Modeling Prediction of the Nosocomial Pneumonia with a Multistate model

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1 Modeling Prediction of the Nosocomial Pneumonia with a Multistate model M.Nguile Makao 1 PHD student Director: J.F. Timsit 2 Co-Directors: B Liquet 3 & J.F. Coeurjolly 4 1 Team 11 Inserm U823-Joseph Fourier University 2 Team 11 Chu-Inserm U823-Joseph Fourier University 3 Team Biostatistic Inserm U897-Victor Segalen Bordeaux 2 University 4 Team SAGAG-Pierre Mendès-FranceUniversity September 23, 2009 M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm September U823-Joseph 23, 2009 Fourier 1 / 29Un

2 VAP definition and Epidemiology 1 Ventilator-Associated nosocomial Pneumonia (VAP): lung infection occurring within 48 hours or more after hospital admission. 2 Incidence 8% to 28% Patients receiving mechanical ventilation (MV). Consequences 1 Increases the length of the stay in ICU (5 days). 2 Mortality Increasing vs decreasing (controversial literature). Possible reasons: Definition and diagnosis of VAP is problematic. Heterogeneity (observable or unobservable). 3 Increasing the cost of expenditures. M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm September U823-Joseph 23, 2009 Fourier 2 / 29Un

3 Problematic of the VAP 1 Using anti-microbial problematic 2 Prediction of the VAP Figure: Monitoring of a patient in ICU Figure: Prediction of VAP in ICU 3 Identification of patients with hight risk to contract VAP. M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm September U823-Joseph 23, 2009 Fourier 3 / 29Un

4 OUTCOMEREA database 1 November 1996 to April 2009 / 16 French ICUs. 2 Data collected daily by senior physicians. 3 patients Information at admission and during the stay (Iatrogenic events,simplified Acute Physiology Score...) Selection criteria of study population Being in ICU at least 48h. Receiving MV since the first 48h after admission. Stop observation at discharge (48 h after MV) or death. Study population : 2871, 433(15.1%) VAP, 470(16.4%) Death without VAP and 1968(68.5%) Discharge. Among 433 VAP, 119(27.5%) Death with VAP and 314(72.5%) Discharge with VAP. M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm September U823-Joseph 23, 2009 Fourier 4 / 29Un

5 Movement of ICU patient Figure: Multistate of ICU patient Transition n (%) Min.time Max.time Median Mean (15.1) (16.4) (68.5) (27) (73) M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm September U823-Joseph 23, 2009 Fourier 5 / 29Un

6 Definition and notation 1 Basic definitions {X(t), t T = [0, τ]} a stochastic process on the finite state space E = {0, 1, 2, 3}. F t σ-algebra integrating all past of process until the time t. E := {0 1, 0 2, 0 3, 1 2, 1 3}. 2 Transition probability 3 Transition intensity P hj (s, t; F s ) = Pr(X(t) = j X(s) = h; F s ) (1) P hj (t, t + t; F α hj (t; F t ) = lim t ) t 0 t (2) M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm September U823-Joseph 23, 2009 Fourier 6 / 29Un

7 Different Model Different model assumptions can be made about the dependence of the transition rate (2) on time. These include: 1 Time homogeneous models: The intensities are constant over time. Kay R. Markov model for analysing cancer markers and disease states in survival studies Biometrics 1986;42: Markov Models: The transition intensities only depend on the history of the process through the current state. α hj (t; F t ) = α hj (t) Cox DR, Miller HD. the theory of stochastic processes. Chapman & Hall Semi-Markov Model: Future evolution not only depends on the occurent state h, but also on the entry time T h in to h. α hj (t; F t ) = α hj (t, T h ) = α hj (t T h ) Andersen PK, Esbjerg S,Sorensen TIA. Multistate models for bleeding episodes and mortality in liver cirrhosis. Statistics in Medicine 2001;19: M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm September U823-Joseph 23, 2009 Fourier 7 / 29Un

8 Use of covariates in multistates model 1 General regression model Z hj (t) = set of covariates for the transition h j. 2 Cox Model ( Z1 hj,..., Z p hj hj, Z 1 hj(t),..., Z q hj hj (t) ) T α hj (t; Z hj (t)) = Φ ( α hj0 (t; µ hj ); β T hjz hj (t) ) (3) Φ(u(.); υ) = u(.) exp(υ) α hj (t; Z hj (t)) = α hj0 (t; µ hj ) exp(β T hjz hj (t)) Assumption of Cox model The influence of a covariates is constant over time (Proportionality). Continuous covariates are log-linear All individuals have the same base line intensity. ( β hj = β1 hj,..., β p hj hj, β hj 1,..., β ) q hj T; hj µhj m hj vector of distribution parameters M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm September U823-Joseph 23, 2009 Fourier 8 / 29Un

9 Use of covariates in multistate model Z h (t) = {Z hj (t); j E, h j E} 1 Integrated intensity A hj (t; Z hj (t)) = 2 Survival function h j t S hj (t; Z hj (t)) = exp( 3 Survival function in state h S h (t; Z h (t)) = 0 exp(β T hjz hj (u))α hj0 (u)du. t 0 exp(β T hjz hj (u))α hj0 (u)du). S hj (t; Z hj (t)). h j E M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm September U823-Joseph 23, 2009 Fourier 9 / 29Un

10 Likelihood of multistate model I h set of patient in the state h. d h duration, t h entry time, τ h release time. d i h = τ h i ti h. θ parameters vector of model. 1 Likelihood of multistate model using the paths L (θ) 2 Likelihood of survival model L hj (θ hj) 3 Important result where θ hj = L (θ) = ( β T hj, µ T hj) T. (h j) E L hj(θ hj ), = M, SM M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 10 / 29Un

11 Parametric estimation Âhj(.; Z hj (.)) h j E 1 Likelihood of transition h j with j {1, 2, 3} L 0j(θ 0j ) = i I 0 Where N 0j (τ i 0 ) = N 0j(τ i 0) N 0j (τ i 0 ) τ i 0 = d i 0, t i 0 = 0. ( α0j (d i 0; Z i 0j(d i 0)) ) N 0j(τ i 0 ) S 0j (τ i 0; Z i 0j(τ i 0)). τ i S 0j (τ0; i Z i 0j(τ0)) i 0 = exp( exp(β T 0jZ i 0j(u))α 0j0 (u; µ 0j )du). 0 M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 11 / 29Un

12 Parametric estimation Âhj(.; Z hj (.)) h j E 1 Likelihood Markov L M 1j(θ hj ) = Semi-Markov i I h L SM 1j (θ hj ) = i I h ( α1j (τ i 1; Z i 1j(τ i 1)) ) N 1j(τ i 1 ) S 1j (τ i 1; Z i 1j(τ i 1)) S 1j (t i 1 ; Zi 1j(t i 1 )) ( α1j (d i 1; Z i 1j(d i 1)) ) N 1j(d i 1 ) S 1j (d i 1; Z i 1j(d i 1)) 2 Practical Notes t i h s 0,..., s ki τh i the time when covariates value change for the ith patient. Integrals used in previous equation are equal to: τ i h t i h exp(β T hjz i hj(u))da hj (u) = k i 1 l=0 exp(β T hjz i hj(s l )) A hj (s l ). (4) M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 12 / 29Un

13 Parametric estimation Âhj(.; Z hj (.)) h j E 1 Optimization of L(θ hj ) θ hj (Method Quasi-Newton) 2 Asymptotic results n( θhj θ hj ) converges to a zero-mean normal distribution with a covariance matrix that is estimated by 1 n F( θ hj ) 1. Variance of the parameters var( θ hj ) = diag( 1 n F( θ hj ) 1 ) (5) Under the hypothesis θ hj = θ hj the Wald statistic defined by: ( θ hj θ hj ) T F( θ hj )( θ hj θ hj ) is approximately distributed as the chi-square distribution with p hj + q hj + m hj degrees freedom. M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 13 / 29Un

14 Non-parametric estimation Âhj(.; Z hj (.)) h j E N hj (t) = i I h N i hj (t), Y h(t) = i I h Y i h (t) counting process. 1 Markov 1 j with j {2, 3} N i 1j (t) = 1 {t i 1 τ i 1 t} Y i 1 (t) = 1 {t i 1 t τ i 1}. P L(β hj ) = i I h τh i 0 exp(β T hjz i hj(τ i h)) Yh(τ l h) i exp(β T hjz l hj(τh)) i l I h 2 Semi-Markov 1 j with j {2, 3} N i hj (t) = 1 {d i h t} Y i h (t) = 1 {d i h t}. P L(β hj ) = i I h d i h 0 exp(β T hjz i hj(d i h)) Yh(d l i h) exp(β T hjz l hj(d i h)) l I h N i hj (τ i h ) N i hj (di h ), (6), (7) M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 14 / 29Un

15 Non-parametric estimation Âhj(.; Z hj (.)) h j E 1 Optimization of P L(β hj ) β hj. 2 Breslow estimator  hj0 (t) = t i I 0 h dnhj i (u) Yh l(u) exp( β T. hjz l hj(u)) l I h 3 Estimator of integrated intensity  hj (t; Z hj (t)) = T exp( β hjz hj (s i hj)) Âhj 0 (s i hj) s i hj t 4 Asymptotic results n(âhj 0 (t) A hj0 (t)) converges to a zero-mean Gaussian process and the covariance function can be estimated (see Andersen et al. 1992) M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 15 / 29Un

16 Non-parametric estimation α hj (.; Z hj (.)) h j E 1 Estimator of transition intensity α hj (t; Z hj (t)) = Âhj(t; Z hj (t)) t 2 Smoothing estimator of transition intensity αhj (t; Z hj (t)) = ( ) 1 b exp( β T s i hjz hj (s i hj ))K hj t Âhj b 0 (s i hj ). s i hj t Gaussian K(t) = 1 2π exp( 1 2 t2 ). U(t) = U(t) U(t ). b : bandwidth M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 16 / 29Un

17 Analyze of database 1 Selection Covariates step 1: Test the log-linearity of continuous covariates (Using Poisson regression) step 2: Test the Proportionality of covariates (the time dependent coefficients and Schoenfeld residuals) 2 Non-parametric Model (Cox) Use stepwise selection with Cox model where the entry threshold is equal to 0.25 and the stay threshold is equal to The model of each transition is validated by the C index defined by C t = AUC(t) = t weigths t 0 0 ROCI/D t AUC(u)w t (u)du (p)dp, ROC I/D t (p) is the true-positive rate, w t (u) are M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 17 / 29Un

18 Application 1: OUTCOMEREA database Result 1: estimation of the covariates effects M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 18 / 29Un

19 Application 1: OUTCOMEREA database 1 C index Transition M 13M 12SM 13SM C index Selection of model distribution M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 19 / 29Un

20 Estimation α hj0 (.) Markov/semi-Markov M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 20 / 29Un

21 Application 1: Description of profile M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 21 / 29Un

22 Estimator α 01 (., Z hj (.)) by profile M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 22 / 29Un

23 Definition of individualized prediction Figure: Prediction of VAP in ICU H hj (s, t) = {Z hj (x); x [s, t]} the covariates history over the interval [s, t] associated to the transition h j. H h (s, t) = {Z h (x); x [s, t]} the covariate history for all transitions from the state h and H i h(s, t) = { Z i h(x); x [s, t] }. M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 23 / 29Un

24 Definition of individualized prediction The prediction of the VAP for the ith individual over time interval [t + k, t + k + l] for all l, k > 0 ϕ i (t, k, l; H i 0(x(k), x(k + l))) = x(k+l) x(k) P 00 (x(k), u; H i 0(x(k), u)) da 010 (u) exp(β T 01Z i 01(u)). ϕ i (t, k, l; H i 0(x(k), x(k + l))) = 1 S 0 (x(k); Z i 0(x(k))) With x(v) = t + v x(k+l) x(k) S 0 (u; Z i 0(u)) exp(β T 01Z i 01(u))dA 010 (u) M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 24 / 29Un

25 Estimation of the profile 1 problematic: missing values in the prediction interval Figure: history of covariate of time-dependent 2 hypothetical solution The prediction of the VAP for ith individual over time interval [t + k, t + k + l] for all l, k > 0 we pose Z i hj(t) = Z i hj,t and we define the profile of ith patient by Z i hj,t(x) = Z i hj(x)1 [0,t] + Z i hj,t1 ]t,+ [, M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 25 / 29Un

26 Estimation of prediction Parametric estimator of prediction ϕ i (t, k, l; Z i h,t (x)) = exp( β T 01Z i 01,t) Ŝ 0 (t + k; Z i 0,t(t + k)) t+k+l t+k Ŝ 0j (u; Z i 0,t) α 010 (u)du. Non-parametric estimator of prediction exp( β T 01Z i 01,t) ϕ i (t, k, l; Z i h,t(x)) = Ŝ 0 (t + k; Z i 0,t(t + k)) Smoothed estimator of prediction ϕ i (t, k, l; Z i 1 h,t(x)) = Ŝ 0 (t + k; Z i 0,t(t + k)) t+k s i t+k+l s i t+k s i t+k+l s i Ŝ 0 (s i ; Z i 0,t) Â01 0 (s i ). S 0 (u; Z i 0,t) α hj (s i ; Z i 0j,t) s i. M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 26 / 29Un

27 Application:2 OUTCOMEREA database Base prediction M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 27 / 29Un

28 Application 2 OUTCOMEREA database Individualyzed prediction M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 28 / 29Un

29 Comments 1 Assumptions of the Cox model 2 Heterogeneity 3 Model validation Thank you your attention!!!!! M.Nguile Makao ( Team 11 Inserm U823-Joseph Multistate/prediction Fourier University VAP Team 11 Chu-Inserm SeptemberU823-Joseph 23, 2009 Fourier 29 / 29Un