Modlisation des Effets Cumulatifs des Expositions et des

Modélisation des Effets Cumulatifs des Expositions et des Facteurs Pronostiques Michal Abrahamowicz, Ph. D* James Mc. Gill Professor Department of Epidemiology & Biostatistics Mc. Gill University, Montreal, Quebec, CANADA & Marie-Pierre Sylvestre, Ph. D Professeur Adjoint, Universite de Montreal, Quebec, CANADA 1

Outline 1. 2. 3. 4. 5. 6. 7. 8. Objectives Challenges in modeling the effects of Time. Varying 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

Objectives l l 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

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 l How to relate the Entire History of Past Values of the Exposure with Current Risk of the Outcome ?

Challenges in modeling time-varying exposures l Do effects cumulate over time? Subject A l Subject B What is the relative importance of exposures that occurred at different times in the past? Subject C Subject D

Examples of Cumulative Effects l l l 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

Conventional Models for Time-Varying Prognostic Factors l Different models are used to relate the hazard at current time u with a time-varying exposure: i. ii. iii. iv. v. Current value (“dose”): X(u) Total Past Dose (=un-weighted cumulative dose) Total Duration of Past Exposure Any Exposure e. g. in the Last Month Mean Dose e. g. in the Last Month

(Some) Limitations of Conventional Models l 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,

Weighted Cumulative Exposure (WCE) model l 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,

Recency-weighted cumulative exposure (WCE) Abrahamowicz et al. (2006), Breslow et al. (1983), Hauptmann et al. (2000), Thomas (1988), Vacek (1997) (1) 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)*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

Example of a Weight Function From Abrahamowicz et al. (2006, Journal of Clinical Epidemiology, Figure 1)

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)

Need for Flexible Modeling of the Weight Function l 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 l Therefore, it would be Advantageous to Estimate the Weight Function Directly from the Empirical Data l To this end, we need Flexible Assumptions-free methods such as Splines

Flexible spline-based WCE Model [Sylvestre & Abrahamowicz (SIM 2009)] l In our Flexible WCE model, the Weight function is estimated by Cubic Splines: (2) where Bj, j=1, …, m, represent the m functions in the Cubic Spline basis, and j represent the estimable coefficients of the linear combination of the basis splines WCE in (2) is then modeled as a

Examples of Cubic Regression Splines (with 3 Interior Knots) The m functions Bj of the cubic spline basis B 1 B 2 B 3 B 4 Regression spline u-t = (0. 6, 0. 9, 0. 4, 0. 2) = (0. 5, 0. 1, 0. 2, 0. 4, 0. 2, 0. 5) Interior knots

Inference about the Estimated Weight Function ŵ( ) l Quasi-parametric LRT tests* of: i. ii. l H 0 = No Association between X(t) and risk: w(u-t)=0 for 0<t<u (df=k+4) H 0 = 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

Simulations Results: True w(u-t) (white) vs 100 Constrained Estimates [a=180 days] From Sylvestre and Abrahamowicz (2009, Statistics in Medicine, Figure 2)

Simulation Results: Goodness of Fit of Alternative Models l 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 l 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) l 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

Correcting for the “too short” initial support interval (in scenario “f”)

APPLICATION: Flurazepam (a Psychotropic drug) use vs Fall-related Injuries in Elderly l l l l Data from Dr. Robyn Tamblyn from Mc. Gill 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) T 0 = Jan. 1, 1990 (Delayed Entry at the time of the 1 st 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

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)

Flurazepam vs Falls: Constrained WEIGHT function with a=60 (Cubic Spline; 1 knot) From Sylvestre and Abrahamowicz (2009, Statistics in Medicine, Figure 3)

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)

Another Challenge: unknown Dose. Response function for continuous X(t) l l “Classic” WCE model (1), assumes Linear relationship between “dose” X(t) and risk (e. g. , Logit or Log hazard) Yet, many continuous exposures have Non. Linear effects on risk! [e. g. Abrahamowicz et al, AJE 1997] Ø Need to Extend the WCE models to incorporate Flexible Modeling of the possibly Non-linear effects of Continuous Time-Varying Prognostic Factors

Example of NON-LINEARITY of dose-response: Smoking Intensity vs (logit of) Lung Cancer risk Rachet et al (2004)

EXTENSION of our flexible WCE model to Joint Estimation of w( ) and g( ) in Time-to-Event analyses (7) l Model w( ) and g( ) with Cubic regression splines 1 or 2 equidistant interior knots 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

APPLICATION: assessing Relative importance of Past SBP values for CHD risk l l l l Data Source = Framingham Heart Study 30 years of Follow-up with 16 bi-annual exams (SBP and other risk factor measures) Time 0 = 1 st exam Event = 1 st Coronary Heart Disease (CHD) event (mortality/morbidity) 2603 women (762 CHD events) 2128 men (995 CHD events) Adjustment for: current age, serum

SBP vs CHD risk Men: PL(WCE)=6768 vs PL(Cox)=6787 Women: PL(WCE)=5570 vs PL(Cox)=5581 w( ) g( )

Some Limitations of Our Methods l l l 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 timevarying mediating/confounding variables (need to extend to Marginal Structural

Conclusions l 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 l 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 l 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. l 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. l Abrahamowicz M, Mac. Kenzie 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. l Breslow NE, Lubin JH, Marek P, Langholz B. Multiplicative models and cohort analysis. Journal of the American Statistical Society 1983; 78(381): 1– 12. l 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. l Mac. Kenzie T, Abrahamowicz M. Marginal and hazard ratio specific random data generation: applications to semiparametric bootstrapping. Statistics and Computing. 2002; 12(3): 245 -352. l 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. l 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. l Sylvestre MP, Abrahamowicz M. Comparisons of algorithms to generate event times conditional on time-dependent covariates. Statistics in Medicine. 2008; 27(14): 2618 -34. l 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. l Thomas DC. Models for exposure-time-response relationships with applications to cancer epidemiology. Annual Reviews of Public Health 1988; 9: 451 -82. l Vacek PM. Assessing the effect of intensity when exposure varies over time. Statistics in Medicine. 1997; 16(5): 505 -13.

MERCI ! Michal. Abrahamowicz@mcgill. ca

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)

Sensitivity analysis: impact of Changing the “Time Window” 39

Identifiability problems l Because w(u-t) is multiplied by g[X(t)], the 2 functions “share the scale” for r ≠ 0: which implies: i. Identifiability problems [Abrahamowicz & Mac. Kenzie (2007)] ii. Non-linearity of model (7) in its parameters 40

2 -step iterative Alternating Conditional Estimation (ACE) (maximum Partial Likelihood (PL)) l 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) l l l We continue Iterations (through steps 1 & 2) until “Convergence” (ΔPL<0. 01) At step 1) of the 1 st iteration g( ) = αX(t) [Linear] At step 1) of each iteration ŵ( ) is standardized: 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

Flexible WCE Model [Sylvestre & Abrahamowicz (2009)] l WCE in (2) is then modeled as a Time-Dependent Covariate in Cox’s model: (3) where: h 0(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

Use of Artificial Time-Dependent covariates permits implementation with standard software l To estimate w( ), at step 1 of ACE, construct artificial timedependent covariates: l and fit the following Cox’s model: 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)

Simulations Design l l l l Hypothetical Pharmaco-epi study of the Adverse Effects of a drug Cohort of N=500 new drug users t 0 = time of 1 st 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): Øsubjects repeatedly stop and re-start drug use Øboth Inter- & Intra-subject Variation in the length of consecutive Periods of use/non-use & in the Dose Ødose constant within each period of use but varied from one period to another

Use of Artificial Time-Dependent covariates permits implementation with standard software l To estimate g( ), at step 2 of ACE, calculate: l and fit the following Cox’s model: 47

Simulations l i. The Design and Methods of Simulations were Similar to slides 24 -25, Except: 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.

Estimates of w( ) [N=600; ~300 events]

Generation of Event Times Conditional on the WCE via novel ‘Permutational Algorithm’ A. B. C. Generate Individual exposure vectors xs[t], s=1, …, N Generate Observed Event/Censoring Times: 1) 2) • Marginal distribution of Event times Ti ~ U[1, 365] days, i=1, …, N 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 [Mac. Kenzie 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) If ti* = censored time, assign ti* to one of the xs[t] at risk by simple random sampling 50

Constrained WCE Model l l 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: Ø 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) Ø (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 51

Model Selection l l 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: where BOTH β & j need to be estimated. To Avoid Identifiability Problems, we define: (4) & construct Artificial Time-varying Covariates: (5) for j=1, …, m 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) l 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] l ** Cubic splines ensure that w(u-t) and its 1 st & 2 nd Derivatives are Continuous 54

ESTIMATION of the Flexible WCE Model through Artificial Time-varying Covariates Given (4) & (5), the Cox’s model in (3) becomes: (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

Simulation Results

More Simulation Results 57