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

marker longitudinal structure latent time to event

−2 2 4 6 5 10 15

Years since the end of EBRT log(PSA+0.1)

0.00 0.25 0.50 0.75 1.00 5 10 15

Years since the end of RT Survival probability

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

marker longitudinal structure latent time to 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

marker longitudinal structure latent time to event

Mixed model (usually linear) Survival model (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)

marker longitudinal structure latent time to event

Shared random-effects distribution : bi ∼ N (µ, B) Linear mixed model for the biomarker trajectory :

Yi(tij) = Yi(tij)∗ + ǫij = Zi(tij)Tbi + XLi(tij)Tβ + ǫij with ǫij ∼ N

• 0, σ2

ǫ

• Proportional hazard model including marker trajectory characteristics :

λ(t | bi) = λ0(t)eXSi(t)Tδ+Wi(bi,β,t)Tη

→ 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)

marker longitudinal structure latent time to event

Shared latent class (ci) membership :

πig = P(ci = g|Xpi) = eξ0g+XCi⊤ξ1g G

l=1 eξ0l+XCi/topξ1l with ξ0G = 0 & ξ1G = 0

Class-specific linear mixed model for the biomarker trajectory :

Yi(tij) |ci=g= Zi(tij)Tbig + XLi(tij)⊤βg + ǫij with big ∼ N (µg, Bg) , ǫij ∼ N

• 0, σ2

ǫ

• Class-specific proportional hazard model :

λ(t | ci = g) = λ0g(t)eXTi(t)δg

→ 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 :

◮ prognostic factors at

diagnosis (T-stage, Gleason, dose of RT, ...)

◮ repeated measures of

PSA (prostate specific antigen) collected in routine

2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 time since end of RT log(PSA+0.1) 2 4 6 8 10 −2 −1 1 2 3 4 2 4 6 8 10 −2 −1 1 2 3 4 2 4 6 8 10 −2 −1 1 2 3 4 2 4 6 8 10 −2 −1 1 2 3 4 2 4 6 8 10 −2 −1 1 2 3 4 2 4 6 8 10 −2 −1 1 2 3 4 2 4 6 8 10 −2 −1 1 2 3 4 2 4 6 8 10 −2 −1 1 2 3 4 2 4 6 8 10 −2 −1 1 2 3 4 2 4 6 8 10 −2 −1 1 2 3 4 2 4 6 8 10 −2 −1 1 2 3 4 2 4 6 8 10 −2 −1 1 2 3 4 2 4 6 8 10 −2 −1 1 2 3 4 2 4 6 8 10 −2 −1 1 2 3 4

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

(S` ene et al., SMMR 2014) :

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) Time (years) since end of EBRT 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 log(PSA + 0.1) PSA measures time of prediction

• Init. now

In 1 year In 2 years If no HT

0.0 0.2 0.4 0.6 0.8 1.0 Probability of recurrence From M4b From M2c C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 10 / 45

Dynamic prediction of clinical recurrence of any type

(S` ene et al., SMMR 2014) :

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) Time (years) since end of EBRT 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 log(PSA + 0.1) PSA measures time of prediction

• Init. now

In 1 year In 2 years If no HT

0.0 0.2 0.4 0.6 0.8 1.0 Probability of recurrence From M4b From M2c C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 10 / 45

Dynamic prediction of clinical recurrence of any type

(S` ene et al., SMMR 2014) :

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) Time (years) since end of EBRT 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 log(PSA + 0.1) PSA measures time of prediction

• Init. now

In 1 year In 2 years If no HT

0.0 0.2 0.4 0.6 0.8 1.0 Probability of recurrence From M4b From M2c C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 10 / 45

Multiple types of progression

Clinical progression is a multistate process :

treatment end therapy hormon. death local recurr. recurr. distant years logPSA

Importance of distinguishing the different types to :

◮ clarify the impact of PSA dynamics and other prognostic factors on each

transition

◮ predict type-specific progressions C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 11 / 45

The joint longitudinal and multistate model (Ferrer et al., arXiv

2015)

Notations : multi-state process

◮ Ei = {Ei(t), Ti0 ≤ t ≤ Ci} is a non-homogeneous Markov process ⋆ Ei(t) takes values in the finite state space S = {0, 1, . . . , M} ⋆ Ti0 the left truncation time, Ci the right censoring time ◮ Ti = (Ti1, . . . , Timi)⊤ the mi observed times with Tir < Ti(r+1), ∀r ∈ S ◮ δi = (δi1, . . . , δimi)⊤ the vector of observed transition indicators

longitudinal process

◮ Yi = (Yi1, . . . , Yini)⊤ the ni measures of the marker collected at times

ti1, . . . , tini, with tini ≤ Ti1

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 12 / 45

The joint longitudinal and multistate model (Ferrer et al., arXiv

2015) (cont’d)

Longitudinal part : mixed model Yij = Y∗

i (tij) + ǫij

= XLi(tij)⊤β + Zi(tij)⊤bi + ǫij

◮ bi ∼ N(0, D),

ǫi = (ǫi1, . . . , ǫini)⊤ ∼ N(0, σ2Ini), bi | = ǫi

Survival part : multistate model λi

hk(t|bi)

= limdt→0 Pr(Ei(t + dt) = k|Ei(t) = h; bi) dt = λhk,0(t) exp(X⊤

Thk,iγhk + Whk,i(bi, t)⊤ηhk), for h, k ∈ S,

◮ λhk,0(t) parametric baseline intensity, XThk,i prognostic factors ◮ Whk,i(bi, t) the dependence structure (S`

ene et al., J sfds 2014)

⋆ Whk,i(bi, t) = Y∗

i (t)

− →

(true current level)

⋆ Whk,i(bi, t) = ∂Y∗

i (t)/∂t

− →

(true current slope)

⋆ Whk,i(bi, t) =

• Y∗

i (t), ∂Y∗ i (t)/∂t

⊤ − →

(both)

⋆ . . . C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 13 / 45

Maximum Likelihood Estimation

Likelihood function using Yi | =

bi Ti,

L(θ) =

N

• i=1
• Rq fY(Yi|bi; θ) fT(Ti, δi|bi; θ) fb(bi; θ) dbi

with :

◮ Random effects part :

bi ∼ Nq(0, D)

◮ Longitudinal part :

Yi |bi ∼ Nni(X⊤

Li β + Z⊤ i bi, σ2Ini)

◮ Multi-state part :

fT(Ti, δi|bi; θ) =

mi−1

• r=0
• Pi

Ei(Tir),Ei(Tir)(Tir, Ti(r+1)|bi)×

λi

Ei(Tir),Ei(Ti(r+1))(Ti(r+1)|bi) δi(r+1)

• with Phh(s, t) = exp

t

s λhh(u) du

• = exp
• −

k=h

t

s λhk(u) du

• C´

ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 14 / 45

Implementation under R

Relies on JM package Implementation procedure decomposed into four steps :

• 1. lme() function (nlme package) to initialise the parameters in the

longitudinal sub-model ;

• 2. mstate package to adapt the data to the multi-state framework ;
• 3. coxph() function (survival package) to initialise the parameters in the

multi-state sub-model ;

• 4. JMstateModel() function (extension of jointModel()) to estimate all

the parameters of the joint multi-state model.

Likelihood computed and optimised using :

◮ numerical integration algorithms (Gaussian quadratures) ◮ optimisation algorithms (EM + quasi-Newton) C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 15 / 45

Application

2 cohorts of men with localized prostate cancer treated by radiotherapy (N=1474) Repeated measures of PSA

−2 2 4 6 5 10 15

Years since the end of EBRT log(PSA+0.1)

10 (3, 21) measurements per patient (50th, (5th, 95th) %iles)

Multi-state representation of the clinical progressions

End EBRT Local Recurrence

1

Hormonal Therapy

2

Distant Recurrence

3

Death

4 λ02(t) λ12(t) λ23(t) λ24(t) λ03(t) λ13(t) λ34(t) λ01(t) λ14(t) λ04(t)

Υ =      533 144 227 47 523 20 90 10 24 106 33 178 13 77 802      matrix of direct transitions

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 16 / 45

Specification of the joint model (1/2)

Longitudinal sub-model specification Yij = Y∗

i (tij) + ǫij

=

• β0 + X⊤

L0iβ0,cov + bi0

• +
• β1 + X⊤

L1iβ1,cov + bi1

• × f1(tij) +
• β2 + X⊤

L2iβ2,cov + bi2

• × f2(tij) + ǫij

◮ f1(t) = (1 + t)−1.2 − 1

and f2(t) = t

◮ bi = (bi0, bi1, bi2)⊤ ∼ N (0, D),

D unstructured, ǫi ∼ N(0, σ2Ini)

◮ XL0i, XL1i and XL2i were obtained using a backward stepwise procedure. C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 17 / 45

Specification of the joint model (2/2)

Multi-state sub-model specification λi

hk(t|bi) = λhk,0(t) exp

• X⊤

T,hk,iγhk +

• Y∗

i (t)

∂Y∗

i (t)/∂t

⊤ ηhk,level ηhk,slope

• ◮ Log-baseline intensities approximated by B-splines

◮ Proportionality assumptions between several baseline intensities ◮ Backward stepwise procedure to select the prognostic factors ◮ Dependence function chosen by Wald tests C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 18 / 45

Results

Estimates of the association parameters between the longitudinal and multi-state processes

Value StdErr p-value Level : 01 0.37 0.09 < 0.001 Level : 02 0.51 0.07 < 0.001 Level : 03 0.45 0.11 < 0.001 Level : 04 −0.17 0.05 0.001 Level : 12 −0.16 0.10 0.110 Level : 13 −0.41 0.20 0.042 Level : 14 0.10 0.14 0.487 Level : 23 −0.15 0.09 0.120 Level : 24 0.00 0.05 0.412 Level : 34 0.04 0.08 0.609 . . . . . . . . . . . . . . . . . . . . . Slope : 01 2.54 0.31 < 0.001 Slope : 02 3.04 0.25 < 0.001 Slope : 03 2.43 0.49 < 0.001 Slope : 04 1.03 0.32 0.001 Slope : 12 2.01 0.61 0.001 Slope : 13 3.18 0.80 < 0.001 Slope : 14 −0.20 1.27 0.873 Slope : 23 0.97 0.67 0.150 Slope : 24 0.29 0.52 0.583 Slope : 34 −0.79 0.78 0.313

Multi-state process

End EBRT Local Recurrence

1

Hormonal Therapy

2

Distant Recurrence

3

Death

4 λ02(t) λ12(t) λ23(t) λ24(t) λ03(t) λ13(t) λ34(t) λ01(t) λ14(t) λ04(t)

Prognostic factors : advanced initial stage not always associated with intensities of transitions between health states after adjustment on PSA

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 19 / 45

Results

Estimates of the association parameters between the longitudinal and multi-state processes

Value StdErr p-value Level : 01 0.37 0.09 < 0.001 Level : 02 0.51 0.07 < 0.001 Level : 03 0.45 0.11 < 0.001 Level : 04 −0.17 0.05 0.001 Level : 12 −0.16 0.10 0.110 Level : 13 −0.41 0.20 0.042 Level : 14 0.10 0.14 0.487 Level : 23 −0.15 0.09 0.120 Level : 24 0.00 0.05 0.412 Level : 34 0.04 0.08 0.609 . . . . . . . . . . . . . . . . . . . . . Slope : 01 2.54 0.31 < 0.001 Slope : 02 3.04 0.25 < 0.001 Slope : 03 2.43 0.49 < 0.001 Slope : 04 1.03 0.32 0.001 Slope : 12 2.01 0.61 0.001 Slope : 13 3.18 0.80 < 0.001 Slope : 14 −0.20 1.27 0.873 Slope : 23 0.97 0.67 0.150 Slope : 24 0.29 0.52 0.583 Slope : 34 −0.79 0.78 0.313

Multi-state process

End EBRT Local Recurrence

1

Hormonal Therapy

2

Distant Recurrence

3

Death

4 λ02(t) λ12(t) λ23(t) λ24(t) λ03(t) λ13(t) λ34(t) λ01(t) λ14(t) λ04(t)

Prognostic factors : advanced initial stage not always associated with intensities of transitions between health states after adjustment on PSA

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 19 / 45

Results

Estimates of the association parameters between the longitudinal and multi-state processes

Value StdErr p-value Level : 01 0.37 0.09 < 0.001 Level : 02 0.51 0.07 < 0.001 Level : 03 0.45 0.11 < 0.001 Level : 04 −0.17 0.05 0.001 Level : 12 −0.16 0.10 0.110 Level : 13 −0.41 0.20 0.042 Level : 14 0.10 0.14 0.487 Level : 23 −0.15 0.09 0.120 Level : 24 0.00 0.05 0.412 Level : 34 0.04 0.08 0.609 . . . . . . . . . . . . . . . . . . . . . Slope : 01 2.54 0.31 < 0.001 Slope : 02 3.04 0.25 < 0.001 Slope : 03 2.43 0.49 < 0.001 Slope : 04 1.03 0.32 0.001 Slope : 12 2.01 0.61 0.001 Slope : 13 3.18 0.80 < 0.001 Slope : 14 −0.20 1.27 0.873 Slope : 23 0.97 0.67 0.150 Slope : 24 0.29 0.52 0.583 Slope : 34 −0.79 0.78 0.313

Multi-state process

End EBRT Local Recurrence

1

Hormonal Therapy

2

Distant Recurrence

3

Death

4 λ02(t) λ12(t) λ23(t) λ24(t) λ03(t) λ13(t) λ34(t) λ01(t) λ14(t) λ04(t)

Prognostic factors : advanced initial stage not always associated with intensities of transitions between health states after adjustment on PSA

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 19 / 45

Results

Estimates of the association parameters between the longitudinal and multi-state processes

Value StdErr p-value Level : 01 0.37 0.09 < 0.001 Level : 02 0.51 0.07 < 0.001 Level : 03 0.45 0.11 < 0.001 Level : 04 −0.17 0.05 0.001 Level : 12 −0.16 0.10 0.110 Level : 13 −0.41 0.20 0.042 Level : 14 0.10 0.14 0.487 Level : 23 −0.15 0.09 0.120 Level : 24 0.00 0.05 0.412 Level : 34 0.04 0.08 0.609 . . . . . . . . . . . . . . . . . . . . . Slope : 01 2.54 0.31 < 0.001 Slope : 02 3.04 0.25 < 0.001 Slope : 03 2.43 0.49 < 0.001 Slope : 04 1.03 0.32 0.001 Slope : 12 2.01 0.61 0.001 Slope : 13 3.18 0.80 < 0.001 Slope : 14 −0.20 1.27 0.873 Slope : 23 0.97 0.67 0.150 Slope : 24 0.29 0.52 0.583 Slope : 34 −0.79 0.78 0.313

Multi-state process

End EBRT Local Recurrence

1

Hormonal Therapy

2

Distant Recurrence

3

Death

4 λ02(t) λ12(t) λ23(t) λ24(t) λ03(t) λ13(t) λ34(t) λ01(t) λ14(t) λ04(t)

Prognostic factors : advanced initial stage not always associated with intensities of transitions between health states after adjustment on PSA

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 19 / 45

Diagnostics for the parametric assumptions

Goodness-of-fit plots for the longitudinal process

◮ Conditional standardized residuals

versus fitted values

−2 −1 1 2 3 −2 2 4

Fitted Values Conditional Standardized Residuals

◮ Observed and predicted values of

the biomarker

0.00 0.25 0.50 0.75 1.00 0.0 2.5 5.0 7.5 10.0 12.5

Years since the end of EBRT log(PSA+0.1)

Observed Predicted C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 20 / 45

Diagnostics for the parametric assumptions

Goodness-of-fit plots for the longitudinal process Goodness-of-fit plots for the multi-state process

◮ Predicted transition probabilities from the joint multi-state model and

non-parametric probability transitions

From state 0 From state 1 From state 2 From state 3 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 5 10 15 5 10 15

Years since the end of EBRT Transition Probabilities

To state 1 2 3 4 Estimators Par. Non−par. 95% CI

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 20 / 45

Natural history of Alzheimer’s disease, dementias and cognitive aging

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 21 / 45

Cognitive aging and dementia

Dementia (e.g. Alzheimer’s disease) characterized by a progressive decline of cognition Most interest in

◮ the natural history of dementia ◮ risk factors of cognitive decline and dementia ◮ dynamic individual prediction of dementia C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 22 / 45

Cognitive aging and dementia

Dementia (e.g. Alzheimer’s disease) characterized by a progressive decline of cognition Most interest in

◮ the natural history of dementia ◮ risk factors of cognitive decline and dementia ◮ dynamic individual prediction of dementia

→ 1st complexity : elderly pathology

◮ delayed entry ◮ dementia in competition with death ◮ diagnosis at pre-established visit times

→ 2nd complexity : cognition is not directly observed

◮ cognitive process (trait) defined in continuous time ◮ repeated psychometric tests measured in discrete times ⋆ multiple cognitive functions (langage, memory, attention,...) ⋆ noisy measures of overall cognition ⋆ limited statistical properties C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 22 / 45

Latent process mixed model

time t covariates X(t) true PSA level underlying Y*(t) PSA Y at T

1

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 23 / 45

Latent process mixed model

time t covariates X(t) Λ(t) process cognitive test 1 Y at T

1 11

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 23 / 45

Latent process mixed model

time t covariates X(t) Λ(t) process ... ... Y at T Y at T

1 K

Y at T

k 1k 11 1K

cognitive test 1 test k test K

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 23 / 45

Latent process mixed model

time t covariates X(t) Λ(t) process ... ... Y at T Y at T

1 K

Y at T

k 1k 11 1K

cognitive test 1 test k test K

← Λi(t) = XL1i(t)⊤β + Zi(t)⊤bi + wi(t)

◮ bi ∼ MVN(µ, B) ◮ wi(t) autocorrelated process ◮ bi0 ∼ N(0, 1) for identifiability C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 23 / 45

Latent process mixed model

time t covariates X(t) Λ(t) process ... ... Y at T Y at T

1 K

Y at T

k 1k 11 1K

cognitive test 1 test k test K

← Λi(t) = XL1i(t)⊤β + Zi(t)⊤bi + wi(t)

◮ bi ∼ MVN(µ, B) ◮ wi(t) autocorrelated process ◮ bi0 ∼ N(0, 1) for identifiability

← Ykij = ζ1k + ζ2k˜ Ykij ˜ Ykij = Λi(tijk) + ǫkij

◮ ǫkij ∼ N(0, σ2

ǫk)

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 23 / 45

Latent process mixed model

time t covariates X(t) Λ(t) process ... ... Y at T Y at T

1 K

Y at T

k 1k 11 1K

cognitive test 1 test k test K

← Λi(t) = XL1i(t)⊤β + Zi(t)⊤bi + wi(t)

◮ bi ∼ MVN(µ, B) ◮ wi(t) autocorrelated process ◮ bi0 ∼ N(0, 1) for identifiability

← Ykij = ζ1k + ζ2k˜ Ykij ˜ Ykij = Λi(tijk) + XL2i(t)⊤γk + αki + ǫkij

◮ αki ∼ N(0, σ2

αk)

◮ ǫkij ∼ N(0, σ2

ǫk)

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 23 / 45

Latent process mixed model

time t covariates X(t) Λ(t) process ... ... Y at T Y at T

1 K

Y at T

k 1k 11 1K

cognitive test 1 test k test K

← Λi(t) = XL1i(t)⊤β + Zi(t)⊤bi + wi(t)

◮ bi ∼ MVN(µ, B) ◮ wi(t) autocorrelated process ◮ bi0 ∼ N(0, 1) for identifiability

← linear link function

Y noisy latent process

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 23 / 45

Latent process mixed model involving nonlinear link functions (Proust-Lima et al., 2015)

time t covariates X(t) Λ(t) process ... ... Y at T Y at T

1 K

Y at T

k 1k 11 1K

cognitive test 1 test k test K

← Λi(t) = XL1i(t)⊤β + Zi(t)⊤bi + wi(t)

◮ bi ∼ MVN(µ, B) ◮ wi(t) autocorrelated process ◮ bi0 ∼ N(0, 1) for identifiability

← nonlinear link function

Y noisy latent process

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 24 / 45

Latent process mixed model involving nonlinear link functions (Proust-Lima et al., 2015)

time t covariates X(t) Λ(t) process ... ... Y at T Y at T

1 K

Y at T

k 1k 11 1K

cognitive test 1 test k test K parameterized link functions

← Λi(t) = XL1i(t)⊤β + Zi(t)⊤bi + wi(t)

◮ bi ∼ MVN(µ, B) ◮ wi(t) autocorrelated process ◮ bi0 ∼ N(0, 1) for identifiability

← Ykij = Hk( ˜ Ykij ; ζk) ˜ Ykij = Λi(tijk) + XL2i(t)⊤γk + αki + ǫkij

◮ αki ∼ N(0, σ2

αk)

◮ ǫkij ∼ N(0, σ2

ǫk)

Hk = flexible parameterized transformation for outcome k → linear, standardised Beta CDF , quadratic I-splines, thresholds, ...

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 24 / 45

Joint model for multivariate cognitive measures, dementia and death

Λ process latent ...

1 K

Y at t

k

Y at t Y at t

• utcome
• utcome
• utcome

... ( (t))

j1 jk jK

u effects random

cognitive measures :

◮ latent process mixed

model

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 25 / 45

Joint model for multivariate cognitive measures, dementia and death

Λ process latent ...

1 K

(Τ, δ ) Y at t

k

Y at t Y at t

• utcome
• utcome
• utcome

... ( (t))

j1 jk jK

u effects random times to event

cognitive measures :

◮ latent process mixed

model dementia and death ?

◮ option 1 : first event in competing setting ◮ option 2 : multistate model C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 25 / 45

Joint model for multivariate cognitive measures, dementia and death

Λ process latent ...

1 K

(Τ, δ ) Y at t

k

Y at t Y at t

• utcome
• utcome
• utcome

... ( (t))

j1 jk jK

latent classes c u effects random times to event

cognitive measures :

◮ latent process mixed

model shared latent quantity = latent classes

◮ heterogeneous trajectories ◮ no assumption on the association

dementia and death ?

◮ option 1 : first event in competing setting ◮ option 2 : multistate model C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 25 / 45

Option 1 : competing setting (Proust-Lima et al., ArXiv 2014)

Times to events = Time to P competing events

◮ Ti = min(censoring ˜

Ti, and cause-spec. times T∗

i1, ..., T∗ iP),

◮ δi = 0 for censored, δi = p otherwise

class-specific cause-specific proportional hazard models λp(t)|ci=g = λ0p(t; νpg) exp(X⊤

Ti ζpg)

◮ λ0p parametric (splines, Gompertz, Weibull,...) C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 26 / 45

Maximum Likelihood Estimation

Likelihood function using Yi | =

ci Ti,

L(θ) =

N

• i=1

G

• g=1

fY(Yi|XLi, Zi, ci = g; θ) fT(Ti, δi|XTi, ci = g; θ) P(ci = g|XCi; θ)

with :

◮ fY(Yi|XLi, Zi, ci = g; θ) from the latent process mixed model ⋆ closed form if only continuous markers :

= multivariate normal × Jacobian of (Hk)k=1,...,K

⋆ by numerical integration otherwise ... ◮ fT(Ti, δi|XTi, ci = g; θ) from the cause-specific model

= overall survival Sig × instantaneous risk for cause p in g if δi = p

◮ P(ci = g|XCi; θ) from a multinomial logistic model

Left truncation (entry at T0i) : lT0(θ) = log

• L(θ)

N

i=1 Si(T0i; θ)

• C´

ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 27 / 45

Implementation, model selection and evaluation

Iterative (Marquardt) algorithm for a given G

◮ implemented in HETMIXSURV V2 parallel Fortran90 program ◮ validated in simulation studies (with for instance splines and threshold link

functions)

◮ implemented in Jointlcmm (R) for 1 marker

Posterior selection of the optimal number G of latent classes

◮ Information measures : AIC, BIC ◮ Score Test for conditional independence assumption :

→ longitudinal and survival parts are independent conditionally on the latent classes

Further evaluation of the model using :

◮ Posterior classification

stemmed from P(ci = g|Xi, Yi, (Ti, δi); ˆ θ)

◮ Longitudinal/Survival predictions versus observations

→ posterior-probability-weighted means over time intervals

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 28 / 45

Conditional independence assumption :

Class-specific cause-specific proportional hazard model λp(t) = λ0p(t; νpg) exp(X⊤

Ti ζpg)

Λ process latent ...

1 K

latent effects c classes random u (Τ, δ ) Y at t

k

Y at t Y at t

• utcome
• utcome
• utcome

... time to event multiple cause ( (t))

j1 jk jK

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 29 / 45

Conditional independence assumption : alternative

Class-specific cause-specific proportional hazard model λp(t)H1 = λ0p(t; νpg) exp(X⊤

Ti ζpg + uigκp)

→ Score test for H0 : κp = 0 and H0 : κ = (κ⊤

1 , ..., κ⊤ P )⊤ = 0

Λ process latent ...

1 K

latent effects c classes random u (Τ, δ ) Y at t

k

Y at t Y at t

• utcome
• utcome
• utcome

... time to event multiple cause ( (t))

j1 jk jK

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 29 / 45

Application : Semantic memory decline in the elderly

Clinical background

◮ semantic memory (verbal fluency, ...) affected long before dementia

diagnosis

◮ could play a role for early prediction of dementia

Objective

◮ describe profiles of semantic memory decline in association with dementia

and death

◮ predict the risk of dementia from semantic memory history

PAQUID cohort data :

◮ population-based cohort on cerebral aging ◮ 65 years and older ◮ 22 years of follow-up every 2 or 3 years ◮ subpopulation with genetic information : ApoE4

→ 588 subjects

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 30 / 45

Dynamics of semantic memory

2 longitudinal measures :

◮ Isaacs Set Test (IST15) (discrete quantitative in {0-40}) ◮ Wechsler similarities test (WST) (ordinal in {0-10})

Trajectory according to age (natural history)

◮ age at entry (ageT0), sex (sex), education (EL), apoE4 (E4)

In each latent class g : Λ(age)|ci=g = b0ig + β1sex + β2EL + β3E4 + β4ageT0+ (b1ig + β5sex + β6EL + β7E4)×age (b2ig + β8sex + β9EL + β10E4)×age2 + wi(age) IST15ij = H1( Λ(age1ij) + α1i + ǫ1ij ; η1) WSTij = H2( Λ(age2ij) + α2i + ǫ2ij ; η2)

with big ∼ N(µg, B), wi ∼ Brownian motion α2i ∼ N(0, σ2

α), ǫki ∼ N(0, σk 2), k = 1, 2

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 31 / 45

Preliminary analysis : link functions H1 and H2

In separated analyses, Splines with 3 internal knots = good balance between estimation difficulty and goodness-of-fit

−10 −5 5 10 20 30 40

linear AICd=13866.2 splines (23,27,31) AICd=13842.4 thresholds AICd=13857.7

IST underlying latent process Isaacs Set Test (IST) −4 −2 2 2 4 6 8 10

linear AICd=10738.3 splines(6,8,9) AICd=10448.3 thresholds AICd=10409.4

WST underlying latent process Wechsler Similarities Test (WST)

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 32 / 45

Risk of dementia in presence of death

2-cause censored time-to-event with delayed entry : age at entry in the cohort age at dementia (in between a negative and positive diagnosis) age at death in the two years after a negative dementia diagnosis In each latent class g : λp(t)|ci=g = λ0pg(t; )eζ1psex+ζ2pEL+ζ3pE4 , p = 1, 2 λ0pg(t; ) parametric hazards among Gompertz, Weibull, M-splines (5 knots) and piecewise constant (5 knots) In separated analyses, Weibull hazards = good balance between estimation difficulty and goodness-of-fit

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 33 / 45

Selection of the number of classes in the joint model

1 2 3 4 5 27080 27120 27160 number of latent classes BIC

BIC

1 2 3 4 5 20 40 60 80 120 number of latent classes Score Test Statistic 5% significance level Global score test 1 2 3 4 5 20 40 60 80 120 Score Test Statistic number of latent classes 5% significance level Dementia score test 1 2 3 4 5 10 20 30 40 50 Score Test Statistic number of latent classes 5% significance level Death without dementia score test C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 34 / 45

4 profiles of semantic decline, dementia & death

65 70 75 80 85 90 95 10 20 30 40 age in years Isaacs Set Test 65 70 75 80 85 90 95 2 4 6 8 10 age in years Wechsler Similarities Test 65 70 75 80 85 90 95 0.0 0.2 0.4 0.6 0.8 1.0 age in years

• f dementia

cumulative incidence 65 70 75 80 85 90 95 0.0 0.2 0.4 0.6 0.8 1.0 age in years

• f free−dementia death

cumulative incidence Class 1 (12.1%) Class 2 (52.2%) Class 3 (11.2%) Class 4 (24.5%)

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 35 / 45

Weighted predictions versus weighted observations

65 70 75 80 85 90 95 10 20 30 40 age in years Isaacs Set Test

mean observation mean subject−specific prediction

65 70 75 80 85 90 95 2 4 6 8 10 age in years Wechsler Similarities Test 65 70 75 80 85 90 95 0.0 0.2 0.4 0.6 0.8 1.0 age in years cumulative incidence

• f dementia

non−parametric estimate 95% bands prediction

65 70 75 80 85 90 95 0.0 0.2 0.4 0.6 0.8 1.0 age in years cumulative incidence

• f free−dementia death

Class 1 (12.1%) Class 2 (52.2%) Class 3 (11.2%) Class 4 (24.5%)

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 36 / 45

Individual dynamic prediction : the principle

From a landmark age s

◮ history of the markers Y(s)

i

= {Ykij such as tkij ≤ s}

◮ history of the covariates X(s)

i

= {XL1i(tkij), Zi(tkij), XL2i(tkij) such as tkij ≤ s}

◮ other time-independent covariates Xi

Cumulative incidence for cause p at an horizon of t years Ppi(s, t) = P(Ti ≤ s + t, δi = p|Ti > s, Y(s)

i

, X(s)

i , XTi, Xci; θ)

(1)

◮ computed using the Bayes theorem

and the conditional independence assumption

◮ with class-specific and cause-specific cumulative incidence

approximated by Gauss-Legendre

Monte-Carlo approximation of the posterior distribution of Ppi(s, t)

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 37 / 45

Example of individual dynamic prediction

65 70 75 80 85 90 10 20 30 40 cognitive measures IST WST

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 38 / 45

Example of individual dynamic prediction

65 70 75 80 85 90 10 20 30 40 cognitive measures 0.2 0.4 0.6 0.8 1 probabilities of event Age in years IST WST dementia death

at 80 years old 5-year probability of dementia (%) : 13.0 [7.7,21.0] 5-year probability of death (%) : 25.1 [18.1,36.5]

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 38 / 45

Example of individual dynamic prediction

65 70 75 80 85 90 10 20 30 40 cognitive measures 0.2 0.4 0.6 0.8 1 probabilities of event Age in years IST WST dementia death

at 80 years old at 85 years old 5-year probability of dementia (%) : 13.0 [7.7,21.0] 16.4 [9.1,29.4] 5-year probability of death (%) : 25.1 [18.1,36.5] 36.0 [25.9,46.3]

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 38 / 45

Option2 : multistate model with interval censoring

(Rouanet et al., ArXiv 2015) Λ process latent ...

1 K

Y at t

k

Y at t Y at t

• utcome
• utcome

... ( (t))

j1 jk jK

latent classes c u effects random

• utcome

ill healthy dead

Transition intensity from state k to state l for subject i in class g : αklig(t) = α0

klg(t) eXTi⊤γklg

α0

klg class-specific baseline intensity

γklg class-specific regression parameters

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 39 / 45

Maximum Likelihood Estimation

Likelihood function using Yi | =

ci Ti,

L(θ) =

N

• i=1

G

• g=1

fY(Yi|ci = g; θ) fD(Di, δi|ci = g; θ) P(ci = g|Xi; θ)

◮ P(ci = g|Xi; θ) from a multinomial logistic model ◮ fY(Yi|ci = g; θ) from the latent process mixed model ◮ fD(Di, δi|ci = g; θ) from the multistate model with interval censoring

D⊤

i = (T0i, Li, Ri, δA i , Ti, δD i )

with Ri = +∞ if δA

i = 0.

= e−A01g(T)−A02g(T) α02g(T) + ⊤

Li

e−A01g(u)−A02g(u) α01g(u) e−(A12g(T)−A12g(u)) α12g(T)du

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 40 / 45

Application

From PAQUID study (N=3777)

◮ Change over time of Isaacs set test (verbal fluency) ◮ in association with dementia and death

Mixed model : Λ(t)|ci=g = b0ig + β1gsex + β2gEL+ (b1ig + β3gEL)×t (b2ig + β4gEL)×t2 Yij = Λi(tij) + ǫij with big ∼ N(µg, B)&ǫi ∼ N(0, σ2) Proportional transition intensities of the illness-death model : αklig(t) = α0

klg(t) eγkls sexi+γkle ELi

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 41 / 45

Four-class Markovian model : poorly educated men

N EL=0 EL=1 men women Class 1 7.3% 42.4 57.6 42.8 57.2 Class 2 8.3% 25.0 75.0 43.1 56.9 Class 3 34.2% 29.3 70.7 39.4 60.6 Class 4 50.5% 37.6 62.4 43.9 56.1

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 42 / 45

Goodness-of-fit assessment

Class-specific weighted predicted trajectories vs. observed predicted class-specific cumulative incidences vs. semi-parametric estimator

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 43 / 45

Concluding remarks

Joint model methodology

◮ extended to multivariate longitudinal markers ◮ extended to multistate process for events

→ useful for different purposes in chronic diseases Different assumptions for the shared quantity

◮ depends on the data ◮ depends on the objective

Parametric assumptions

◮ flexible and/or selected distributions according to the data ◮ progressive construction of the models, goodness-of-fit (graphs, measures,

tests) :

◮ ... score tests for conditional independence assumptions (between events

and marker, between events)

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 44 / 45

Funding and references

Fundings : INCa/Inserm MULTIPLE, Inserm/Region PhD grant Further details in :

Ferrer Lo¨ ıc et al. (2015). arXiv :1506.07496 [stat]. Rouanet Ana¨ ıs et al. (2015). arXiv :1506.07415 [stat]. Proust-Lima C´ ecile et al. (2014). arXiv :1409.7598 [stat]. Proust-Lima C´ ecile et al. (2014). Statistical Methods in Medical Research, 23(1), 74-90. S` ene Mb´ ery et al. (2014). Journal de la Soci´ et´ e Franc ¸aise de Statistique, 155(1), 134-155. (in english) S` ene Mb´ ery et al. (2014). Statistical Methods in Medical Research. (online)

C´ ecile Proust-Lima (INSERM) Joint models for multiple outcomes July 2015 45 / 45