ηϛύϥϝτϦοΫਪ࿦ͷجૅͷ෮श Daisuke Yoneoka September 29, 2014

Notations جຊతʹ Tsiatis,2006 ʹै͏. Θ͔Μͳ͔ͬͨΒࣗ෼Ͱௐ΂ͯͶ! ϕΫτϧ΋ߦྻ΋ଠࣈʹͯ͠ͳ͍͚Ͳ, ͦ͜͸ࣗ෼Ͱิ͍ͬͯͩ͘͞. σʔλ͸ i.i.d Ͱ Zi = (Zi1, . . . , Zim) ∈ Rm αϯϓϧαΠζ͸ n ਓ. i.e., Z1, . . . , Zn φ(Z) ͸Өڹؔ਺ u(Zi, θ) ͸ਪఆؔ਺ Լ෇͖ࣈͷ eff ͸ (઴ۙ) ༗ޮ (efficient) ͱ͍͏ҙຯ

ηϛύϥϝτϦοΫਪ࿦ͱ͸ʁ Zi ͷີ౓ؔ਺͕ηϛύϥϝτϦοΫϞσϧʹै͏ͱ͸ S = {p(z : θ, η)|θ ∈ Θ ⊂ Rr, η ∈ H} θ ͸༗ݶ࣍ݩͷڵຯ͋ΔύϥϝλͰ, η ͸ແݶ࣍ݩͷͲ͏Ͱ΋͍͍ύ ϥϝλ (ہ֎ (nuisance) ύϥϝʔλʔ). ηϛύϥϝτϦοΫਪ࿦: ͜ͷ΋ͱͰ θ ͷ࠷ྑͷਪఆྔ (RAL ਪఆ ྔ) Λ΋ͱΊΔ͜ͱ

Өڹؔ਺ θ ͸ͳΜͰ΋͍͍͔Β࠷ྑΛݟ͚ͭΔͱ͍͏ͷ͸ແཧήʔ → Ϋϥε Λݶఆͯͦ͜͠Ͱݟ͚ͭΔ! (౷ܭͰ͸Α͘΍ΔΑͶ) Өڹؔ਺: ਪఆྔ ˆ θ ͷӨڹؔ਺ͱ͸, (Ϟʔϝϯτʹ੍໿͕͋Δ) √ n(ˆ θ − θ) = 1 √ n n i=1 φ(Zi, θ, η) + op(1) Λຬͨ͢ϕΫτϧ஋ؔ਺. ˆ θ ͸઴ۙઢܗਪఆྔͱݺͼ n → ∞ ͰҰகੑ ͱ઴ۙਖ਼نੑ͕͋Δ √ n(ˆ θ − θ) → N 0, E[φ(Zi, θ, η)φ(Zi, θ, η)T ] Πϝʔδతʹ͸͋Δσʔλ͕ͲΕ͚ͩਪఆʹӨڹΛ༩͍͑ͯΔ͔Λ දݱͨ͠΋ͷ

ਪఆؔ਺ͱ M ਪఆ ਪఆํఔࣜ n i=1 u(Zi, θ) ਪఆؔ਺ = 0 ͷղͱͯ͠ಘΒΕΔ΋ͷΛ M ਪఆྔ ͱݺͿ. Α͘ݟΔ score ؔ਺ͳΜ͔΋ίϨ. ͨͩ͠, E[φ(Zi, θ)] = 0 ظ଴஋͸ 0 , E[∥φ(Zi, θ)∥2] < ∞ ෼ࢄతͳ΋ͷ͸ൃࢄ͠ͳ͍ . ͋ͱ΋͏গ͚ͩ͠৚݅͋Δ. Ұகੑͱ઴ۙਖ਼نੑΛ࣋ͭ √ n(ˆ θ − θ) = 1 √ n n i=1 E[ ∂u(Zi, θ) ∂θ ] −1 u(Zi, θ) ͕͜͜Өڹؔ਺ʹͳ͍ͬͯΔ +op(1) → N 0, E[ ∂u(Zi, θ) ∂θ ] −1 E[u(Zi, θ)u(Zi, θ)T ] E[ ∂u(Zi, θ) ∂θ ] −T ] ͜ͷ઴ۙ෼ࢄͷਪఆྔΛαϯυΠονਪఆྔͱݺΜͩΓ͢Δ

RAL ਪఆྔ ઴ۙઢܥਪఆྔ͸ͳΜ͔ྑͦ͞͏ʂͰ΋ super efficiency ͷ໰୊ (Hodges) ͕࢒Δʂ Super efficiency: ઴ۙతʹ Cramer-Rao ͷԼݶΑΓ΋ྑ͍΋ͷ͕Ͱ͖ Δ໰୊ͷ͜ͱ ͜ͷ໰୊Λղܾͨ͠ͷ͕ RAL (Regular asymptotic linear) ਪఆྔ. ͦͷਖ਼ଇ৚݅͸ۃݶ෼෍͕ LDGP (local data generating process) ʹґ ଘ͠ͳ͍͜ͱ (ৄ͘͠͸ Tsiatis, 2006) ηϛύϥਪ࿦͸͜ͷ RAL ਪఆྔͷӨڹؔ਺ΛٻΊΔ͜ͱΛߟ͑Δ

Parametric submodel ηϛύϥϝτϦοΫϞσϧ S ͷ֤఺ʹର͠ p(z; θ, η) ∈ Ssub ⊂ S Λຬͨ͢ύϥϝτϦοΫϞσϧ Ssub = {p(z; θ, γ)|θ ∈ Θ ⊂ Rr, γ ∈ Γ ⊂ Rs, s ∈ N} ΛύϥϝτϦοΫαϒϞσϧͱݺͿ.

Nuisance tangent space (ہ֎઀ۭؒ) ηϛύϥϝτϦοΫϞσϧ S ͷ֤఺ʹର͠, ύϥϝτϦοΫαϒϞσϧ Ssub ͷہ֎઀ۭؒΛ TN θ,γ (Ssub) = {BT sγ(z, θ, γ)|B ∈ Rs} ͱ͢Δ. γ ͸ p(z; θ, η) ʹରԠ͢Δ΋ͷͰ sγ(z, θ, γ) = ∂ ∂γ log p(z; θ, γ) Ͱ ද͞ΕΔ nuisance score ؔ਺. ͜ͷઢܗۭؒ͸͜ͷ nuisance score vector ʹ ΑͬͯுΒΕ͍ͯΔ. ͜ͷͱ͖ TN θ,η (S) = Ssub TN θ,γ (Ssub) Λ S ্ͷ఺ p(z; θ, η) ʹ͓͚Δہ֎઀ۭؒͱΑͿ. ͪͳΈʹ, ಺ ͸಺ଆͷू ߹ʹؔͯ͠ closure ΛͱΔԋࢉࢠ. Note:͜ͷۭؒ͸େ੾Ͱޙʹ, RAL ਪఆྔͷӨڹؔ਺͸͜ͷۭؒʹ௚ަۭͨؒ͠ʹ ଐ͢Δ͜ͱ͕ॏཁʹͳͬͯ͘Δʂ

ઢܗ෦෼ۭؒ΁ͷࣹӨͷزԿͱϐλΰϥεͷఆཧ

RAL ਪఆྔͷӨڹؔ਺ͷॏཁͳఆཧ ηϛύϥϝτϦοΫ RAL ਪఆྔ β ͷӨڹؔ਺ φ(Z) ͸ҎԼͷ৚݅Λຬ଍ ͢Δ. Corollary1 E[φ(Z)sβ] = E[φ(Z)sT efficient (Z, β0, η0)] = I. ͨͩ͠, s ͸είΞؔ਺Ͱ, sT efficient ͸༗ޮείΞؔ਺ Corollary2 φ(Z) ͸ہ֎઀ۭؒʹ௚ަ͍ͯ͠Δ. ༗ޮӨڹؔ਺͸্ͷ 2 ͭͷ৚݅Λຬͨ͠, ͦͷ෼ࢄߦྻ͸, ޮ཰ݶքΛୡ ੒ͦ͠Ε͸ φeffi(Z, β0, η0) = E[seff (Z, β0, η0)sT eff (Z, β0, η0)] −1 seff (Z, β0, η0)

ηϛύϥ઀ۭؒͷఆཧ ύϥϝτϦοΫαϒϞσϧͷ৔߹ͷ RAL ਪఆྔͷӨڹؔ਺ͱ઀ۭؒͱͷؔ܎͸ Tsiatis, 2006 ͷ Ch4.3 ͋ͨΓΛݟͯͶʂ ఆཧ 1 RAL ਪఆྔͷӨڹؔ਺͸ {φ(Z) + TN θ,η (S)⊥} ͱ͍͏ۭؒʹؚ·ΕΔ. ͨͩ͠, φ(Z) ͸೚ҙͷ RAL ਪఆྔͷӨڹؔ਺Ͱ, TN θ,η (S)⊥ ͸ηϛύϥϝτϦο Ϋ઀ۭؒͷ௚ަิۭؒ ఆཧ 2 ηϛύϥϝτϦοΫ༗ޮͳਪఆྔ͸, ͦͷӨڹؔ਺͕Ұҙʹ well-defined Ͱܾఆ͞ Ε,φefficient = φ(Z) − {φ(Z)|TN θ,η (S)⊥} ͷཁૉ. ͪͳΈʹ, (h|U) ͸ projection of h ∈ H(಺ੵΛಋೖͨ͠ώϧϕϧτۭؒ) onto the space U (ઢܗۭؒ)

GEE ʹ͍ͭͯͷ Remarks Liang-Zeger ͷ GEE ͷηϛύϥϝτϦοΫϞσϧ (੍໿ϞʔϝϯτϞσϧ: 1 ࣍ͱ 2 ࣍ͷϞʔϝϯτʹ੍͚ͩ໿Λஔ͍ͨϞσϧ) ͸ҎԼͷಛ௃Λ΋ͭ. ہॴ (઴ۙ༗) ޮਪఆྔ: ෼ࢄؔ਺ͷԾఆ͕ਖ਼͚͠Ε͹, ༗ޮਪఆྔ Robustness: ແݶ࣍ݩͷύϥϝʔλਪఆ͕ඞཁ͕ͩ, ෼ࢄؔ਺Λ misspecify ͨ͠ͱͯ͠΋Ұகੑͱ઴ۙਖ਼نੑ͸อ࣋ GEE ͷຊΛಡΊ͹Θ͔Δ͚Ͳ, Working covariance matrix Λؒҧ͑ͯ ΋༗ޮੑ͸ࣦΘΕΔ͕, ͦͷଞͷ޷·͍͠ੑ࣭ (઴ۙਖ਼نੑͱҰகੑ) ͸อ࣋Ͱ͖Δͬͯ͜ͱ