Statistical Methods in Epi II (171:242)
The Cox Regression Model: Functional
Form of the Covariates
Brian J. Smith, Ph.D.
February 24, 2003
Functional Form of the Covariates
In this section we will cover different parameterizations of the
covariates included in the regression model
 t , x i   0 t exp βx i 
 0 t exp 1 x1i     K xKi 
Implicitly, we assume that the predictors are continuous covariates.
Although the relative risk component is generally written as an
additive function of the predictors, interaction terms and
transformed versions of the covariates can be included in the
model.
Interaction: Categorical Variables
Recall the bone marrow transplant study of Non-Hodgkin’s and
Hodgkin’s lymphoma patients. The primary goal was to compare
diseases-free survival between allogeneic or autologous transplant
recipients. Also measured, were the patient’s Karnofsky score and
waiting time to receive the transplant. The data are summarized in
Table 1 and Table 2.
Table 1. Summary of the categorical variables in the lymphoma
study.
Variable
Levels
Event Indicator 0 = Disease-Free
1 = Death or Relapse
Graft
0 = Allogeneic
1 = Autologous
Disease
0 = Non-Hodgkin’s
1 = Hodgkin’s
1
N
17
26
16
27
23
20
Percents
40%
60%
37%
63%
53%
47%
Table 2. Summary of the continuous variables in the lymphoma
study.
Variable
Disease-Free
Survival in Days
Karnofsky Score
Wait Time
Mean
Std. Dev.
Min
Max
429.7
570.5
2
2144
76.3
37.7
21.0
33.6
20
5
100
171
A Cox regression model was fit with main effects for graft, type of
disease, Karnofsky score, and wait time; the resulting estimates are
ˆ
i
 
se ̂ i
p-value
Graft
-0.2332
0.4430
0.60
Disease
0.9806
0.5226
0.061
Karnofsky
-0.05584
0.01216
< 0.0001
Wait
-0.00786
0.00788
0.32
No interactions between variables were specified in the model.
Thus, the effect of a covariate, say graft, is the same across all
levels of the remaining variables. For instance, the estimated risk
of relapse for autologous patients, relative to allogeneic patients is




 t, x1  1 exp ˆ1  1

 exp ˆ1
 t, x1  0 exp ˆ1  0
 
 exp  0.2332  0.792
.
Because only main effects were included in the model, it is
assumed that this hazard ratio is the same among both NonHodgkin’s and Hodgkin’s patients. However, in our stratified logrank analysis, we saw that the hazard rates were reversed in order
when going from one disease type to the other. Disease was, in
fact, found to be an effect modifier that interacted with graft type in
its effect on survival. Therefore, an interaction term is needed in
the Cox model.
2
If an interaction term is formed by multiplying together the indicator
variables for graft and disease, the result is
Variable
Graft*Disease
Levels
0 = Otherwise
1 = Autologous
and Hodgkin’s
N
28
15
Percents
65%
35%
and the model becomes
 t , x i   0 t exp 1 x1i   2 x2i  1, 2 x1i x2i   3 x3i   4 x4i  .
The resulting estimates are given below.
ˆi
 
se ̂ i
p-value
Graft
0.6394
0.5937
0.2800
Disease
2.7603
0.9474
0.0036
Graft*Disease Karnofsky
-2.3709
-0.0495
1.0355
0.0124
0.0220
0.0001
Wait
-0.0166
0.0102
0.1000
The interaction term is highly significant (p = 0.0001). Notice the
differences in the parameter estimates, particularly for graft and
disease, as compared to the model without an interaction term. In
particular, the coefficient for graft type has changed sign, and the
coefficient for disease is about three times larger than before. With
the interaction term in the model, the hazard ratio for graft type is
allowed to vary between the two disease types.
Autologous vs. Allogeneic (Non-Hodgkin’s):




 t ,  x1  1, x2  0 exp ˆ1  1  ˆ2  0  ˆ1, 2  0

 exp ˆ1
ˆ
ˆ
ˆ
 t ,  x1  0, x2  0 exp 1  0   2  0  1, 2  0
 exp 0.6394   1.895
Autologous vs. Allogeneic (Hodgkin’s):
3
 




 t ,  x1  1, x2  1 exp ˆ1  1  ˆ2  1  ˆ1, 2  1

 exp ˆ1  ˆ1, 2
 t ,  x1  0, x2  1 exp ˆ1  0  ˆ2  1  ˆ1, 2  0


 exp 0.6394  2.3709   0.177
Interaction: Continous Variables
We could likewise have an interaction term that involves a
continuous variable. Consider the breast-feeding example. We
originally estimated coefficients for the following main effects.
White
ˆ1
Black Poverty Smoke Alcohol
ˆ 2
-0.305 -0.111
̂ 3
ˆ 4
̂ 5
-0.211
0.249
0.168
Care
Age
Educ
̂ 6
̂ 7
̂ 8
-0.027 0.020 -0.056
Again, the main effects model specifies that the effect of a given
covariate is the same across all levels of the remaining variables.
For example, the effect of education in this model is the same for
both blacks and whites. However, differences in educational
environments might suggest that an incremental increase in years
of education has a different effect for whites and blacks. Thus, we
might construct two interaction terms by separately multiplying the
education variable by the white and black race indicators. The
resulting model is
 t , x i   0 t exp 1 x1i   2 x2i    8 x8i  1,8 x1i x8i   2,8 x2i x8i  .
The regression estimates for the race and education variables, with
the interaction terms in the model, are
ˆi
 
se ̂ i
p-value
White
0.1018
0.5030
0.840
Black
-0.2853
0.9441
0.760
Educ
-0.0336
0.0373
0.370
4
White*Educ Black*Educ
-0.0344
0.0125
0.0421
0.0769
0.410
0.870
The likelihood ratio test for the two interaction terms indicates that
they are not significant (p = 0.6172). Nevertheless, we can use
these estimates to illustrate hazard ratio estimation in the presence
of a continuous interaction term.
Education 16 years vs. 12 years (White Mothers):


 exp 4ˆ  ˆ 
 t ,  x1  1, x8  16 exp ˆ1  1  ˆ8  16  ˆ1,8  16

 t ,  x1  1, x8  12  exp ˆ1  1  ˆ8  12  ˆ1,8  12
8


1,8
 exp 4 0.0336  0.0334 
 0.762
Education 16 years vs. 12 years (Black Mothers):


 exp 4ˆ  ˆ 
 t ,  x2  1, x8  16 exp ˆ2  1  ˆ8  16  ˆ2,8  16

 t ,  x2  1, x8  12  exp ˆ2  1  ˆ8  12  ˆ2,8  12
8


2 ,8
 exp 4 0.0336  0.0125
 0.919
Indicator Variables for Categorical Covariates
Indicator variables should be used to quantify the effects of a
categorical predictor in the model. An indicator variable is
constructed for each level of the categorical predictor. For level L,
the indicator takes on a value of one if a subject falls within that
level; otherwise, the indicator is zero. The variables “white” and
“black” in the breast-feeding example are indicator variables for the
categorical predictor race (white, black, or other). We did not,
however, include in the model another indicator for mothers of race
“other”. The reason for this is that a variable cannot be included if it
5
is a linear function of other variables already in the model. Since
the indicator variable other = 1 – black – white, it cannot be
included. In effect, it becomes the reference group in the model.
Transformations for Continuous Covariates
Veteran’s Administration Lung Cancer Trial
The effectiveness of a new drug for lung cancer was tested by the
Verteran’s Administrations. A summary of the variables collected in
the study is given in Table 3 and Table 4.
Table 3. Summary of the categorical variables in the VA lung
cancer study.
Variable
Event Indicator
Treatment
Cell Type
* Small Cell
* Adenocarcinoma
* Large Cell
Prior Therapy
Levels
0 = Censored
1 = Death
1 = Standard
2 = Test
1 = Squamous
2 = Small Cell
3 = Adenocarcinoma
4 = Large Cell
0 = No
1 = Yes
0 = No
1 = Yes
0 = No
1 = Yes
0 = No
1 = Yes
N
9
128
69
68
35
48
27
27
48
89
27
110
27
110
97
40
* Indicator variables for use in regression models.
6
Percents
7%
93%
50%
50%
26%
35%
20%
20%
71%
29%
Table 4. Summary of the continuous variables in the VA lung
cancer study.
Variable
Survival in Days
Age in years
Karnofsky Score
Months from Diagnosis
to Study Entry
Mean
121.6
58.3
58.6
Std. Dev.
157.8
10.5
20.0
Min
1
34
10
Max
999
81
99
8.77
1.07
1
87
Assessing Linearity
For illustrative purposes, a Cox regression model was fit with age
as the only predictor in the model. The estimates are as follows:
Variable
Age

ˆ
se ˆ
0.0075
0.00957
p-value
0.43
One might be tempted to conclude that age does not have a
significant effect on survival. However, the model that was fit
 t, xi   0 t exp xi 
assumes that the relative risk component is linear in the continuous
variable age. It is very important that we try to determine whether
the effect of age is truly linear, or whether there is some nonlinearity
to it.
Categorical Method
One method of checking for departures from linearity is to
categorize x into three or more categories. For example, the age
variable could be dived into categories defined as
7
Categorical Age
Variables
Age1
Age2
Age3
Levels
N
1 = (Age  45)
0 = Otherwise
1 = (45 < Age  60)
0 = Otherwise
1 = (Age > 60)
0 = Otherwise
23
114
37
100
77
60
Two of these indicator variables could be used in the Cox model
instead of the continuous variable for age. The likelihood ratio test
indicates that age is significant in the Cox model as a categorical
variable (p = 0.0466); whereas, the Wald test is not significant for
age as a linear variable in the model (p = 0.43). This indicates only
that the relative risk is not linearly related to age. It does not imply
that age has no effect on survival. The small p-value for the
categorical measure of age and the non-significance of the linear
effect suggests that a nonlinear relationship exists.
Graphical Method
Another method for detecting nonlinearity is to construct a scatter
plot of the survival times by age. A smoothing function can be
employed to fit a line to the data. Curves in the fitted line suggest
departures from linearity, as is the case in Figure 1.
8
1000
800
600
Survival Time
400
200
0
40
50
60
70
80
Age
Figure 1. Scatter plot of the survival times by age in the VA lung
cancer trial.
Adding Polynomial Terms
To address the problem of nonlinearity, we can try adding
polynomial terms to the model. The previous scatter plot suggests
the quadratic relationship
 t , xi   0 t exp 1 xi   2 xi2  .
which gives the estimates
Variable
Age
Age2
 
ˆi
se ̂ i
-0.2190
0.00205
0.0819
0.000734
p-value
0.0075
0.0053
The likelihood ratio test indicates a significant effect of age as a
quadratic effect in the model (p = 0.0258).
9
0.0
0.2
0.4
x^2
0.6
0.8
1.0
Great care must be exercised in adding polynomial terms to a
regression model. Polynomial terms are highly correlated (see
Figure 2), which proves to be problematic in the parameter
estimation routines.
0.0
0.2
0.4
0.6
0.8
1.0
x
Figure 2
In linear regression we address this problem by centering the
covariate about its mean to get xi  x . There is very little
correlation between polynomial terms computed from these
centered variables (see Figure 3). Centering the age
measurements about the mean age of 58.3 and refitting the survival
model gives
Variable
Age – 58.3
(Age – 58.3)2
 
ˆi
se ̂ i
0.01963
0.00205
0.00944
0.000734
10
p-value
0.0380
0.0053
-0.4
-0.2
x - mean(x)
0.0
Figure 3
11
0.2
0.4
0.00
0.05
0.15
(x - mean(x))^2
0.10
0.20
0.25