1. Lecture 12: Cox Proportional Hazards Model
Introduction

2. Cox Proportional Hazards Model
Names
Cox regression
Semi-parametric proportional hazards
Proportional hazards model
Multiplicative hazards model
When?
1972
Why?
Allows for adjustment of covariates (continuous and categorical) in a survival setting
Allows prediction of survival based on a set of covariates
Analogous to linear and logistic regression in many ways

3. Cox PHM Notation (K & M) Data on n individuals:
Tj : time on study for individual j
dj : event time indicator for individual j
Zj : vector of covariates for individual j
More complicated: Zj(t)
Covariates are time dependent
May change with time/age

4. Basic Model

5. Comments on the Basic Model
h0(t):
Arbitrary baseline hazard
Notice that it varies by t b:
Regression coefficient vector
Interpretation is a log hazard ratio
Semi-parametric form
Non-parametric baseline hazard
Parametric form assumed for covariate effects only

6. Linear Model Formulation
Usual formulation
Coding of covariates similar to linear and logistic (and other generalized linear models)

7. Refresher of Coding Covariates
Should be nothing new
Two kinds of “independent” variables
Quantitative
Qualitative
Quantitative are continuous
Need to determine scale
Units
Transformations?
Qualitative are generally categorical
Ordered
Nominal
Coding affects interpretation

8. Why Proportional Hazard ratio
Does not depend on t (i.e. it is constant over time)
But, it is proportional (constant multiplicative factor)
Also referred to (sometimes) as the relative risk

9. Simple Example One covariate: Hazard ratio:
Interpretation: exp{btrt}is the risk of having the event in the new treatment group vs. the standard treatment group
Interpretation: At any point in time, the risk of the event in the new treatment group is exp{btrt} time the risk in the standard treatment group

10. Fig 3. Cantù M G et al. JCO 2002;20:1232-1237
©2002 by American Society of Clinical Oncology

11. Hazard Ratios Hazard function = P(die at time t| survived to time t)
Assumption: “proportional hazards”
The risk does not depend on time
That is “the risk is constant over time”
But that is still vague…
Hypothetical example: Assume the hazard ratio is 0.5
Patient in new therapy group are at half the risk of death as those in the standard treatment, at any given point in time
Hazard function = P(die at time t| survived to time t)

12. Hazard Ratios Hazard ratio = Makes assumption that this ratio is
constant over time

13. Interpretation Again
For any fixed point in time, individuals in the new treatment group have half the risk of death as the standard treatment group.

14. A Slightly More Complicated Example
What if we had 2 binary covariates?
How is the hazard ratio estimated in this case?
What about the proportional hazards assumption?

15. A Slightly More Complicated Example
Consider a model that includes

16. A Slightly More Complicated Example
Our model looks like:

17. A Slightly More Complicated Example
From this we can estimate 4 possible hazard rates

18. A Slightly More Complicated Example
And if we “compare” the different hazards by taking the ratio we get

19. A Slightly More Complicated Example
But what does this mean in terms of proportional hazards?

20. Hazard ratio is not always valid…

21. Let’s Think About the Likelihood…

22. Let’s Think About the Likelihood…

23. Let’s Think About the Likelihood…

24. Let’s Think About the Likelihood…

25. Partial Likelihood The partial likelihood is defined as Where
j = 1, 2, …, n
No ties
t1 < t2 < … < tD
Z(i)k is the kth covariate associated with the individual whose failure time is ti
R(ti) =Yi is the risk set at time ti

26. Things to Notice
Numerator only depends on information from a patient who experiences the event
The denominator incorporates information across all patients in the risk set

27. Constructing the Likelihood
Without Censoring…
Say we have the following data on n = 5 subjects
Observed times and even indicators:
ti = 11, 12, 14, 16, 21
di = 1, 1, 1, 1, 1
And a single binary covariate
zi = 0, 1, 0, 1, 1

28. Constructing the Likelihood
First let’s construct our risk set for each unique time

29. Constructing the Likelihood
Now, we can construct our likelihood…

30. Constructing the Likelihood
Now, we can construct our likelihood…

31. Constructing the Likelihood
But what if we have censoring?
Consider the revised data:
Observed times and even indicators:
ti = 11, 12, 14, 16, 21
di = 1, 1, 0, 1, 0
And a single binary covariate
zi = 0, 1, 0, 1, 1

32. Constructing the Likelihood
Again let’s construct our risk set for each unique time

33. Constructing the Likelihood
And again we can construct our likelihood…

34. Estimation The log-likelihood
Maximize log-likelihood to solve for estimates of b

35. Estimation Maximize log-likelihood to solve for estimates of b
Score equations and information matrices are found using standard approaches
Solving for estimates can be done numerically (e.g. Newton-Raphson)

36. Tests of the Model Testing that bk = 0 for all k = 1, 2, …, p
Three main tests
Chi-square/ Wald test
Likelihood ratio test
Score test
All three have chi-square distribution with p degrees of freedom

37. Example: CGD
Study examining the impact of gamma interferon treatment on infection in people with chronic granulotomous disease
203 subject
Main variable of interest is treatment
Placebo
Gamma interferon
Other variables
Demographics (age, height, weight)
Steroid use
Pattern of inheritance
Treatment center …
Outcome: Time to first major infection

39. Cox PHM Approach
> data(cgd) > st<-Surv(cgd$time, cgd$infect) > reg1<-coxph(st~cgd$treat) > reg1 Call: coxph(formula = st ~ cgd$treat) coef exp(coef) se(coef) z p cgd$treatrIFN-g e-05 Likelihood ratio test=18.9 on 1 df, p=1.36e-05 n= 203, number of events= 76 > attributes(reg1) $names [1] "coefficients" "var" "loglik" "score" "iter" "linear.predictors" [7] "residuals" "means" "concordance" "method" "n" "nevent" [13] "terms" "assign" "wald.test" "y" "formula" "xlevels" "contrasts" "call" $class [1] "coxph"

40. Results
> summary(reg1) Call: coxph(formula = st ~ cgd$treat) n= 203, number of events= 76 coef exp(coef) se(coef) z Pr(>|z|) cgd$treatrIFN-g e-05 *** --- Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 exp(coef) exp(-coef) lower .95 upper .95 cgd$treatrIFN-g Likelihood ratio test= on 1 df, p=1.364e-05 Wald test = on 1 df, p=4.933e-05 Score (logrank) test = on 1 df, p=2.124e-05

41. Fitting More Covariates in R
> reg2<-coxph(st~treat+steroids+inherit+hos.cat+sex+age+weight, data=cgd) > reg2 Call: coxph(formula = st ~ treat + steroids + inherit + hos.cat + sex + age + weight, data = cgd) coef exp(coef) se(coef) z p treatrIFN-g e-05 steroids e-03 inheritautosomal e-02 hos.catUS:other e-01 hos.catEurope:Amsterdam e-01 hos.catEurope:other e-01 sexfemale e-01 age e-02 weight e-02 Likelihood ratio test=41.2 on 9 df, p=4.65e-06 n= 203, number of events= 76

42. Next Time
More on constructing our hypothesis tests next time…