Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Using restricted cubic splines to approximatecomplex hazard functions.
Mark J. Rutherford 1 Michael J. Crowther 1
Paul C. Lambert 1,2
1Department of Health Sciences,University of Leicester, UK.
2MEB, Karolinska Institutet,Stockholm.
Survival Analysis for Junior Researchers. 3rd April, 2012.
Mark Rutherford Leicester. 3rd April, 2012. 1 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Introduction
We want a good approximation to the underlying hazardfunction.
These functions can often be complex; with, for example,multiple turning points or sharp changes over short periodsover time; particularly early in follow-up.
Cox models have been used to circumvent the estimationof the baseline hazard function; only get relative effects.
However, it is often of interest to have the baseline hazardin order to report absolute estimates of risk (e.g.differences in mortality rates) and to more easilyincorporate time-dependent effects.
Mark Rutherford Leicester. 3rd April, 2012. 2 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Cubic Splines
0
20
40
60y
0 20 40 60 80 100x
Scatter Plot
Mark Rutherford Leicester. 3rd April, 2012. 3 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Cubic Splines
0
20
40
60y
0 20 40 60 80 100x
No Constraints
Mark Rutherford Leicester. 3rd April, 2012. 3 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Cubic Splines
0
20
40
60y
0 20 40 60 80 100x
Forced to Join at Knots
Mark Rutherford Leicester. 3rd April, 2012. 3 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Cubic Splines
0
20
40
60y
0 20 40 60 80 100x
Continuous First Derivatives
Mark Rutherford Leicester. 3rd April, 2012. 3 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Cubic Splines
0
20
40
60y
0 20 40 60 80 100x
Continuous First & Second Derivatives
Mark Rutherford Leicester. 3rd April, 2012. 3 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Cubic Splines
0
20
40
60y
0 20 40 60 80 100x
Restricted Cubic Splines
Mark Rutherford Leicester. 3rd April, 2012. 3 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Flexible Parametric Models
Starting from a Weibull survival curve:
S(t) = exp (−λtγ) (1)
Converting to the log-cumulative hazard scale:
ln [H(t)] = ln(λ) + γ ln(t). (2)
This function is linear in ln(t), we can relax this linearityby using restricted cubic splines for ln(t).
We can also introduce covariates, x, to obtain aproportional hazards model where βββ are log-hazard ratios.
ln {H(t|x)} = s (ln(t)|γ, k0) + xβββ, (3)
Mark Rutherford Leicester. 3rd April, 2012. 4 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Splines for hazard function?
Increased flexibility can be introduced by increasing thedegrees of freedom of the spline functions.
It is easy to transform to the hazard, and survival function.
No need for numerical integration techniques ortime-splitting.
Flexible parametric modelling is becoming increasinglypopular.
To date, there hasn’t been a simulation study toinvestigate the performance of the spline functions forcapturing the shape.
Mark Rutherford Leicester. 3rd April, 2012. 5 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
Female breast cancer patients (age< 50) diagnosed inEngland and Wales between 1986 and the end of 1990.
Deprivation status based on quantiles of the Carstairsdeprivation index (variable with 5 levels from leastdeprived up to most deprived).
Restrict to patients in the least deprived group and themost deprived group to provide a direct comparison and abinary covariate.
All-cause survival for the remaining 9,721 patients withfollow-up restricted to 10 years post-diagnosis.
Mark Rutherford Leicester. 3rd April, 2012. 6 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
Cox Smoothed
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 7 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
Cox Smoothed Weibull
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 7 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
Cox Smoothed WeibullMixture-Weibull
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 7 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
Cox Smoothed WeibullMixture-Weibull Flex Para DF=5
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 7 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
Cox Smoothed WeibullMixture-Weibull Flex Para DF=5Flex Para DF=10
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 7 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0.00
0.25
0.50
0.75
1.00
Pro
port
ion
Rel
apse
-fre
e
0 2 4 6 8 10Relapse-free Interval (years)
WeibullMixture-WeibullFlex Para DF=5Flex Para DF=10Cox
Baseline Survival Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 8 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
DF=1
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 9 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
DF=1 DF=2
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 9 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
DF=1 DF=2DF=3
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 9 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
DF=1 DF=2DF=3 DF=4
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 9 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
DF=1 DF=2DF=3 DF=4DF=5
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 9 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
DF=1 DF=2DF=3 DF=4DF=5 DF=6
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 9 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
DF=1 DF=2DF=3 DF=4DF=5 DF=6DF=7
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 9 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
DF=1 DF=2DF=3 DF=4DF=5 DF=6DF=7 DF=8
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 9 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
DF=1 DF=2DF=3 DF=4DF=5 DF=6DF=7 DF=8DF=9
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 9 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0
20
40
60
80
100
Mor
talit
y ra
te (
per
1,00
0 pe
rson
-yea
rs)
0 2 4 6 8 10Relapse-free Interval (years)
DF=1 DF=2DF=3 DF=4DF=5 DF=6DF=7 DF=8DF=9 DF=10
Baseline Hazard Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 9 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
0.00
0.25
0.50
0.75
1.00
Pro
port
ion
Rel
apse
-fre
e
0 2 4 6 8 10Relapse-free Interval (years)
DF=1 DF=2DF=3 DF=4DF=5 DF=6DF=7 DF=8DF=9 DF=10
Baseline Survival Function(Age=35, Dep Level=Least Deprived)
Mark Rutherford Leicester. 3rd April, 2012. 10 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
ModelDeprivation Age
AIC BIClog HR (SE) log HR (SE)
Cox 0.2147 (0.0355) -0.0163 (0.0030) - -Mixture Weibull 0.2147 (0.0355) -0.0163 (0.0030) 26416.42 26466.69
Weibull 0.2141 (0.0355) -0.0165 (0.0030) 26620.41 26645.10FPM df=2 0.2135 (0.0355) -0.0163 (0.0030) 26542.77 26573.64FPM df=3 0.2150 (0.0355) -0.0163 (0.0030) 26424.16 26461.19FPM df=4 0.2148 (0.0355) -0.0163 (0.0030) 26406.65 26449.86FPM df=5 0.2147 (0.0355) -0.0162 (0.0030) 26401.81 26451.19
FPM df=10 0.2148 (0.0355) -0.0163 (0.0030) 26406.59 26486.83
Mark Rutherford Leicester. 3rd April, 2012. 11 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
ModelDeprivation Age
AIC BIClog HR (SE) log HR (SE)
Cox 0.2147 (0.0355) -0.0163 (0.0030) - -Mixture Weibull 0.2147 (0.0355) -0.0163 (0.0030) 26416.42 26466.69
Weibull 0.2141 (0.0355) -0.0165 (0.0030) 26620.41 26645.10FPM df=2 0.2135 (0.0355) -0.0163 (0.0030) 26542.77 26573.64FPM df=3 0.2150 (0.0355) -0.0163 (0.0030) 26424.16 26461.19FPM df=4 0.2148 (0.0355) -0.0163 (0.0030) 26406.65 26449.86FPM df=5 0.2147 (0.0355) -0.0162 (0.0030) 26401.81 26451.19
FPM df=10 0.2148 (0.0355) -0.0163 (0.0030) 26406.59 26486.83
Mark Rutherford Leicester. 3rd April, 2012. 11 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
ModelDeprivation Age
AIC BIClog HR (SE) log HR (SE)
Cox 0.2147 (0.0355) -0.0163 (0.0030) - -Mixture Weibull 0.2147 (0.0355) -0.0163 (0.0030) 26416.42 26466.69
Weibull 0.2141 (0.0355) -0.0165 (0.0030) 26620.41 26645.10FPM df=2 0.2135 (0.0355) -0.0163 (0.0030) 26542.77 26573.64FPM df=3 0.2150 (0.0355) -0.0163 (0.0030) 26424.16 26461.19FPM df=4 0.2148 (0.0355) -0.0163 (0.0030) 26406.65 26449.86FPM df=5 0.2147 (0.0355) -0.0162 (0.0030) 26401.81 26451.19
FPM df=10 0.2148 (0.0355) -0.0163 (0.0030) 26406.59 26486.83
Mark Rutherford Leicester. 3rd April, 2012. 11 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
ModelDeprivation Age
AIC BIClog HR (SE) log HR (SE)
Cox 0.2147 (0.0355) -0.0163 (0.0030) - -Mixture Weibull 0.2147 (0.0355) -0.0163 (0.0030) 26416.42 26466.69
Weibull 0.2141 (0.0355) -0.0165 (0.0030) 26620.41 26645.10FPM df=2 0.2135 (0.0355) -0.0163 (0.0030) 26542.77 26573.64FPM df=3 0.2150 (0.0355) -0.0163 (0.0030) 26424.16 26461.19FPM df=4 0.2148 (0.0355) -0.0163 (0.0030) 26406.65 26449.86FPM df=5 0.2147 (0.0355) -0.0162 (0.0030) 26401.81 26451.19
FPM df=10 0.2148 (0.0355) -0.0163 (0.0030) 26406.59 26486.83
Mark Rutherford Leicester. 3rd April, 2012. 11 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Motivating Example
ModelDeprivation Age
AIC BIClog HR (SE) log HR (SE)
Cox 0.2147 (0.0355) -0.0163 (0.0030) - -Mixture Weibull 0.2147 (0.0355) -0.0163 (0.0030) 26416.42 26466.69
Weibull 0.2141 (0.0355) -0.0165 (0.0030) 26620.41 26645.10FPM df=2 0.2135 (0.0355) -0.0163 (0.0030) 26542.77 26573.64FPM df=3 0.2150 (0.0355) -0.0163 (0.0030) 26424.16 26461.19FPM df=4 0.2148 (0.0355) -0.0163 (0.0030) 26406.65 26449.86FPM df=5 0.2147 (0.0355) -0.0162 (0.0030) 26401.81 26451.19
FPM df=10 0.2148 (0.0355) -0.0163 (0.0030) 26406.59 26486.83
Mark Rutherford Leicester. 3rd April, 2012. 11 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Simulation Strategy
Simulation to evaluate the use of splines for complexhazard shapes.
Simulate complex hazard shapes using mixture Weibulldistribution (4 different “complex” shapes).
Simulate the dataset with a continuous covariate and abinary covariate, with proportional hazards for each.
Fit 1 to 10 DF flexible parametric model to 1000 datasets.
Fit for 3 different sample sizes (300, 3000, 30000).
Also fit mixture Weibull model (true model) and Coxmodel for comparison.
Compare area differences of hazard curves and comparehazard ratios.
Mark Rutherford Leicester. 3rd April, 2012. 12 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Simulated Shapes - Hazard
0.0
0.5
1.0
1.5
2.0
2.5H
azar
d ra
te
0 2 4 6 8 10Time Since Diagnosis (Years)
Scenario 1
0.0
0.5
1.0
1.5
2.0
2.5
Haz
ard
rate
0 2 4 6 8 10Time Since Diagnosis (Years)
Scenario 2
0.0
0.5
1.0
1.5
2.0
2.5
Haz
ard
rate
0 2 4 6 8 10Time Since Diagnosis (Years)
Scenario 3
0.0
0.5
1.0
1.5
2.0
2.5
Haz
ard
rate
0 2 4 6 8 10Time Since Diagnosis (Years)
Scenario 4
Mark Rutherford Leicester. 3rd April, 2012. 13 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Comparison - Scenario 3
0
1
2
3
4
5H
azar
d F
unct
ion
0 2 4 6 8 10Follow-up Time (Years)
True function
0.0
0.2
0.4
0.6
0.8
1.0
Sur
viva
l Fun
ctio
n
0 2 4 6 8 10Follow-up Time (Years)
True function
Mark Rutherford Leicester. 3rd April, 2012. 14 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Comparison - Scenario 3
0
1
2
3
4
5H
azar
d F
unct
ion
0 2 4 6 8 10Follow-up Time (Years)
True function
Weibull model
0.0
0.2
0.4
0.6
0.8
1.0
Sur
viva
l Fun
ctio
n
0 2 4 6 8 10Follow-up Time (Years)
True function
Weibull model
Mark Rutherford Leicester. 3rd April, 2012. 14 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Comparison - Scenario 3
0
1
2
3
4
5H
azar
d F
unct
ion
0 2 4 6 8 10Follow-up Time (Years)
Integral Area
True function
Weibull model
0.0
0.2
0.4
0.6
0.8
1.0
Sur
viva
l Fun
ctio
n
0 2 4 6 8 10Follow-up Time (Years)
Integral Area
True function
Weibull model
Mark Rutherford Leicester. 3rd April, 2012. 14 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
% Area Difference for Hazard
0
10
20
30
40
50
60
Per
cent
age
of T
otal
Are
a D
iffer
ence
on th
e H
azar
d S
cale
1 2 3 4 5 6 7 8 9 10Degrees of Freedom
Sample Size 300Sample Size 3000Sample Size 30,000
Scenario 1
0
10
20
30
40
50
60
Per
cent
age
of T
otal
Are
a D
iffer
ence
on th
e H
azar
d S
cale
1 2 3 4 5 6 7 8 9 10Degrees of Freedom
Sample Size 300Sample Size 3000Sample Size 30,000
Scenario 3
Note: The dashed lines represent the average area difference achieved by the mixture Weibull model for each sample size.
Hazard Scale
Mark Rutherford Leicester. 3rd April, 2012. 15 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
% Area Difference for Hazard
0
10
20
30
40
50
60
Per
cent
age
of T
otal
Are
a D
iffer
ence
on th
e H
azar
d S
cale
1 2 3 4 5 6 7 8 9 10Degrees of Freedom
Sample Size 300Sample Size 3000Sample Size 30,000
Scenario 1
0
10
20
30
40
50
60
Per
cent
age
of T
otal
Are
a D
iffer
ence
on th
e H
azar
d S
cale
1 2 3 4 5 6 7 8 9 10Degrees of Freedom
Sample Size 300Sample Size 3000Sample Size 30,000
Scenario 3
Note: The dashed lines represent the average area difference achieved by the mixture Weibull model for each sample size.
Hazard Scale
Mark Rutherford Leicester. 3rd April, 2012. 15 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
% Area Difference for Hazard
0
10
20
30
40
50
60
Per
cent
age
of T
otal
Are
a D
iffer
ence
on th
e H
azar
d S
cale
1 2 3 4 5 6 7 8 9 10Degrees of Freedom
Sample Size 300Sample Size 3000Sample Size 30,000
Scenario 1
0
10
20
30
40
50
60
Per
cent
age
of T
otal
Are
a D
iffer
ence
on th
e H
azar
d S
cale
1 2 3 4 5 6 7 8 9 10Degrees of Freedom
Sample Size 300Sample Size 3000Sample Size 30,000
Scenario 3
Note: The dashed lines represent the average area difference achieved by the mixture Weibull model for each sample size.
Hazard Scale
Mark Rutherford Leicester. 3rd April, 2012. 15 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
% Area Difference for Survival
0
5
10
15
20
Per
cent
age
of T
otal
Are
a D
iffer
ence
on th
e S
urvi
val S
cale
1 2 3 4 5 6 7 8 9 10Degrees of Freedom
Sample Size 300Sample Size 3000Sample Size 30,000
Scenario 1
0
5
10
15
20
Per
cent
age
of T
otal
Are
a D
iffer
ence
on th
e S
urvi
val S
cale
1 2 3 4 5 6 7 8 9 10Degrees of Freedom
Sample Size 300Sample Size 3000Sample Size 30,000
Scenario 3
Note: The dashed lines represent the average area difference achieved by the mixture Weibull model for each sample size.
Survival Scale
Mark Rutherford Leicester. 3rd April, 2012. 16 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Hazard Ratios for age (red) and treatment (blue)
CoxModel
0.005 0.010 0.015 0.0200.005 0.010 0.015 0.0200.005 0.010 0.015 0.0200.005 0.010 0.015 0.0200.005 0.010 0.015 0.020
0.005
0.010
0.015
0.020
0.005 0.010 0.015 0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
MixtureWeibull
0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
WeibullModel
0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
FlexibleParametric
df(2)0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
FlexibleParametric
df(3)0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
FlexibleParametric
df(5)0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
-0.65-0.60-0.55-0.50-0.45-0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40
FlexibleParametric
df(10)
Scenario 3 - Sample Size 3000
Mark Rutherford Leicester. 3rd April, 2012. 17 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Hazard Ratios for age (red) and treatment (blue)
CoxModel
0.005 0.010 0.015 0.0200.005 0.010 0.015 0.0200.005 0.010 0.015 0.0200.005 0.010 0.015 0.0200.005 0.010 0.015 0.020
0.005
0.010
0.015
0.020
0.005 0.010 0.015 0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
MixtureWeibull
0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
WeibullModel
0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
FlexibleParametric
df(2)0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
FlexibleParametric
df(3)0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
FlexibleParametric
df(5)0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
-0.65-0.60-0.55-0.50-0.45-0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40
FlexibleParametric
df(10)
Scenario 3 - Sample Size 3000
Mark Rutherford Leicester. 3rd April, 2012. 17 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Hazard Ratios for age (red) and treatment (blue)
CoxModel
0.005 0.010 0.015 0.0200.005 0.010 0.015 0.0200.005 0.010 0.015 0.0200.005 0.010 0.015 0.0200.005 0.010 0.015 0.020
0.005
0.010
0.015
0.020
0.005 0.010 0.015 0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
MixtureWeibull
0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
WeibullModel
0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
FlexibleParametric
df(2)0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
FlexibleParametric
df(3)0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
FlexibleParametric
df(5)0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
-0.65-0.60-0.55-0.50-0.45-0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40
FlexibleParametric
df(10)
Scenario 3 - Sample Size 3000
Mark Rutherford Leicester. 3rd April, 2012. 17 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Hazard Ratios for age (red) and treatment (blue)
CoxModel
0.005 0.010 0.015 0.0200.005 0.010 0.015 0.0200.005 0.010 0.015 0.0200.005 0.010 0.015 0.0200.005 0.010 0.015 0.020
0.005
0.010
0.015
0.020
0.005 0.010 0.015 0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
MixtureWeibull
0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
WeibullModel
0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
FlexibleParametric
df(2)0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
FlexibleParametric
df(3)0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
FlexibleParametric
df(5)0.005
0.010
0.015
0.020
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
-0.65-0.60-0.55-0.50-0.45-0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40
FlexibleParametric
df(10)
Scenario 3 - Sample Size 3000
Mark Rutherford Leicester. 3rd April, 2012. 17 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
Discussion
Using restricted cubic splines appears to perform well.
The number of knots (degrees of freedom) needs to bechosen for the spline function.
Provided that a sufficient number of knots are chosen;flexible parametric models perform well.
Selection criteria can be used as a guide for selecting thedegrees of freedom.
Hazard ratio values are largely insensitive to the poorspecification of the baseline.
No reason not to get a good approximation to the baselineas well as HR estimates; plus this can be useful for:
Reporting absolute risks.Dealing appropriately with time-dependent effects.
Mark Rutherford Leicester. 3rd April, 2012. 18 / 19
Restrictedcubic splinesfor hazards
Introduction
Splines
Using Splines
MotivatingExample
Simulation
References
References
S. Durrelman, and R. Simon.Flexible regression models with cubic splines.Statistics in Medicine, 8 :551–561, 1989.
P. Royston and M. K. B. Parmar.Flexible parametric proportional-hazards and proportional-odds models forcensored survival data, with application to prognostic modelling andestimation of treatment effects.Statistics in Medicine, 21 :2175–2197, 2002.
P.C. Lambert, and P Royston.Further Development of Flexible Parametric Models for Survival Analysis.Stata Journal, 9 :265–290, 2009.
R. Bender, T. Augustin, and M. Blettner.Generating survival times to simulate Cox proportional hazards model.Statistics in Medicine, 24 :1713–1723, 2005.
M.J. Rutherford, M.J. Crowther and P.C. Lambert.Using restricted cubic splines to approximate complex hazard functions inthe analysis of time-to-event data.Statistics in Medicine, (submitted), 2012.
Mark Rutherford Leicester. 3rd April, 2012. 19 / 19