estimating survival from gray's exible model outline · p (y y < y + yjy y) y; (1) where p (:)...

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Estimating survival from Gray’sflexible model

Zdenek Valenta

Department of Medical InformaticsInstitute of Computer Science

Academy of Sciences of the Czech Republic


Outline

I. Introduction

II. Semi–parametric survival models

III. Gray’s model introduction

IV. Survival estimates based on semi-parametric models

V. Estimating survival from Gray’s model (with examples)

VI. Impact of misspecifying the survival model – simulation studyresults

VII. Discussion


I. Introduction

• Let Y be a random variable capturing time to occurrence of acertain event of interest. The hazard function h(y) is at timey formally defined as follows:

h(y) = lim∆y→0

P (y ≤ Y < y +∆y|Y ≥ y)∆y

, (1)

where P (.) denotes conditional probability, that an event ofinterest would occur immediately after time y, given it did notprior to this time.

• It follows from (1) that the hazard function may only take non–negative values.


I. Introduction

• Let F (Y ) denote a cumulative distribution function of the ran-dom variable Y , i.e. F (Y ) = P (Y ≤ y). We assume that Yis absolutely continuous with density f (y). The expression (1)may be then written as:

h(y) = lim∆y→0

P (y ≤ Y < y +∆y|Y ≥ y)∆y

= lim∆y→0

P (y ≤ Y < y +∆y)P (Y ≥ y)∆y

=1

P (Y ≥ y)d

dyF (y) =

f (y)P (Y ≥ y)

=f (y)S(y)

,

(2)

where S(y) denotes the value of the survival functionat time y.


I. Introduction

• We define a cumulative hazard function H(t) at time t as:

H(t) =∫ t0

h(y)dy (3)

It follows from (2) that S(.) and H(.) capture equivalent infor-mation:

H(t) = − ln (S(t)) (4)

• Furthermore, it follows from (4) that we can determine thevalue of the survival function S(t) at time t whenever we areable to evaluate the cumulative hazard function H(t):

S(t) = exp {−H(t)} (5)



• Multiplicative models:– Cox PH model:

h(y|Z) = h0(y) · exp (β ′Z) (6)

– Gray’s flexible model:

h(y|Z) = h0(y) · exp {β(y)′Z} (7)

• Additive models:

– Aalen’s linear model:

h(y|Z) = h0(y) + β(y)′Z (8)



Note: Aalen’s linear model (8) may be embedded in the class ofmultiplicative models:

• Aalen’s model:

h(y|Z) = h0(y) + β(y)′Z

exp (h(y|Z)) = exp{h0(y) + β(y)

′Z}

h1(y|Z) = h10(y) · exp{β(y)′Z

}(9)

The class of multiplicative models represented by the Cox PH andGray’s flexible model includes the whole class of models pro-posed by Aalen.


III. Gray’s model introduction

• Let us recall the definition of Gray’s flexible model: (7):

h(y|Z) = h0(y) · exp {β(y)′Z}

• Gray’s model uses penalized B-splines for modelling time-varying effects β(y). B-splines allow for flexible modelling ofthe covariate effects β(y) and the hazard function over time.

• In the context of Gray’s model using piecewise-constanttime-varying regression coefficients the β(y) remain con-stant for y ∈ [τj−1, τj). We can thus write β(y) = βj =(βj1, βj2, . . . , βjp), where p denotes the number of model co-variates and j = 1, . . . ,M + 1 indexes the intervals on timeaxis. Here τj denote the knots that allow for a change of theregression coefficients βj.


Piecewise-constant vs. quadraticpenalized splines

500 1000 2000

−4

−3

−2

−1

0

1

2

Intervention

Days of Follow−up

Log

Haz

ard

Rat

io

500 1000 2000

−0.1

0.0

0.1

0.2

Age

Days of Follow−upLo

g H

azar

d R

atio

500 1000 2000

−1

0

1

2

3

Diabetes Mellitus

Days of Follow−up

Log

Haz

ard

Rat

io

500 1000 2000

−6

−4

−2

0

2

Intervention

Days of Follow−up

Log

Haz

ard

Rat

io

500 1000 2000−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

Age

Days of Follow−up

Log

Haz

ard

Rat

io

500 1000 2000

−1

0

1

2

3

4

Diabetes Mellitus

Days of Follow−up

Log

Haz

ard

Rat

io


Piecewise-constant vs. cubicpenalized splines

500 1000 2000

−4

−3

−2

−1

0

1

2

Intervention

Days of Follow−up

Log

Haz

ard

Rat

io

500 1000 2000

−0.1

0.0

0.1

0.2

Age

Days of Follow−up

Log

Haz

ard

Rat

io

500 1000 2000

−1

0

1

2

3

Diabetes Mellitus

Days of Follow−up

Log

Haz

ard

Rat

io

500 1000 2000

−5

0

5

Intervention

Days of Follow−up

Log

Haz

ard

Rat

io

500 1000 2000

−0.4

−0.2

0.0

0.2

0.4

Age

Days of Follow−up

Log

Haz

ard

Rat

io

500 1000 2000

−2

0

2

4

6

8

Diabetes Mellitus

Days of Follow−up

Log

Haz

ard

Rat

io


IV. Survival estimates based onsemi-parametric models

• Cox PH model:

S(t|Z) = exp{−

∫ t0

h0(y) · exp (β ′Z) dy}

= exp {−H0(t) · exp (β ′Z)}= [S0(t)]

exp(β′Z), (10)

where S0(t) represents baseline survival function estimateat time t.


IV. Survival estimates based onsemi-parametric models

• Aalen’s linear model:

h(y|Z) = h0(y) + β(y)′Z

h(y|Z) =∼β(y)′

∼Z,

(11)

while∼β(y) = (h0(y), β(y)) a

∼Z = (1, Z).

• Survival function estimates based on Aalen’s model use cumu-lative regression coefficients

∼B(t), where

∼Bi(t) =

∫ t0

∼βi(y)dy.

Estimating survival based on Aalen’s model may then proceedas follows:

S (t|Z) = exp{−

∼B(t)′

∼Z

}(12)


V. Estimating survival from Gray’smodel

• Survival function estimate based on Gray’s modela us-ing piecewise–constant penalized splines may be obtained asfollows:

S (t|Z) = exp

{−

M+1∑j=1

H0j (t) · exp(β ′jZ

)}, (13)

where Z denotes p-dimensional vector of patient’s characteris-tics, and

H0j(t) =∫

[τj−1,τj)

I (u ≤ t) dH0(u) (14)

represents a contribution to the cumulative baselinehazard function H0(t) on the interval [τj−1, τj), j =1, . . . ,M + 1.

aValenta Z et al, Statistics in Medicine 2002.



• Derivation of confidence limits for the survival function esti-mate based on Gray’s model uses the Delta method.

• Recall the Taylor formulae for a function f (X) of a randomvariable X with expectation µ:

f (X) =n∑

k=0

f (k)(µ)k!(X − µ)k +Rn (15)

• Delta method:Var(f (X)) ≈ Var [f (µ) + f ′(µ)(X − µ)]

= [f ′(µ)]2 · Var(X)(16)



• If X is a random vector the Delta method G(X) takes theform:

Var(G(X)) ≈ OG′(µ) · Var(X) · OG(µ), (17)where OG(µ) is a column vector of first partial derivatives of G.

• Confidence limits estimatesa were derived for a log– andlog(- log)–transformed survival function S(t) and are reportedsimultaneously in R.

aValenta Z et al, Statistics in Medicine 2002.


“coxspline” package in R


“cox.spline” R-routine


“cox.spline” R-routine (cont.)


R-function “gsurv.R”


R-function “gsurv.R” (cont.)


Example 1: survival estimates from Gray’s model with95% C.L. (log-transf.)

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Follow−up time (in days)

Est

imat

ed s

urvi

val w

ith 9

5% C

L

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Time−varying hazards dataIncreasing hazards (0.2,1,2)Decreasing hazards (2,1,0.2)Constant hazard (1,1,1)

Hazard change points are 0.6 and 0.8 years


Example 1: survival estimates from Gray’s model with95% C.L. (log-log-transf.)

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Follow−up time (in days)

Est

imat

ed s

urvi

val w

ith 9

5% C

L

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Time−varying hazards dataIncreasing hazards (0.2,1,2)Decreasing hazards (2,1,0.2)Constant hazard (1,1,1)

Hazard change points are 0.6 and 0.8 years


Example 2: Secondary prevention trial of CHD inLitomerice men after MI (Cox PH model results)

Cox

's s

urvi

val p

roba

bilit

y

500 1000 1500 2000

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Litomerice StudyIntervention GroupControl Group

Men 45 years of age

500 1000 1500 2000

0.4

0.5

0.6

0.7

0.8

0.9

1.0


Men 50 years of age

500 1000 1500 2000

0.4

0.5

0.6

0.7

0.8

0.9

1.0


Men 56 years of age

Follow−up time in days

500 1000 1500 2000

0.4

0.5

0.6

0.7

0.8

0.9

1.0


Men 60 years of age


Example 2: Secondary prevention trial of CHD inLitomerice men after MI (Gray’s model results)

500 1000 1500 2000

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Gra

y's

surv

ival

pro

babi

lity

500 1000 1500 2000

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Litomerice studyIntervention GroupControl Group

Men 45 years of age

500 1000 1500 2000

0.4

0.5

0.6

0.7

0.8

0.9

1.0


Men 50 years of age

500 1000 1500 2000

0.4

0.5

0.6

0.7

0.8

0.9

1.0


Men 56 years of age

Follow−up time in days

500 1000 1500 2000

0.4

0.5

0.6

0.7

0.8

0.9

1.0


Men 60 years of age



• Implementation of the “coxspline” package for R statisti-cal system is available from the Dr. Gray’s website (HarvardUniversity and Dana-Farber Cancer Institute, Boston, USA).

• Web address:

http://biowww.dfci.harvard.edu/~gray/

• Package “coxspline”, version 1.0-2, implements Gray’s modelin R, including the survival function estimation using R-function“gsurv.R”.

• Current version of the “coxspline” package is compatiblewith the latest release of R 2.3.1. (2006-06-01):

http://www.r-project.org/


VI. Impact of misspecifying thesurvival modela

• In three simulation studies we generated right-censored survivaldata that would satisfy exactly one of the semi-parametric sur-vival models under consideration (i.e. Aalen, Cox, Gray).

• The data obtained were subsequently analyzed using each ofthe three models considered.

• The performance of each model was assessed using the BiasandMean Square Error (MSE) of the estimated (conditional)survival distribution.

aValenta Z et al, Model misspecification effect in univariable regression models for right-censored survival data, Proceedings of the 2002 Joint Statistical Meeting of the AmericanStatistical Society.


VI. Impact of misspecifying thesurvival model

• Bias of the survival estimator Ŝ(t|Z):

Bias(Ŝ(t|Z)

)=1ns

ns∑i=1

Ŝ(i)(t|Z)− S(t|Z) (18)

• Mean Square Error of the estimated survival Ŝ(t|Z):

MSE(Ŝ(t|Z)

)=1ns

ns∑i=1

(Ŝ(i)(t|Z)− S(t|Z)

)2(19)

• Bias-variance trade-off:

MSE(Ŝ(t|Z)

)= var(Ŝ) +

2

Bias(Ŝ) (20)


Aalen’s model with constant β

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20Time

Bia

s0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

●●●●●●●●●●●●

●

●

●

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

●●●●●●●●●●●● ●

●

●

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

Aalen

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

Time

Bia

s

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

●●●●●●●●● ●●● ●●

●

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

●●●●●●●●●

●●●

● ●●

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

Cox

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

Time

Bia

s

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

●●●●●●●●● ●●● ●

●

●

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

●●●●●●●●●

●●●

●●

●

0.0 0.2 0.4 0.6 0.8

−0.

050.

050.

100.

150.

20

Gray

●

●

Percentile of z 1% 10% 25% 50% 75% 90% 99%

Legend

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

Time

MS

E R

elat

ive

to A

alen

's m

odel

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

●●●●●●

●●●

●●●

●

●●

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

●●

●

●

●●

●●●

●●

●

●

● ●

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

Cox

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

Time

MS

E R

elat

ive

to A

alen

's m

odel

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

●●

●

●●●●●● ●

●●●

●●

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

●

●

●

●

●●●●●

●●

●

●

● ●

0.0 0.2 0.4 0.6 0.8

−0.

50.

00.

51.

01.

5

Gray


Aalen’s model with time-varying β(t)

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

Time

Bia

s

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

●●●●●● ●● ● ●● ●

● ●

●

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

●●●●●● ● ● ● ●● ● ●

●

●

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

Aalen

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

TimeB

ias

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

●●●●●● ● ● ● ●● ● ●●

●

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

●●●

●●● ● ●●

●●●

● ●●

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

Cox

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

Time

Bia

s

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

●●●●●● ● ● ● ●●● ● ●

●

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

●●●

●●●● ● ●

●●●

●● ●

0.0 0.1 0.2 0.3 0.4

−0.

050.

000.

050.

100.

15

Gray

●

●

Percentile of z 1% 10% 25% 50% 75% 90% 99%

Legend

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

Time

MS

E R

elat

ive

to A

alen

's m

odel

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

●●●●●●

●● ●

●● ●

●

●

●

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

●●

●

●

●●

●● ●

●●

●

●

●●

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5Cox

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

Time

MS

E R

elat

ive

to A

alen

's m

odel

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

●●●

●●● ● ● ● ●

● ● ●

●

●

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

●

●

●

●

●● ● ● ●

●●

●

●

●●

0.0 0.1 0.2 0.3 0.4

−0.

50.

00.

51.

01.

5

Gray

censoring limit −>

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

Time

Bia

s

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

●●

●●●● ● ●

●●●●

● ● ●

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

●

●

●●●

●

●

●● ●●● ● ● ●

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

Cox

median survivaltime −>


0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

Time

Bia

s

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

● ● ● ●●● ● ● ● ●●● ● ● ●

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

●● ● ●●

● ●● ● ●●● ● ● ●

0.05 0.10 0.15

−0.

10.

10.

20.

30.

40.

5

Gray


0.05 0.10 0.15

010

2030

4050

Time

MS

E R

elat

ive

to G

ray'

s m

odel

0.05 0.10 0.15

010

2030

4050

● ●● ●●● ● ● ● ●●● ● ● ●

0.05 0.10 0.15

010

2030

4050

0.05 0.10 0.15

010

2030

4050

0.05 0.10 0.15

010

2030

4050

0.05 0.10 0.15

010

2030

4050

0.05 0.10 0.15

010

2030

4050

●● ● ●●●

● ● ●●●

●

●

●

0.05 0.10 0.15

010

2030

4050

Aalen



0.05 0.10 0.15

010

2030

4050

Time

MS

E R

elat

ive

to G

ray'

s m

odel

0.05 0.10 0.15

010

2030

4050

●● ● ●●● ● ● ● ●●● ● ● ●

0.05 0.10 0.15

010

2030

4050

0.05 0.10 0.15

010

2030

4050

0.05 0.10 0.15

010

2030

4050

0.05 0.10 0.15

010

2030

4050

0.05 0.10 0.15

010

2030

4050

● ● ● ●●●●

●

●

●

●

●

0.05 0.10 0.15

010

2030

4050

Cox



●

●

Percentile of z 1% 10% 25% 50% 75% 90% 99%

Legend


VII. Discussion

• When the data satisfied Aalen’s linear model, both Cox’s aGray’s model rendered biased survival estimates. They have,however, often shown a lower MSE than the native model forthe data at hand.

• When analyzing Cox PH model data using the Gray’s rou-tine, we observed no dramatic increase in bias and MSE relativeto native model, while using the same criteria the survival esti-mates based on Aalen’s model appeared to be highly distorted.

• When the data followed Gray’s model with time-varying co-variate effects, both Cox’s and Aalen’s model rendered in termsof bias and MSE highly unreliable estimates of the conditionalsurvival distribution. In other words, there was no alternativeto using the native model in this instance.


References

[1] Cox DR: Regression Models and Life Tables (with discussion), Journal of theRoyal Statistical Society, 1972, Vol. 34, pp. 187–220.

[2] Aalen OO: A linear regression model for the analysis of life times, Statistics inMedicine, 1989, Vol. 8, pp. 907–925.

[3] Gray RJ: Flexible methods for analyzing survival data using splines, with ap-plication to breast cancer prognosis, Journal of the American Statistical Associ-ation, 1992, Vol. 87, pp. 942–951.

[4] Gray RJ: Spline-based tests in survival analysis, Biometrics, 1994, Vol. 50., pp.640–652.

[5] Valenta Z and Weissfeld LA: Estimation of the Survival Function for Gray’sPiecewise-Constant Time-Varying Coefficients Model, Statistics in Medicine,2002, Vol. 21(5), pp. 717-727.


Thank you for your attention!

estimating survival from gray's exible model outline · p (y y < y + yjy y) y; (1) where p (:)...

Documents