log T = β T Z + ɛ Zi Z(u; β) } dn i (ue βzi ) = 0,

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1 Accelerated failure time model: log T = β T Z + ɛ β estimation: solve where S n ( β) = n i=1 { Zi Z(u; β) } dn i (ue βzi ) = 0, Z(u; β) = j Z j Y j (ue βz j) j Y j (ue βz j)

2 How do we show the asymptotics of β? asymptotic linear approximation 1. find a neighborhood of β 0 2. approximate S n (β) by a close enough S n (β) linear in β 3. solve S n (β ) = 0: n 1/2 (β β 0 ) D N (0, σ ) 2 4. then show: n 1/2 ( β β ) P 0

3 in summary of approximation Term I: S n (0) Term II: nβa(0) S n (β): is a linear function of β U(0) denote S(β ) = 0, i.e. therefore S n (β) = S n (0) nβa(0) n 1/2 β = n 1/2 S n (0) A(0) n 1/2 β D N (0, σ 2 )

4 Weighted estimation S W n (β) = n i=1 W n (u; β) { Z i Z(u; β) } dn i (ue βz i) = 0, W n (u; β 0 ) is F u -measurable and converges uniformly in probability to a deterministic function w(u) W n (u) = j Y j (u)/n: Gehan-Wilcoxon W n (u) = ŜKM(u ): Prentice-Wilcoxon W n (u) = {ŜKM(u )} ρ : G ρ -family

5 Asymptotic properties on weighted estimators β W n for a general β 0 n 1/2 ( β W n β 0) D N (0, σ 2 (w)/a(w) 2 ) σ 2 W = 0 w(u; β 0 ) 2 A(u, β 0 )λ(u Z)du A(w) = 0 w(u; β 0 )A(u, β 0 ){λ(u Z)u} du By Cauchy-Schwarz inequality, optimal weight function should be proportional to {λ(u Z)u} λ(u Z) = {λ 0(ue βz )u} λ 0 (ue βz ) [ {λ = 0 (ue βz )ue βz ] } λ 0 (ue βz + 1 Z ) = d { log λ0 (ue βz ) + βz } dβ log λ(t Z) = β

6 Partial likelihood for AFTM data: (X i, i, Z i ) e i (β) = log X i +βz i = min{log T i βz i,log C i βz i } = min(ɛ i, d i ) i = 1 for ɛ i d i and 0 for ɛ i > d i for a known β, data reformulated as (e i (β), i, Z i ). when β = β 0, log T i + β 0 Z i follows same distribution as if for γ = 0 λ(t Z i ) = λ 0 (t)exp(γz i ) partial likelihood score function for γ: n i=1 i [Z i j Z j I{e j (β 0 ) e i (β 0 )} j I{e j (β 0 ) e i (β 0 )} ]

7 Time-dependent covariates Z(t) = {Z(s), 0 s < t} hazard-based model [ t ] λ(t Z(t)) = λ 0 exp{βz(u)}du exp{βz(t)} 0 equivalently, T 0 = T 0 exp{βz(u)}du when Z(u) Z, log T = βz + log T 0 interpretation of β: time expansion/contraction Parameter estimation Parametric approach: Cox & Oakes (1984, p.66) Rank-type approach: Robins & Tsiatis (1992, Bmka) QPS approach: Lin & Ying (1995, JSPI)

8 Buckley-James estimation log-likelihood function l(β) = n i=1 {δ i log f(x i ; β) + (1 δ i )log S(x i ; β)} score function: l (θ) = n i=1 { log S(x i ; β) β δ i log f(x i ; β) β = E + (1 δ i ) log S(x } i; β) β [ log f(ti ; β) β T i > x i ]

9 in general, we can substitute log f(x i;β) α(u; β), because β with any mean-zero quantity E[δ i α(x i ; β) + (1 δ i )E{α(T i ; β) T i > X i }] =E[α(T i ; β)] = 0 Buckley-James (1979, Bmka): α(u; β) = log u + βz i assuming that E(ɛ i ) = 0 to estimate β, consider estimating equations n i=1 δ i α(x i ; β) + (1 δ i )E{α(T i ; β) T i > X i } = 0, where E{α(T i ; β) T i > X i } needs to be appropriately estimated

10 Buckely-James approach: where Ê{α(T; β) T > x i } = 1 Ŝ(x i ) Ŝ( ) is the Kaplan-Meier estimator x i α(u)d F(u) therefore, we can use estimating equations S n (β) = to solve for β n i=1 δ i α(x i ; β) + (1 δ i )Ê{α(T i; β) T i > X i } = 0

11 Additional thoughts on Buckley-James: self-consistency representation of Symp) Ŝ (Efron, 1967, Proc. 5th Berk. d F(u) = n 1 n {δ i di(u x i ) + (1 δ i )d F(u)I(u > x i )/Ŝ(x i)} i=1 considering Ŝ(u ) Ŝ(u) therefore Ê{α(T)} = α(u)d F(u) = S n (β)/n solving S n (β) = 0 is equivalent to solving α(u; β)d F(u) = 0

12 Functional ud F(u) major difficulty is in the right-tail with censored data Susarla & Van Ryzin (1980, Ann. Stat.): integration with upper limit that goes to infinity at appropriate rate Reid (1981, Ann. Stat.): various functional forms Gill (1983, Ann. Stat.): regularity conditions on censoring distributions

13 Summary on AFT Model model setup and interpretation estimation extending QPS approach local linear approximation Buckley-James estimation

14 3. Alternative regression models Two major classes: hazard models based on rates: proportional hazards model failure time models based on actual failure times: accelerated failure time model

15 General relative risk model (Prentice & Self, 1983, Ann. Stat.) λ(t Z) = λ 0 (t)r(βz) exponential form: r(βz) = exp(βz) linear form: r(βz) = 1 + βz

16 Additive hazards model (Lin & Ying, 1994, Bmka): λ(t Z) = λ 0 (t) + βz interpretation: additive covariate effect embedded constraint: λ 0 (t) + βz > 0 QPS model estimation E[dN i (t) Y i (t)βz i dt F t ] = Y i (t)λ 0 (t)dt invariant in marginalization extension: additive-multiplicative hazards model (Lin & Ying, 1994, Ann. Stat.) λ(t Z 1, Z 2 ) = λ 0 (t)exp(βz 1 ) + γz 2

17 Accelerated hazards model: λ(t Z) = λ 0 {texp(βz)} parameter interpretation: accelerated/decelerated risk progression parameter estimation: QPS approach E[dN i (te β 0Z i ) F t ; β 0 ] = Y i (te β 0Z i )e β 0Z i λ 0 (t)dt

18 alternative approach: for any γ, transform T i = T i e γz i. The hazard function for transformed time becomes: λ i (t Z i) = λ i (te γz i )e γz i = λ 0 (te (β γ)z i )e γz i when γ = β, then we obtain the proportional hazards model λ i (t Z i) = λ 0 (t)e βz i algorithm motivated to find an estimate Chen & Wang (2000, JASA)

19 General class of hazards model: λ(t Z) = λ 0 {texp(βz)}exp(γz) a general class to include PHM, AFTM and AHM as sub classes identifiability: exponential distribution model estimation: QPS approach semiparametric efficiency Chen & Jewell (2001, Bmka)

20 Proportional odds model log Bennett (1983, Stat. Med.) Murphy (1997, JASA) Yang & Prentice (1999, JASA) S(t Z) 1 S(t Z) = log S 0(t) 1 S 0 (t) + βz Open questions: does QPS work? how does it compare with other approaches? what s the optimal weight function for log-rank statistic when alternative is the proportional odds model?

21 Generalized model g{s(t Z)} = g 0 (t) + βz g( ) known decreasing function log-log link: proportional hazards model logit link: proportional odds model generalized odds-rate model g(s) = log ( 1 λ 1 s λ s λ ) I(λ > 0) + log( logs)i(λ = 0)

22 Linear transformation model h(t) = βz + ɛ h( ) is unknown ɛ s distribution is known extreme value distribution: proportional hazards model standard logistic distribution: proportional odds model Cheng & Wei (1995, Bmka)

23 Proportional mean residual life model model interpretation m(t Z) = m 0 (t)exp(βz) model constraint: m 0 (t)exp(βz) + t shall be monotonically nondecreasing connection with hazard functions under renewal processes model estimation: QPS approach Oakes (1990, Bmka) Chen & Cheng (2005, Bmka)

24 Additive mean residual life model m(t Z) = m 0 (t) + βz model interpretation: additional life expectancy model constraint: m 0 (t) + βz 0 model estimation: QPS and Buckley-James semiparametric efficiency Chen & Cheng (2006, Bmka)

25 Summary alternative regression models developed to address various limitations of the proportional hazards model no perfect model software yet to be developed still an active research area

STAT 331. Accelerated Failure Time Models. Previously, we have focused on multiplicative intensity models, where

STAT 331. Accelerated Failure Time Models. Previously, we have focused on multiplicative intensity models, where STAT 331 Accelerated Failure Time Models Previously, we have focused on multiplicative intensity models, where h t z) = h 0 t) g z). These can also be expressed as H t z) = H 0 t) g z) or S t z) = e Ht