Marginal Structural models
Mechanistic model
Comparison
Prospects
Mechanistic versus marginal structural models for
estimating causal effects. Application to the effect
of HAART on CD4 counts
Mélanie Prague, Rodolphe Thiébaut and Daniel Commenges
Cambridge, MRC BSU , March, 25th 2014
D. Commenges
Mechanistic versus marginal structural models
2 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Data analysis background
→ Analyse of data coming from clinical trials and observational studies
→ Longitudinal data Yijk : patient i, time j and biomarker k
D. Commenges
Mechanistic versus marginal structural models
3 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Problem and Objectives
Indication bias
Observational studies : Attribution of treatment depends on process
of interest (here : CD4 counts)
Patients under treatment tend to have worst health status (here :
lower CD4 counts)
Naive regression methods are biased and may find a deleterious
effect of the treatment
D. Commenges
Mechanistic versus marginal structural models
4 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Problem and Objectives
Indication bias
Observational studies : Attribution of treatment depends on process
of interest (here : CD4 counts)
Patients under treatment tend to have worst health status (here :
lower CD4 counts)
Naive regression methods are biased and may find a deleterious
effect of the treatment
From Marginal Structural models ...
... To mechanistic models
Comparison of these methods on clinical observational
data from a cohort.
D. Commenges
Mechanistic versus marginal structural models
4 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Marginal structural models
Proposed by J Robins.
Hernan et al., (2002), Stat Med
Cole et al. (2005) Am J Epidemiol
D. Commenges
Mechanistic versus marginal structural models
5 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Marginal structural model : counterfactuals
Notations
Yt , t = 0, 1, . . . : CD4 counts
A = (At , t = 0, 1, . . .) : treatment (0 : no HAART ; 1 : HAART)
Pt
cum(Āt ) = k=0 Ak
Lt , t = 0, 1, . . . : time-varying variables including Yt
D. Commenges
Mechanistic versus marginal structural models
6 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Marginal structural model : counterfactuals
Notations
Yt , t = 0, 1, . . . : CD4 counts
A = (At , t = 0, 1, . . .) : treatment (0 : no HAART ; 1 : HAART)
Pt
cum(Āt ) = k=0 Ak
Lt , t = 0, 1, . . . : time-varying variables including Yt
The counterfactual approach
Yat , t = 0, 1, . . . : each subject is supposed to have a counterfactual
response trajectory Yat if submitted to treatment regime A = a.
"Causal" model in term of counterfactual response :
E(Yat ) =
β0 + β2 t
|{z}
|{z}
Intercept
Time effect
+ β1 cum(āt−1 )
|
{z
}
Treatment effect
Models with two slopes are also considered.
D. Commenges
Mechanistic versus marginal structural models
6 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Marginal structural model (MSM) : parameters estimation
MSM mimics a randomized trial by putting dynamic weights on
observations of each subject
Weights take into account the probability of receiving the treatment
given the information at time t.
Qt
k |Āik−1 ,L0 )
; requires a model for treatment
Wt = k=0 ff(A
(Ak |Āk−1 ,L̄k )
attribution
the weights are used in a weighted GEE approach (since this is a
marginal model)
-> Without confounders : Only use information of CD4 in L
-> With confounders : Also adjust for other time-varying variables like
Viral load in L.
D. Commenges
Mechanistic versus marginal structural models
7 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Dynamical models
Commenges and Gégout-Petit (2009) JRSS-B
Aalen et al. (2012) JRSS-A
D. Commenges
Mechanistic versus marginal structural models
8 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Linear increment model
Assumptions :
-> Write Doob decomposition and only keep the previsible process :
i
= Y0i + ΛY i ,t + MY i ,t − Y0i − ΛY i ,t−1 − MY i ,t−1
Yti − Yt−1
-> E(MY,t − MY,t−1 |Ft−1 ) = 0.
Modeling :
-> Without confounders : Only use information from CD4 in L
E(Yt − Yt−1 |Ft−1 )
= β2 + β1 At−1 + β3 Yt−1
-> With confounders : Also adjust for Viral load in L.
E(Yt − Yt−1 |Ft−1 ) = β0 + β1 At−1 + β2 Yt−1 + β3 Vt−1
E(Vt − Vt−1 |Ft−1 ) = α0 + α1 At−1 + α3 Yt−1 + α4 Vt−1
D. Commenges
Mechanistic versus marginal structural models
9 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
The fundamental framework of mechanistic
modeling
Real systems live in continuous time
Observations are in discrete time
D. Commenges
Mechanistic versus marginal structural models
10 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
HIV mechanistic model
Biological Compartments
Q
T
T∗
V
D. Commenges
Concentration of
Quiescent CD4
Target CD4
Infected CD4
Virus
Mechanistic versus marginal structural models
11 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
HIV mechanistic model
Target cells model [Guedj et al. 2007 ; Drylewicz et al. 2010 ; Prague et
al. 2012]








dQ
dt
=
λ − µQ Q − αQ + ρT
dT
dt
=
αQ − ρT − µT T − γV T
dT
dt
=
γV T − µT ∗ T ∗
dV
dt
=
πT ∗ − µV V
∗







D. Commenges
Mechanistic versus marginal structural models
12 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Statistical model
Mixed Effects model on parameters (γ, µT ∗ and λ) :
Fixed effects : drugs, other covariates
Random effects : Inter-individual variability
ξ˜i
ξ˜i
= log αi , λi , ..., γ0 i , µiV
ui
∼
l
= φl + dil (t)βl +
|
{z
}
Fixed effects
D. Commenges
uil
|{z}
Random effects
N (0, Iq )
Mechanistic versus marginal structural models
13 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Observational Model
Viral Load :
Total CD4 count :
Yij1 = log10 (V ) + ij1
Yij2 = (Q + T + T ∗ )0.25 + ij2
2
ijm ∼ N (0, σm
)
D. Commenges
Mechanistic versus marginal structural models
14 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
NIMROD : A program of inference in ODE
Parameters of interest :
h
i
θ = (φl )l=1...nb , (βl )l=1...ne , (ωl )l=1...q , (σl )l=1...M
Estimaton tool :
Likelihood and penalized likelihood maximization,
Normal approximation of the posterior,
Software Distribution with parallel computing.
Prague M. et al. (2013) Computer Methods and Programs in Biomedicine
111 :447-458 , NIMROD : A Program for Inference via Normal Approximation of the
Posterior in Models with Random effects based on Ordinary Differential Equations.
D. Commenges
Mechanistic versus marginal structural models
15 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Illustration
D. Commenges
Mechanistic versus marginal structural models
16 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Data Aquitaine cohort [same as Cole et al. 2002]
All = 6580 (156958 person-­‐visit) Patients included after April 1996 Visits after April 2005 Visits before April 1996 All = 2100 (47444 person-­‐visit) Visits with ARV (no HAART) prescibed All = 1970 (36431 person-­‐visit) Visits with missing values for CD4 Visits with missing values for Viral load All = 1900 (29601 person-­‐visit) Patients with treatment before April 1996 All = 685 (8689 person-­‐visit) Patients without half-­‐year visits +/-­‐ 2.5 months Patients with less than 1 year follow-­‐up (2 visits) All = 212 (1510 person-­‐visit) D. Commenges
Mechanistic versus marginal structural models
17 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Description of the sample
Characteristics
Age (yr.)
Women(%)
AIDS (%)
HAART (%)
CD4 count (%)
<200
200-350
>350
Mean CD4 count
Viral load(%)
<500
501-10000
>10000
Mean VL in log10
D. Commenges
Baseline
n=212 pat
36.7
30.2
6.1
0.0
End of follow-up
n= 1298 pat-visits
40.6
28.5
12.3
61.2
15.1
11.3
73.1
490 [69 ;1047]
7.6
18.7
73.5
508 [168 ;1092]
15.1
33.5
51.4
4.0 [2.7 ;5.4]
55.1
26.0
18.9
2.7 [1.7 ;4.8]
Mechanistic versus marginal structural models
18 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Application
The naive model : Yt = γ0 + γ1 cum(Āt−1 ) + γ2 t
Model 1 : Linear mixed effect model
D. Commenges
Mechanistic versus marginal structural models
19 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Application
The naive model : Yt = γ0 + γ1 cum(Āt−1 ) + γ2 t
Model 1 : Linear mixed effect model
The MSM : E(Yat ) = β0 + β1 cum(āt−1 ) + β3 cum(āt−3 ) + β2 t
Model 2 : MSM, weighted GEE depending only on CD4 counts
Model 3 : MSM, weighted GEE for treatment and censoring,
depending on CD4, viral load . . .
D. Commenges
Mechanistic versus marginal structural models
19 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Application
The naive model : Yt = γ0 + γ1 cum(Āt−1 ) + γ2 t
Model 1 : Linear mixed effect model
The MSM : E(Yat ) = β0 + β1 cum(āt−1 ) + β3 cum(āt−3 ) + β2 t
Model 2 : MSM, weighted GEE depending only on CD4 counts
Model 3 : MSM, weighted GEE for treatment and censoring,
depending on CD4, viral load . . .
Linear increment models : E(Yt − Yt−1 |Ft−1 ) = β2 + β1 At−1 + β3 At−3
Model 4 : CD4 linear increment
Model 5 : CD4 and VL linear increment
D. Commenges
Mechanistic versus marginal structural models
19 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Application
The naive model : Yt = γ0 + γ1 cum(Āt−1 ) + γ2 t
Model 1 : Linear mixed effect model
The MSM : E(Yat ) = β0 + β1 cum(āt−1 ) + β3 cum(āt−3 ) + β2 t
Model 2 : MSM, weighted GEE depending only on CD4 counts
Model 3 : MSM, weighted GEE for treatment and censoring,
depending on CD4, viral load . . .
Linear increment models : E(Yt − Yt−1 |Ft−1 ) = β2 + β1 At−1 + β3 At−3
Model 4 : CD4 linear increment
Model 5 : CD4 and VL linear increment
Mechanistic models : dY (t)/dt
Model 6 : Target cells model
D. Commenges
Mechanistic versus marginal structural models
19 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Results : Treatment effect quantification
D. Commenges
Mechanistic versus marginal structural models
20 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Results
Table: Effect of treatment on CD4 count : Model 1 : naive regression ; Model
2 : MSM with simple weights ; Model 3 : MSM with well-adjusted weights ;
Model 4 : CD4 linear increment ; Model 5 : CD4/VL linear increment ; Model
6 : mechanistic model .
Model
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
D. Commenges
≤1
>1
≤1
>1
≤1
>1
year
year
year
year
year
year
CD4
log Viral load
Infectivity
Treat. effect
-25.6
-52.1
-10.7
63.4
44.0
40.5
27.1
4.96
-0.12
-2.18
Stand. error
9.7
19.6
8.4
37.2
30.4
9.5
6.8
9.70
0.05
0.29
Mechanistic versus marginal structural models
Z
-2.64
-2.66
-1.27
1.71
1.44
4.27
3.95
0.51
-2.36
-7.41
21 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Non- or badly- adjusted regression models are misleading
Table: Effect of treatment on CD4 count
Model 1 : Yt = γ0 + γ1 cum(Āt−1 ) + γ2 t ;
Model 2 : MSM E(Yat ) = β0 + β1 cum(āt−1 ) + β3 cum(āt−3 ) + β2 t
with simple weights ;
Model
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
D. Commenges
≤1
>1
≤1
>1
≤1
>1
year
year
year
year
year
year
CD4
log Viral load
Infectivity
Treat. effect
-25.6
-52.1
-10.7
63.4
44.0
40.5
27.1
4.96
-0.12
-2.18
Stand. error
9.7
19.6
8.4
37.2
30.4
9.5
6.8
9.70
0.05
0.29
Mechanistic versus marginal structural models
Z
-2.64
-2.66
-1.27
1.71
1.44
4.27
3.95
0.51
-2.36
-7.41
22 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
linear increment models seem more sensitive
Table: Effect of treatment on CD4 count
Model 3 : MSM E(Yat ) = β0 + β1 cum(āt−1 ) + β3 cum(āt−3 ) + β2 t
with well-adjusted weights ;
Model 4 : E(Yt − Yt−1 |Ft−1 ) = β2 + β1 At−1 + β3 At−3 ;
Model
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
D. Commenges
≤1
>1
≤1
>1
≤1
>1
year
year
year
year
year
year
CD4
log Viral load
Infectivity
Treat. effect
-25.6
-52.1
-10.7
63.4
44.0
40.5
27.1
4.96
-0.12
-2.18
Stand. error
9.7
19.6
8.4
37.2
30.4
9.5
6.8
9.70
0.05
0.29
Mechanistic versus marginal structural models
Z
-2.64
-2.66
-1.27
1.71
1.44
4.27
3.95
0.51
-2.36
-7.41
23 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Dynamical CD4/VL : More power and explanatory strenght
Model 5 :
Model
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
D. Commenges
Table: Effect of treatment on CD4 count
E(Yt − Yt−1 |Ft−1 ) = β0 + β1 At−1 + β2 Yt−1 + β3 Vt−1
;
E(Vt − Vt−1 |Ft−1 ) = α0 + α1 At−1 + α3 Yt−1 + α4 Vt−1
Model 6 : mechanistic model .
≤1
>1
≤1
>1
≤1
>1
year
year
year
year
year
year
CD4
log Viral load
Infectivity
Treat. effect
-25.6
-52.1
-10.7
63.4
44.0
40.5
27.1
4.96
-0.12
-2.18
Stand. error
9.7
19.6
8.4
37.2
30.4
9.5
6.8
9.70
0.05
0.29
Mechanistic versus marginal structural models
Z
-2.64
-2.66
-1.27
1.71
1.44
4.27
3.95
0.51
-2.36
-7.41
24 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Results : Biomarkers trajectories
D. Commenges
Mechanistic versus marginal structural models
25 / 29
Marginal Structural models
Mechanistic model
Comparison
Prospects
Figure: Predicted trajectories of CD4 count
Model 1 : Yt = γ0 + γ1 cum(Āt−1 ) + γ2 t ;
Model 3 : MSM E(Yat ) = β0 + β1 cum(āt−1 ) + β3 cum(āt−3 ) + β2 t
E(Yt − Yt−1 |Ft−1 ) = β0 + β1 At−1 + β2 Yt−1 + β3 Vt−1
Model 5 :
;
E(Vt − Vt−1 |Ft−1 ) = α0 + α1 At−1 + α3 Yt−1 + α4 Vt−1
Model 6 : mechanistic model .
D. Commenges
Mechanistic versus marginal structural models
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Marginal Structural models
Mechanistic model
Comparison
Prospects
Superiority of mechanistic models : individual predictions
We are able to predict the Viral load and CD4 count trajectories (here, a
rebound for viral load) for patient who changed treatment molecule.
D. Commenges
Mechanistic versus marginal structural models
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Marginal Structural models
Mechanistic model
Comparison
Prospects
Conclusion and prospects
D. Commenges
Mechanistic versus marginal structural models
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Marginal Structural models
Mechanistic model
Comparison
Prospects
Conclusion and prospects
-> There exists a continuum :
MSM model -> Incremental model -> mechanistic models.
Naive regression model –> deleterious effect of treatment
MSM with simple weights did not correct
MSM with elaborate weights did correct but did not reach always
significance (especially, lack of power when confounders are missing)
Discrete time dynamic models did correct and had a higher power
The Mechanistic model had very high power, could use much more
data, but is hard to implement (NIMROD program).
-> Ongoing work : Make simulations to confirm these results ; repeat
the analysis on the Swiss cohort
D. Commenges
Mechanistic versus marginal structural models
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