Transcript (WCE) model

Modélisation des Effets
Cumulatifs des Expositions
et des Facteurs Pronostiques
Michal Abrahamowicz, PhD*
James McGill Professor
Department of Epidemiology & Biostatistics
McGill University, Montreal, Quebec, CANADA
&
Marie-Pierre Sylvestre, PhD
Professeur Adjoint, Universite de Montreal, Quebec,
CANADA
1
Outline
1.
2.
3.
4.
5.
6.
7.
8.
Objectives
Challenges in modeling the effects of TimeVarying exposures/treatments/prognostic factors
Flexible estimation of Cumulative Effects
Evaluation via Simulations
Pharmaco-epidemiologic Application
Inclusion of Non-linear Effects
Application: Impact of Past SBP on CVD risks
Conclusions
2
Objectives




To emphasize the Importance of accounting for
Cumulative Effects of Past
Exposures/Prognostic Factors
To introduce New Flexible Methods for
Modeling Cumulative Effects
To Evaluate proposed methods in Simulations
To Illustrate Real-Life Applications
in Etiologic (Pharmaco-epidemiologic) and
Prognostic Studies
3
2 Examples of Time-varying
Prognostic Factors/Exposures
Doses of Flurazepam
Days
Systolic Blood Pressure
Exams
4
Main Challenge in Modeling Effects of
Time-Varying Exposures/Risk Factors

How to relate the
Entire History of Past Values of the Exposure
with
Current Risk of the Outcome ?
5
Challenges in modeling time-varying
exposures

Do effects cumulate over time?
Subject A

Subject B
What is the relative importance of exposures that occurred at different
times in the past?
Subject C
Subject D
6
Examples of Cumulative Effects



Adverse Effects of Psychotropic Medications:
Short-term Cumulative Effects may occur until
the drug is entirely eliminated from the Plasma
(but Recent doses have the highest impact)
Impact of Smoking on Lung Cancer Risk:
an example of Long-Term Cumulative Effect
(Impact of Past Smoking cumulates over
Lifetime but there is a Lag of about 5-15 years
and current/recent exposures have
No Impact on Current Risk)
7
Conventional Models for Time-Varying
Prognostic Factors

Different models are used to relate the
hazard at current time u with a time-varying
exposure:
i. Current value (“dose”): X(u)
u
ii. Total Past Dose (=un-weighted
 X (t )
cumulative dose)
t0
iii. Total Duration of Past Exposure
iv. Any Exposure e.g. in the Last Month
v. Mean Dose e.g. in the Last Month
8
(Some) Limitations of Conventional
Models
All “conventional models” impose
strong A Priori assumptions about the
Relative Importance of Past Doses (X(t), t<u)
for determining Current Risk at time u.
e.g.:

i.
ii.
the Current Dose model assumes that
ONLY the Current Dose X(u) does matter
In contrast,
the Total Dose model assumes that
ALL Past Doses have Equal Impact on the current risk,
regardless of their Timing
9
Weighted Cumulative Exposure (WCE)
model

All conventional models (i)-(iii) are special cases
of a much more General Model,
based on the Concept of:
recency-Weighted Cumulative Exposure (WCE),
where the Cumulative Effect is modeled as a
Weighted Sum ** of all Past Doses,
[** these Weights describe how
the Relative Importance of Doses (Exposures)
changes as a function of the Time-since-Exposure]
10
Recency-weighted cumulative exposure
(WCE)
Abrahamowicz et al. (2006), Breslow et al. (1983),
Hauptmann et al. (2000), Thomas (1988), Vacek (1997)
WCE u  
 w u  t   X t 
(1)
tu
where:
u= current time (when Risk is being assessed)
WCE(u)= Weighted Cumulative Effect of the Past
Exposures on Risk at time u
X(t)= exposure intensity (dose) at time t(t≤u)
u-t= time elapsed since exposure X(t)
w(u-t)= Weight (Relative Importance) assigned to
exposure X(t) as a function of Time-since-Exposure (u-t)
11
Hypothetical example of calculating
cumulative weighted dose WCE (u=30)
t
X(t)
u-t
w(u-t)
w(u-t)*X(t)
23
0.5
7
0.50
0.25
24
0.5
6
0.60
0.30
25
0.5
5
0.70
0.35
26
0.5
4
0.80
0.40
27
0.5
3
0.88
0.44
28
0.5
2
0.94
0.47
29
1.5
1
0.99
1.48
30
1.5
0
1.00
1.50
=5.19
12
Example of a Weight Function
From Abrahamowicz et al. (2006, Journal of Clinical Epidemiology, Figure 1)
13
Variation over Time of Dose X(t) [Upper graph] & the resulting
WCE(u) calculated using w(u-t) on slide 12 [Lower graph])
From Abrahamowicz et al. (2006, Journal of Clinical Epidemiology, Figure 2)
14
Need for Flexible Modeling of the
Weight Function

In most real-life prognostic studies, there is little
A priori knowledge about the (i) Exact Shape,
and (ii) the Exact Values of the Weight Function

Therefore, it would be Advantageous to Estimate
the Weight Function Directly from the Empirical
Data

To this end, we need Flexible Assumptions-free
methods such as Splines
15
Flexible spline-based WCE Model
[Sylvestre & Abrahamowicz (SIM 2009)]

In our Flexible WCE model, the Weight
function is estimated by Cubic Splines:
m
w (u  t ) 

j
B j (u  t )
(2)
j  1 the m functions in the Cubic
where Bj, j=1,…,m, represent
Spline basis, and j represent the estimable coefficients of
the linear combination of the basis splines
WCE in (2) is then modeled as a
Time-Dependent Covariate in Cox’s model
16
Examples of Cubic Regression Splines
(with 3 Interior Knots)
The m functions Bj of the cubic spline basis
1.0
B4
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
Regression spline
B3
B2
0.8
1.0
B1
0.0
0.2
0.4
u-t
0.6
0.8
0.0
1.0
0.2
0.4
0.6
0.8
1.0
x
u-t
x
 = (0.6, 0.6, 0.6, 0.9, 0.4, 0.2, 0.2)
 = (0.5, 0.1, 0.2, 0.4, 0.2, 0.5, 0.5)
Interior knots
17
Inference about the Estimated
Weight Function ŵ( )


Quasi-parametric LRT tests* of:
i. H0 = No Association between X(t) and risk:
w(u-t)=0 for 0<t<u (df=k+4)
ii. H0 = Equal Weighting of All past values:
w(u-t)=V for 0<t<u (df=k+3)
Bootstrap-based pointwise confidence bands
around the estimated Weight function
18
Simulations Results: True w(u-t) (white) vs
100 Constrained Estimates [a=180 days]
From Sylvestre and Abrahamowicz (2009, Statistics in Medicine, Figure 2)
19
Simulation Results: Goodness of Fit of
Alternative Models

In scenarios (a) – (d) [where the True weight function
decayed to 0], the Constrained models had the best
fit (lowest BIC) in > 90% of the simulated samples

In scenario (e) the (conventional) UN-weighted Total
Cumulative Dose (corresponding to the “true” model)
had the lowest BIC in in > 90% of the simulated
samples (this Eliminated concerns about the Over-fit
bias of the weighted estimates)

In scenario (f) the Un-constrained model fitted better
than the constrained estimate and the former
suggested that doses taken > 180 days ago are still
relevant
20
Correcting for the “too short” initial
support interval (in scenario “f”)
21
APPLICATION: Flurazepam (a Psychotropic
drug) use vs Fall-related Injuries in Elderly







Data from Dr. Robyn Tamblyn from McGill University
Prospective study based on Administrative Health
Data (Prescription Claims database) from the
Canadian province of Quebec
Cohort of N=4,666 elderly New Users of
Flurazepam (started use in 1990-1994)
T0 = Jan. 1, 1990 (Delayed Entry at the time of the
1st Flurazepam prescription)
Event = Fall-related Injury [252 events]
Available Data for Each Prescription: (i) Duration
& (ii) Daily Standardized Dose
3 “time windows”: a = 60, 90 & 180 days
22
Flurazepam use vs Fall-related Injuries:
Comparison of BIC of Alternative Models
Reference
Hazard
ratio*
95%
confidence
interval
BIC**
Current use
Non-user
1.34
(0.97-1.85)
3769.9
Current dose
Non-user
1.67
(1.14-2.46)
3766.7
Unweighted duration of use (30 days)
Non-user
0.99
(0.97-1.23)
3772.4
Unweighted cumulative dose (30 days)
Non-user
0.99
(0.97-1.03)
3772.7
Pattern of use
Conventional models
BIC-optimal WCE model
+
3763.5
* All models were adjusted for age at first benzodiazepine prescription, sex, and a binary indicator of the history of
injury in the baseline year (1989).
** 1-knot constrained spline-based WCE model.
+ BIC of the corresponding model, with lower BIC indicating better fit to data.
From Table 1 in Sylvestre and Abrahamowicz (2009, Statistics in Medicine)
23
Flurazepam vs Falls: Constrained WEIGHT
function with a=60 (Cubic Spline; 1 knot)
From Sylvestre and Abrahamowicz (2009, Statistics in Medicine, Figure 3)
24
Flurazepam use vs Fall-related Injuries:
adjusted HR’s for different Patterns of Use
Adjusted hazard ratios, with 95% confidence intervals, for the association
between various patterns of Flurazepam and fall-related injuries
Reference
Hazard
ratio*
95%
confidence
interval
Current user, dose=1, for 1 day
Non-user
1.07
(0.70-1.26)
Current user, dose=1, for 7 days
Non-user
1.68
(0.68-3.05)
Past user, dose=1, 8-14 days ago
Non-user
1.59
(1.19-4.35)
Past user, dose=1, 15-21 days ago
Non-user
1.23
(0.85-1.77)
Past user, dose=1, 22-28 days ago
Non-user
0.92
(0.49-1.14)
Current user, dose=1, for 30 days
Non-user
2.83
(1.45-4.34)
Current user, dose=1, for 7 days
Past user, dose=1, for
7 days, between 14
and 7 days ago
1.37
(0.43-3.45)
Pattern of use
* All models were adjusted for age at first benzodiazepine prescription, sex, and a binary indicator of the history of
injury in the baseline year (1989).
From Table 1 in Sylvestre and Abrahamowicz (2009, Statistics in Medicine)
25
Another Challenge: unknown DoseResponse function for continuous X(t)

“Classic” WCE model (1),
WCE u  

 w u  t   X t ,
tu
assumes Linear relationship
between “dose” X(t)
and risk (e.g., Logit or Log hazard)
Yet, many continuous exposures have NonLinear effects on risk! [e.g. Abrahamowicz et al, AJE
1997]
 Need to Extend the WCE models to incorporate
Flexible Modeling of the possibly
Non-linear effects of Continuous Time-Varying
Prognostic Factors
26
Example of NON-LINEARITY of dose-response:
Smoking Intensity vs (logit of) Lung Cancer risk
Rachet et al (2004)
27
EXTENSION of our flexible WCE model to Joint
Estimation of w( ) and g( ) in Time-to-Event analyses
WCE u  

u

w u  t  g  X t 
(7)
t
Model w( ) and g( ) with Cubic regression splines
w (u  t ) 
  j B j (u  t )
1 or 2 equidistant
interior knots
j
g ( X [ t ]) 
  p B ( X [ t ])
*
p
p
2 interior knots at
tertiles of overall
distribution of X[t]
28
SIMULATIONS: Estimates of NON-Linear function
[N=600; 300 events] at selected times in the past
29
APPLICATION: assessing Relative
importance of Past SBP values for CHD risk







Data Source = Framingham Heart Study
30 years of Follow-up with 16 bi-annual exams
(SBP and other risk factor measures)
Time 0 = 1st exam
Event = 1st Coronary Heart Disease (CHD) event
(mortality/morbidity)
2603 women (762 CHD events)
2128 men (995 CHD events)
Adjustment for: current age, serum cholesterol,
glucose, smoking
30
SBP vs CHD risk
Men:
PL(WCE)=6768 vs PL(Cox)=6787
Women:
PL(WCE)=5570 vs PL(Cox)=5581
w( )
w( )
g( )
g( )
31
Some Limitations of Our Methods



Our methods require
frequent measurements of the prognostic
factor X(t)
[≥15 observations during the follow-up]
The method performs well in Large Databases
(> 300 events)
The current methods do not handle time-varying
mediating/confounding variables (need to extend
to Marginal Structural Models)
32
Conclusions

Flexible Weighted Cumulative
Exposure (WCE) model offers New
Insights about :
1) Relative Importance of values of
Exposures and Risk Factor observed
at different times in the Past and
their Cumulative Effects on the
Current Risk
2) Non-Linear Exposure-Risk
relations for continuous variables
33
Conclusions

The flexible WCE model
fits data from many
Longitudinal Studies
better than conventional models
and
may be useful in All Studies with
Repeated-over-Time Measures of
Treatments, Exposures, Risk factors
or Prognostic factors
34
References

Abrahamowicz M, Bartlett G, Tamblyn R, du Berger R. Modeling cumulative dose and exposure duration provided
insights regarding the associations between benzodiazepines and injuries. Journal of Clinical Epidemiology.
2006;59(4):393-403.

Abrahamowicz M, du Berger R, Grover SA. Flexible modeling of the effects of serum cholesterol on coronary heart
disease mortality. American Journal of Epidemiology 1997;145(8):714-729.

Abrahamowicz M, MacKenzie T, Esdaile JM. Time-dependent hazard ratio: modeling and hypothesis testing with
application in lupus nephritis. Journal of the American Statistical Association. 1996;91(436):1432-39.

Breslow NE, Lubin JH, Marek P, Langholz B. Multiplicative models and cohort analysis. Journal of the American
Statistical Society 1983; 78(381):1–12.

Hauptmann M, Wellmann J, Lubin JH, Rosenberg PS, Kreienbrock L. Analysis of exposure-time-response relationships
using a spline weight function. Biometrics. 2000;56(4):1105-8.

MacKenzie T, Abrahamowicz M. Marginal and hazard ratio specific random data generation: applications to semiparametric bootstrapping. Statistics and Computing. 2002;12(3):245-352.

Mahmud M, Abrahamowicz M, Leffondré K, Chaubey YP. Selecting the optimal transformation of a continuous
covariate in Cox’s regression: Implications for hypothesis testing. Communication in Statistics 2006;35(1):27-45.

Rachet B, Siemiatycki J, Abrahamowicz M, Leffondré K. A flexible modeling approach to estimating the component
effects of smoking behavior on lung cancer. Journal of Clinical Epidemiology 2004;57(10):1076-1085.

Sylvestre MP, Abrahamowicz M. Comparisons of algorithms to generate event times conditional on time-dependent
covariates. Statistics in Medicine. 2008; 27(14):2618-34.

Sylvestre MP, Abrahamowicz M. Flexible modeling of the cumulative effects of time-dependent exposures on the
hazard. Statistics in Medicine. 2009; 28(27):3437-53.

Thomas DC. Models for exposure-time-response relationships with applications to cancer epidemiology. Annual
Reviews of Public Health 1988; 9:451-82.

Vacek PM. Assessing the effect of intensity when exposure varies over time. Statistics in Medicine. 1997;16(5):505-13.
35
MERCI !
[email protected]
36
Additional material
37
Accuracy of the Estimated log HR for
the Weighted Cumulative Exposure
*
*
* In these two scenarios: (i) the constraint w(u-a)=0 was incorrect, as the
true weight function remained positive beyond the support interval [0; a];
(ii) the un-constrained models fitted better so the bias of the constrained
estimates was practically irrelevant.
From Sylvestre and Abrahamowicz (2009, Statistics in Medicine, Table 1)
38
Sensitivity analysis:
impact of Changing the “Time Window”
39
Identifiability problems
Because w(u-t) is multiplied by g[X(t)], the 2 functions “share
the scale”

for r ≠ 0:
 r    u  t  
j
j
 p 



 r  p  X t   


 
j






u

t



 p p  X t 
j
which implies:
i.
Identifiability problems [Abrahamowicz & MacKenzie (2007)]
ii.
Non-linearity of model (7) in its parameters
40
2-step iterative Alternating Conditional Estimation (ACE)
(maximum Partial Likelihood (PL))

We propose an ACE algorithm that iterates through 2
alternating steps:
Estimate w( ) given previous estimate of ĝ( )
2) Estimate g( ) given previous estimate of ŵ( )
1)



We continue Iterations (through steps 1 & 2) until
“Convergence” (ΔPL<0.01)
At step 1) of the 1st iteration g( ) = αX(t) [Linear]
At step 1) of each iteration ŵ( ) is standardized:
 w ( u  t )dt
1
41
Performance of the ACE algorithm in a
randomly selected simulated sample
Main iteration #
Step
PL*
PL
1
1 (a)
2592.181
---
1
2 (b)
2589.140
-3.041
2
1
2588.991
-0.149
2
2
2588.985
-0.006**
** Convergence: |PL|<0.01
*PL = negative partial log likelihood of the WCE model with 1 interior knot
PL = improvement in PL relative to the previous step
(a) = at step 1, the weight function w(u-t) is estimated
(b) = at step 2, the non-linear dose-response function g[X(t)] is estimated
42
Flexible WCE Model
[Sylvestre & Abrahamowicz (2009)]

WCE in (2) is then modeled as a Time-Dependent
Covariate in Cox’s model:
q
 u

h ( u | X ( u ), Z ( u ))  h 0 ( u ) exp    w ( u  t ) X ( t )    s Z s ( u )  (3)
s 1
 t

where:
h0(u) is the baseline hazard,
X(u)={X(t),0≤t≤u} represents the time-vector of the past
exposures,
Zs(u), s=1,…,q, are the values of the fixed-in-time or
time-dependent covariates relevant for time u
43
Use of Artificial Time-Dependent covariates
permits implementation with standard software

To estimate w( ), at step 1 of ACE, construct artificial timedependent covariates:
D j (u ) 
B
j
( u  t ) gˆ ( X [ t ])
u  at u

and fit the following Cox’s model:


h ( u | X ( t ))  h 0 ( u ) exp    j D j ( u ) 
 j

44
Simulations Results: True w(u-t) (white) vs
100 Un-constrained Estimates [a=180 days]
From Sylvestre and Abrahamowicz (2009, Statistics in Medicine, Figure 1)
45
Simulations Design







Hypothetical Pharmaco-epi study of the Adverse Effects of
a drug
Cohort of N=500 new drug users
t0 = time of 1st prescription
Follow-up duration up to 1 year
Random right censoring (about 50%)
6 simulated scenarios, each with a different ‘true’ weight
function
Time-varying Patterns of Drug Use/Dose X(t):
 subjects repeatedly stop and re-start drug use
 both Inter- & Intra-subject Variation in the length of
consecutive Periods of use/non-use & in the Dose
 dose constant within each period of use but varied from
one period to another
46
Use of Artificial Time-Dependent covariates
permits implementation with standard software

To estimate g( ), at step 2 of ACE, calculate:
G p (u ) 

*
wˆ ( u  t ) B p ( X [ t ])
u  at u

and fit the following Cox’s model:


h ( u | X ( t ))  h 0 ( u ) exp    p G p ( u ) 
 p

47
Simulations
The Design and Methods of Simulations were Similar to
slides 24-25, Except:

i.
ii.
“dose” X(t) was generated from a Continuous Distribution
U(0, 3);
X(t) was assumed to have different Non-linear relationships with
the logarithm of the hazard.
48
Estimates of w( ) [N=600; ~300 events]
49
Generation of Event Times Conditional on the WCE
via novel ‘Permutational Algorithm’
Generate Individual exposure vectors xs[t], s=1,…,N
Generate Observed Event/Censoring Times:
A.
B.
C.
1)
Marginal distribution of Event times Ti ~ U[1, 365] days, i=1,…,N
2)
Marginal distribution of Censoring times Ci ~ U[1, 365] days
•
Observed times ti* = min(Ti, Ci)
MATCH Exposure Vectors (step A) with Observed Times (step B) via Permutational
algorithm [MacKenzie and Abrahamowicz (2002); Sylvestre and Abrahamowicz (2008)]
1)
If ti* = event, assign ti* to one of the xs[t] at risk by sampling xs[t] with probability
proportional to hazard at ti*:
2)

*

   assign t * to one of the x [t] at risk by simple random
 exp WCE t | x [ t ] 
*
i
time,
p t
s
If ti* = censored
sampling

exp WCE t i | x s [ t ]
i
k Ri
s
*
i
k
50
Constrained WCE Model


In many applications, it may be A Priori evident that
the Weight Function w(u-t) should asymptotically decay to
0 at the either end of the support interval [0;a]
This can be easily achieved by Constraining the WCE model:
 1=0 & 2=0 ensures, respectively, that w(0)=0 &
w’(0)=0
so that Current Value X(u) has No Impact on
Current Risk at u (e.g., Current Smoking is
Irrelevant for Current Cancer Risk)
 (k+4)=0 & (k+3)=0 ensures that w(a)=0 & w’(a)=0
so that the Value of X(u-a) has No Impact on the
Current Risk (e.g., Drug Use a days/weeks ago is
Irrelevant for Current Risk of Adverse Events)
51
Model Selection




We fit models with k=1, 2 or 3 ‘interior knots’
(Uniformly Distributed within [0; a] support interval)
(in addition, 4 ‘exterior knots’ are placed at both u=0
and u=a)
The resulting Cubic Spline has, respectively, 5, 6 or 7
functional segments, i.e. model (6) [slide 21] requires
estimating k+4 = 5-7 coefficients j
In some applications, the users may also want to
consider Sensitivity Analyses with respect to a
(= the Upper Limit of the Support Interval [0; a])
* BIC is used to select the Best-fitting of the
Alternative Spline Models (with Different k and/or a)
52
ESTIMATION of the Flexible WCE Model
through Artificial Time-varying Covariates
From (1), (2) & (3), the effect of WCE is modeled as:
WCE ( u ) 
 w (u  t ) * X (t )    
tu
t
j
B j (u  t ) * X (t )
j
where BOTH β & j need to be estimated.
To Avoid Identifiability Problems, we define:
 j  
(4)
j
& construct Artificial Time-varying Covariates:
u
D j (u ) 
for j=1,…,m
B
j
(u  t ) X (t )
(5)
t
53
Cubic Regression Spline Basis


Weight function in (2) [on slide 16] is estimated using
Cubic** Regression Splines with Fixed Knots (= points
where consecutive Cubic Polynomials join each other)
Spline Basis is defined over a Limited Support Interval
[0; a] where:
a = (user-specified) maximum length of the
‘etiologically relevant exposure time window’
[past values of X(t) at t<u-a are a priori considered
irrelevant for the risk at time u and, thus, are assigned
the weight=0]
** Cubic splines ensure that w(u-t) and its 1st & 2nd Derivatives are
Continuous
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ESTIMATION of the Flexible WCE Model
through Artificial Time-varying Covariates
Given (4) & (5), the Cox’s model in (3) becomes:
q
 m

h ( u | X ( u ), Z ( u ))  h 0 ( u ) exp    j D j ( u )    s Z s ( u ) 
s 1
 j 1

(6)
Once Dj(u), j=1,…m, are calculated for each u =
un-censored event time, the model in (6) can
be implemented using standard software for
Cox’s model with time-dependent covariates
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Simulation Results
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More Simulation Results
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