1

Using dynamic path analysis to estimate direct and indirect effects of treatment and other fixed covariates in the presence of an

internal time-dependent covariate

Ørnulf BorganDepartment of Mathematics

University of Oslo

Based on joint work with Odd Aalen, Egil Ferkingstad and Johan Fosen

2

Motivating example:- Randomized trial of survival for patients with liver cirrhosis

- Randomized to placebo or treatment with prednisone (a hormone)

- Consider only the 386 patients without ascites (excess fluid in the abdomen).

- Treatment: 191 patients, 94 deaths Placebo: 195 patients, 117 deaths

- A number of covariates registered at entry

- Prothrombin index was also registered at follow-up visits throughout the study

3

Cox regression analysis:

CovariateTreatment

Sex

Age

Acetylcholinesterase

Inflammation

Baseline prothrombin

Current prothrombin

Model II-0.06 (0.14)

0.31 (0.15)

0.043 (0.008)

-0.0015 (0.0006)

-0.43 (0.15)

-0.054 (0.004)

Model I-0.28 (0.14)

0.27 (0.16)

0.041 (0.008)

-0.0019 (0.0007)

-0.47 (0.15)

-0.014 (0.007)

Model I gives an estimate of the total treatment effect

CodingPlacebo=0; Prednisone=1

Female=0; Male=1

Years (range 27-77)

Range 42-556

Absent=0; Present=1

Percent of normal

Percent of normal

Model II shows the importance of prothrombin

4

Purpose:

Get a better understanding of how treatment (and other fixed covariates) partly have a direct effect on survival and partly an indirect effect operating via the internal time-dependent covariate (current prothrombin)

This will be achieved by a combining classical path analysis with Aalen's additive regression model to obtain a dynamic path analysis for censored survival data

5

- Brief introduction to counting processes, intensity processes and martingales

- Brief review of Aalen's additive regression model for censored survival data

- Dynamic path analysis explained by means of the cirrhosis example

Outline:

- Concluding comments

6

Counting processes

Have censored survival data (Ti , Di)

t

Ni(t)

0

1

Ni(t) counts the observed number of failures for indivdual i as a function of (study) time t

censored survival time

status indicator (censored=0; failure=1)

t

Ni(t)

0

1

Two possible outcomes for Ni(t):

Ti Di= 1

Ti Di= 0

7

Denote by Ft- all information available to the researcher "just before" time t (on failures, censorings, covariates, etc.)

The intensity process i(t) of Ni(t) is given by

where dNi(t) is the increment of Ni over [t,t+dt)

Cumulative intensity process:

( ) ( ( ) 1| )i i tt dt P dN t F ( ( ) | )i tE dN t F

0( ) ( )

t

i it s ds

Intensity processes and martingales

8

Introduce Mi(t) = Ni(t) – i(t).

Mi(t) is a martingale

( ( ) | )i tE dM t F

( ) ( )) ( i i it dMdtdN t t

( ( ) | ) ( )i t iE dN t F t dt

( ( ) ( ) | )i i tE dN t t dt F

( ) ( ) 0i it dt t dt

Note that

observation signal noise

For statistical modeling we focus on i(t)

9

Aalen's additive regression model

Intensity process for individual i

( )( )) (i ii t tt Y hazard at risk indicator

0 1 1( ) ( ) ( ) ( ) ( ) ( )i i p ipt t t x t t x t

(possibly) time dependent covariates for individual i (assumed predictable)

1( ), , ( )i ipx t x t

Aalen's non-parametric additive model:

baseline hazard excess risk at t per unit increase of xip(t)

10

At each time s we have a linear model conditional on "the past" Fs-

It is difficult to estimate the j(t) non-parametrically, so we focus on the cumulative regression functions:

Estimate the increments by ordinary least squares at each time s when a failure occurs

( )jdB s

0

( ) ( ) ( ( ))) (jp

ij

ij iix s Y dB ssdN ss dM

parameterscovariatesobservation noise

0with ( )( ) 1ix t

0( ) ( )

t

j jB t s ds

11

The vector of the is a multivariate "Nelson-Aalen type" estimator

The statistical properties can be derived using results on counting processes, martingales, and stochastic integrals (e.g. Andersen et al. Springer, 1993)

ˆ ( )jB t

Estimate by adding the estimated increments at all times s up to time t

( )jB t

Software: "aareg" in Splus version 6.1 for Windows (not in R)"addreg" for Splus and R at www.med.uio.no/imb/stat/addreg/"aalen" for Splus and R at www.biostat.ku.dk/~ts/timereg.html

12

Illustration of additive regression model: Survival after operation from malignant melanoma (cf. Andersen et al, 1993)

205 patients operated from malignant melanoma in Odense, Denmark, 1962-77

126 females, 28 deaths79 males, 29 deaths

Fit additive model with the fixed covariates:- Sex (0 = female, 1 = male)- Tumour thickness – 2.92

13

Years after operation

0 2 4 6 8

0.0

0.1

0.2

0.3

0.4

0.5

Years after operation

0 2 4 6 8

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Years after operation

0 2 4 6 8

0.0

0.02

0.04

0.06

0.08

0.10

0.12

Baseline

Sex Thickness

14

Fit additive models: (i) with treatment as only covariate (marginal model)

Cirrhosis example: only treatment and current prothrombin for a start

Treatment dNi(t)1( )d t

(ii) with treatment and current prothrombin (dynamic model)

Current prothrombin

Treatment dNi(t)1( )dB t

2 ( )dB t

15

Years since randomization

Cum

ulat

ive

regr

essi

on fu

nctio

n

0 2 4 6 8

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

Years since randomization

Cum

ulat

ive

regr

essi

on fu

nctio

n

0 2 4 6 8

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

Years since randomization

Cum

ulat

ive

regr

essi

on fu

nctio

n

0 2 4 6 8

-0.2

0-0

.15

-0.1

0-0

.05

0.0

0.05

(i) marginal model: (ii) dynamic model:

Treatment: Treatment:

Current prothrombin:Total effect of treatment is underestimated in the dynamic model

Current prothrombin has a strong effect on mortality

1ˆ ( )t 1

ˆ ( )B t

2ˆ ( )B t

16

Treatment

dNi(t)1( )dB t

2 ( )dB t( )t

By a dynamic path analysis we may see how the two analyses fit together Current

prothrombin

1 1 2ˆ ˆ( ) ˆˆ ( ) ( ( ) )tdB t dBd tt

total effect direct effect indirect effect

Due to additivity and least squares estimation:

Treatment on prothrombin: ˆ ( )t(least squares at each failure time) Treatment increases

prothrombin, and high prothrombin reduces mortality

Part of the treatment effect is mediated through prothrombin

17

Cirrhosis example: a dynamic path analysis with all covariates

Block I Block II

Block III

Block IV

18

Direct effects on block II variables (ordinary least squares):

Direct effects on current prothrombin (ordinary least squares):

Age on acetyl Sex on inflammation

Treatment Acetyl Baseline proth

19

Direct cumulative effects on death (additive regression):

Age

Inflammation

Current prothrombin

20

Treatment

Baseline prothrombin

Indirect cumulative effects on death:

Sex Age

21

Conclusions of the cirrhosis example:

- The dynamic path analysis gives a detailed picture of how the effect of treatment operates via current prothrombin and how the effect is largest the first year

- By not treating all fixed covariates on an equal footing we are able to distinguish between "basic" covariates (block I) and covariates that are a measure of progression of the disease (block II)

22

General conclusions:Aalen's additive regression model is a useful supplement to Cox's regression model

Additivity and least squares estimation make dynamic path analysis feasible, including the concepts direct, indirect and total treatment effects

Dynamic path analysis may be extended to recurrent event data with e.g. the previous number of events as an internal time-dependent covariate

Much methodological work remains to be done on dynamic path analysis, e.g. on methods for model selection