Week 4 Lecture Slides (1)

1045 actuarial statistics week 4 the cox regression

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Unformatted text preview: erence due to covariates and NOT the baseline hazard - we can ignore λo (t ) and concentrate on the function g (Z ) only (β terms in Cox). 10/45 Actuarial Statistics – Week 4: The Cox Regression Model On the proportionality of hazard rates Example The following data for each patient have been recorded: 0 for females 1 if patient attended Hospital B Z1 = Z2 = 1 for males 0 otherwise Z3 = 1 0 if patient attended Hospital C otherwise T Suppose the force of mortality at time t is modelled by λ0 (t )e β Z ˆ and the parameter values have been estimated to be β1 = 0.031, ˆ2 = −0.025 and β3 = 0.011. ˆ β Compare the force of mortality for a female patient who attended Hospital A with that of: 1 2 11/45 a female patient who attended Hospital B a male patient who attended Hospital C Actuarial Statistics – Week 4: The Cox Regression Model On the proportionality of hazard rates Solution 1 The hazard rate at time t for a female who attended Hospital A is λfemale ,A (t ) = λ0 (t ) and the hazard rate at time t for a female who attended Hospital B is λfemale ,B (t ) = λ0 (t )e −0.025 The ratio is λfemale ,A (t ) = e 0.025 = 1.0253 λfemale ,B (t ) 12/45 2 So we estimate that the hazard rate for a female who attended Hospital A is 2.53% higher than that of a female who attended Hospital B. ratio: e −0.042 = 0.9589 Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the regression parameters β 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 12/45 Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the regression parameters β Partial likelihood estimation of β To estimate β = (β1 , β2 , . . . βp ), we will select the ones that maximize the partial likelihood under the following assumptions: non-informative censoring lives are independent Notation: Ordered times of observed death are t1 < t2 < · · · < tk dj : number of deaths occuring at time tj (1 ≤ j ≤ k ) We will label all the lives as 1, · · · , n. R (tj ) is the set of the label of all lives who are still under study at a time just prior to tj . (Note R (tj ) and nj are not identical) Use (j ) to denote the label of the life who dies at tj 13/45 Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the regression parameters β Two cases We will distinguish two cases: 1 assume that only one death occurs at each tj (dj = 1 for 1 ≤ j ≤ k ) 2 Relax that assumption. In practice there might be ties in the data, that is 1 2 some dj > 1; or some observations are censored at an observed lifetime. This can significantly complicate the partial likelihood as one needs to include the lives censored at time tj in the risk set R (tj ) and all permutations of simultaneous events. 14/45 Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the regression parameters β Case 1 - one death at each observed lifetime The partial likelihood for the death at tj is the first death of the lives Pr the individual (j ) dies at tj in R (tj ) occurs at tj and only one death at tj λ tj ; Z(j ) exp − tj 0 tj 0 λ(s , Zi )ds i =(j ) = λ (tj ; Zi ) exp − = i λ tj ; Z(j ) = λ (tj ; Zi ) R (tj ) tj 0 λ (s ; Zi ) ds i R (tj ) i R (tj ) 15/45 exp − λ(s , Z(j ) )ds λ0 (tj ) exp β Z(T) j λ0 (tj )...
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