Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox...

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Survival Analysis for Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008

Transcript of Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox...

Page 1: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Survival Analysis for Randomized Clinical Trials

II Cox Regression

Ziad Taib

Biostatistics

AstraZeneca

February 26, 2008

Page 2: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Plan• Introduction to the proportional hazard

model (PH)

• Partial likelihood

• Comparing two groups

• A numerical example

• Comparison with the log-rank test

Page 3: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

The model

)exp()()( 110 ikkii zzthth ββ ++=

Understanding “baseline hazard” h0(t)

)]()(exp[)(

)(111 jkikkji

j

i zzzzth

th −++−= ββ

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Cox Regression Model• Proportional hazard. • No specific distributional assumptions (but

includes several important parametric models as special cases).

• Partial likelihood estimation (Semi-parametric in nature).

• Easy implementation (SAS procedure PHREG).• Parametric approaches are an alternative, but they

require stronger assumptions about h(t).

Page 5: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Cox proportional hazards model, continued

• Can handle both continuous and categorical predictor variables (think: logistic, linear regression)

• Without knowing baseline hazard ho(t), can still calculate coefficients for each covariate, and therefore hazard ratio

• Assumes multiplicative risk—this is the proportional hazard assumption– Can be compensated in part with interaction terms

Page 6: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Cox Regression

• In 1972 Cox suggested amodel for survival data that would make it possible to take covariates into account. Up to then it was customary to discretise continuos variables and build subgroups.

• Cox idea was to model the hazard rate function

tthtTttTtP ∆=≥∆+<≤ )()|(

where h(t) is to be understood as an intensity i.e. a probability by time unit. Multiplied by time we get a probability. Think of the analogy with speed as distance by time unit. Multiplied by time we get distance.

Page 7: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Cox’ suggestion is to model h using

( )( ) vector.covariate the,...,

vector;parameter the,...,

;)()(

1

1

0

iki

kT

i

ZZ

eththT

==

=

i

Z

β

i

ββ

where each parameter is a measure of the importance of the corresponding variable.

A consequence of this is that two individuals with different covariate values will have hazard rate functions which differ by a multiplicative term which is the same for all values of t.

Page 8: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

The covariates can be discrete, continuous or even categorical. It is also possible to generalise the model to allow time varying covariates.

Cthth

Ceeth

eth

th

th

iethth

T

T

T

T

i

).()(

;)(

)(

)(

)(

;2,1 ;)()(

21

)

0

0

2

1

0

=

===

==

21

2

1

i

Z-(Zβ

Page 9: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Example

Assume we have a situation with one covariate that takes two different values 0 and 1. This is the case when we wish to compare two treatments

β

ββ

ethth

thth

k

).()(

);()(

;1

;;1

02

01

1

=

===

===

21 Z 0;Z

β

h1(t)

h2(t)

t

Page 10: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

From the definition we

see immediately that

( )

[ ] iT

e

iT

t

t

iT

t

i

tS

e

e

e

tS

e

duuh

eduuh

duuh

i

)(0

)(

)(

)(

0

0

0

0

0

=

∫=

∫=

∫=

• One of the advantages of the PH model is that we can estimate the parameters and make comparisons between treatments without having to estimate the baseline.

• The baseline can be interpreted as the case corresponding to an individual with covariate values zero.

• The epitate semi-parametric is due to the fact that we do not model h0 explicitley. Usual likelihood theory does not apply.

Comments

Page 11: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Partial likelihood• The key to understanding partial likelihood is the following

conditional probability:P(patient j dies under ∆t | one patient dies during ∆t)

=P(patient j dies under ∆t)/P(one patient dies during ∆t)

=P(patient j dies under ∆t) / Σi P( one patient dies under ∆t | patient i dies during ∆t) P(patient i dies during ∆t)

=P(patient j dies under ∆t) / Σi P(patient i dies during ∆t)

∑∑===

∩=

ii

j

iii

jjjj AP

AP

APABP

AP

BP

AP

BP

BAPBAP

)(

)(

)()|(

)(

)(

)(

)(

)()|(

)|( BAP j

)(

)(

)(

)(

BP

AP

BP

BAP jj =∩

∑i

ii

j

APABP

AP

)()|(

)(

∑i

i

j

AP

AP

)(

)(

Page 12: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Therefore P(patient j dies under ∆t | one patient dies during ∆t)

(*))(

)(

)(

)(

0

0 ==∆

∆=∆

∆=

∑∑∑iii

i

j

iT

T

iT

jT

e

e

teth

teth

tth

tthZβ

Zβ j

• Let R(t)= # of patients at risk at time t (i.e alive and under observation at t).

• Let further Rj=R(tj).

tthtTttTtP iii ∆=≥∆+<≤ )()|(P(patient i dies under ∆t) =

Page 13: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

From this it follows that we can base our inference on the likelihood function

( )

( ) ( ) (**)

)(

)(),(

11

10

====

∆∆

=

∏∏ ∑

∏ ∑

==∈

=∈

ββ

β

LLe

e

tth

tththL

J

jj

J

j

Ri

J

jRi

i

j

j

iT

jT

j

∑∈

∆∆

=

jRii

j

tth

tth

)(

)(

P(patient j dies under ∆t | one patient dies during ∆t)

Then

Page 14: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Comments• If we treat this expression as an ordinary likelihood function we

can get parameter estimates.• Due to non-linearity we cannot hope finding explicit formulas

for the estiamtes. We have to rely on numerical maximisation methods such as Newton-Raphson.

• On the top of the usual score functions, we can consider the second derivative to compute the information and use this to estimate the covariance matrix of the parameter vector .

• As for the ordinary ML method, the estimates are asymptotically unbiased and and multivariate normal with mean and covariance matrix that can be estimated using and the inverse of the covariance matrix.

β̂

Page 15: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Let L(β) stand for the likelihood function and β for some parameter (vector) of interest. Then, the maximum likleihood estimates are found by solving the set of equations

It can be shown that the estimate to have an asymptotically normal distribution with means βi and variance/covariance matrix

Maximum Likelihood estimates

Define the information matrix, I(β) to have elements

,...,2,1 ,0)(log ==

∂∂

iL

iββ

.)(log

)(2

∂∂

∂−=ji

ij

LEI

ββββ

[ ] 1)( −βI

Page 16: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

• We take logLj = lj

( )

( )

( )( )∑

∑ ∑

=

= ∈

=

=

−=

=

J

jj

J

j Rij

T

J

jj

l

eLog

LogL

LogL

j

iT

1

1

1

β

β

β

Page 17: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Score and information( )

−−=∂∂

−=∂∂

=

22

2

2

j

iT

j

iT

j

iT

j

iT

ki

k

j

iT

j

iT

Ri

Riki

Ri

Rij

Ri

Riki

kjk

jk

e

eZ

e

eZl

e

eZ

Zl

U

β

β

β

• The two derivatives can be used to get parameter estimates and information .

• One difficulty we avoided so far is the occurence of ties.

Page 18: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Ties• dj = the # of failures at tj.• sj = the sum of the covariate values for patients who die at time tj.• Notice that sum (arbitrary) order must be imposed on the ties.

( ) ( )∑∑ ∑== ∈

=

−=

J

jj

J

j Rijj

T leLogdLogLj

iT

11

Zβsββ

−−=∂∂

−=∂∂

22

2

2

j

iT

j

iT

j

iT

j

iT

ki

k

j

iT

j

iT

Ri

Riki

Ri

Ri

jj

Ri

Riki

jkjk

j

e

eZ

e

eZ

dl

e

eZ

dsl

β

β

Page 19: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

A numerical ExampleTIME TO RELIEF OF ITCH SYMPTOMS FOR PATIENTS USING A STANDARD AND EXPERIMENTAL CREAM

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What about a t-test?

The mean difference in the time to “cure” of 1.2 days is not statistically significant between the two groups.

Page 21: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Using SASPROC PHREG ; MODEL RELIEFTIME * STATUS(0) = DRUG ;

Page 22: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Note that the estimate of the DRUG variable is -1.3396 with a p value of 0.0346. The negative sign indicates a negative association between the hazard of being cured and the DRUG variable. But the variable DRUG is coded 1 for the new drug and coded 2 for the standard drug. Therefore the hazard of being cured is lower in the group given the standard drug. This is an awkward but accurate way of saying that the new drug tends to produce a cure more quickly than the standard drug. The mean time to cure is lower in the group given the new drug. There is an inverse relationship between the average time to an event and the hazard of that event.

Page 23: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

• At each time point the cure rate of the standard drug is about 25% of that of the new drug. Put more positively we might state that the cure rate is 1.8 times higher in the group given the experimental cream compared to the group given the standard cream.

ˆ 2ˆ(2 1) 1.33962

ˆ 11

( )0.262

( )

h t ee e

h t e

ββ

β

β− −

−= = = =

The ratio of the hazards is given by

Page 24: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Comparing two groups in general

Let • nAj be the number at risk in group 1 at time tj• nBj be the number at risk in group 2 at time tjthen

( )

( ) ∑

=

=

+−

+=

+−=

J

j BA

A

BA

A

j

J

j BA

A

jj

jj

j

jj

j

jj

j

nen

en

nen

endI

nen

endsU

1

2

1

β

β

β

β

β

β

β

β ( )( ) ;0

0 22

χ≅=I

UQ

The score test

Page 25: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

A numerical example• Assume that a study aiming at comparing two

groups resulted in the following data

• U(0)=-10.2505

• I(0)=6.5957

• ;509192.1ˆ −=β 2211.0

;)(

)(

)(

)(

;2,1 ;)()(

150919ˆ

0.0

0

2

1

0

==

==

==

−ee

eeth

eth

th

th

ieththT

i

β

ββ

β .1

Zβ i

The treatment reduces the hazard rate to almost ¼ compared with placebo.

Page 26: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Log-rank vs the score test (PHREG)

• The two methods give very similar answers, why?• On one hand the score test can be based on the

statistic( )( )

;

0

0

2

12

2

1

2

χ≅

=

=

=

=

J

j

BAj

J

j j

Aj

A

j

jj

j

j

n

nnd

n

ndd

I

UQ

( )

( )

;

groups;both in risk at #

groups;both in events #

;0

;0

12

1

j

jj

jj

j

jj

j

j

Aj

BAj

BAj

J

j

BAj

J

j j

Aj

A

ds

nnn

ddd

n

nndI

n

nddU

=

=+=

=+=

=

−=

=

=

Page 27: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

nAjnAj-dAjdAjDRUG

njnj-djdjTOTAL

nBjnBj-dBjdBjCONTROL

NO CURECURE

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• Compare this to the Log-rank statistic

[ ]21

2

χ≅=− UVar

UQ HM

[ ] [ ] ( )( )

1

D-Tar ;

2tt2

ttt

tttt

t

tttt TT

MNDAV

T

NDAE

−==== σµ

( )∑=

−=k

tttAU

1

µ

[ ] [ ] ( )( )

1ar ;

22

nn

nn-dnddVσ

n

nddE

j

jj

j

j

j

j

BAjjj

Ajj

Aj

Aj −====µ

( )( )

;

1

2

12

2

1χ≅

=

=

=

− J

j j

BAjjj

J

j j

Aj

A

HM

j

jj

j

j

nn

nn-dnd

n

ndd

Q

Page 29: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

;2

12

2

1χ≅

=

=

=

J

j

BAj

J

j j

Aj

A

j

jj

j

j

n

nnd

n

ndd

Q ;

)1(

)(2

12

2

1χ≅

=

=

=

− J

j j

BAjjj

J

j j

Aj

A

HM

j

jj

j

j

nn

nn-dnd

n

ndd

Q

LOG-RANK TESTSCORE TEST

• dj = 1 i.e. no ties and the two formulas are identical. In general the two formulas are approximately the same when nj is large.

• For the example in a case with ties there is a slight difference

- Q=15.930

- QM-H=16.793

Page 30: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

Generalizations of Cox regression

1. Time dependent covariates

2. Stratification

3. General link function

4. Likelihood ratio tests

5. Sample size determination

6. Goodness of fit

7. SAS

Page 31: Survival Analysis for Randomized Clinical Trials II Cox ... · Randomized Clinical Trials II Cox Regression Ziad Taib Biostatistics AstraZeneca February 26, 2008. Plan • Introduction

References

• Cox & Oakes (1984) “Analysis of survival data”. Chapman & Hall.

• Fleming & Harrington (1991) “Counting processes and survival analysis”. Wiley & Sons.

• Allison (1995). “Survival analysis using the SAS System”. SAS Institute.

• Therneau & Grambsch (2000) “Modeling Survival Data”. Springer.

• Hougaard (2000) “Analysis of Multivariate survival data”. Springer.

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Questions or Comments?