Examples of joint models for multivariate longitudinal and - PowerPoint PPT Presentation

Examples of joint models for multivariate longitudinal and multistate processes in chronic diseases C´ ecile Proust-Lima works with Lo¨ ıc Ferrer, Ana¨ ıs Rouanet and H´ el` ene Jacqmin-Gadda INSERM U897, Epidemiology and Biostatistics, Bordeaux, France Univ. Bordeaux, ISPED, Bordeaux, France cecile.proust-lima@inserm.fr Workshop on Flexible Models for Longitudinal and Survival Data with Applications in Biostatistics C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 1 / 45

Joint modelling principle Simultaneous modelling of correlated longitudinal and survival data latent structure longitudinal time to marker event 6 1.00 4 0.75 Survival probability log(PSA+0.1) 2 0.50 0 0.25 −2 0.00 0 5 10 15 0 5 10 15 Years since the end of EBRT Years since the end of RT C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 2 / 45

Joint modelling principle Simultaneous modelling of correlated longitudinal and survival data latent structure longitudinal time to marker event Objectives : ◮ describe the longitudinal process stopped by the event ◮ predict the risk of event ajusted for the longitudinal process ◮ explore the association between the two processes C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 2 / 45

2 main families of joint models latent structure longitudinal time to marker event Mixed model Survival model (usually linear) (usually proportional hazards) Link with the latent structure : ◮ random effects from the mixed model (shared random effect models) ◮ latent class structure (joint latent class models) C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 3 / 45

Shared random-effect model (SREM) (Rizopoulos, 2012) latent structure longitudinal time to marker event Shared random-effects distribution : b i ∼ N ( µ, B ) Linear mixed model for the biomarker trajectory : Y i ( t ij ) = Y i ( t ij ) ∗ + ǫ ij = Z i ( t ij ) T b i + X Li ( t ij ) T β + ǫ ij with ǫ ij ∼ N � 0 , σ 2 � ǫ Proportional hazard model including marker trajectory characteristics : λ ( t | b i ) = λ 0 ( t ) e X Si ( t ) T δ + W i ( b i ,β, t ) T η → JM, JMBayes in R, stjm in Stata, JMFit in SAS C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 4 / 45

Joint latent class model (JLCM) (Proust-Lima et al., 2014) latent structure longitudinal time to marker event Shared latent class ( c i ) membership : e ξ 0 g + X Ci ⊤ ξ 1 g π ig = P ( c i = g | X pi ) = l = 1 e ξ 0 l + X Ci / top ξ 1 l with ξ 0 G = 0 & ξ 1G = 0 � G Class-specific linear mixed model for the biomarker trajectory : Y i ( t ij ) | c i = g = Z i ( t ij ) T b ig + X Li ( t ij ) ⊤ β g + ǫ ij with b ig ∼ N ( µ g , B g ) , ǫ ij ∼ N 0 , σ 2 � � ǫ Class-specific proportional hazard model : λ ( t | c i = g ) = λ 0 g ( t ) e X Ti ( t ) δ g → lcmm in R C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 5 / 45

Remarks (Proust-Lima et al., 2014) Shared random effect models : ◮ extension of the standard time-to-event models ◮ assessment of specific associations (surrogacy) ◮ quantification of the association Joint latent class models : ◮ heterogeneous population ◮ no assumption on the association ◮ useful for predictive tools C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 6 / 45

Remarks (Proust-Lima et al., 2014) Shared random effect models : ◮ extension of the standard time-to-event models ◮ assessment of specific associations (surrogacy) ◮ quantification of the association Joint latent class models : ◮ heterogeneous population ◮ no assumption on the association ◮ useful for predictive tools In any case, most developments for : ◮ a Gaussian longitudinal marker ◮ a right-censored time to event → but more complex data in most cohort studies on chronic diseases C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 6 / 45

In chronic diseases Longitudinal part : ◮ multiple markers of progression ◮ markers of different nature ◮ Gaussian, binary, poisson ◮ ordinal ◮ continuous but non Gaussian Survival part : ◮ competing risks ◮ recurrent events ◮ multiple events ◮ succession of different events C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 7 / 45

In chronic diseases Longitudinal part : ◮ multiple markers of progression ◮ markers of different nature ◮ Gaussian, binary, poisson ◮ ordinal ◮ continuous but non Gaussian Survival part : ◮ competing risks ◮ recurrent events ◮ multiple events ◮ succession of different events 3 examples of developments through the study of ◮ progression of localized Prostate cancer after treatment ◮ natural history of Alzheimer’s disease C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 7 / 45

Progression of localized Prostate Cancer after a treatment by radiation therapy C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 8 / 45

Localized prostate cancer Monitoring of patients after radiation therapy for a localized Prostate cancer : 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 ◮ prognostic factors at diagnosis (T-stage, 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) log(PSA+0.1) Gleason, dose of RT, ...) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ◮ repeated measures of PSA (prostate specific −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 antigen) collected in −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 routine 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT time since end of RT Interest in predicting the risk of progression ◮ multiple types : local recurrence, distant recurrence, death ◮ problem of initiation of new treatment : hormonal treatment C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 9 / 45

Dynamic prediction of clinical recurrence of any type ene et al., SMMR 2014) : (S` Individualized probability of clinical recurrence : ◮ in the next three years ◮ for a man naive of HT ◮ according to hypothetical times of initiation of HT (time-dependent covariate) 2.0 1.0 PSA measures From M4b time of prediction From M2c 0.8 1.5 Probability of recurrence log ( PSA + 0.1 ) 0.6 1.0 0.4 0.5 0.2 0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Init. now In 1 year In 2 years If no HT Time (years) since end of EBRT C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 10 / 45

Dynamic prediction of clinical recurrence of any type ene et al., SMMR 2014) : (S` Individualized probability of clinical recurrence : ◮ in the next three years ◮ for a man naive of HT ◮ according to hypothetical times of initiation of HT (time-dependent covariate) 2.0 1.0 PSA measures From M4b time of prediction From M2c 0.8 1.5 Probability of recurrence log ( PSA + 0.1 ) 0.6 1.0 0.4 0.5 0.2 0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Init. now In 1 year In 2 years If no HT Time (years) since end of EBRT C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 10 / 45