Week 4 Lecture Slides (1)

If j 1 actuarial statistics week 4 the cox regression

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Unformatted text preview: ikelihood estimate of β . (for a justification, see K&M, section 8.8, in conjunction with Technical Note 2 on page 258) Finally, since ˆ ˆ S0 (t ) = exp{−Λ0 (t )} we have 35/45 T ˆ ˆ S (t ) = S0 (t )exp(bZ ) . Actuarial Statistics – Week 4: The Cox Regression Model Diagnostics for the Cox regression model 1 Introduction 2 Main assumptions 3 On the proportionality of hazard rates 4 Estimation of the regression parameters β 5 Hypothesis tests on the β ’s 6 Estimation of the full survival function 7 Diagnostics for the Cox regression model 35/45 Actuarial Statistics – Week 4: The Cox Regression Model Diagnostics for the Cox regression model Diagnostics for the Cox model We will discuss two types of diagnostics: 1 goodness-of-fit via Cox-Snell residuals 2 appropriateness of the proportionality assumption via graphica techniques Notation: Cj : right censoring time δj = 0 means censoring, that is, Xj = 36/45 Cj Tj if δj = 0 . if δj = 1 Actuarial Statistics – Week 4: The Cox Regression Model Diagnostics for the Cox regression model Cox-Snell residuals The Cox-Snell residuals are defined as p ˆ ˆ ej = − log(S (Xj ; zj )) = Λ0 (Xj ) exp( T bk zjk ) k =1 where b is the MLE of β , and ˆ Λ0 (t ) is Breslow’s estimator of the cumulative baseline hazard rate. If the model is correct and the b ’s (MLEs) are close to the true values of β , then the Cox-Snell residuals ej ’s behave as a censored sample from a unit exponential distribution. 37/45 Actuarial Statistics – Week 4: The Cox Regression Model Diagnostics for the Cox regression model Proof 1 − S (Tj ; Zj ) ∼ Uniform (0,1) So S (Tj ; Zj ) ∼ Uniform (0,1) Define Ej = − log(S (Tj ; Zj )). Then Ej ∼ Exponential(1) Define C j = − log(S (Cj ; Zj )). Notice that − log(S (Xj ; Zj )) = − log(S (Tj ; Zj )) if Tj ≤ Cj i .e . Ej ≤ C j − log(S (Cj ; Zj )) if Tj > Cj i .e . Ej > C j Consider Ej as a new lifetime r.v. and C j the corresponding censoring time. Then − log(S (Xj ; Zj )) is just a right censored, exponentially distributed version of Ej with censoring time C j . 38/45 Actuarial Statistics – Week 4: The Cox Regression Model Diagnostics for the Cox regression model To check the goodness of fit of a Cox regression model, we need to check wether the Cox-Snell residuals: p ˆ ˆ ej = − log(S (Xj ; zj )) = Λ0 (Xj ) exp( T bk zjk ) k =1 behave as samples from a unit exponential random variable. To check whether ej behaves as a sample from a unit exponential, compute the Nelson-Aalen estimator of the cumulative hazards ˆ rate of Ej s : ΛE (ej ) ˆ A plot of ΛE (ej ) versus ej should be roughly straight line through the origin with a slope of 1 39/45 Actuarial Statistics – Week 4: The Cox Regression Model Diagnostics for the Cox regression model Example 2.5 2.0 1.5 1.0 0.5 0.0 estimated cumulative hazard function of the residuals 3.0 For the Cox regression model, a Cox-Snell residual plot is given as below. Comment on the overall fit of the Cox model. 0.0 40/45 0.5 1.0 1.5 residuals 2.0 2.5 3.0 Actuarial Statistics – Week 4: The Cox Regression Model Diagnostics for the Cox regression model Assessment of the proportionality assumption To check for proportional haza...
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This document was uploaded on 04/03/2014.

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