Z i ) = 1 Z i Gamma(1, ν) Y i Negbin(ν, µ i /ν) E(Y i ) = µ i V ar(y i ) = µ i + µ 2 i /ν quadratic c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

11 Test for overdispersion Dean (1992) Assume Y i P oisson(µ i ) with µ i = e x i t β θ i = ln(µ i ) = x t i β To model overdispersion we assume that the canonical parameters θ i are not fixed but random quantities θ i with E(θ i ) = θ i V ar(θ i ) = τk i(θ i ) > 0 for τ 0 and k i (θ i ) differentiable c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

12 To test for overdispersion, want to test H 0 : τ = 0 versus H 1 : τ > 0 Example: θi = xt i β + Z i with Z i i.i.d. E(Z i ) = 0 V ar(z i ) = τ < E(θi ) = θ i V ar(θi ) = τ i.e. k i(θ i ) = 1 Compare to: Y i Z i P oisson(µ i Z i ) E(Z i ) = 1 θ i = ln(µ iz i ) = ln(µ i ) + ln(z i ) E(ln(Z i )) 0 A score test will now be developed for H 0 : τ = 0 versus H 1 : τ > 0. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

13 General score test Let Y 1,..., Y n ind. r.v. with densities f i (y, θ) θ Ω and H 0 : θ Ω 0 H 1 : θ Ω 1 Ω 0 + Ω 1 = Ω R d Ω 0 Ω1 = Loglikelihood: I(θ) := E l(θ) = n i=1 ( [ l(θ) θ l i (θ) l i (θ) = ln f i (y i ; θ) ] [ ] ) t l(θ) θ R d d Fisher information = E ( ) l 2 (θ) θ θ t under regularity conditions c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

14 We are interested in the following composite hypothesis for θ = (β t, τ) t : H 0 : τ = τ 0 versus H 1 : τ τ 0 Consider the score statistic T s = [ ] t θ l(θ) θ=ˆθ0 I 1 (ˆθ 0 ) [ ] θ l(θ) θ=ˆθ0 where ˆθ 0 is the MLE of θ in model θ = (β t, τ 0 ) t, i.e. ˆθ 0 = (ˆβ t, ˆτ 0 ) t satisfies θ l(θ) = ( l(β, τ) t, β ) t l(β, τ) τ θ=ˆθ0 = 0 Under regularity conditions we have T S χ 2 dimω dimω 0 under H 0. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

15 Remark: There are regularity condition under which the standard asymptotic remains valid if τ 0 is at the boundary of Ω, as for example at H 0 : τ = 0 versus H 1 : τ > 0. Here: θ = (β t, τ) t For τ = 0 we have a Poisson GLM and one can show that ˆθ 0 = (ˆβ 0t, 0) t where ˆβ 0 is the MLE of the Poisson GLM. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

16 Score test for overdispersion in count regression Y i θ i P oisson(eθ i ) with E(θ i ) = xt i β and V ar(θ i ) = τk i(θ i ) Let f(y i, θi ) be the conditional density of Y i given θi, then with Taylor we have f(y i, θ i ) = f(y i, θ i ) + θ i f(y i, θ i ) θ i =θ i (θ i θ i) θi 2 R = remainder term f(y i, θ i ) θ i =θ i (θ i θ i) 2 + R c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

17 f M (Y i, θ i ) = marginal likelihood of Y i = f(y i, θ i )f(θ i )dθ i = E θ i (f(y i, θ i )) = f(y i, θ i ) θi 2 [ = f(y i, θ i ) τk i(θ i )f 1 (Y i, θ i ) 2 f(y i, θi ) θi =θ i E[(θ i θ i ) 2 ] +R }{{} V ar(θi )=τk i(θ i ) θi 2 ] f(y i, θi ) θi =θ + Rf 1 (Y i i, θ i ) c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

18 Marginal log likelihood l(θ) = n i=1 l i (θ i ) = n i=1 = n [ln f(y i, θ i ) i=1 + ln l(θ) θ ln f M (Y i, θ i ) { τk i(θ i ) f 1 (Y i, θ i ) 2 θ=( ˆβ0t,0) t = l(β,τ) β l(β,τ) τ θi 2 β= ˆβ0,τ=0 β= ˆβ0,τ=0 } f(y i, θi ) θi =θ + Rf 1 (Y i i, θ i ) ] = 0 l(β,τ) τ β= ˆβ0,τ=0 θ i = x t i β ˆθ 0 i := x i tˆβ0 c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

19 We assume E((θ i θ i) r ) =: d r with d r = o(τ) for r 3 l(β,τ) τ β= ˆβ0,τ=0 n i=1 1 2 k i(θ i ) f 1 (Y i,θ i ) 2 θ i τ k i(θ i ) f 1 (Y i,θ i ) 2 f(y i,θ i ) θ i =θ i θ i 2 f(y i,θi ) θ i =θ i β= ˆβ0,τ=0 = n i=1 1 2 k i(ˆθ i 0 ) f 1 (Y i, ˆθ i 0 ) θi 2 f(y i, θi ) θ i =ˆθ i 0 }{{} 2 =:T i (ˆθ 0 i ) (denominator = 1, since τ=0) c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

20 Additionally I(ˆθ 0 ) = T s = ( ) Iββ I βτ I βτ I ττ [ ˆθ 0 = (ˆβ 0t, 0) t θ l(θ) θ=( ˆβ0,0)] t I 1 (ˆβ 0, 0) [ ] θ l(θ) θ=( ˆβ0,0) = ( l(β,τ) τ ) 2 ( ) β= ˆβ0,τ=0 I 1 (ˆβ 0, 0) ττ = [ n i=1 ] 2 ) T i (ˆθ i (I 0) 1 (ˆβ 0, 0) ττ since components corresponding to β are zero. Further I 1 (ˆβ 0, 0) = (I ττ I t βτ I 1 ββ I βτ) 1 (partioned matrixes) T S = [ n i=1 T i (ˆθ 0 i ) ] 2 / V 2, V 2 = (I ττ I t βτ I 1 ββ I βτ) c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

21 Overdispersion test Reject H 0 : τ = 0 versus H 1 : τ > 0 at level α T S > χ 2 1,1 α c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

22 Ex.: Poisson model with random effects θ i = xt i β + Z i E(Z i ) = 0 V ar(z i ) = τ Y i θ i P oisson(eθ i ) θi = ln µ i = x t i β E(Y i ) = E θ i (E(Y i θ i )) = E θ i (eθ i ) Eθ i (1 + θ i ) = 1 + θ i = 1 + ln µ i 1 + µ i 1 = µ i V ar(y i ) = E θ i (V ar(y i θi )) + V ar }{{} θ i (E(Y i θi )) }{{} E(Y i θi ) = E(Y i ) + (e θ i) 2 V ar(e Z i ) }{{} V ar(1+z i )=V ar(z i )=τ µ i + µ 2 i τ can model overdispersion e θ i c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

23 Overdispersion test in Poisson model with random effect Model: Y i θ i P oisson(eθ i ) θ i = x t i β + Z i E(Z i ) = 0 V ar(z i ) = σ 2 Score test: Reject H 0 : τ = 0 versus H 1 : τ > 0 n i=1 T i (ˆθ i 0 ) V = 1 2 n [(Y i ˆµ 0 i )2 ˆµ 0 i] n i=1 1 2 i=1 ˆµ 02 i > Z 1 α/2 c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

24 References Dean, C. (1992). Testing for overdispersion in Poisson and binomial regression models. JASA 87(418), McCullagh, P. and J. Nelder (1989). Generalized linear models. Chapman & Hall. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

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