Using restricted cubic splines to approximate complex

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


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.



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References

Cubic Splines

0

20

40

60y

0 20 40 60 80 100x

Scatter Plot



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References

Cubic Splines

0

20

40

60y

0 20 40 60 80 100x

No Constraints



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References

Cubic Splines

0

20

40

60y

0 20 40 60 80 100x

Forced to Join at Knots



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References

Cubic Splines

0

20

40

60y

0 20 40 60 80 100x

Continuous First Derivatives



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References

Cubic Splines

0

20

40

60y

0 20 40 60 80 100x

Continuous First & Second Derivatives



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References

Cubic Splines

0

20

40

60y

0 20 40 60 80 100x

Restricted Cubic Splines



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)



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.



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.



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)



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)


Cox Smoothed Weibull




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)


Cox Smoothed WeibullMixture-Weibull




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)


Cox Smoothed WeibullMixture-Weibull Flex Para DF=5




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)


Cox Smoothed WeibullMixture-Weibull Flex Para DF=5Flex Para DF=10




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


WeibullMixture-WeibullFlex Para DF=5Flex Para DF=10Cox

Baseline Survival Function(Age=35, Dep Level=Least Deprived)



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)


DF=1




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)


DF=1 DF=2




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)


DF=1 DF=2DF=3




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)


DF=1 DF=2DF=3 DF=4




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)


DF=1 DF=2DF=3 DF=4DF=5




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)


DF=1 DF=2DF=3 DF=4DF=5 DF=6




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)


DF=1 DF=2DF=3 DF=4DF=5 DF=6DF=7




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)


DF=1 DF=2DF=3 DF=4DF=5 DF=6DF=7 DF=8




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)


DF=1 DF=2DF=3 DF=4DF=5 DF=6DF=7 DF=8DF=9




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)


DF=1 DF=2DF=3 DF=4DF=5 DF=6DF=7 DF=8DF=9 DF=10




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


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)



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



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References

Motivating Example





FPM df=10 0.2148 (0.0355) -0.0163 (0.0030) 26406.59 26486.83



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References

Motivating Example





FPM df=10 0.2148 (0.0355) -0.0163 (0.0030) 26406.59 26486.83



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References

Motivating Example





FPM df=10 0.2148 (0.0355) -0.0163 (0.0030) 26406.59 26486.83



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References

Motivating Example





FPM df=10 0.2148 (0.0355) -0.0163 (0.0030) 26406.59 26486.83



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.



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


Scenario 2

0.0

0.5

1.0

1.5

2.0

2.5

Haz

ard

rate


Scenario 3

0.0

0.5

1.0

1.5

2.0

2.5

Haz

ard

rate


Scenario 4



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


True function



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References


0

1

2

3

4

5H

azar

d F

unct

ion


True function

Weibull model

0.0

0.2

0.4

0.6

0.8

1.0

Sur

viva

l Fun

ctio

n


True function

Weibull model



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References


0

1

2

3

4

5H

azar

d F

unct

ion


Integral Area

True function

Weibull model

0.0

0.2

0.4

0.6

0.8

1.0

Sur

viva

l Fun

ctio

n


Integral Area

True function

Weibull model



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



Scenario 3

Note: The dashed lines represent the average area difference achieved by the mixture Weibull model for each sample size.

Hazard Scale



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References


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



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



Scenario 3


Hazard Scale



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References


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



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



Scenario 3


Hazard Scale



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



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



Scenario 3


Survival Scale



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



Introduction

Splines

Using Splines

MotivatingExample

Simulation

References


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)




Introduction

Splines

Using Splines

MotivatingExample

Simulation

References


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)




Introduction

Splines

Using Splines

MotivatingExample

Simulation

References


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)




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.



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.


Documents

Using restricted cubic splines to approximate complex