1. Assessing Survival: Cox Proportional Hazards Model
Statistics for Health Research
Assessing Survival:
Cox Proportional Hazards Model
Peter T. Donnan
Professor of Epidemiology and Biostatistics

2. Objectives of Workshop
Understand the general form of Cox PH model
Understand the need for adjusted Hazard Ratios (HR)
Implement the Cox model in SPSS
Understand and interpret the output from SPSS

3. Modelling: Detecting signal from background noise

4. Survival Regression Models
Expressed in terms of the hazard function formally defined as:
The instantaneous risk of event (mortality) in next time interval t, conditional on having survived to start of the interval t

5. What is hazard?
Hazard rate is an instantaneous rate of events as a function of
time
Survival is one measure of outcome but cancer studies also use hazards. Hazards intuitively sounds like the inverse of survival which is true but it is not quite as simple as that as they are related through the exponential function
The hazard is an instantaneous rate of events.
The hazard ratio (HR) (see later) is the ratio of hazards in two groups, often trial arms and is a measure of the efficacy in a cancer trial. A HR = 1 indicates no difference between the two arms, a HR > 1 indicates a greater hazard for the numerator arm and a HR < 1 indicates a lower hazard. Often in cancer trials of a new drug the trialists would be seeking a HR < 1 indicating a survival benefit by reducing deaths relative to the comparator.

6. Plot of hazard Note that the hazard changes over time denoted by h(t)
The hazard rate is a function of time and tends to increase over time.
Think of the risk of death from birth to old age which is high just after birth, declines through childhood and increases from middle-age.
time
Birth
Old age

7. Survival Regression Models
The Cox model expresses the relationship between the hazard and a set of variables or covariates
These could be arm of trial, age, gender, social deprivation, Dukes stage, co-morbidity, etc….

8. How is the relationship formulated?
Simplest equation is:
Age in years
Hazard
h is the hazard
K is a constant e.g. 0.3 per Person-year

9. How is the relationship formulated?
Next Simplest is linear equation:
h is the outcome; a is the intercept;
β is the slope related to x the explanatory variable and;
e is the error term or ‘noise’

10. Linear model of hazard
Age in years
Hazard

11. Cox Proportional Hazards Model (1972)
h0 is the baseline hazard;
r ( β, x) function reflects how the hazard function changes (β) according to differences in subjects’ characteristics (x)

12. Exponential model of hazard
Age in years

13. What is Hazard Ratio?
Hazard Ratio (HR) is ratio of hazards in two groups
e.g. men vs women, new drug vs. BSC
N.B. It is the improvement in one group over the other in terms of rate at which events will occur from a particular time point to another time point
Note that although the hazard is changing over time the ratio of the hazard in two groups is often assumed to be constant for most of the models we use in cancer research. This is the property of proportional hazards.

14. What is Hazard Ratio?
Hazard Ratio (HR) is ratio of hazards in two groups and remains constant over time (n.b. survival curve widens)
Survival
Constant ratio of hazards means the survival curves gradually widen. Hazard and survival are related through the log/ exponential functions. The HR is a summary of the difference in effect across the whole time course but at any time point the HR is the same if we have proportional hazards.
Time

15. Interpretation of HR comparing two groups
HR = 1 ; Do NOT reject null hypothesis (i.e. no difference)
HR < 1; Reduction in Hazard relative to comparator (e.g. HR = is 40% reduction)
HR > 1; Increase in Hazard relative to comparator (e.g. HR = 1.7 is 70% increase)
A HR = 1 indicates no difference between the two groups. HR < 1 is generally what we aim for in cancer trials as mainly seeking to reduce events such as death. A HR > 1 indicates a factor that increases the hazard in relation to the comparator.

16. Cox Proportional Hazards Model: Hazard Ratio
Consider hazard ratio for men vs. women, then -

17. Cox Proportional Hazards Model: Hazard Ratio
If coding for gender is x=1 (men) and x=0 (women) then:
where β is the regression coefficient for gender

18. Hazard ratios in SPSS
SPSS gives hazard ratios for a binary factor coded (0,1) automatically from exponentiation of regression coefficients (95% CI are also given as an option)
Note that the HR is labelled as EXP(B) in the output

19. Fitting Gender in Cox Model in SPSS

20. Output from Cox Model in SPSS
p-value
Standard error
Degrees of freedom
Variable in model
HR for men vs. women
Regression Coefficient
Test Statistic
( β/se(β) )2

21. Logrank Test: Null Hypothesis
The Null hypothesis for the logrank test:
Hazard Rate group A =
Hazard Rate for group B
= HR = OA / EA = 1
OB / EB

22. Wald Test: Null Hypothesis
The Null hypothesis for the Wald test:
Hazard Ratio = 1
Equivalent to regression coefficient β=0
Note that if the 95% CI for the HR includes 1 then the null hypothesis cannot be rejected

23. Hazard ratios for categorical factors in SPSS
Enter factor as before
Click on ‘categorical’ and choose the reference category (usually first or last)
E.g. Duke’s staging may choose Stage A as the reference category
HRs are now given in output for survival in each category relative to Stage A
Hence there will be n-1 HRs for n categories

24. Fitting a categorical variable: Duke’s Staging
Reference category
B vs. A
C vs. A
D vs. A
UK vs. A

25. One Solution to Confounding
Use multiple Cox regression with both predictor and confounder as explanatory variables i.e fit:
x1 is Duke’s Stage and x2 is Age

26. Fitting a multiple regression: Duke’s Staging and Age
Age adjusted for Duke’s Stage

27. Interpretation of the Hazard Ratio
For a continuous variable such as age, HR represents the incremental increase in hazard per unit increase in age i.e HR=1.024, increase 2.4% for a one year increase in age
For a categorical variable the HR represents the incremental increase in hazard in one category relative to the reference category i.e. HR = 6.66 for Stage D compared with A represents a 6.7 fold increase in hazard

28. First steps in modelling
What hypotheses are you testing?
If main ‘exposure’ variable, enter first and assess confounders one at a time
Assess each variable on statistical significance and clinical importance.
It is acceptable to have an ‘important’ variable without statistical significance

29. Summary
The Cox Proportional Hazards model is the most used analytical tool in survival research
It is easily fitted in SPSS
Model assessment requires some thought
Next step is to consider how to select multiple factors for the ‘best’ model

30. Check assumption of proportional hazards (PH)
Proportional hazards assumes that the ratio of hazard in one group to another remains the same throughout the follow-up period
For example, that the HR for men vs. women is constant over time
Simplest method is to check for parallel lines in the Log (-Log) plot of survival

31. Check assumption of proportional hazards for each factor
Check assumption of proportional hazards for each factor. Log minus log plot of survival should give parallel lines if PH holds
Hint: Within Cox model select factor as CATEGORICAL and in PLOTS select log minus log function for separate lines of factor

32. Check assumption of proportional hazards for each factor
Check assumption of proportional hazards for each factor. Log minus log plot of survival should give parallel lines if PH holds
Hint: Within Cox model select factor as CATEGORICAL and in PLOTS select log minus log function for separate lines of factor

33. Proportional hazards holds for Duke’s Staging
Categorical Variable Codings(b)
Frequency (1) (2) (3) (4)
dukes(a) 0=A
1=B
2=C
3=D
9=UK
a Indicator Parameter Coding
b Category variable: dukes (Dukes Staging)

34. Proportional hazards holds for Duke’s Staging

35. Summary
Selection of factors for Multiple Cox regression models requires some judgement
Automatic procedures are available but treat results with caution
They are easily fitted in SPSS
Check proportional hazards assumption
Parsimonious models are better

36. Practical
Read in Colorectal.sav and try to fit a multiple proportional hazards model
Check proportional hazards assumption