Cox Regression
Cox Regression Models
Questions with Answers Worked Example
An investigation is carried out into popularity of new cars being bought in the showroom of a Mercedes dealer. Data recorded for each car included colour, engine size and car type. A Cox proportional hazards model was fitted to the data and the results are given below:
Write down the Cox hazard function according to this model. With regards to the model you have written down above state the following: • To which class of car does the baseline hazard refer • Which car has the lowest hazard rate • Which car has the highest hazard rate
1
Exercises
1. Suppose the lifetime, the age and sex of a number of lives are recorded and the Cox model is fitted by maximising the partial likelihood. The fitted parameters are βf = −0.5 where f denotes a female life and βa = 0.01 denoted the is the age effect in years. Taking a male aged 40 as the reference hazard calculate the ratio of the hazards for (a) a male aged 20, (b) a male aged 60, (c) a female aged 20, (d) a female aged 60. 2. A study has been carried out to investigate the factors affecting the survival times of mice suffering from leukaemia. A sample of 42 mice was divided into two equal groups - a treatment and a control group. The data available in respect of each mouse are: • survival time in weeks • group (0=treatment, 1=control) • sex (0=male, 1=female) • log WBC (WBC=white blood count) The results of two particular fitted Cox models are given: Variable Group Sex Log WBC Group x Sex Group x Log WBC Model 1 Coefficient 0.405 -1.070 1.610 2.013 Standard Error 0.561 0.725 0.332 0.883 Model 2 Coefficient 0.894 -1.012 1.693 1.952 -0.151 Log-likelihood -69.51 Standard Error 1.815 0.752 0.441 0.907 0.531
Log-likelihood -69.55
(a) Write down expressions for the hazard rate using models 1 and 2. (b) Use model 1 to calculate the ratio of hazards for the following two mice (1) male, in treatment group logWBC=2.32 and (2) female, in control group, logWBC=2.15
Questions with Answers Worked Example
An investigation is carried out into popularity of new cars being bought in the showroom of a Mercedes dealer. Data recorded for each car included colour, engine size and car type. A Cox proportional hazards model was fitted to the data and the results are given below:
Write down the Cox hazard function according to this model. With regards to the model you have written down above state the following: • To which class of car does the baseline hazard refer • Which car has the lowest hazard rate • Which car has the highest hazard rate
1
Exercises
1. Suppose the lifetime, the age and sex of a number of lives are recorded and the Cox model is fitted by maximising the partial likelihood. The fitted parameters are βf = −0.5 where f denotes a female life and βa = 0.01 denoted the is the age effect in years. Taking a male aged 40 as the reference hazard calculate the ratio of the hazards for (a) a male aged 20, (b) a male aged 60, (c) a female aged 20, (d) a female aged 60. 2. A study has been carried out to investigate the factors affecting the survival times of mice suffering from leukaemia. A sample of 42 mice was divided into two equal groups - a treatment and a control group. The data available in respect of each mouse are: • survival time in weeks • group (0=treatment, 1=control) • sex (0=male, 1=female) • log WBC (WBC=white blood count) The results of two particular fitted Cox models are given: Variable Group Sex Log WBC Group x Sex Group x Log WBC Model 1 Coefficient 0.405 -1.070 1.610 2.013 Standard Error 0.561 0.725 0.332 0.883 Model 2 Coefficient 0.894 -1.012 1.693 1.952 -0.151 Log-likelihood -69.51 Standard Error 1.815 0.752 0.441 0.907 0.531
Log-likelihood -69.55
(a) Write down expressions for the hazard rate using models 1 and 2. (b) Use model 1 to calculate the ratio of hazards for the following two mice (1) male, in treatment group logWBC=2.32 and (2) female, in control group, logWBC=2.15