Title: Assessing Survival:
1Statistics for Health Research
Assessing Survival Cox Proportional Hazards
Model
Peter T. Donnan Professor of Epidemiology and
Biostatistics
2Objectives 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
3Modelling Detecting signal from background noise
4Survival 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
5What is hazard?
- Hazard rate is an instantaneous rate of events as
a function of - time
6Plot of hazard
- Note that the hazard changes over time denoted
by h(t)
h(t)
time
Old age
Birth
7Survival 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.
8How is the relationship formulated?
Simplest equation is
h is the hazard K is a constant e.g. 0.3 per
Person-year
9How is the relationship formulated?
Next Simplest is linear equation
h is the outcome a is the intercept ß is the
slope related to x the explanatory variable
and e is the error term or noise
10Linear model of hazard
Hazard
11Cox Proportional Hazards Model (1972)
h0 is the baseline hazard r ( ß, x) function
reflects how the hazard function changes (ß)
according to differences in subjects
characteristics (x)
12Exponential model of hazard
Hazard
Age in years
13What 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
14What is Hazard Ratio?
- Hazard Ratio (HR) is ratio of hazards in two
groups and remains constant over time (n.b.
survival curve widens)
Survival
Time
15Interpretation of HR comparing two groups
- HR 1 Do NOT reject null hypothesis (i.e. no
difference) - HR lt 1 Reduction in Hazard relative to
comparator (e.g. HR 0.6 is 40 reduction) - HR gt 1 Increase in Hazard relative to
comparator (e.g. HR 1.7 is 70 increase)
16Cox Proportional Hazards Model Hazard Ratio
Consider hazard ratio for men vs. women, then -
17Cox Proportional Hazards Model Hazard Ratio
If coding for gender is x1 (men) and x0 (women)
then
where ß is the regression coefficient for gender
18Hazard 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
19Fitting Gender in Cox Model in SPSS
20Output from Cox Model in SPSS
p-value
Standard error
Degrees of freedom
Variable in model
HR for men vs. women
Regression Coefficient
Test Statistic ( ß/se(ß) )2
21Logrank 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
22Wald Test Null Hypothesis
- The Null hypothesis for the Wald test
- Hazard Ratio 1
- Equivalent to regression coefficient ß0
- Note that if the 95 CI for the HR includes 1
then the null hypothesis cannot be rejected
23Hazard ratios for categorical factors in SPSS
- Enter factor as before
- Click on categorical and choose the reference
category (usually first or last) - E.g. Dukes 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
24Fitting a categorical variable Dukes Staging
Reference category
B vs. A
C vs. A
D vs. A
UK vs. A
25One Solution to Confounding
Use multiple Cox regression with both predictor
and confounder as explanatory variables i.e fit
x1 is Dukes Stage and x2 is Age
26Fitting a multiple regression Dukes Staging and
Age
Age adjusted for Dukes Stage
27Interpretation 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 HR1.024, 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
28First 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
29Summary
- 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
30Check 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
31Check 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
32Check 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
33Proportional hazards holds for Dukes Staging
Categorical Variable Codings(b)
Frequency (1) (2) (3) (4) dukes(a) 0A 18 1 0 0
0 1B 107 0 1 0 0 2C 188 0 0 1 0
3D 123 0 0 0 1 9UK 40 0 0 0 0 a Indicator
Parameter Coding b Category variable dukes
(Dukes Staging)
34Proportional hazards holds for Dukes Staging
35Summary
- 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
36Practical
- Read in Colorectal.sav and try to fit a multiple
proportional hazards model - Check proportional hazards assumption