Survival Analysis Math 434 – Fall 2011 Part III: Chap. 2.5,2.6 & 12 Jimin Ding Math Dept. www.math.wustl.edu/ jmding/math434/index.html Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 1/14 Parametric Models ● Outlines ● Exponential Distribution ● Weilbull Distribution ● Lognormal Distribution ● Gamma Distribution ● Log-logistic Distribution ● Gompertz Distribution ● Parametric Regression Parametric Models Models with Covariates ● Accelerated Failure-Time (AFT) Model ● Proportional Hazards Model ● Proportional Odds Model ● Model Comparison Using Akaikie Information Criterion (AIC) Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 2/14 Outlines Parametric Models ■ Exponential distribution ; ● Outlines exp(λ),λ> 0 ● Exponential Distribution ● Weilbull Distribution ● Lognormal Distribution ● Gamma Distribution ● Log-logistic Distribution ● Gompertz Distribution ● Parametric Regression Models with Covariates ● Accelerated Failure-Time (AFT) Model ● Proportional Hazards Model ● Proportional Odds Model ● Model Comparison Using Akaikie Information Criterion (AIC) Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 3/14 Outlines Parametric Models ■ Exponential distribution ; ● Outlines exp(λ),λ> 0 ● Exponential Distribution ■ ● Weilbull Distribution Weiblull distribution W eibull(λ, α),λ> 0; ● Lognormal Distribution ● Gamma Distribution ● Log-logistic Distribution ● Gompertz Distribution ● Parametric Regression Models with Covariates ● Accelerated Failure-Time (AFT) Model ● Proportional Hazards Model ● Proportional Odds Model ● Model Comparison Using Akaikie Information Criterion (AIC) Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 3/14 Outlines Parametric Models ■ Exponential distribution ; ● Outlines exp(λ),λ> 0 ● Exponential Distribution ■ ● Weilbull Distribution Weiblull distribution W eibull(λ, α),λ> 0; ● Lognormal Distribution ■ 2 ● Gamma Distribution Lognormal distribution logN(µ, σ ),σ> 0; ● Log-logistic Distribution ● Gompertz Distribution ● Parametric Regression Models with Covariates ● Accelerated Failure-Time (AFT) Model ● Proportional Hazards Model ● Proportional Odds Model ● Model Comparison Using Akaikie Information Criterion (AIC) Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 3/14 Outlines Parametric Models ■ Exponential distribution ; ● Outlines exp(λ),λ> 0 ● Exponential Distribution ■ ● Weilbull Distribution Weiblull distribution W eibull(λ, α),λ> 0; ● Lognormal Distribution ■ 2 ● Gamma Distribution Lognormal distribution logN(µ, σ ),σ> 0; ● Log-logistic Distribution ● Gompertz Distribution ■ Gamma distribution Gamma(λ, β),λ> 0; ● Parametric Regression Models with Covariates ● Accelerated Failure-Time (AFT) Model ● Proportional Hazards Model ● Proportional Odds Model ● Model Comparison Using Akaikie Information Criterion (AIC) Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 3/14 Outlines Parametric Models ■ Exponential distribution ; ● Outlines exp(λ),λ> 0 ● Exponential Distribution ■ ● Weilbull Distribution Weiblull distribution W eibull(λ, α),λ> 0; ● Lognormal Distribution ■ 2 ● Gamma Distribution Lognormal distribution logN(µ, σ ),σ> 0; ● Log-logistic Distribution ● Gompertz Distribution ■ Gamma distribution Gamma(λ, β),λ> 0; ● Parametric Regression Models with Covariates ■ ● Accelerated Failure-Time Log-logistic distribution loglogit(λ, α),λ> 0; (AFT) Model ● Proportional Hazards Model ● Proportional Odds Model ● Model Comparison Using Akaikie Information Criterion (AIC) Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 3/14 Outlines Parametric Models ■ Exponential distribution ; ● Outlines exp(λ),λ> 0 ● Exponential Distribution ■ ● Weilbull Distribution Weiblull distribution W eibull(λ, α),λ> 0; ● Lognormal Distribution ■ 2 ● Gamma Distribution Lognormal distribution logN(µ, σ ),σ> 0; ● Log-logistic Distribution ● Gompertz Distribution ■ Gamma distribution Gamma(λ, β),λ> 0; ● Parametric Regression Models with Covariates ■ ● Accelerated Failure-Time Log-logistic distribution loglogit(λ, α),λ> 0; (AFT) Model ● Proportional Hazards Model ■ * Gompertz distribution Gompertz(θ, α),α> 0. ● Proportional Odds Model ● Model Comparison Using Akaikie Information Criterion (AIC) Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 3/14 Outlines Parametric Models ■ Exponential distribution ; ● Outlines exp(λ),λ> 0 ● Exponential Distribution ■ ● Weilbull Distribution Weiblull distribution W eibull(λ, α),λ> 0; ● Lognormal Distribution ■ 2 ● Gamma Distribution Lognormal distribution logN(µ, σ ),σ> 0; ● Log-logistic Distribution ● Gompertz Distribution ■ Gamma distribution Gamma(λ, β),λ> 0; ● Parametric Regression Models with Covariates ■ ● Accelerated Failure-Time Log-logistic distribution loglogit(λ, α),λ> 0; (AFT) Model ● Proportional Hazards Model ■ * Gompertz distribution Gompertz(θ, α),α> 0. ● Proportional Odds Model ● Model Comparison Using Akaikie Information Criterion For each parametric model, we will discuss the distribution (AIC) properties and parameter estimation. Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 3/14 Exponential Distribution Parametric Models Survival ● Density Function Outlines Function ● Exponential Distribution ● Weilbull Distribution 1.0 2.0 lambda= 0.5 lambda= 0.5 ● Lognormal Distribution lambda= 1 lambda= 1 0.8 lambda= 2 lambda= 2 ● Gamma Distribution 1.5 ● Log-logistic Distribution 0.6 ● Gompertz Distribution 1.0 f(t) ● Parametric Regression S(t) 0.4 Models with Covariates ● Accelerated Failure-Time 0.5 (AFT) Model 0.2 ● Proportional Hazards Model ● Proportional Odds Model 0.0 0.0 ● Model Comparison Using 0 1 2 3 4 5 0 1 2 3 4 5 Akaikie Information Criterion (AIC) t t Hazard Function 2.5 lambda= 0.5 lambda= 1 2.0 lambda= 2 1.5 h(t) 1.0 0.5 0.0 0 1 2 3 4 5 t Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 4/14 Exponential Distribution Properties 4-1 Parameter Estimation in Exponential Model 4-2 Weilbull Distribution Parametric Models Extension of exponential distribution: ● Outlines ● Exponential Distribution ● Weilbull Distribution Survival Density ● Lognormal Distribution Function Function ● Gamma Distribution ● Log-logistic Distribution 1.0 2.0 ● Gompertz Distribution 0.8 ● Parametric Regression 1.5 Models with Covariates ● Accelerated Failure-Time 0.6 1.0 (AFT) Model f(t) S(t) ● Proportional Hazards Model 0.4 ● Proportional Odds Model 0.5 ● Model Comparison Using 0.2 Akaikie Information Criterion (AIC) 0.0 0.0 0 1 2 3 4 5 0 1 2 3 4 5 t t Hazard Function 2.0 1.5 lambda= 0.5 , alpha= 0.02 lambda= 0.5 , alpha= 1 lambda= 0.5 , alpha= 2 1.5 lambda= 1 , alpha= 0.02 1.0 lambda= 1 , alpha= 1 lambda= 1 , alpha= 2 1.0 0 f(t) lambda= 2 , alpha= 0.02 lambda= 2 , alpha= 1 0.5 lambda= 2 , alpha= 2 0.5 0.0 0.0 Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 5/14 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 t 0 Weilbull Distribution Properties 5-1 Parameter Estimation in Weilbull Model 5-2 Lognormal Distribution Parametric Models Exponential of normal distribution: ● Outlines ● Exponential Distribution ● Weilbull Distribution ● Lognormal Distribution Survival Function Density Function ● Gamma Distribution ● Log-logistic Distribution 1.0 2.0 ● Gompertz Distribution 0.8 ● Parametric Regression 1.5 Models with Covariates ● Accelerated Failure-Time 0.6 1.0 (AFT) Model f(t) S(t) ● Proportional Hazards Model 0.4 ● Proportional Odds Model 0.5 ● Model Comparison Using 0.2 Akaikie Information Criterion (AIC) 0.0 0.0 0 1 2 3 4 5 0 1 2 3 4 5 t t Hazard Function 5 1.5 mu= −1 , sigma= 0.1 mu= −1 , sigma= 1 4 mu= −1 , sigma= 2 mu= 0 , sigma= 0.1 1.0 mu= 0 , sigma= 1 3 mu= 0 , sigma= 2 0 f(t) mu= 1 , sigma= 0.1 2 mu= 1 , sigma= 1 0.5 mu= 1 , sigma= 2 1 0 0.0 Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 6/14 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 t 0 Lognormal Distribution Properties 6-1 Parameter Estimation in Lognormal Model 6-2 Gamma Distribution Parametric Models ● Outlines Survival Function Density Function ● Exponential Distribution ● Weilbull Distribution ● Lognormal Distribution ● Gamma Distribution ● Log-logistic Distribution ● Gompertz Distribution f(t) ● Parametric Regression S(t) Models with Covariates ● Accelerated Failure-Time (AFT) Model ● Proportional Hazards Model ● 0.0 0.5 1.0 1.5 2.0 Proportional Odds Model 0.0 0.2 0.4 0.6 0.8 1.0 ● Model Comparison Using 0 1 2 3 4 5 0 1 2 3 4 5 Akaikie Information Criterion (AIC) t t Hazard Function 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 0 f(t) scale= 2 , shape= 0.5 scale= 2 , shape= 1 scale= 2 , shape= 2 0 1 2 3 4 0.0 0.5 1.0 1.5 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 t 0 Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 7/14 Gamma Distribution Properties 7-1 Parameter Estimation in Gamma Model 7-2 Log-logistic Distribution Parametric Models ● Outlines Survival Function Density Function ● Exponential Distribution ● Weilbull Distribution 1.0 2.0 ● Lognormal Distribution 0.8 ● Gamma Distribution 1.5 ● Log-logistic Distribution 0.6 ● Gompertz Distribution 1.0 f(t) ● Parametric Regression S(t) 0.4 Models with Covariates ● Accelerated Failure-Time 0.5 (AFT) Model 0.2 ● Proportional Hazards Model ● Proportional Odds Model 0.0 0.0 ● Model Comparison Using 0 1 2 3 4 5 0 1 2 3 4 5 Akaikie Information Criterion (AIC) t t Hazard Function 4 1.5 mu= −1 , sigma= 0.5 mu= −1 , sigma= 1 mu= −1 , sigma= 2 3 mu= 0 , sigma= 0.5 1.0 mu= 0 , sigma= 1 mu= 0 , sigma= 2 2 0 f(t) mu= 1 , sigma= 0.5 mu= 1 , sigma= 1 0.5 mu= 1 , sigma= 2 1 0