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13_AS_3_lec_a - Actuarial Statistics Module 3...

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Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model Actuarial Statistics Benjamin Avanzi c University of New South Wales (2013) School of Risk and Actuarial Studies [email protected] Module 3: Semi-parametric methods: Cox Regression Model 1/45
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Actuarial Statistics – Module 3: Semi-parametric methods: 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 2/45
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Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model Introduction 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 2/45
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Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model Introduction Why a regression model? Central question: How can we model separately different factors that are likely to impact the observed event, so that we can isolate their individual effects? Heterogeneity: lives with very different characteristics (eg males and females, smokers and non-smokers) have different level of mortality. In other words, different factors (covariates) may have different effects on the risk. One method is to construct a model including the effects of the covariates on survival directly: a regression model. The most widely used regression model is the proportional hazards model (the Cox model) 3/45
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Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model Introduction Covariates In many survival analysis problems, covariates are of the following types: Demographic / Societal (eg age, gender, education) Behavioral (eg smoking, physical activity level, alcohol) Physiological (eg blood pressure, heart rate) Covariates can be continuous measurements (weight) discrete measurements (age last birthday) indicators (1 for smoking, 0 for non-smoking) qualitative indicators (5 severe disease to 0 no symptoms) see K&M for a discussion of how to ‘code’ variables/factors 4/45
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Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model Introduction Notation For the i th life we will denote the covariates ( risk factors ) by a 1 × p vector Z i = ( z i 1 , z i 2 , . . . , z ip ) Example: Consider the covariate vector Z i = (sex, age, weight, symptoms) . If the 3rd life is a 68 year old male, weighing 74kg, with mild symptoms of the condition under investigation (graded as 1 on a scale of 0 to 5), then we have Z 3 = (0 , 68 , 74 , 1) . 5/45
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Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model Main assumptions 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 5/45
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