Statistics for Health Research
Assessing Survival:
Cox Proportional
Hazards Model
Peter T. Donnan
Professor of Epidemiology and Biostatistics
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
Modelling: Detecting signal
from background noise
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
What is hazard?
Hazard rate is an instantaneous rate
of events as a function of
time
Plot of hazard
Note that the hazard changes over time
denoted by h(t)
h(t)
Birth
time
Old age
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….
How is the relationship
formulated?
Simplest equation is:
h = k
H
a
z
a
r
d
Age in years
h is the hazard
K is a constant e.g. 0.3 per Person-year
How is the relationship
formulated?
Next Simplest is linear equation:
h = a + βx + ei
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’
Linear model of hazard
Hazard
Age in years
Cox Proportional Hazards
Model (1972)
h( t ) = h0 ( t ) × r(β, x)
h0 is the baseline hazard;
r ( β, x) function reflects how the
hazard function changes (β)
according to differences in
subjects’ characteristics (x)
Exponential model of
hazard
Hazard
Age in years
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
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
Time
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 = 0.6
is 40% reduction)
HR > 1; Increase in Hazard relative to
comparator (e.g. HR = 1.7 is
70% increase)
Cox Proportional Hazards
Model: Hazard Ratio
r(β, x) = exp(βx)
Consider hazard ratio for men vs.
women, then -
h(t )
HR 
h(t )
men
women
h (t )r ( x )

h (t )r ( x )
0
0
men
women
Cox Proportional Hazards
Model: Hazard Ratio
If coding for gender is x=1 (men)
and x=0 (women) then:
r ( βxmen )
exp(β)
HR =
=
r ( βxwomen )
exp(0 )
= exp(β)
where β is the regression coefficient for gender
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
Fitting Gender in Cox Model
in SPSS
Output from Cox Model in
SPSS
Variables in the Equation
SEXNUM
B
-.038
SE
.121
Standard
error
Variable
in model Regression
Coefficient
Wald
.097
df
Degrees
of
freedom
Test Statistic
( β/se(β) )2
1
Sig.
.755
pvalue
Exp(B)
.963
HR for
men vs.
women
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
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
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
Fitting a categorical variable:
Duke’s Staging
Categorical Variable Codingsa,b
Reference
category
DUKES
Freqency
18
107
188
123
40
0= A
1= B
2= C
3= D
9= UK
(1)
.000
1.000
.000
.000
.000
(2)
.000
.000
1.000
.000
.000
(3)
.000
.000
.000
1.000
.000
(4)
.000
.000
.000
.000
1.000
a. Indi cator Parameter Codi ng
b. Category vari able: DUKES
(Dukes Stag ing )
Variables in the Equation
B vs. A
C vs. A
D vs. A
UK vs. A
B
DUKES
DUKES(1)
DUKES(2)
DUKES(3)
DUKES(4)
.066
.716
1.753
1.328
SE
.441
.421
.420
.446
Wald
105.703
.022
2.893
17.379
8.875
df
4
1
1
1
1
Sig .
.000
.882
.089
.000
.003
Exp(B)
1.068
2.047
5.769
3.775
95.0% CI for Exp(B)
Lower
Upper
.450
.897
2.531
1.575
2.536
4.672
13.151
9.046
One Solution to Confounding
Use multiple Cox regression with
both predictor and confounder
as explanatory variables i.e fit:
h(t) = h0 ( t) exp(β1x1 + β2 x2 )
x1 is Duke’s Stage and x2 is Age
Fitting a multiple regression:
Duke’s Staging and Age
Variables in the Equation
B
AGE
.019
SE
.006
Wald
9.181
df
Sig .
.002
1
Exp(B)
1.019
95.0% CI for Exp(B)
Lower
Upper
1.007
1.032
Variables in the Equation
B
DUKES
DUKES(1)
DUKES(2)
DUKES(3)
DUKES(4)
AGE
.159
.822
1.896
1.321
.024
SE
.442
.422
.422
.446
.007
Wald
111.400
.130
3.800
20.181
8.773
13.761
df
4
1
1
1
1
1
Sig .
.000
.719
.051
.000
.003
.000
Age adjusted for Duke’s Stage
Exp(B)
1.172
2.276
6.662
3.748
1.024
95.0% CI for Exp(B)
Lower
Upper
.493
.996
2.913
1.564
1.011
2.788
5.203
15.238
8.986
1.038
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
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
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
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
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
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
Proportional hazards holds
for Duke’s Staging
Categorical Variable Codings(b)
Frequency
(1)
(2)
(3)
(4)
18
1
0
0
0
1=B
107
0
1
0
0
2=C
188
0
0
1
0
3=D
123
0
0
0
1
9=UK
40
0
0
0
0
dukes(a) 0=A
a Indicator Parameter Coding
b Category variable: dukes (Dukes Staging)
Proportional hazards holds
for Duke’s Staging
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
Practical
• Read in Colorectal.sav and try to fit
a multiple proportional hazards model
• Check proportional hazards
assumption