Anais Rouanet

Transcript

1. Dynamic predictions from joint models for multiple longitudinal markers correlated to competing events with interval censoring Anaïs Rouanet Joint work with Hélène Jacqmin-Gadda MRC, Biostatistics Unit, Cambridge University, UK INSERM, Centre INSERM U1219 - Bordeaux Population Health, France SAM Conference 2017
2. Introduction Model Dynamic predictions Application Discussion Introduction Dementia : Syndrome implying deterioration in cognitive functions with impairment of the social and occupational functioning. Stakes of current research Natural history of dementia Tools for early diagnosis Anaïs Rouanet SAM Conference 2017 17/03/2017 1 / 14
3. Introduction Model Dynamic predictions Application Discussion Introduction Dementia : Syndrome implying deterioration in cognitive functions with impairment of the social and occupational functioning. Stakes of current research Natural history of dementia Tools for early diagnosis Methodological challenges Heterogeneity in cognitive decline Correlation between cognitive decline and occurrence of dementia Competing risk of death Interval censoring of time-to-dementia onset Anaïs Rouanet SAM Conference 2017 17/03/2017 1 / 14
4. Introduction Model Dynamic predictions Application Discussion Visitk Visitk+1 T0i Censoring interval Visitk Visitk+1 T0i Dementia Censoring interval Ti Dementia Anaïs Rouanet SAM Conference 2017 17/03/2017 2 / 14
5. Introduction Model Dynamic predictions Application Discussion Visitk Visitk+1 T0i Censoring interval Visitk Visitk+1 T0i Dementia Censoring interval Ti Dementia Objective Develop a dynamic predictive tool for dementia occurrence from repeated cognitive tests, accounting for heterogeneity of the data competing risk of death interval censoring Anaïs Rouanet SAM Conference 2017 17/03/2017 2 / 14
6. Introduction Model Dynamic predictions Application Discussion Joint latent class illness-death model - Rouanet et al. (2016), Biometrics Class-specific mixed model Marker Latent process Class-specific transition intensities Parametric transformation H Health (0) Dementia (1) Death (2) α 01g (t) α 02g (t) α 12g (t) Multinomial Logistic model Latent class Conditional independence assumption between the marker and the times-to-events, given the latent classes. Anaïs Rouanet SAM Conference 2017 17/03/2017 3 / 14
7. Introduction Model Dynamic predictions Application Discussion Membership probability : pig = P(ci = g|Xpi ) Anaïs Rouanet SAM Conference 2017 17/03/2017 4 / 14
8. Introduction Model Dynamic predictions Application Discussion Membership probability : pig = P(ci = g|Xpi ) → Latent process Λi , given class g : Λi (tij |ci = g) = XT ij βg + ZT ij uig βg : class-specific parameters uig ∼ N(0, σ2 g B) Zij sub-vector of Xij Anaïs Rouanet SAM Conference 2017 17/03/2017 4 / 14
9. Introduction Model Dynamic predictions Application Discussion Membership probability : pig = P(ci = g|Xpi ) → Latent process Λi , given class g : Λi (tij |ci = g) = XT ij βg + ZT ij uig βg : class-specific parameters uig ∼ N(0, σ2 g B) Zij sub-vector of Xij → Transformed gaussian score ˜ Y : ˜ Yij = ψ(Yij ; η) = Λi (tij ) + ij ij ∼ N(0, σ2 e ) ψ(.; η) : Parametric transformation Anaïs Rouanet SAM Conference 2017 17/03/2017 4 / 14
10. Introduction Model Dynamic predictions Application Discussion Membership probability : pig = P(ci = g|Xpi ) → Latent process Λi , given class g : Λi (tij |ci = g) = XT ij βg + ZT ij uig βg : class-specific parameters uig ∼ N(0, σ2 g B) Zij sub-vector of Xij → Transformed gaussian score ˜ Y : ˜ Yij = ψ(Yij ; η) = Λi (tij ) + ij ij ∼ N(0, σ2 e ) ψ(.; η) : Parametric transformation Health (0) α 01g (t) Dementia (1) α 12g (t) Death (2) α 02g (t) → Transition intensity from state k to state l for subject i in class g : αklig (t) = α0 klg (t) eXei γklg α0 klg : class-specific baseline intensity γklg : class-specific regression parameters Anaïs Rouanet SAM Conference 2017 17/03/2017 4 / 14
11. Introduction Model Dynamic predictions Application Discussion Log-likelihood L(θG ) = N i=1 log G g=1 pig f (Yi |ci = g; θG )P(Di |ci = g; θG ) − N i=1 log G g=1 pig e−A01ig (T0i ;θG )−A02ig (T0i ;θG ) f (Yi |ci = g; θG ), gaussian density Di = (T0i , Li , Ri , δA i , Ti , δD i ) with Ri = +∞ if δA i = 0 Visitk =Li Visitk+1 T0i Dementia ? Interval censoring Ti Anaïs Rouanet SAM Conference 2017 17/03/2017 5 / 14
12. Introduction Model Dynamic predictions Application Discussion Log-likelihood L(θG ) = N i=1 log G g=1 pig f (Yi |ci = g; θG )P(Di |ci = g; θG ) − N i=1 log G g=1 pig e−A01ig (T0i ;θG )−A02ig (T0i ;θG ) f (Yi |ci = g; θG ), gaussian density Di = (T0i , Li , Ri , δA i , Ti , δD i ) with Ri = +∞ if δA i = 0 Visitk =Li Visitk+1 T0i Dementia ? Interval censoring Ti Anaïs Rouanet SAM Conference 2017 17/03/2017 5 / 14
13. Introduction Model Dynamic predictions Application Discussion Log-likelihood L(θG ) = N i=1 log G g=1 pig f (Yi |ci = g; θG )P(Di |ci = g; θG ) − N i=1 log G g=1 pig e−A01ig (T0i ;θG )−A02ig (T0i ;θG ) f (Yi |ci = g; θG ), gaussian density Di = (T0i , Li , Ri , δA i , Ti , δD i ) with Ri = +∞ if δA i = 0 Visitk =Li Visitk+1 T0i Dementia ? Interval censoring Ti Anaïs Rouanet SAM Conference 2017 17/03/2017 5 / 14
14. Introduction Model Dynamic predictions Application Discussion Log-likelihood L(θG ) = N i=1 log G g=1 pig f (Yi |ci = g; θG )P(Di |ci = g; θG ) − N i=1 log G g=1 pig e−A01ig (T0i ;θG )−A02ig (T0i ;θG ) f (Yi |ci = g; θG ), gaussian density Di = (T0i , Li , Ri , δA i , Ti , δD i ) with Ri = +∞ if δA i = 0 Visitk =Li Visitk+1 T0i Dementia ? Interval censoring Ti Anaïs Rouanet SAM Conference 2017 17/03/2017 5 / 14
15. Introduction Model Dynamic predictions Application Discussion Log-likelihood L(θG ) = N i=1 log G g=1 pig f (Yi |ci = g; θG )P(Di |ci = g; θG ) − N i=1 log G g=1 pig e−A01ig (T0i ;θG )−A02ig (T0i ;θG ) f (Yi |ci = g; θG ), gaussian density Di = (T0i , Li , Ri , δA i , Ti , δD i ) with Ri = +∞ if δA i = 0 Visitk =Li Visitk+1 T0i Dementia ? Interval censoring Ti Anaïs Rouanet SAM Conference 2017 17/03/2017 5 / 14
16. Introduction Model Dynamic predictions Application Discussion Dynamic predictions of dementia occurrence X Landmark time s Marker Follow-up time X 0 1-πi (s,t) Anaïs Rouanet SAM Conference 2017 17/03/2017 6 / 14
17. Introduction Model Dynamic predictions Application Discussion Dynamic predictions of dementia occurrence X X Horizon t Landmark time s Prediction time s+t 1-πi (s,t) 0 1-πi (s,t) Marker Follow-up time Anaïs Rouanet SAM Conference 2017 17/03/2017 6 / 14
18. Introduction Model Dynamic predictions Application Discussion Dynamic predictions of dementia occurrence X X X Horizon t Landmark time s Prediction time s+t 0 1-πi (s,t) Marker Follow-up time Anaïs Rouanet SAM Conference 2017 17/03/2017 6 / 14
19. Introduction Model Dynamic predictions Application Discussion Dynamic predictions of dementia occurrence Horizon t Landmark time s Prediction time s+t 1-πi (s,t) 0 1-πi (s,t) Marker Follow-up time X X X Anaïs Rouanet SAM Conference 2017 17/03/2017 6 / 14
20. Introduction Model Dynamic predictions Application Discussion Dynamic probabilities of dementia occurrence Landmark probabilities : πi (s, t) = P(s < TA i s + t, TD i > TA i |TA i > s, TD i > s, Yi (s), Xi ) s landmark time, t prediction horizon TA i the time to dementia onset, TD i the time to death Yi (s) = {Yij , tij s} Anaïs Rouanet SAM Conference 2017 17/03/2017 7 / 14
21. Introduction Model Dynamic predictions Application Discussion Assessment criteria - (Blanche et al., 2015) Dynamic Area under the ROC curve : AUC(s, t) = P(πi (s, t) > πj (s, t)|Di (s, t) = 1, Dj (s, t) = 0, TA i > s, TD i > s, TA j > s, TD j > s) with Di (s, t) = 1(s<TA i s+t,TD i >TA i ) TA i the time to dementia onset, TD i the time to death Dynamic Brier’s Score : BS(s, t) = E D(s, t) − π(s, t) 2 |TA > s, TD > s Anaïs Rouanet SAM Conference 2017 17/03/2017 8 / 14
22. Introduction Model Dynamic predictions Application Discussion Assessment criteria - (Blanche et al., 2015) Dynamic Area under the ROC curve : AUC(s, t) = P(πi (s, t) > πj (s, t)|Di (s, t) = 1, Dj (s, t) = 0, TA i > s, TD i > s, TA j > s, TD j > s) with Di (s, t) = 1(s<TA i s+t,TD i >TA i ) TA i the time to dementia onset, TD i the time to death Dynamic Brier’s Score : BS(s, t) = E D(s, t) − π(s, t) 2 |TA > s, TD > s Anaïs Rouanet SAM Conference 2017 17/03/2017 8 / 14
23. Introduction Model Dynamic predictions Application Discussion Assessment criteria - (Blanche et al., 2015) Dynamic Area under the ROC curve : AUC(s, t) = P(πi (s, t) > πj (s, t)|Di (s, t) = 1, Dj (s, t) = 0, TA i > s, TD i > s, TA j > s, TD j > s) with Di (s, t) = 1(s<TA i s+t,TD i >TA i ) TA i the time to dementia onset, TD i the time to death Anaïs Rouanet SAM Conference 2017 17/03/2017 8 / 14
24. Introduction Model Dynamic predictions Application Discussion Assessment criteria - (Blanche et al., 2015) Dynamic Area under the ROC curve : AUC(s, t) = P(πi (s, t) > πj (s, t)|Di (s, t) = 1, Dj (s, t) = 0, TA i > s, TD i > s, TA j > s, TD j > s) with Di (s, t) = 1(s<TA i s+t,TD i >TA i ) TA i the time to dementia onset, TD i the time to death Dynamic Brier’s Score : BS(s, t) = E D(s, t) − π(s, t) 2 |TA > s, TD > s Anaïs Rouanet SAM Conference 2017 17/03/2017 8 / 14
25. Introduction Model Dynamic predictions Application Discussion Objective : Compare the predictive abilities of Isaacs Set Test (IST), Benton Visual Retention Test and their combination. Training sample : Paquid cohort - (Letenneur et al., 1994) 3328 subjects from Dordogne and Gironde, aged 65 and over Visits every 2/3 years during 25 years Validation sample : 3C cohort - (3C Study Group, 2003) 8809 subjects from 3 French cities, aged 65 and over Visits every 2/3 years during 12 years Selection in both samples : Dementia-free at baseline Performed at least once : IST [0-40], Benton [0-15] (and MMSE [0-30]) Anaïs Rouanet SAM Conference 2017 17/03/2017 9 / 14
26. Introduction Model Dynamic predictions Application Discussion Specification of the 3 models : IST, Benton, IST & Benton Conditional Mixed Model : Λi (tij ) =β0g + u(0) ig + β0,age Agei0 + β0,CEP CEPi + β0,learn 1(tij =0) + (β1g + u(1) ig + β1,age Agei0 ) × tij + (β2g + u(2) ig + β2,age Agei0 ) × t2 ij , uig ∼ N(0, σg B) Anaïs Rouanet SAM Conference 2017 17/03/2017 10 / 14
27. Introduction Model Dynamic predictions Application Discussion Specification of the 3 models : IST, Benton, IST & Benton Conditional Mixed Model : Λi (tij ) =β0g + u(0) ig + β0,age Agei0 + β0,CEP CEPi + β0,learn 1(tij =0) + (β1g + u(1) ig + β1,age Agei0 ) × tij + (β2g + u(2) ig + β2,age Agei0 ) × t2 ij , uig ∼ N(0, σg B) Parametric transformation : ψ(Yij ; η) = Λi (tij ) + ij ij ∼ N(0, σ2) Anaïs Rouanet SAM Conference 2017 17/03/2017 10 / 14
28. Introduction Model Dynamic predictions Application Discussion Specification of the 3 models : IST, Benton, IST & Benton Conditional Mixed Model : Λi (tij ) =β0g + u(0) ig + β0,age Agei0 + β0,CEP CEPi + β0,learn 1(tij =0) + (β1g + u(1) ig + β1,age Agei0 ) × tij + (β2g + u(2) ig + β2,age Agei0 ) × t2 ij , uig ∼ N(0, σg B) Parametric transformation : ψ(Yij ; η) = Λi (tij ) + ij ij ∼ N(0, σ2) Conditional Illness-death Model : α01ig (t) = α0 01g (t) exp(γ01g,age Agei0 + γ01g,CEP CEPi ) α 2ig (t) = α0 2g (t) exp(γ 2g,age Agei0 + γ 2g,Sex Sexi + γ 2g,CEP CEPi ), = 0, 1 Anaïs Rouanet SAM Conference 2017 17/03/2017 10 / 14
29. Introduction Model Dynamic predictions Application Discussion Specification of the 3 models : IST, Benton, IST & Benton Conditional Mixed Model : Λi (tij ) =β0g + u(0) ig + β0,age Agei0 + β0,CEP CEPi + β0,learn 1(tij =0) + (β1g + u(1) ig + β1,age Agei0 ) × tij + (β2g + u(2) ig + β2,age Agei0 ) × t2 ij , uig ∼ N(0, σg B) Parametric transformation : ψk (Yijk ; ηk ) = Λi (tij ) + β(k) age Agei0 + β(k) CEP CEPi + αik + ijk αik ∼ N(0, σ2 αk ), ijk ∼ N(0, σ2 k ) , k=1,...,K Conditional Illness-death Model : α01ig (t) = α0 01g (t) exp(γ01g,age Agei0 + γ01g,CEP CEPi ) α 2ig (t) = α0 2g (t) exp(γ 2g,age Agei0 + γ 2g,Sex Sexi + γ 2g,CEP CEPi ), = 0, 1 Anaïs Rouanet SAM Conference 2017 17/03/2017 10 / 14
30. Introduction Model Dynamic predictions Application Discussion Comparison results : IST vs Benton 0 0.5 0.6 0.7 0.8 0.9 1.0 estimates of AUC(s,t) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 IST3 Benton3 0 4 0.02 0.10 estimates of BS(s,t) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.05 0 4 0.00 0.15 estimates of diff AUC(s,t) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.05 0.1 0 4 0.000 estimates of diff BS(s,t) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −0.005 0.005 Landmark time s (years) AUC Expected Brier Score Anaïs Rouanet SAM Conference 2017 17/03/2017 11 / 14
31. Introduction Model Dynamic predictions Application Discussion Comparison results : IST vs IST & Benton 0 0.5 0.6 0.7 0.8 0.9 1.0 estimates of AUC(s,t) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 IST3 IST_Benton3 0 4 0.02 0.10 estimates of BS(s,t) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.05 0 4 0.00 0.15 estimates of diff AUC(s,t) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.05 0.1 0 4 0.000 estimates of diff BS(s,t) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −0.005 0.005 Landmark time s (years) AUC Expected Brier Score Anaïs Rouanet SAM Conference 2017 17/03/2017 12 / 14
32. Introduction Model Dynamic predictions Application Discussion Individual dynamic predictions of 6 subjects x x x 0 1 2 3 4 5 0 10 20 30 40 years Y − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − 0 0.2 0.4 0.6 0.8 1 Probability of recurrence x x x 0 1 2 3 4 5 0 10 20 30 40 years Y − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − 0 0.2 0.4 0.6 0.8 1 Probability of recurrence x x 0 1 2 3 4 5 0 10 20 30 40 years Y − − − − − − − − − − − − − − − − − − 0 0.2 0.4 0.6 0.8 1 Probability of recurrence x x x 0 1 2 3 4 5 0 10 20 30 40 years Y − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − 0 0.2 0.4 0.6 0.8 1 Probability of recurrence x x x 0 1 2 3 4 5 0 10 20 30 40 years Y − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − 0 0.2 0.4 0.6 0.8 1 Probability of recurrence x x 0 1 2 3 4 5 0 10 20 30 40 years Y − − − − − − − − − − − − − − − 0 0.2 0.4 0.6 0.8 1 Probability of recurrence Anaïs Rouanet SAM Conference 2017 17/03/2017 13 / 14
33. Introduction Model Dynamic predictions Application Discussion Discussion Dynamic tool to predict dementia occurrence, handling interval censoring, competing risk of death and heterogeneity among the data. IST has better AUC and BS than BENTON at s = 5 years. Combination of tests does not improve predictions when the common latent process catches a smaller variability. Perspectives : - Apply on other tests : MMSE, dependency scores... Anaïs Rouanet SAM Conference 2017 17/03/2017 14 / 14
34. References Rouanet A., Joly, P., Dartigues J-F., Proust-Lima C. and Jacqmin-Gadda H. (2016). Joint Latent Class Model for Longitudinal Data and Interval-Censored Semi-Competing Events : Application to Dementia. Biometrics, 72(4) :1123-1135. Proust-Lima C., Dartigues J-F. and Jacqmin-Gadda H. (2016). Joint modeling of repeated multivariate cognitive measures and competing risks of dementia and death : a latent process and latent class approach. Statist. Med., 35 : 382-398. Letenneur, L., Commenges, D., Dartigues, J.-F. and Barberger- Gateau, P. (1994). Incidence of dementia and Alzheimers disease in elderly community residents of south-western France. International Journal of Epidemiology 23 : 1256-1261. Blanche P., Proust-Lima C., Loubere L., Berr C., Dartigues J-F and Jacqmin-Gadda H. (2015). Quantifying and Comparing Dynamic Predictive Accuracy of Joint Models for Longitudinal Marker and Time-to-Event in Presence of Censoring and Competing Risks. Biometrics, 71(1) :102-13.
35. Imputation rule for handling interval censoring on 3C sample : - if diagnosed with dementia, Ti = Li +Ri 2 - if dead and Ti − Li < 2 years, Ti = Ti - if dead and Ti − Li > 2 years, Ti = Li Choice of models - BIC criterion G IST BENTON IST BENTON 1 91678 66210 146132 2 91232 65953 145601 3 91060 65922 145373 4 91075 65972 145378 Anaïs Rouanet SAM Conference 2017 17/03/2017 14 / 14
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