Week 4 Lecture Slides (1)

p are the regression parameters zi 1 zi 2 zip

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Unformatted text preview: j zij j =1 where: λo (t ) is the baseline hazard β1 , β2 , . . . βp are the regression parameters zi 1 , zi 2 , . . . zip are the covariates for the i th subject Note: In this formulation only λo (t ) depends on time but is independent of the covariates p conversely, exp j =1 βj zij is independent of t but dependent on the covariates 6/45 Actuarial Statistics – Week 4: The Cox Regression Model Main assumptions Interpretation - sign of β If βj is positive, the hazard rate increases with the j th covariate, ie there is a positive correlation between hazard rate and the j th covariate If βj is negative, the hazard rate decreases with the j th covariate, ie there is a negative correlation between hazard rate and the j th covariate 7/45 Discussions: If obese individuals are more likely to suffer from major heart disease, what’s the sign of the regression covariate associated with the covariate representing weight? If individuals who drink a high volume of non-alcoholic liquids are less likely to suffer from liver disease, what’s the sigh sigh of the regression parameter associated with the covariate representing liquid intake? Actuarial Statistics – Week 4: The Cox Regression Model Main assumptions Interpretation - magnitude of β the sheer magnitude of the β does not say much (as this depends on how the covariates have been defined) so need of hypothesis test to check that β = 0 at a significant level if β is estimated with standard techniques then it is easy to check their level of significance (more in the model building section) 8/45 Actuarial Statistics – Week 4: The Cox Regression Model On the proportionality of hazard rates 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 8/45 Actuarial Statistics – Week 4: The Cox Regression Model On the proportionality of hazard rates Relative risk The ratio of hazard rates of two different lives x and y , p p βj zxj exp j =1 λ (t ; Zx ) = = exp βj {zxj − zyj } p λ (t ; Zy ) exp βz j =1 j yj j =1 is constant at all times, which explains the qualification of proportional hazards model. This ratio is also called the relative risk of an individual with risk factor Zx as compared to an individual with risk factor Zy . 9/45 Actuarial Statistics – Week 4: The Cox Regression Model On the proportionality of hazard rates Advantages of proportionality Under the Cox model, differences of hazard rates of different groups (different covariates) are accounted for via the exponential term (linear function inside the exponential), which leads to a simple expression for the relative risk The Cox model is not the only model with proportional hazards; one can generalise Cox to (see Exercise 3.2) λ(t ; Zi ) = λ0 (t )g (Zi ) where g (Z ) is any function of Z , but not t . What additional properties should g (Z ) have? If we are only interested in the di...
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This document was uploaded on 04/03/2014.

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