Parametric modelling of the cumulative incidence … · Parametric modelling of the cumulative incidence function in competing risks models ... 2 Department of Medical Epidemiology

Parametric modelling of the cumulative

incidence function in competing risks models

Paul C Lambert1,2

1Department of Health Sciences,University of Leicester, UK

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

Fundamental Problems in Survival AnalysisLondon 4/06/2014

Modelling with competing risks data

The two main modelling approaches.

Modelling cause specific hazards

hk(t|x) = hk,0(t) exp (xβ)

CIFk(t) =

∫ t

0

hk(u) exp

(−∫ u

0

K∑i=1

hi(s)ds

)du

Modelling subhazard

h∗k(t) =d log (1− CIFk(t))

dt

h∗k(t|x) = h∗k,0(t) exp (xβ)

My preference is to use (flexible) parametric models.

Paul C Lambert Parametric Cumulative Incidence London 4/06/2014 2

Modelling with competing risks data

The two main modelling approaches.

Modelling cause specific hazards

hk(t|x) = hk,0(t) exp (xβ)

CIFk(t) =

∫ t

0

hk(u) exp

(−∫ u

0

K∑i=1

hi(s)ds

)du

Modelling subhazard

h∗k(t) =d log (1− CIFk(t))

dt

h∗k(t|x) = h∗k,0(t) exp (xβ)

My preference is to use (flexible) parametric models.Paul C Lambert Parametric Cumulative Incidence London 4/06/2014 2

Outline of the rest of the talk

1 Flexible parametric models

2 Modelling cause-specific hazards (brief)

3 Modelling cumulative incidence functions.


European Blood and Marrow Transplantation Data

1977 patients from the European Blood and MarrowTransplantation (EBMT) registry who received an allogeneicbone marrow transplantation[1].

Events are death and relapse

836 censored456 relapse685 died

One covariate of interest, the EBMT risk score, which has beencategorized into 3 groups (low, medium and high risk).

Proportional hazards / proportional subhazards not appropriatefor relapse.


Cumulative Incidence Functions

0.0

0.1

0.2

0.3

0.4

0.5

CIF

0 2 4 6Years since transplantation

Low Risk

Medium Risk

High Risk

Relapse

0.0

0.1

0.2

0.3

0.4

0.5

CIF


Low Risk

Medium Risk

High Risk

Died


Parametric Models

General link function[2].

g [S(t|xi)] = ηi = s (t|γ) + xiβ

s (t|γ) is a smooth parametric function of time.

Proportional hazards model

g(x) = log[− log(x)]

Proportional odds model

g(x) = log[(1− x)/x ]

Aranda-Ordaz link function

g(x |θ) = log

(x−θ − 1

θ

)


Parametric Models







g(x) = log[(1− x)/x ]


g(x |θ) = log

(x−θ − 1

θ

)


Parametric Models







g(x) = log[(1− x)/x ]


g(x |θ) = log

(x−θ − 1

θ

)


Parametric Models







g(x) = log[(1− x)/x ]


g(x |θ) = log

(x−θ − 1

θ

)Paul C Lambert Parametric Cumulative Incidence London 4/06/2014 6

Parametric models

I will concentrate on models on the log cumulative hazard scale.

Hi(t) = H0(t) exp(xiβ)

ln(Hi(t)) = ln(H0(t)) + xiβ

Need to decide on functional form for ln(H0(t)).

ln(H0(t)) can take standard parametric form (e.g. Weibull), butalso more general non-linear function such as splines [3, 4].

Also can model on log hazard scale.

hi(t) = h0(t) exp(xiβ) ln(hi(t)) = ln(h0(t)) + xiβ

Need numerical integration or Poisson regression for complexfunctions (e.g. splines) [5, 6].


Parametric models

I will concentrate on models on the log cumulative hazard scale.

Hi(t) = H0(t) exp(xiβ)

ln(Hi(t)) = ln(H0(t)) + xiβ

Need to decide on functional form for ln(H0(t)).

ln(H0(t)) can take standard parametric form (e.g. Weibull), butalso more general non-linear function such as splines [3, 4].

Also can model on log hazard scale.

hi(t) = h0(t) exp(xiβ) ln(hi(t)) = ln(h0(t)) + xiβ

Need numerical integration or Poisson regression for complexfunctions (e.g. splines) [5, 6].


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 assuming linearity with ln(t) flexible parametricmodels use restricted cubic splines for ln(t).


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 assuming linearity with ln(t) flexible parametricmodels use restricted cubic splines for ln(t).


Flexible parametric models: incorporating splines

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 baseline

cumulative hazard

+ xiβ︸︷︷︸log hazard

ratios

We are fitting a linear predictor on the log cumulative hazardscale.







cumulative hazard


ratios








cumulative hazard


ratios








cumulative hazard


ratios



Survival and Hazard Functions

We can transform to the survival function

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 feed into the likelihood. No needfor numerical integration or time-splitting.







dtexp(ηi)










dtexp(ηi)





Simulation Study (Rutherford et al.)[7]

Generate data assuming a mixture Weibull distribution,

S(t) = π exp(−λ1tγ1) + (1− π) exp(−λ2t

γ2)

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 1

0.0

0.5

1.0

1.5

2.0

2.5

Haz

ard

rate


Scenario 2

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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

Fit models using restricted cubic splines.


Simulation Study (Rutherford et al.)[7]

Generate data assuming a mixture Weibull distribution,

S(t) = π exp(−λ1tγ1) + (1− π) exp(−λ2t

γ2)

0.0

0.5

1.0

1.5

2.0

2.5H

azar

d ra

te


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

Fit models using restricted cubic splines.Paul C Lambert Parametric Cumulative Incidence London 4/06/2014 11

Scenario 3 comparison of Log Hazard Ratios

-.6

-.55

-.5

-.45

-.4C

ox M

odel

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


Scenario 3 comparison of Log Hazard Ratios

-.6

-.55

-.5

-.45

-.4C

ox M

odel

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


Restricted cubic splines vs true model

0

5

10

15

20

25

Perc

enta

ge o

f T

ota

l A

rea D

iffe

rence

on t

he S

urv

ival S

cale

1 2 3 4 5 6 7 8 9 10Degrees of Freedom

Sample Size 300

Sample Size 3000

Sample Size 30,000

Scenario 3

More details in Rutherford et al.[7].Paul C Lambert Parametric Cumulative Incidence London 4/06/2014 13

Modelling Cause-specific hazards

ln[Hk(t|xi)] = ln [Hk,0(t)] + xiβ

Fit all K causes simultaneously.

Can share parameters across causes.

Can incorporate time dependent effects for some covariates.

Parametric functions for hazard/survival functions.

CIF (and standard error) obtained using numerical integration.

CIFk(t) =

∫ t

0

hk(u) exp

(−

K∑i=1

Hi(u)

)du

See Hinchliffe et al. [8]


Modelling Cause-specific hazards

ln[Hk(t|xi)] = ln [Hk,0(t)] + xiβ

Fit all K causes simultaneously.

Can share parameters across causes.

Can incorporate time dependent effects for some covariates.

Parametric functions for hazard/survival functions.

CIF (and standard error) obtained using numerical integration.

CIFk(t) =

∫ t

0

hk(u) exp

(−

K∑i=1

Hi(u)

)du

See Hinchliffe et al. [8]


Estimated hazard functions

0

200

400

600

800

Rel

apse

Rat

e (p

er 1

000

py)


Low Risk

Medium Risk

High Risk

Relapse

0

200

400

600

800

1000

1200

Mor

talit

y R

ate

(per

100

0 py

)


Low Risk

Medium Risk

High Risk

Death


CIFs from Cox and Flexible Parametric PH Models

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0.1

0.2

0.3

0.4

0.5

CIF


Low Risk

Medium Risk

High Risk

Relapse

0.0

0.1

0.2

0.3

0.4

0.5

CIF


Low Risk

Medium Risk

High Risk

Died


CIFs from Cox and Flexible Parametric PH Models

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0.1

0.2

0.3

0.4

0.5

CIF


Low Risk

Medium Risk

High Risk

Relapse

0.0

0.1

0.2

0.3

0.4

0.5

CIF


Low Risk

Medium Risk

High Risk

Died


CIFs from PH model vs Empirical

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0.2

0.3

0.4C

IF


Low RiskMedium RiskHigh Risk

Relapse


CIFs from PH model vs Empirical

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0.2

0.3

0.4C

IF



Relapse


Incorporating time-dependent effects

Relapse

ln[H1(t|xi)] = s (ln(t)|γ1, k1,0) + xiβ1 + s (ln(t)|δ1, k1,1) xi

Death

ln[H2(t|xi)] = s (ln(t)|γ2, k2,0) + xiβ2


Incorporating time-dependent effects

Relapse

ln[H1(t|xi)] = s (ln(t)|γ1, k1,0) + xiβ1 + s (ln(t)|δ1, k1,1) xi

Death

ln[H2(t|xi)] = s (ln(t)|γ2, k2,0) + xiβ2


Hazard functions for relapse

0

200

400

600

800R

elap

se R

ate

(100

0 py

)



Relapse


Hazard functions for relapse

0

200

400

600

800R

elap

se R

ate

(100

0 py

)



Relapse


CIFs from TVC model (for relapse) vs Empirical

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0.4C

IF



Relapse


CIFs from TVC model (for relapse) vs Empirical

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0.4C

IF



Relapse


Model Cumulative Incidence Functions

Now switch to modelling the CIF directly

Aim:

Predict CIF for individual subjects.Direct covariate effects on CIF.Use different link functions.Model non-proportional effects.Be able to fit to large data sets.


Geskus (Biometrics 2011) Approach

Geskus (2011) showed that weighted versions of standardsurvival analysis procedures can be used to estimate the CIF[9].

Uses inverse probability of censoring weights.

Weights are time-dependent so data is expanded for subjectswith competing events.

After data expansion

Weighted Kaplan-Meier gives product-limit estimate of CIF.Weighted Cox model gives Fine and Gray model.

crprep in R and stcrprep in Stata.

Using similar weights in a likelihood based setting enablesstandard parametric models to be used to directly model the CIF.





























Data expansion and weighting

Define event of interest.

Subjects that have a competing event are kept in the risk set tothe end of follow-up.

However, there is a a chance that they would be censored aftertheir competing event.

Estimate censoring distribution.

Weights depend on conditional probability of not being censoredafter competing event.


Data Expansion for Competing Events

9

8

7

6

5

4

3

2

1

Time

Censored Cancer Other



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5

4

3

2

1

Time




w81 w82 w83 w84

w41 w42

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4

3

2

1

Time




w81 w82 w83 w84

w41 w42

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3

2

1

Time



Large datasets

I analyse registry data, which is often large.

Datasets of 500,000 is not unusual.

Method needs to be usable in these large datasets.


Parametric approach

For those with competing events, allow to be at risk to end ofpotential follow-up.Split follow-up from competing event into (small) time-intervals.Apply weights to each interval.

Likelihood

ln Li = d1i ln [h∗(ti)] + (1− d2i) ln [1− CIF (ti)] +

d2i

Ji∑j=1

wij

(ln [1− CIF (tij)]− ln

[1− CIF (ti(j−1))

])Need to specify parametric form of CIF for event of interest, butnot for competing events.Also need weighting function. Could be obtained through KMestimate or modelling.


Splitting

Censored

Event 2

Event 1

Time


Splitting

Censored

Event 2

Event 1

Time


Splitting

Censored

Event 2

Event 1

Time


Splitting

w1 w2 w3 w4 w5

Censored

Event 2

Event 1

Time


The censoring distribution

Geskus used KM estimate for censoring distribution (withpossibility to stratify by covariates of interest).

We choose to fit parametric model with option to include avariety of covariates.

Need to decide where to evaluate censoring distribution (numberof split points) for weighted likelihood.

0.0

0.2

0.4

0.6

0.8

1.0

S(t

)

0 2 4 6 8Years since transplantation

Kaplan-MeierParametric Model






0.0

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0.6

0.8

1.0

S(t

)








0.0

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0.4

0.6

0.8

1.0

S(t

)




Flexible parametric models

Possible to use any parametric approach that allows for delayedentry and weights.

We use flexible parametric survival models that uses restrictedsplines to model the baseline using stpm2 in Stata. [10, 4].

g [1− CIFk(t|xi)] = ηi = s (ln(t)|γ, k0) + xiβ


Link Functions

When using weights with expanded data

proportional sub hazards

log(− log (1− CIF (t|xi))) = s (ln(t)|γ, k0) + xiβ

proportional odds

log

(CIF (t|xi)

1− CIF (t|xi)

)= s (ln(t)|γ, k0) + xiβ

relative absolute risk

log (CIF (t|xi)) = s (ln(t)|γ, k0) + xiβ


Simulation

When using splines we are never fitting the ‘true’ model.

When assessing performance using simulation we simulate froma complex hazard function and then compare the truth to ourapproximation using splines.

For this study we simulated data as follows,

2 types of events. Event 1 of primary interest.Subhazard for event 1 from mixture Weibull distribution.Cause-specific hazard for event 2 from mixture Weibulldistribution.Using a combination of numerical integration and root finding itis possible to obtain the cause-specific hazard for event 1.


Simulation

When using splines we are never fitting the ‘true’ model.

When assessing performance using simulation we simulate froma complex hazard function and then compare the truth to ourapproximation using splines.

For this study we simulated data as follows,

2 types of events. Event 1 of primary interest.Subhazard for event 1 from mixture Weibull distribution.Cause-specific hazard for event 2 from mixture Weibulldistribution.Using a combination of numerical integration and root finding itis possible to obtain the cause-specific hazard for event 1.


Simulation

Given cause-specific hazards for both causes use methods ofBeyersmann[11] to simulate data.

Based on general survival simulation algorithm, Crowther et al.2013 [12]

Follow-up restricted to 5 years.

Censoring distribution generated from Weibull distribution.

Probability of censoring by 5 years 0.3-0.4.

1000 simulated datasets for each scenario.

Flexible Parametric models with 3, 5 and 7 df for baseline fitted.

Splits every 0.01, 0.1, 0.2, 0.5 and 1 years used for weighting.


Scenario 4

0.00

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0.30

0.40

CIF

0 1 2 3 4 5Time

x=0

x=1

Cause 1: CIF

0.00

0.05

0.10

0.15

caus

e-sp

ecifi

c su

bhaz

ard

rate

0 1 2 3 4 5Time

x=0

x=1

Cause 1: subhazard rate


Scenario 4

0.00

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0.20

0.30

0.40

CIF

0 1 2 3 4 5Time

x=0

x=1

Cause 1: CIF

0.00

0.05

0.10

0.15

caus

e-sp

ecifi

c su

bhaz

ard

rate

0 1 2 3 4 5Time

x=0

x=1


0.040

0.060

0.080

0.100

0.120

caus

e-sp

ecifi

c ha

zard

rat

e

0 1 2 3 4 5Time

x=0

x=1

Cause 2: cause-specific hazard


Scenario 4

0.00

0.10

0.20

0.30

0.40

CIF

0 1 2 3 4 5Time

x=0

x=1

Cause 1: CIF

0.00

0.05

0.10

0.15

caus

e-sp

ecifi

c su

bhaz

ard

rate

0 1 2 3 4 5Time

x=0

x=1


0.040

0.060

0.080

0.100

0.120

caus

e-sp

ecifi

c ha

zard

rat

e

0 1 2 3 4 5Time

x=0

x=1


0.000

0.050

0.100

0.150

caus

e-sp

ecifi

c ha

zard

rat

e

0 1 2 3 4 5Time

x=0

x=1



Scenario 4 Bias: Simulation Results

Bias (3 df for splines)Splits per year

0.01 0.1 0.2 0.5 1β 0.00022 0.00022 0.00006 0.0072 -0.00038CIF1 -0.00022 -0.00022 -0.00023 -0.00026 -0.00029CIF2 -0.00036 -0.00037 -0.00037 -0.00044 -0.00067CIF3 -0.00039 -0.00039 -0.00042 0.00072 -0.00182CIF4 -0.00079 -0.00079 -0.00085 -0.00129 -0.00315CIF5 -0.00123 -0.00123 -0.00130 -0.00191 -0.00456

Very similar results for other scenarios.

Coverage close to 95%.


Scenario 4 Bias: Simulation Results

Bias (3 df for splines)Splits per year

0.01 0.1 0.2 0.5 1β 0.00022 0.00022 0.00006 0.0072 -0.00038CIF1 -0.00022 -0.00022 -0.00023 -0.00026 -0.00029CIF2 -0.00036 -0.00037 -0.00037 -0.00044 -0.00067CIF3 -0.00039 -0.00039 -0.00042 0.00072 -0.00182CIF4 -0.00079 -0.00079 -0.00085 -0.00129 -0.00315CIF5 -0.00123 -0.00123 -0.00130 -0.00191 -0.00456

Very similar results for other scenarios.

Coverage close to 95%.


Comparison of Models

Group Spline ModelsFG log(− log) logit log

RelapseLow Risk - - - -Medium Risk 1.34 1.34 1.42 1.27

(1.04,1.72) (1.04,1.72) (1.08,1.88) (1.03,1.56)High Risk 1.88 1.88 2.18 1.65

(1.31,2.70) (1.31,2.70) (1.44,3.30) (1.24,2.19)

DeathLow Risk - - - -Medium Risk 1.78 1.78 1.95 1.62

(1.42,2.22) (1.42,2.22) (1.52,2.52) (1.26,2.08)High Risk 2.70 2.70 3.37 2.17

(2.00,3.64) (2.00,3.65) (2.33,4.87) (1.63,2.90)





(1.04,1.72) (1.04,1.72) (1.08,1.88) (1.03,1.56)High Risk 1.88 1.88 2.18 1.65

(1.31,2.70) (1.31,2.70) (1.44,3.30) (1.24,2.19)


(1.42,2.22) (1.42,2.22) (1.52,2.52) (1.26,2.08)High Risk 2.70 2.70 3.37 2.17

(2.00,3.64) (2.00,3.65) (2.33,4.87) (1.63,2.90)





(1.04,1.72) (1.04,1.72) (1.08,1.88) (1.03,1.56)High Risk 1.88 1.88 2.18 1.65

(1.31,2.70) (1.31,2.70) (1.44,3.30) (1.24,2.19)


(1.42,2.22) (1.42,2.22) (1.52,2.52) (1.26,2.08)High Risk 2.70 2.70 3.37 2.17

(2.00,3.64) (2.00,3.65) (2.33,4.87) (1.63,2.90)





(1.04,1.72) (1.04,1.72) (1.08,1.88) (1.03,1.56)High Risk 1.88 1.88 2.18 1.65

(1.31,2.70) (1.31,2.70) (1.44,3.30) (1.24,2.19)


(1.42,2.22) (1.42,2.22) (1.52,2.52) (1.26,2.08)High Risk 2.70 2.70 3.37 2.17

(2.00,3.64) (2.00,3.65) (2.33,4.87) (1.63,2.90)


Predicted Cumulative Incidence Functions

Fine and Gray vs Parametric (proportional sub hazards)

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0 1 2 3 4 5Years since transplantation

Relapse

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Death

Low Risk Medium Risk High Risk


Predicted Cumulative Incidence Functions

Fine and Gray vs Parametric (proportional sub hazards)

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Relapse

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Death



Evidence of non-proportionality

Empirical vs Parametric (proportional subhazards)

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Relapse

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Death





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Relapse

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ulat

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Death





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Relapse

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Time-dependent effects

Time-dependent effects fitted by forming an interaction betweenspline terms and covariate of interest.

Generally use fewer knots for time-dependent effects than forbaseline.

Can be used with any link function.



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Low RiskMedium RiskHigh Risklog(-log) link

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Software

Time (in seconds).

EBMT data ×10 (19770 subjects): no ties

stcrreg - 2814stcrprep - 922stcox - 49

stcrprep only needs to be run once!

Parametric model: splits every 0.25 years

stpm2 - 167


Discussion

Parametric models give essentially same hazard/subhazard ratiosas Cox / F&G models.Useful to have parametric function for hazard or CIF.Modelling time-dependent effects.My preference is to model cause-specific hazards

When modelling CIF, no need to model other competingevent(s).Alternative would be to directly model CIF for all eventtypes[13].Splits for censoring distribution can be fairly crude (importantfor large datasets).Range of link funtions.In flexible parametric models knots choice not crucial.Software available crprep in R and stcrprep in Stata, then usestandard survival methods.


Discussion

Parametric models give essentially same hazard/subhazard ratiosas Cox / F&G models.Useful to have parametric function for hazard or CIF.Modelling time-dependent effects.My preference is to model cause-specific hazardsWhen modelling CIF, no need to model other competingevent(s).Alternative would be to directly model CIF for all eventtypes[13].

Splits for censoring distribution can be fairly crude (importantfor large datasets).Range of link funtions.In flexible parametric models knots choice not crucial.Software available crprep in R and stcrprep in Stata, then usestandard survival methods.


Discussion

Parametric models give essentially same hazard/subhazard ratiosas Cox / F&G models.Useful to have parametric function for hazard or CIF.Modelling time-dependent effects.My preference is to model cause-specific hazardsWhen modelling CIF, no need to model other competingevent(s).Alternative would be to directly model CIF for all eventtypes[13].Splits for censoring distribution can be fairly crude (importantfor large datasets).Range of link funtions.

In flexible parametric models knots choice not crucial.Software available crprep in R and stcrprep in Stata, then usestandard survival methods.


Discussion

Parametric models give essentially same hazard/subhazard ratiosas Cox / F&G models.Useful to have parametric function for hazard or CIF.Modelling time-dependent effects.My preference is to model cause-specific hazardsWhen modelling CIF, no need to model other competingevent(s).Alternative would be to directly model CIF for all eventtypes[13].Splits for censoring distribution can be fairly crude (importantfor large datasets).Range of link funtions.In flexible parametric models knots choice not crucial.Software available crprep in R and stcrprep in Stata, then usestandard survival methods.


References

[1] de Wreede L, Fiocco M, Putter H. mstate: An r package for the analysis of competingrisks and multi-state models. Journal of Statistical Software 2011;38.

[2] Younes N, Lachin J. Link-based models for survival data with interval and continuous timecensoring. Biometrics 1997;53:1199–1211.

[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] Royston P, Lambert PC. Flexible parametric survival analysis in Stata: Beyond the Coxmodel . Stata Press, 2011.

[5] Carstensen B. Demography and epidemiology: Practical use of the Lexis diagram in thecomputer age or: Who needs the Cox model anyway? Tech. rep., Department ofBiostatistics, University of Copenhagen, 2006.

[6] Crowther MJ, Lambert PC. stgenreg: A stata package for general parametric survivalanalysis. Journal of Statistical Software 2013;53:1–17.

[7] Rutherford MJ, Crowther MJ, Lambert PC. The use of restricted cubic splines toapproximate complex hazard functions in the analysis of time-to-event data: a simulationstudy. Journal of Statistical Computation and Simulation 2014 (in press);.


References 2

[8] Hinchliffe SR, Lambert PC. Flexible parametric modelling of cause-specific hazards toestimate cumulative incidence functions. BMC Medical Research Methodology 2013;13:13.

[9] Geskus RB. Cause-specific cumulative incidence estimation and the fine and gray modelunder both left truncation and right censoring. Biometrics 2011;67:39–49.

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

[11] Beyersmann J, Latouche A, Buchholz A, Schumacher M. Simulating competing risks datain survival analysis. Stat Med 2009;28:956–971.

[12] Crowther MJ, Lambert PC. Simulating biologically plausible complex survival data.Statistics in Medicine 2013;32:41184134.

[13] Jeong JH, Fine JP. Parametric regression on cumulative incidence function. Biostatistics2007;8:184–196.


Sensitivity Analysis

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df(4) dftvc(3)df(3) dftvc(3)df(5) dftvc(3)df(5) dftvc(4)df(7) dftvc(3)


Documents

Parametric modelling of the cumulative incidence … · Parametric modelling of the cumulative incidence function in competing risks models ... 2 Department of Medical Epidemiology