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 0.0
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. 8/14 Log-logistic Distribution Properties
8-1 Parameter Estimation in Logistic Model
8-2 Gompertz 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
10 1.5 theta= 0.5 , alpha= 0.5 theta= 0.5 , alpha= 1 8 theta= 0.5 , alpha= 2 theta= 1 , alpha= 0.5 1.0 theta= 1 , alpha= 1 6 theta= 1 , alpha= 2 0
f(t) theta= 2 , alpha= 0.5 4 theta= 2 , alpha= 1 0.5 theta= 2 , alpha= 2
2
0 0.0
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. 9/14 Parametric Regression Models with Covariates
Parametric Models Denote as the survival time of interest, Z as ● Outlines X =(Z1, ··· ,Zp) ● Exponential Distribution the vector of covariates. Consider a linear model for modeling ● Weilbull Distribution ● Lognormal Distribution Y = log(X), namely, ● Gamma Distribution ● Log-logistic Distribution T ● Gompertz Distribution Y = µ + γ Z + σW, ● Parametric Regression Models with Covariates ● Accelerated Failure-Time T (AFT) Model where γ =(γ1, ··· , γp) is a vector of regression coefficients ● Proportional Hazards Model ● Proportional Odds Model and W is the error distribution. ● Model Comparison Using Akaikie Information Criterion Common choices for W : (AIC) ■ W ∼ N(0, 1) ⇒ X ∼ logN(µ + γT Z, σ2); ■ W ∼ extreme value distribution ⇒ X ∼ Weibull. ■ W ∼ standard logistic ⇒ X ∼ log-logistic.
Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 10/14 Accelerated Failure-Time (AFT) Model
Parametric Models The above regression models are special cases of accelerated ● Outlines ● Exponential Distribution failure-time model (AFT). AFT model states that the survival ● Weilbull Distribution ● Lognormal Distribution function of an individual with covariate Z could be written as ● Gamma Distribution ● Log-logistic Distribution the survival function of exp(µ + σW ) with scaled age ● Gompertz Distribution T ● Parametric Regression x exp(−γ Z). Mathematically we have Models with Covariates ● Accelerated Failure-Time (AFT) Model Z T Z ● Proportional Hazards Model S(x| )= S0(x exp(−γ )), ● Proportional Odds Model ● Model Comparison Using Akaikie Information Criterion where S0(x) is the survival function of the random variable (AIC) exp(µ + σW ) and referred as the baseline survival function. When γ is negative, the time is accelerated by a constant. The AFT property is independent of error distribution. A general AFT model relax the error distribution assumption and belongs semiparametric models. (The (γ) is a p-dim parameter and the distribution of W is an infinite dimensional parameter.)
Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 11/14 Proportional Hazards Model
Parametric Models Another common model for survival time assumes ● Outlines ● Exponential Distribution ● Weilbull Distribution Z T Z ● Lognormal Distribution h(x| )= h0(x)exp(β ), ● Gamma Distribution ● Log-logistic Distribution where is called the baseline hazard function. This type of ● Gompertz Distribution h0(x) ● Parametric Regression model has constant hazard ratios over time. Models with Covariates ● Accelerated Failure-Time (AFT) Model ● If the form of h0(x) is known, then it is corresponding to a Proportional Hazards Model − ● Proportional Odds Model parametric model. For example α 1, the ● Model Comparison Using h0(x)= αλt Akaikie Information Criterion proportional hazards model is same as Weibull model. Note (AIC) that β = −γ/σ. In fact, Weibull distribution is the only distribution satisfies both proportional Hazards assumption and AFT assumption.
If the form of h0(x) is unknown and need to be estimated, then it is a semiparametric model. This model was first proposed by Cox (1972) and is one of the most popular survival models currently.
Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 12/14 Proportional Odds Model
Parametric Models S(t) ● Outlines Define odds(t)= 1−S(t) . The proportional odds model ● Exponential Distribution ● Weilbull Distribution assumes: ● Lognormal Distribution ● Gamma Distribution Z Z T Z ● Log-logistic Distribution odds(t| )= odds(t| = 0)exp(−β ), ● Gompertz Distribution ● Parametric Regression Models with Covariates and odds(t|Z = 0) is called the baseline odds function. This ● Accelerated Failure-Time (AFT) Model type of model has constant odds ratios over time. ● Proportional Hazards Model ● Proportional Odds Model ● Model Comparison Using Similar as the proportional hazard model, if the form of Akaikie Information Criterion (AIC) odds0(t)= odds(t|Z = 0) is unknown, this is a semiparametric model. Log-logistic distribution is the only distribution satisfies both proportional odds assumption and AFT assumption. The parameters of log-logistic distribution are
λ = exp(µ/σ), α = 1/σ, βi = −γi/σ.
Jimin Ding, October 4, 2011 Survival Analysis, Fall 2011 - p. 13/14 Model Comparison Using Akaikie Information
Parametric Models To compare nested parametric models, we could use Wald or ● Outlines ● Exponential Distribution Likelihood ratio tests. ● Weilbull Distribution ● Lognormal Distribution ● Gamma Distribution In general, to compare two parametric models, one may use ● Log-logistic Distribution ● Gompertz Distribution AIC defined as, ● Parametric Regression Models with Covariates ● Accelerated Failure-Time AIC = −2 log (Likelihood) + 2p, (AFT) Model ● Proportional Hazards Model ● Proportional Odds Model where p is the total number of parameters in the model. For ● Model Comparison Using Akaikie Information Criterion example, p = 1 for the exponential model, p = 2 for the Weibull (AIC) model and p = 3 for the generalized gamma model. The smaller AIC, the better the model is. Example 12.1 on P407.
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