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Introduction Residuals Testing Example
Testing the proportional hazards
assumption
Maja Pohar Perme1,
Janez Stare1, Robin Henderson2
1IBMI, Faculty of Medicine, Ljubljana, Slovenia
2University of Newcastle, UK
Stockholm, September 2009
Introduction Residuals Testing Example
Proportional excess hazard assumption
Additive model
λO = λP + λE
λO(t |Z ) = λP(t) + λ0(t)eβZ
Notation
λO observed hazard
λP populationhazard
λE excess hazard =disease specifichazard
λ0 baseline excesshazard
Z covariates
β regressioncoefficients
Introduction Residuals Testing Example
Proportional excess hazard assumption
Additive model
λO = λP + λE
λO(t |Z ) = λP(t) + λ0(t)eβZ
PH assumption: β is constant in time
Notation
λO observed hazard
λP populationhazard
λE excess hazard =disease specifichazard
λ0 baseline excesshazard
Z covariates
β regressioncoefficients
Introduction Residuals Testing Example
Proportional excess hazard assumption
Additive model
λO = λP + λE
λO(t |Z ) = λP(t) + λ0(t)eβZ
PH assumption: β is constant in time
Presentation resume
Present a method for both graphically and
formally testing the PH assumption.
Notation
λO observed hazard
λP populationhazard
λE excess hazard =disease specifichazard
λ0 baseline excesshazard
Z covariates
β regressioncoefficients
Introduction Residuals Testing Example
Motivational example
Colon cancer, additive model, follow-up time 5 years
Estimate Std. Error z value p
sex -0.091 0.051 -1.767 0.077
age 0.006 0.002 2.772 0.006
Introduction Residuals Testing Example
Motivational example
Colon cancer, additive model, follow-up time 5 years
Estimate Std. Error z value p
sex -0.091 0.051 -1.767 0.077
age 0.006 0.002 2.772 0.006
Age effect allowed to change after the first year
Estimate Std. Error z value p
sex -0.096 0.051 -1.878 0.060
age1 0.041 0.007 6.281 < 0.001
age2 0.002 0.002 0.676 0.499
Likelihood ratio test p < 0.0001
Introduction Residuals Testing Example
Motivational example - Questions
Age effect allowed to change after the first year
Estimate Std. Error z value p
sex -0.096 0.051 -1.878 0.060
age1 0.041 0.007 6.281 < 0.001
age2 0.002 0.002 0.676 0.499
It seems that the effect of age varies in time, but
Introduction Residuals Testing Example
Motivational example - Questions
Age effect allowed to change after the first year
Estimate Std. Error z value p
sex -0.096 0.051 -1.878 0.060
age1 0.041 0.007 6.281 < 0.001
age2 0.002 0.002 0.676 0.499
It seems that the effect of age varies in time, but
Why did we get suspicious?
Introduction Residuals Testing Example
Motivational example - Questions
Age effect allowed to change after the first year
Estimate Std. Error z value p
sex -0.096 0.051 -1.878 0.060
age1 0.041 0.007 6.281 < 0.001
age2 0.002 0.002 0.676 0.499
It seems that the effect of age varies in time, but
Why did we get suspicious?
How do we choose the follow-up intervals?
Introduction Residuals Testing Example
Motivational example - Questions
Age effect allowed to change after the first year
Estimate Std. Error z value p
sex -0.096 0.051 -1.878 0.060
age1 0.041 0.007 6.281 < 0.001
age2 0.002 0.002 0.676 0.499
It seems that the effect of age varies in time, but
Why did we get suspicious?
How do we choose the follow-up intervals?
Is the model now sensible (does it fit well?)
Introduction Residuals Testing Example
Proportional excess hazard assumption
Additive model
λO = λP + λE
λO(t |Z ) = λP(t) + λ0(t)eβZ
Notation
λO observed hazard
λP populationhazard
λE excess hazard =disease specifichazard
λ0 baseline excesshazard
Z covariates
β regressioncoefficients
Introduction Residuals Testing Example
Proportional excess hazard assumption
Additive model
λO = λP + λE
λO(t |Z ) = λP(t) + λ0(t)eβZ
PH assumption: β is constant in time
Notation
λO observed hazard
λP populationhazard
λE excess hazard =disease specifichazard
λ0 baseline excesshazard
Z covariates
β regressioncoefficients
Introduction Residuals Testing Example
Proportional excess hazard assumption
Additive model
λO = λP + λE
λO(t |Z ) = λP(t) + λ0(t)eβZ
PH assumption: β is constant in time
Cox model
λO(t |Z ) = λ0(t)eβZ
Schoenfeld residuals present a standard
method for checking the PH assumption
Notation
λO observed hazard
λP populationhazard
λE excess hazard =disease specifichazard
λ0 baseline excesshazard
Z covariates
β regressioncoefficients
Introduction Residuals Testing Example
Definition of residuals
Cox model
Schoenfeld residuals
Zi
additive model
partial residuals
Zi
Notation
Z covariate
ti ith eventtime
0 1 2 3 4 5
45
50
55
60
65
Time
Age
Introduction Residuals Testing Example
Definition of residuals
Cox model
Schoenfeld residuals
Zi E(Z , ti)
additive model
partial residuals
Zi E(Z , ti)
Notation
Z covariate
ti ith eventtime
0 1 2 3 4 5
45
50
55
60
65
Time
Age
Introduction Residuals Testing Example
Definition of residuals
Cox model
Schoenfeld residuals
Zi − E(Z , ti)
additive model
partial residuals
Zi − E(Z , ti)
Notation
Z covariate
ti ith eventtime
0 1 2 3 4 5
45
50
55
60
65
Time
Age
Introduction Residuals Testing Example
Definition of residuals
Cox model
Schoenfeld residuals
Ui = Zi − E(Z , ti)
additive model
partial residuals
Ui : = Zi − E(Z , ti)
Notation
Z covariate
ti ith eventtime
Ui residual
0 1 2 3 4 5
45
50
55
60
65
Time
Age
Introduction Residuals Testing Example
Definition of residuals
Cox model
Schoenfeld residuals
Ui = Zi − E(Z , ti)
= Zi −
X
j∈Ri
Zjλj
P
k∈Ri
λk
additive model
partial residuals
Ui : = Zi − E(Z , ti)
= Zi −
X
j∈Ri
Zj
λPj + λEjP
k∈Ri
(λPk + λEk )
Notation
Z covariate
ti ith eventtime
Ui residual
Ri risk set attime i
λj hazard forperson j
λP populationhazard
λE excesshazard
0 1 2 3 4 5
45
50
55
60
65
Time
Age
Introduction Residuals Testing Example
Graphical inspection
Time
Be
ta(t
) fo
r x
0.00077 0.0029 0.0079 0.019 0.042 0.086 0.2 0.63
−4
−2
02
46
Introduction Residuals Testing Example
Graphical inspection
Time
Be
ta(t
) fo
r x
0.00077 0.0029 0.0079 0.019 0.042 0.086 0.2 0.63
−4
−2
02
46
Time
Be
ta(t
) fo
r x
0.00096 0.0092 0.021 0.19 0.47 0.96 2.4
−2
02
4
Introduction Residuals Testing Example
Graphical inspection
Time
Be
ta(t
) fo
r x
0.00077 0.0029 0.0079 0.019 0.042 0.086 0.2 0.63
−4
−2
02
46
Time
Be
ta(t
) fo
r x
0.00096 0.0092 0.021 0.19 0.47 0.96 2.4
−2
02
4
β0(ti) ≃ β +(
∂∂β{E(Z |ti , β)}
)−1E [U(β, t)]
Introduction Residuals Testing Example
Graphical inspection
Time
Be
ta(t
) fo
r x
0.00077 0.0029 0.0079 0.019 0.042 0.086 0.2 0.63
−4
−2
02
46
Time
Be
ta(t
) fo
r x
0.00096 0.0092 0.021 0.19 0.47 0.96 2.4
−2
02
4
plot(rs.zph(fit))
Introduction Residuals Testing Example
Testing the PH assumption
The cumulative sum of standardized residuals
Sum up the standardized residuals in time
If the null hypothesis is true, the average is 0
If the null hypothesis is true, the cumulative sum
oscillates around 0
0.0 0.2 0.4 0.6 0.8 1.0
−3 e−
04
−1 e−
04
1 e
−04
Time
sta
ndard
ized r
esid
uals
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
time
bro
wnia
n m
otion
Introduction Residuals Testing Example
Testing the PH assumption
The cumulative sum of standardized residuals
Sum up the standardized residuals in time
If the null hypothesis is true, the average is 0
If the null hypothesis is true, the cumulative sum
oscillates around 0
If we know the true β, this process converges to
Brownian motion
0.0 0.2 0.4 0.6 0.8 1.0
−3 e−
04
−1 e−
04
1 e
−04
Time
sta
ndard
ized r
esid
uals
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
time
bro
wnia
n m
otion
Introduction Residuals Testing Example
Testing the PH assumption
The Brownian bridge process
Tie down the cumulative sum process at the end
If the null hypothesis is true, the tied down process
(based on the estimated model) can be approximated
with Brownian bridge
0.0 0.2 0.4 0.6 0.8 1.0
−3 e−
04
−1 e−
04
1 e
−04
Time
sta
ndard
ized r
esid
uals
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
time
bro
wnia
n m
otion
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
time
resid
ual
age
Introduction Residuals Testing Example
Testing the PH assumption
The Brownian bridge process
Tie down the cumulative sum process at the end
If the null hypothesis is true, the tied down process
(based on the estimated model) can be approximated
with Brownian bridge
A sensible test statistic is the maximum of the process
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
time
resid
ual
age
Introduction Residuals Testing Example
Brownian motion constructed as the sum of residuals
Cox model & additive model
B(β0,k
n) =
1√
n
kX
i=1
Ui (β0)p
Vi (β0)
n→∞→ Brownian motion
BB(β0,k
n) = B(β0,
k
n)−
k
nB(β0, 1)
n→∞→ Brownian bridge
Notation
Ui Schoenfeld-likeresiduals
V variance
n number ofdeaths
β0 trueregressioncoefficient
0.0 0.2 0.4 0.6 0.8 1.0
−3 e−
04
−1 e−
04
1 e
−04
Time
sta
ndard
ized r
esid
uals
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
time
bro
wnia
n m
otion
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
time
resid
ual
age
Introduction Residuals Testing Example
Brownian motion constructed as the sum of residuals
Cox model & additive model
B(β0,k
n) =
1√
n
kX
i=1
Ui (β0)p
Vi (β0)
n→∞→ Brownian motion
BB(β0,k
n) = B(β0,
k
n)−
k
nB(β0, 1)
n→∞→ Brownian bridge
Notation
Ui Schoenfeld-likeresiduals
V variance
n number ofdeaths
β0 trueregressioncoefficient
0.0 0.2 0.4 0.6 0.8 1.0
−3 e−
04
−1 e−
04
1 e
−04
Time
sta
ndard
ized r
esid
uals
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
time
bro
wnia
n m
otion
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
time
resid
ual
age
Introduction Residuals Testing Example
Brownian bridge example; β changes: −0.5 → 0.5
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Time
sta
ndard
ized r
esid
uals
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
time
resid
ual
Introduction Residuals Testing Example
Tests based on Brownian bridge properties
β0 in time brownian bridge process test statistic
T1
max(abs(BB(t))
T2
max using weighted residuals
T3
Cramer−Von Mises∫ 1
0BB2(t)dt − (
∫ 1
0BB(t)dt)2
Introduction Residuals Testing Example
Example - colon cancer
Graphical presentation
plot(rs.zph(fit))
0 1 2 3 4 5
−2
−1
01
2
Time
Beta
(t)
for
age
Introduction Residuals Testing Example
Example - colon cancer
Graphical presentation
plot(rs.zph(fit))
0 1 2 3 4 5
−2
−1
01
2
Time
Beta
(t)
for
age
Test using the weighted max of the BB process
rs.br(fit) age: p < 0.001
Introduction Residuals Testing Example
Example - colon cancer
Graphical presentation
plot(rs.zph(fit))
0 1 2 3 4 5
−2
−1
01
2
Time
Beta
(t)
for
age
Test using the weighted max of the BB process
rs.br(fit) age: p < 0.001
Test the model with varying effect
rs.br(fit1) age: p = 0.033
Introduction Residuals Testing Example
Discussion
To sum up
for each covariate in the model, we define residuals
(observed value minus predicted value of covariate)
Introduction Residuals Testing Example
Discussion
To sum up
for each covariate in the model, we define residuals
(observed value minus predicted value of covariate)
smoothed average through residuals describes the
behaviour of β in time
Introduction Residuals Testing Example
Discussion
To sum up
for each covariate in the model, we define residuals
(observed value minus predicted value of covariate)
smoothed average through residuals describes the
behaviour of β in time
the null hypothesis can be tested using the cumulative
sums of standardised residuals
Introduction Residuals Testing Example
Discussion
To sum up
for each covariate in the model, we define residuals
(observed value minus predicted value of covariate)
smoothed average through residuals describes the
behaviour of β in time
the null hypothesis can be tested using the cumulative
sums of standardised residuals
Note
The described methods test the proportional excess
hazard assumption, assuming that all the other
assumptions of the model are met. If this is not true, it
might affect the tests.
Introduction Residuals Testing Example
Bibliography
Stare J., Pohar M., Henderson R.Goodness of fit of relative survival models
Statistics in Medicine, 2005
Pohar M., Stare J.Relative survival analysis in R
Computer methods and programs in biomedicine, 2006
Pohar M., Stare J.Making Relative Survival Analysis Relatively Easy
Computers in Biology and Medicine, 2007