Week 4 Lecture Slides (1)

# 6383 16931 1 0213770 0068315 0068965 01386 0068315

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Unformatted text preview: o test the hypothesis that there is no diﬀerence in survival between patients with diﬀerent stages of disease, adjusting for the age of the patient. Recall that that ﬁtting the full model with all four covariates, we ﬁnd MLE: b = (0.0189, 0.1386, 0.6383, 1.6931). Assume that we have found the observed information matrix 5088.5378 −22.8634 −14.0650 25.6149 −22.8634 5.9978 −2.3913 −1.4565 I (b ) = −14.0650 −2.3913 10.9917 −3.3123 25.6149 −1.4565 −3.3123 7.4979 Test the null hypothesis H0 : β2 = β3 = β4 = 0 using the Wald test. 31/45 Actuarial Statistics – Week 4: The Cox Regression Model Hypothesis tests on the β ’s Solution (I (b ))−1 0.000203 0.000802 = 0.000314 −0.000399 0.000802 0.213770 0.068315 0.068965 0.000314 −0.000399 0.068315 0.068965 0.126797 0.068212 0.068212 0.178264 The Wald statistic is (0.1386, 0.6383, 1.6931) −1 0.213770 0.068315 0.068965 0.1386 · 0.068315 0.126797 0.068212 · 0.6383 0.068965 0.068212 0.178264 1.6931 = 0.7586652 32/45 The p -value is Pr(χ2 ≥ 0.7586652) = 0.859(> 0.05). 3 There is no statistical evidence to reject H0 . Actuarial Statistics – Week 4: The Cox Regression Model Hypothesis tests on the β ’s Interaction Sometimes, the eﬀects of some covariates depend on the presence or absence of each other. In this case, there are interactions between these covariates. To test for interaction we need to test the eﬀect of the combination of the covariates. For example, we want to test for interaction between smoking status and high blood pressure. If Z2 represents the smoking status and Z3 represents blood pressure. We need to test the eﬀect of an extra covariate Z3 = Z2 × Z3 , which represents the combination of Z2 and Z3 . Interaction is not required study in this subject. 33/45 Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the full survival function 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 33/45 Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the full survival function Estimation of the survival function To estimate the survival function S (t ; Z ) with covariate vector Z based on a Cox model λ(t ; Z ) = λo (t ) exp{β Z T }) we need not only to ﬁt a proportional hazards model (Cox model) to the data, and obtain the partial maximum likelihood estimates of the parameters β , but also to estimate the baseline cumulative hazard rate t Λo (t ) = 0 λo (s )ds . Note that S (t ) = e − 34/45 t 0 T λ0 (s )e β Z ds = S0 (t )e where S0 (t ) is the baseline survival function. βZ T , Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the full survival function Estimate of the baseline cumulative hazard rate Λ0 (t ) Breslow’s estimator of the baseline cumulative hazard rate Λ0 (t ) is ˆ Λ0 (t ) = tj ≤t dj , T i ∈R (tj ) exp{bZi } where b is the partial maximum l...
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