Week 4 Lecture Slides (1)

# 1 p 0 gives maximum likelihood estimate the maximum

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Unformatted text preview: )T ) + exp (β (1, 68)T ) + exp (β (1, 49)T ) + exp (β (0, 86)T )] 21/45 Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the regression parameters β Properties of the maximum (partial) likelihood estimator The eﬃcient score function is deﬁned by u (β ) = ˆ Solving u β ∂ ln L (β ) ∂ ln L (β ) ,..., ∂β1 ∂βp ˆ = 0 gives maximum likelihood estimate β ˜ The maximum partial likelihood estimator β (of β ) is asymptotically unbiased asymptotically (multivariate) normally distributed with mean ˆ ˆ β and variance (matrix) equal to (I (β ))−1 , where I (β ) is the observed information matrix given by ˆ I (β ) = 22/45 − ∂ 2 ln L (β ) |ˆ ∂βi ∂βj β =β i ,j =1,··· ,p Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the regression parameters β Example I Given the partial likelihood: L= 1 2 3 23/45 Ce β , (e β + 2)(e β + 3) ˆ1 Show that the MLE of β is β = 2 log 6. Compute the asymptotic standard error of the estimator. Construct an approximate 95% conﬁdence interval for the parameter β . Solution: 1 log L = log C + β − log(e β + 2) − log(e β + 3) log eβ eβ Solving d d β L = 1 − e β +2 − e β +3 = 0 =⇒ β = 2 ˆ Noting that d log L | 1 < 0, β = 1 log 6. 2 dβ β = 2 log 6 2 1 2 log 6 Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the regression parameters β Example II 2 d 2 log L d β2 β 2e = − (e β +2)2 − 2 ˆ I (β ) = [− d dlog L ]β =β The ˆ β2 3e β (e β +3)2 ˜ asymptotic variance of β is ˆ ˆ [I (β )]−1 = [ 2e β ˆ (e β + 2)2 Asymptotic standard error 3 + ˆ (e β + 3)2 ˜ Var (β ) = √ −1 = 2.02062 2.02062 = 1.4215 ˜ β is asymptotically normally distributed. A 95% conﬁdence interval for β is ˆ β ±z1−0.05/2 24/45 ˆ 3e β 1 ˜ var (β ) = log 6±1.96×1.4215 = (−1.890, 3.682) 2 Actuarial Statistics – Week 4: The Cox Regression Model Hypothesis tests on the β ’s 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 24/45 Actuarial Statistics – Week 4: The Cox Regression Model Hypothesis tests on the β ’s Rationale Assume you want to test: H0 : β1 = β1,0 for some subset β1 ∈ β . Often we will test β1 = 0, that is, whether the associatied covariates have a signiﬁcant eﬀect. This is also used for model buidling: 1 start with the null model which includes no covariates and add possible covariates one at a time (forward selection); or 2 start with the full model which includes all covariates, and eliminate those of no signiﬁcant eﬀect (backward selection). We will focus on testing the signiﬁcance of the β ’s. Methods for more general local tests can be found in K&M (8.5). 25/45 Actuarial Statistics – Week 4: The Cox Regression Model Hypothesis tests on the β ’s Testing the signiﬁcance of some additional β ’s Assume you want to test whether it is worth adding a set of q β ’s to an existing set of p β ’s. To test the eﬀect of extra covariates (zp+1 , · · · , zp+q ), the nul...
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