Welcome and Introduction to Flexible Parametric Survival ... · Paul C Lambert Flexible Parametric...

Welcome and Introduction to Flexible Parametric

Survival Models

Paul C Lambert1,2

1Department of Health Sciences,University of Leicester, UK

2 Department of Medical Epidemiology and Biostatistics,Karolinska Institutet, Stockholm, Sweden

Workshop on Applications and Developments of

Flexible Parametric Survival Models

Stockholm 10/11/2011

Welcome to the workshop!

This is a satellite meeting to the the Nordic and Baltic StataUsers Group meeting to be held tomorrow.

Thanks to Nicola Orsini, Matteo Bottai and Peter Hedstrom, forallowing us to attach this workshop to the Stata meeting.

AimsTo raise awareness of the models and software.

To present and discuss current applications and developments.

To discuss potential extensions and limitations.

Please ask questions and contribute to the discussion!

Paul C Lambert Flexible Parametric Models Stockholm 10/11/2011 2

Timetable (morning)

09:00 Paul LambertWelcome and introduction toflexible parametric survival models

09:45 Camille Maringe

Using flexible parametric survival modelsfor international comparisons of cancersurvival.

10:10 Coffee

10:40 Edoardo ColzaniPrognosis of Patients With Breast Cancer:Causes of Death and Effects of Time Since Diagnosis,Age, and Tumor Characteristics

11:05 Patrick RoystonRestricted mean survival time: computationand some applications

11:30 Paul Dickman Discussion of morning session.12:00-13:15 Lunch


Timetable (afternoon)

13:15 Anna Johansson Estimation of absolute risks in case-cohort studies.

13:40 Therese AnderssonCure models within the framework of flexiblesurvival models

14:05 Sally Hinchliffe Flexible parametric models for competing risks.

14:30 Sandra Eloranta

Partitioning of excess mortality associatedwith a diagnosis of cancer using flexibleparametric survival models.

14:55 Coffee

15:25 Mark Clements Fitting flexible parametric survival models in R

15:50 Michael CrowtherFlexible parametric joint modelling of longitudinaland survival data

16:15 Patrick Royston Discussion of afternoon session


Why Parametric Models

We have the Cox model so why use parametric models?

Parametric Models have advantages for

Prediction.Extrapolation.Quantification (e.g., absolute and relative differences in risk).Modelling time-dependent effects.Understanding.Complex models in large datasets (time-dependent effects /multiple time-scales)All cause, cause-specific or relative survival.

The estimates we get from flexible parametric survival modelsare incredibly similar to those obtained from a Cox model.


Pregnancy Associated Breast Cancer (Johansson 2011)


The Cox Model I

Web of Science: over 23,300 citations (October 2008).

Has an h-index of 13 from repeat mis-citations1.

hi(t|xi) = h0(t) exp (xiβ)

Estimates (log) hazard ratios.

Advantage: The baseline hazard, h0(t) is not estimated from aCox model.

Disadvantage: The baseline hazard, h0(t) is not estimated froma Cox model.


Quote from David Cox (Reid 1994 [1])

Reid “What do you think of the cottage industry that’s grown uparound [the Cox model]?”

Cox “In the light of further results one knows since, I think Iwould normally want to tackle the problem parametrically.. . . I’m not keen on non-parametric formulations normally.”

Reid “So if you had a set of censored survival data today, youmight rather fit a parametric model, even though there wasa feeling among the medical statisticians that that wasn’tquite right.”

Cox “That’s right, but since then various people have shown thatthe answers are very insensitive to the parametricformulation of the underlying distribution. And if you wantto do things like predict the outcome for a particular patient,it’s much more convenient to do that parametrically.”


Splines

Flexible mathematical functions defined by piecewisepolynomials.

Used in regression models for non-linear effects

The points at which the polynomials join are called knots.

Constraints ensure the function is smooth.

The most common splines used in practice are cubic splines.

However, splines can be of any degree, n.

Function is forced to have continuous 0th, 1st and 2nd

derivatives.


Piecewise hazard function

25

50

100

150

200

Excess M

ort

alit

y R

ate

(1000 p

y’s

)

0 1 2 3 4 5Years from Diagnosis

Interval Length: 1 week


No Continuity Corrections

25

50

100

150

200

Excess M

ort

alit

y R

ate

(1000 p

y’s

)


No Constraints


Function forced to join at knots

25

50

100

150

200

Excess M

ort

alit

y R

ate

(1000 p

y’s

)


Forced to Join at Knots


Continuous first derivative

25

50

100

150

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Excess M

ort

alit

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ate

(1000 p

y’s

)


Continuous 1st Derivatives


Continuous second derivative

25

50

100

150

200

Excess M

ort

alit

y R

ate

(1000 p

y’s

)


Continuous 2nd Derivatives


Restricted Cubic Splines

Restricted cubic splines are splines that are restricted to be linearbefore the first knot and after the last knot [2].

Fitted as a linear function of derived covariates.

For knots, k1, . . . , kK , a restricted cubic spline function can bewritten

s(x) = γ0 + γ1z1 + γ2z2 + . . .+ γK−1zK−1

Issue is to choose the number and location of the knots.


Flexible Parametric Models: Basic Idea

Consider a Weibull survival curve.

S(t) = exp (−λtγ)

If we transform to the log cumulative hazard scale.

ln [H(t)] = ln[− ln(S(t))]

ln [H(t)] = ln(λ) + γ ln(t)

This is a linear function of ln(t)Introducing covariates gives

ln [H(t|xi)] = ln(λ) + γ ln(t) + xiβ

Rather than linearity with ln(t) flexible parametric models userestricted cubic splines (Roston & Parmar 2002 [3]).


Flexible Parametric Models: Incorporating Splines

We thus model on the log cumulative hazard scale.

ln[H(t|xi)] = ln [H0(t)] + xiβ

This is a proportional hazards model.Restricted cubic splines with knots, k0, are used to model thelog baseline cumulative hazard.

ln[H(t|xi)] = ηi = s (ln(t)|γ, k0) + xiβ

For example, with 4 knots we can write

ln [H(t|xi)] = ηi = γ0 + γ1z1i + γ2z2i + γ3z3i︸︷︷︸

log baselinecumulative hazard

+ xiβ︸︷︷︸

log hazardratios


Survival and Hazard Functions

We can transform to the survival scale

S(t|xi) = exp(− exp(ηi))

The hazard function is a bit more complex.

h(t|xi) =ds (ln(t)|γ, k0)

dtexp(ηi)

This involves the derivatives of the restricted cubic splinesfunctions.

These are easy to calculate.

Survival and hazard function used to maximize the likelihood.No need for numerical integration or time-splitting.


Fitting a Proportional Hazards Model

Example: 24,889 women aged under 50 diagnosed with breastcancer in England and Wales 1986-1990.

Compare five deprivation groups from most affluent to mostdeprived.

No information on cause of death, but given their age, mostwomen who die will die of their breast cancer.

Proportional hazards models. stcox dep2-dep5

. stpm2 dep2-dep5, df(5) scale(hazard) eform

The df(5) option implies using 4 internal knots and 2 boundaryknots at their default locations.

The scale(hazard) requests the model to be fitted on the logcumulative hazard scale.Paul C Lambert Flexible Parametric Models Stockholm 10/11/2011 19

Comparison of Hazard Ratios

Cox Proportional Hazards Model

. stcox dep2-dep5,

_t Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]

dep2 1.048716 .0353999 1.41 0.159 .9815786 1.120445dep3 1.10618 .0383344 2.91 0.004 1.03354 1.183924dep4 1.212892 .0437501 5.35 0.000 1.130104 1.301744dep5 1.309478 .0513313 6.88 0.000 1.212638 1.414051

. stpm2 dep2-dep5, df(5) scale(hazard) eform

exp(b) Std. Err. z P>|z| [95% Conf. Interval]

xbdep2 1.048752 .0354011 1.41 0.158 .9816125 1.120483dep3 1.10615 .0383334 2.91 0.004 1.033513 1.183893dep4 1.212872 .0437493 5.35 0.000 1.130085 1.301722dep5 1.309479 .0513313 6.88 0.000 1.212639 1.414052

_rcs1 2.126897 .0203615 78.83 0.000 2.087361 2.167182_rcs2 .9812977 .0074041 -2.50 0.012 .9668927 .9959173_rcs3 1.057255 .0043746 13.46 0.000 1.048715 1.065863_rcs4 1.005372 .0020877 2.58 0.010 1.001288 1.009472_rcs5 1.002216 .0010203 2.17 0.030 1.000218 1.004218


Sensitivity to choice of knots

Hazard Ratios are generally insensitive to the number andlocation of knots.

Too many knots will overfit baseline hazard with local ‘humpsand bumps’.

Too few knots will underfit.

In most situations the choice of knots is not crucial.

We can use the AIC and BIC to help us select how many knotsto use, but a simple sensitivity analysis is recommended.


Example of different knots for baseline hazard

0

25

50

75

100

Pre

dic

ted M

ort

alit

y R

ate

(per

1000 p

y)

0 1 2 3 4 5Time from Diagnosis (years)

1 df: AIC = 53746.92, BIC = 53788.35

2 df: AIC = 53723.60, BIC = 53771.93

3 df: AIC = 53521.06, BIC = 53576.29

4 df: AIC = 53510.33, BIC = 53572.47

5 df: AIC = 53507.78, BIC = 53576.83

6 df: AIC = 53511.59, BIC = 53587.54

7 df: AIC = 53510.06, BIC = 53592.91

8 df: AIC = 53510.78, BIC = 53600.54

9 df: AIC = 53509.62, BIC = 53606.28

10 df: AIC = 53512.35, BIC = 53615.92


Effect of number of knots on hazard ratios

13579

1 1.1 1.2 1.3 1.4

Deprivation Group 2

13579

1 1.1 1.2 1.3 1.4

Deprivation Group 3

13579

1 1.1 1.2 1.3 1.4

Deprivation Group 4

13579

1 1.1 1.2 1.3 1.4

Deprivation Group 5

df

for

Splin

es

Hazard Ratio


Where to place the knots?

The default knots positions tend to work fairly well.

Unless the knots are in silly places then there is usually very littledifference in the fitted values.

The graphs on the following page shows for 5 df (4 interiorknots) the fitted hazard and survival functions with the interiorknot locations randomly selected.


Random knot positions for baseline hazard

0

25

50

75

100

Pre

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(per

1000 p

y)


13.7 55.8 60.5 64.3

6.1 10.9 61.8 68.4

4.5 25.5 55.5 87.1

42.4 52.2 84.1 89.8

21.1 26.5 56.4 94.8

11.8 27.7 40.8 72.2

42.2 46.1 87.2 89.4

5.8 67.6 69.9 71.5

9.8 23.2 35.3 59.5

10.2 10.9 57.7 80.7


Effect of location of knots on baseline survival

.7

.8

.9

1

Pre

dic

ted S

urv

ival


13.7 55.8 60.5 64.3

6.1 10.9 61.8 68.4

4.5 25.5 55.5 87.1

42.4 52.2 84.1 89.8

21.1 26.5 56.4 94.8

11.8 27.7 40.8 72.2

42.2 46.1 87.2 89.4

5.8 67.6 69.9 71.5

9.8 23.2 35.3 59.5

10.2 10.9 57.7 80.7


Why We Need Flexible Models

There are a number of parametric models available, so why can’twe just use these?

For proportional hazards only ‘simple’ models available:Exponential, Weibull, Gompertz.

More complex models such as generalized gamma only availablein accelerated failure form.

These models still may not capture the underlying shape of thedata.

In Stata the most complex parametric survival distributionavailable is the generalized gamma.


Why We Need Flexible Models

0.04

0.05

0.06

0.07

0.08

0.09

Mort

alit

y R

ate


Smoothed hazard function

Hazard (Gamma)

Hazard (stpm2)


Comparison with Cox model

Simulation where true baseline hazard is complex.

‘Truth’ is a mixture of two Weibull distributions. E.g.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Hazard

Function


Model with dichotomous covariate effect, β = −0.5.

Simulate 1000 datasets each with sample size = 3000.


Agreement between parameter estimates

-.6

-.55

-.5

-.45

-.4

Cox M

odel

-.6 -.55 -.5 -.45 -.4Flexible Parametric Model


Agreement between standard errors

.0384

.0386

.0388

.039

.0392

.0394

Cox M

odel

.0384 .0386 .0388 .039 .0392 .0394Flexible Parametric Model


Time dependent effects

An important feature of flexible parametric models is the abilityto model time-dependent effects, i.e., there are non-proportionalhazards

Time-dependent effects are modelled using splines, but willgenerally require fewer knots than the baseline.

This is because we are now modelling deviation from thebaseline hazard rate.

Also possible to split time to estimate hazard ratio in differentintervals.


Time-Dependent Effects

A proportional cumulative hazards model can be written

ln [Hi(t|xi)] = ηi = s (ln(t)|γ, k0) + xiβ

New set of spline variables for each time-dependent effect [4]

If there are D time-dependent effects then

ln [Hi(t|xi)] = s (ln(t)|γ, k0) +D∑

j=1

s (ln(t)|δj , kj)xij + xiβ

The number of spline variables for a particular time-dependenteffect will depend on the number of knots, kj

Interaction between the covariate and the spline variables.


stpm2 and Time-Dependent Effects

Non-proportional effects can be fitted by use of the tvc() anddftvc() options.

Non-proportional hazards models. stpm2 dep5, scale(hazard) df(5) tvc(dep5) dftvc(3)

There is no need to split the time-scale when fittingtime-dependent effects.

When time-dependence is a linear function of ln(t) andN = 50, 000, 50% censored and no ties.

stcox using tvc() - 28 minutes, 24 seconds.stpm2 using dftvc(1) - 0 minutes, 2.5 seconds.


Example of Attained Age as the Time-scale

Study from Sweden[5] comparing incidence of hip fracture of,

17,731 men diagnosed with prostate cancer treated withbilateral orchiectomy.43,230 men diagnosed with prostate cancer not treated withbilateral orchiectomy.362,354 men randomly selected from the general population.

Outcome is femoral neck fractures.

Risk of fracture varies by age.

Age is used as the main time-scale.

Alternative way of “adjusting” for age.

Gives the age specific incidence rates.


Estimates from a PH Model

stset using age as the time-scale. stset dateexit,fail(frac = 1) enter(datecancer) origin(datebirth) ///

id(id) scale(365.25) exit(time datebirth + 100*365.25)

. stcox noorc orc

Cox ModelIncidence rate ratio (no orchiectomy) = 1.37 (1.28 to 1.46)Incidence rate ratio (orchiectomy) = 2.10 (1.93 to 2.28)

. stpm2 noorc orc, df(5) scale(hazard)

Flexible Parametric ModelIncidence rate ratio (no orchiectomy) = 1.37 (1.28 to 1.46)Incidence rate ratio (orchiectomy) = 2.10 (1.93 to 2.28)


Proportional Hazards

.1

1

5

10

25

5075

Incid

ence R

ate

(per

1000 p

y’s

)

40 60 80 100Age

Control

No Orchiectomy

Orchiectomy


Non Proportional Hazards

.1

1

5

10

25

5075

Incid

ence R

ate

(per

1000 p

y’s

)

40 60 80 100Age

Control

No Orchiectomy

Orchiectomy


Incidence Rate Ratio

1

2

5

10

20

Incid

ence R

ate

Ratio

50 60 70 80 90 100Age

horizontal lines from piecewise Poisson model

Orchiectomy vs Control


Incidence Rate Difference

0

10

20

30

Diffe

rence in I

ncid

ence R

ate

s(p

er

1000 p

ers

on y

ears

)

50 60 70 80 90 100Age

Orchiectomy vs Control


Relative Survival

Three of today’s talks fit relative survival models.

Relative survival is a measure used in population based cancerstudies.

Used as unreliable (or missing) cause of death information.

Incorporates expected mortality,

ObservedMortality Rate

=Expected

Mortality Rate+

Excess

Mortality Rate

h(t) = h∗(t) + λ(t)

If we transform to the survival scale,

Relative Survival =Observed Survival

Expected SurvivalR(t) =

S(t)

S∗(t)


Modelling Relative Survival

Flexible parametric survival models extended to relativesurvival[6].

When using these models we estimate

the excess hazard (mortality) rate rather than the hazard rate.the relative survival function rather than the survival function.excess hazard ratios and excess hazard differences.

All cause, cause-specific and relative survival analysed withinsame framework.


Breast Cancer Data

Comparison of breast cancer survival in England and Norway[7, 8].

The data consists of

303,657 women from England.24,919 women from Norway.Year of Diagnosis was between 1996 and 2004.

Model includes

Baseline hazard (splines)Age (splines)CountryAge Country Interaction.Time-dependent effects for age & country (splines).


Relative Survival

0.4

0.6

0.8

1.0

Rela

tive S

urv

ival

0 2 4 6 8Years from Diagnosis

Age 35

0.4

0.6

0.8

1.0

Rela

tive S

urv

ival


Age 45

0.4

0.6

0.8

1.0R

ela

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ival


Age 55

0.4

0.6

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1.0

Rela

tive S

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ival


Age 65

0.4

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1.0

Rela

tive S

urv

ival


Age 75

0.4

0.6

0.8

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Rela

tive S

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ival


Age 85


Excess Mortality Rates

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Excess M

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Age 35

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Age 45

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Age 55

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1000 p

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Age 65

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Excess M

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(per

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Age 75

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Age 85


Excess Mortality Rate Ratios

1

2

3

0 2 4 6 8

Age 35

1

2

3

0 2 4 6 8

Age 45

1

2

3

0 2 4 6 8

Age 55

1

2

3

0 2 4 6 8

Age 65

1

2

3

0 2 4 6 8

Age 75

1

2

3

0 2 4 6 8

Age 85E

xcess M

ort

alit

y R

ate

Ratio

Years from Diagnosis


Differences in Excess Mortality

−10

0

10

20

30

40

50

0 2 4 6 8

Age 35

−10

0

10

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30

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Age 45

−10

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Age 55

−10

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Age 65

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300

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Age 75

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300

0 2 4 6 8

Age 85

Diffe

rence in E

xcess M

ort

alit

y (

1000pys)

Years from Diagnosis

(b)


Software

To install stpm2 using

. ssc install stpm2

Some examples inLambert, P. C., Royston, P. Further development of flexible parametric modelsfor survival analysis. The Stata Journal 2009;9:265-290. [9]

Many more examples plus downloadable do files in


Other Issues

Topics we may discuss todayNumber and locations of knots.

Many time-dependent effects.

Hazard vs cumulative hazard scale.

Model selection (in large studies).

Absolute vs relative effects.

When to use flexible parametric models.

Other alternatives.


References

[1] Reid N. A conversation with Sir David Cox. Statistical Science 1994;9:439–455.

[2] Durrleman S, Simon R. Flexible regression models with cubic splines. Statistics in Medicine

1989;8:551–561.

[3] Royston P, Parmar MKB. Flexible parametric proportional-hazards and proportional-oddsmodels for censored survival data, with application to prognostic modelling and estimationof treatment effects. Statistics in Medicine 2002;21:2175–2197.

[4] Lambert PC, Dickman PW, Nelson CP, Royston P. Estimating the crude probability ofdeath due to cancer and other causes using relative survival models. Statistics in Medicine

2010;29:885 – 895.

[5] Dickman PW, Adolfsson J, Astrm K, Steineck G. Hip fractures in men with prostate cancertreated with orchiectomy. Journal of Urology 2004;172:2208–2212.

[6] Nelson CP, Lambert PC, Squire IB, Jones DR. Flexible parametric models for relativesurvival, with application in coronary heart disease. Statistics in Medicine 2007;26:5486–5498.

[7] Lambert PC, Holmberg L, Sandin F, Bray F, Linklater KM, Purushotham A, et al..Quantifying differences in breast cancer survival between England and Norway. CancerEpidemiology in press 2011;.


References 2

[8] Møller H, Sandin F, Bray F, Klint A, Linklater KM, Purushotham A, et al.. Breast cancersurvival in England, Norway and Sweden: a population-based comparison. InternationalJournal of Cancer 2010;127:2630–2638.

[9] Lambert PC, Royston P. Further development of flexible parametric models for survivalanalysis. The Stata Journal 2009;9:265–290.


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