Part III: Chap. 2.5,2.6 & 12

Transcription

1 Survival Analysis Mah 434 Fall 2011 Par III: Chap. 2.5,2.6 & 12 Jimin Ding Mah Dep. jmding/mah434/index.hml Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 1/14

2 Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 2/14

3 Oulines Exponenial disribuion exp(λ), λ > 0; Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 3/14

4 Oulines Exponenial disribuion exp(λ), λ > 0; Weiblull disribuion W eibull(λ, α), λ > 0; Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 3/14

5 Oulines Exponenial disribuion exp(λ), λ > 0; Weiblull disribuion W eibull(λ, α), λ > 0; Lognormal disribuion logn(µ, σ 2 ), σ > 0; Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 3/14

6 Oulines Exponenial disribuion exp(λ), λ > 0; Weiblull disribuion W eibull(λ, α), λ > 0; Lognormal disribuion logn(µ, σ 2 ), σ > 0; Gamma disribuion Gamma(λ, β), λ > 0; Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 3/14

7 Oulines Exponenial disribuion exp(λ), λ > 0; Weiblull disribuion W eibull(λ, α), λ > 0; Lognormal disribuion logn(µ, σ 2 ), σ > 0; Gamma disribuion Gamma(λ, β), λ > 0; Log-logisic disribuion loglogi(λ, α), λ > 0; Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 3/14

8 Oulines Exponenial disribuion exp(λ), λ > 0; Weiblull disribuion W eibull(λ, α), λ > 0; Lognormal disribuion logn(µ, σ 2 ), σ > 0; Gamma disribuion Gamma(λ, β), λ > 0; Log-logisic disribuion loglogi(λ, α), λ > 0; * Gomperz disribuion Gomperz(θ, α), α > 0. Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 3/14

9 Oulines Exponenial disribuion exp(λ), λ > 0; Weiblull disribuion W eibull(λ, α), λ > 0; Lognormal disribuion logn(µ, σ 2 ), σ > 0; Gamma disribuion Gamma(λ, β), λ > 0; Log-logisic disribuion loglogi(λ, α), λ > 0; * Gomperz disribuion Gomperz(θ, α), α > 0. For each parameric model, we will discuss he disribuion properies and parameer esimaion. Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 3/14

10 Exponenial Disribuion S() Survival Funcion lambda= 0.5 lambda= 1 lambda= 2 f() Densiy Funcion lambda= 0.5 lambda= 1 lambda= 2 Hazard Funcion lambda= 0.5 lambda= 1 lambda= h() 0.5 Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 4/14

11 Exponenial Disribuion Properies 4-1

12 Parameer Esimaion in Exponenial Mo 4-2

13 Weilbull Disribuion Exension of exponenial disribuion: S() Survival Funcion f() Densiy Funcion Hazard Funcion f() lambda= 0.5, alpha= 2 lambda= 0.5, alpha= 1 lambda= 0.5, alpha= 2 lambda= 1, alpha= 2 lambda= 1, alpha= 1 lambda= 1, alpha= 2 lambda= 2, alpha= 2 lambda= 2, alpha= 1 lambda= 2, alpha= 2 Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 5/

14 Weilbull Disribuion Properies 5-1

15 Parameer Esimaion in Weilbull Model 5-2

16 Lognormal Disribuion Exponenial of normal disribuion: S() Survival Funcion f() Densiy Funcion Hazard Funcion f() mu= 1, sigma= 0.1 mu= 1, sigma= 1 mu= 1, sigma= 2 mu= 0, sigma= 0.1 mu= 0, sigma= 1 mu= 0, sigma= 2 mu= 1, sigma= 0.1 mu= 1, sigma= 1 mu= 1, sigma= Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 6/

17 Lognormal Disribuion Properies 6-1

18 Parameer Esimaion in Lognormal Mod 6-2

19 Gamma Disribuion Survival Funcion Densiy Funcion S() f() Hazard Funcion f() scale= 0.5, shape= 0.5 scale= 0.5, shape= 1 scale= 0.5, shape= 2 scale= 1, shape= 0.5 scale= 1, shape= 1 scale= 1, shape= 2 scale= 2, shape= 0.5 scale= 2, shape= 1 scale= 2, shape= Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 7/14 0

20 Gamma Disribuion Properies 7-1

21 Parameer Esimaion in Gamma Model 7-2

22 Log-logisic Disribuion Survival Funcion Densiy Funcion 2.0 S() f() Hazard Funcion f() mu= 1, sigma= 0.5 mu= 1, sigma= 1 mu= 1, sigma= 2 mu= 0, sigma= 0.5 mu= 0, sigma= 1 mu= 0, sigma= 2 mu= 1, sigma= 0.5 mu= 1, sigma= 1 mu= 1, sigma= Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 8/14 0

23 Log-logisic Disribuion Properies 8-1

24 Parameer Esimaion in Logisic Model 8-2

25 Gomperz Disribuion Survival Funcion Densiy Funcion 2.0 S() f() Hazard Funcion f() hea= 0.5, alpha= 0.5 hea= 0.5, alpha= 1 hea= 0.5, alpha= 2 hea= 1, alpha= 0.5 hea= 1, alpha= 1 hea= 1, alpha= 2 hea= 2, alpha= 0.5 hea= 2, alpha= 1 hea= 2, alpha= Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 9/14 0

26 Parameric Regression Models wih Covariae Denoe X as he survival ime of ineres, Z = (Z 1,, Z p ) as he vecor of covariaes. Consider a linear model for modeling Y = log(x), namely, Y = µ + γ T Z + σw, where γ T = (γ 1,, γ p ) is a vecor of regression coefficiens and W is he error disribuion. Common choices for W : W N(0, 1) X logn(µ + γ T Z, σ 2 ); W exreme value disribuion X Weibull. W sandard logisic X log-logisic. Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 10/14

27 Acceleraed Failure-Time The above regression models are special cases of acceleraed failure-ime model (AFT). AFT model saes ha he survival funcion of an individual wih covariae Z could be wrien as he survival funcion of exp(µ + σw) wih scaled age xexp( γ T Z). Mahemaically we have S(x Z) = S 0 (xexp( γ T Z)), where S 0 (x) is he survival funcion of he random variable exp(µ + σw) and referred as he baseline survival funcion. When γ is negaive, he ime is acceleraed by a consan. The AFT propery is independen of error disribuion. A general AFT model relax he error disribuion assumpion and belongs semiparameric models. (The (γ) is a p-dim parameer and he disribuion of W is an infinie dimensional parameer.) Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 11/14

28 Proporional Hazards Model Anoher common model for survival ime assumes h(x Z) = h 0 (x) exp(β T Z), where h 0 (x) is called he baseline hazard funcion. This ype of model has consan hazard raios over ime. If he form of h 0 (x) is known, hen i is corresponding o a parameric model. For example h 0 (x) = αλ α 1, he proporional hazards model is same as Weibull model. Noe ha β = γ/σ. In fac, Weibull disribuion is he only disribuion saisfies boh proporional Hazards assumpion and AFT assumpion. If he form of h 0 (x) is unknown and need o be esimaed, hen i is a semiparameric model. This model was firs proposed by Cox (1972) and is one of he mos popular survival models currenly. Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 12/14

29 Proporional Odds Model Define odds() = S() assumes: 1 S(). The proporional odds model odds( Z) = odds( Z = 0) exp( β T Z), and odds( Z = 0) is called he baseline odds funcion. This ype of model has consan odds raios over ime. Similar as he proporional hazard model, if he form of odds 0 () = odds( Z = 0) is unknown, his is a semiparameric model. Log-logisic disribuion is he only disribuion saisfies boh proporional odds assumpion and AFT assumpion. The parameers of log-logisic disribuion are λ = exp(µ/σ), α = 1/σ, β i = γ i /σ. Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 13/14

30 Model Comparison Using Akaikie Informaion To compare nesed parameric models, we could use Wald or Likelihood raio ess. In general, o compare wo parameric models, one may use AIC defined as, AIC = 2 log (Likelihood) + 2p, where p is he oal number of parameers in he model. For example, p = 1 for he exponenial model, p = 2 for he Weibull model and p = 3 for he generalized gamma model. The smaller AIC, he beer he model is. Example 12.1 on P407. Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall p. 14/14