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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
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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
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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
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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
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- 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
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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
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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
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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)
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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)
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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
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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
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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
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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
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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
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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
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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
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Cirrhosis example: a dynamic path analysis with all covariates
Block I Block II
Block III
Block IV
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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
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Direct cumulative effects on death (additive regression):
Age
Inflammation
Current prothrombin
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Treatment
Baseline prothrombin
Indirect cumulative effects on death:
Sex Age
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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)
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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