Dynamic Predictions in a Joint Model of Two Longitudinal Outcomes - PowerPoint PPT Presentation

Dynamic Predictions in a Joint Model of Two Longitudinal Outcomes and Competing Risk Data. An Application in Heart Valve Data. Eleni-Rosalina Andrinopoulou Department of Biostatistics and CardioThoracic surgery, Erasmus Medical Center e.andrinopoulou@erasmusmc.nl BAYES May 21-23th, 2013

Heart Valve Data • 286 patients who received human tissue valve in aortic position in Erasmus University Medical Center (Department of Cardio-Thoracic Surgery) ◃ Patients were 16 years and older ◃ Echo examinations scheduled at 6 months and 1 year postoperatively and biennially thereafter ◃ Two longitudinal responses: Aortic Gradient (continuous) and Aortic Regurgitation (ordinal) ◃ Time-to-event response: time to death/reoperation (competing risks setting) BAYES May 21-23th, 2013 1

Heart Valve Data - (cont’d) • Aortic gradient and aortic regurgitation are measurements of the valve function ⇓ expecting biologically interrelationship. • Both events (reoperation and death) are highly associated with the disease condition of the patient and correlated to each other. BAYES May 21-23th, 2013 2

Statistical Models - Heart Valve data Joint model to the Heart Valve data: • Linear mixed-effects model for the longitudinal continuous outcome. • Mixed-effects continuation ratio model for the longitudinal ordinal outcome. • Cause-specific hazard model for the competing risk failure time data. BAYES May 21-23th, 2013 3

Statistical Models - Heart Valve data (cont’d) Let Y 1 i and Y 2 i represent the repeated measurements of AG and AR for the i -th patient, i = 1 , . . . , n . • Linear mixed-effects model: Y 1 i = X 1 i β 1 + Z 1 i b 1 i + ϵ i , ϵ i ∼ (0 , V ) ◃ X 1 i β 1 denotes the fixed part ◃ Z 1 i b 1 i denotes the random part BAYES May 21-23th, 2013 4

Statistical Models - Heart Valve data (cont’d) • Mixed-effects continuation ratio model: exp ( θ j + X 2 i β 2 + Z 2 i b 2 i ) π j = P ( Y 2 i = j | Y 2 i ≤ j ) = 1 + exp ( θ j + X 2 i β 2 + Z 2 i b 2 i ) , where j represents the category of the ordinal response ◃ X 2 i β 2 denotes the fixed part ◃ Z 2 i b 2 i denotes the random part BAYES May 21-23th, 2013 5

Statistical Models - Heart Valve data (cont’d) • To account for the correlation between the two longitudinal outcomes we assume multivariate random effects (( 0 )) ( b 1 i ) ) ( Σ b 1 Σ b 12 ∼ N , D = b 2 i 0 Σ b 12 Σ b 2 BAYES May 21-23th, 2013 6

Statistical Models - Heart Valve data (cont’d) Let T i denote the observed failure time for the i -th patient and δ i = 0 , 1 , 2 the event indicator. • Proportional hazard model:  i γ d + ( ˜ β 1 + b 1 i ) ⊤ α 1 d + ( ˜ h d i ( t ) = h d 0 ( t ) exp { w ⊤ β 2 + b 2 i ) ⊤ α 2 d } ,            i γ r + ( ˜ β 1 + b 1 i ) ⊤ α 1 r + ( ˜ h r i ( t ) = h r 0 ( t ) exp { w ⊤ β 2 + b 2 i ) ⊤ α 2 r } ,    where α ⊤ 1 d , α ⊤ 2 d , α ⊤ 1 r , α ⊤ 2 r are the coefficient that link the longitudinal and survival part. Specifically, α ⊤ 1 are the coefficients of the continuous longitudinal outcome and α ⊤ 2 are the coefficients of the ordinal longitudinal outcomes. BAYES May 21-23th, 2013 7

Statistical Models - Heart Valve data (cont’d) • Assumption : full conditional independence, given the random effects P ( T i , δ i , Y 1 i , Y ∗ 2 i | b 1 i , b 2 i ; θ ) = P ( T i , δ i | b 1 i , b 2 i ; θ ) P ( Y 1 i | b 1 i , b 2 i ; θ ) P ( Y ∗ 2 i | b 1 i , b 2 i ; θ ) ∏ P ( Y 1 i | b 1 i ; θ ) = P { Y 1 i ( t ij ) | b 1 i ; θ } j ∏ P ( Y ∗ P { Y ∗ 2 i | b 1 i ; θ ) = 2 i ( t ij ) | b 2 i ; θ } j The random effects b 1 i and b 2 i underlie both the longitudinal and survival processes. BAYES May 21-23th, 2013 8

Statistical Models - Heart Valve data (cont’d) Models for the Heart Valve data • continuous longitudinal submodel ◃ Fixed part: intercept, B-splines for time, age and gender ◃ Random part: B-splines for time • ordinal longitudinal submodel ◃ Fixed part: intercept, time, age and gender ◃ Random part: random intercept • survival submodel ◃ age ◃ piecewise constant baseline hazard BAYES May 21-23th, 2013 9

Bayesian estimation - Heart Valve data • Bayesian formulation for the proposed joint model • Posterior inferences using a Markov chain Monte Carlo (MCMC) algorithm ◃ The posterior distribution is P ( θ | Y 1 i , Y ∗ 2 i , T i , δ i ) ∝ P ( T i , δ i | b 1 i , b 2 i , θ s ) P ( Y 1 i | b 1 i , θ Y 1 ) P ( Y ∗ 2 i | b 2 i , θ Y ∗ 2 ) P ( b 1 i | θ Y 1 ) P ( b 2 i | θ Y ∗ 2 ) P ( θ Y 1 ) P ( θ Y ∗ 2 ) P ( θ s ) BAYES May 21-23th, 2013 10

Bayesian estimation - Heart Valve data (cont’d) • The likelihood for the survival part is P ( T i , δ i | b 1 i , b 2 i ; θ s ) i γ k + ( ˜ β 1 + b 1 i ) ⊤ α 1 k + ( ˜ = [ h 0 k ( T i ) exp { w ⊤ β 2 + b 2 i ) ⊤ α 2 k } ] I ( δ i = k ) m { i γ d + ( ˜ β 1 + b 1 i ) ⊤ α 1 d + ( ˜ ∑ [ h 0 d ( T i ) exp { w ⊤ β 2 + b 2 i ) ⊤ α 2 d } × exp − q =1 } i γ r + ( ˜ β 1 + b 1 i ) ⊤ α 1 r + ( ˜ + h 0 r ( T i ) exp { w ⊤ β 2 + b 2 i ) ⊤ α 2 r } ] T iq BAYES May 21-23th, 2013 11

Bayesian estimation - Heart Valve data (cont’d) • priors ◃ β 1 , β 2 , b 1 , b 2 , α 1 , α 2 ⇒ normal prior distributions ◃ D − 1 ⇒ wishart prior distribution ◃ τ , h d 0 ( t ) , h r 0 ( t ) ⇒ gamma prior distributions • 110000 iterations with 20000 burn in and 20 thinning BAYES May 21-23th, 2013 12

Dynamic predictions - Heart Valve data Increased interest towards dynamic predictions Focus on the prediction of • cumulative incidence function • longitudinal outcomes Useful from the clinical point of view BAYES May 21-23th, 2013 13

Dynamic predictions - Heart Valve data (cont’d) • New patient l that has provided us ˜ ˜ Y 1 l ( t ) = { Y 1 l ( s ) , 0 ≤ s < t } and Y 2 l ( t ) = { Y 2 l ( s ) , 0 ≤ s < t } • The interest lies on lk > t, ˜ Y 1 l ( t ) , ˜ π lk ( u, t ; θ ) = P ( T ∗ lk < u | ∪ K k =1 T ∗ Y 2 l ( t ) , D n ; θ ) , where u > t and D n denotes the sample on which the joint model was fitted BAYES May 21-23th, 2013 14

Dynamic predictions - Heart Valve data (cont’d) • Conditional probabilities π lk ( u, t ; θ ) = ∫ lk > t, ˜ Y 1 l ( t ) , ˜ lk > t, ˜ Y 1 l ( t ) , ˜ P ( T ∗ lk < u | ∪ K k =1 T ∗ Y 2 l ( t ); θ ) × p ( b l | ∪ K k =1 T ∗ Y 2 l ( t ); θ ) db l BAYES May 21-23th, 2013 15

Dynamic predictions - Heart Valve data (cont’d) • Conditional Independence Assumption π lk ( u, t ; θ ) = ∫ lk > t, ˜ Y 1 l ( t ) , ˜ P ( T ∗ lk < u | ∪ K k =1 T ∗ lk > t ; θ ) × p ( b l | ∪ K k =1 T ∗ Y 2 l ( t ); θ ) db l BAYES May 21-23th, 2013 16

Dynamic predictions - Heart Valve data (cont’d) • Conditional Independence Assumption π lk ( u, t ; θ ) = ∫ CIF ( u, t, b l ; θ ) lk > t, ˜ Y 1 l ( t ) , ˜ × p ( b l | ∪ K k =1 T ∗ Y 2 l ( t ); θ ) db l S ( t, b l ; θ ) BAYES May 21-23th, 2013 17

Dynamic predictions - Heart Valve data (cont’d) • Conditional Independence Assumption π lk ( u, t ; θ ) = ∫ CIF ( u, t, b l ; θ ) lk > t, ˜ Y 1 l ( t ) , ˜ × p ( b l | ∪ K k =1 T ∗ Y 2 l ( t ); θ ) db l S ( t, b l ; θ ) • Estimation of these probabilities is based on a Monte Carlo scheme ∫ π lk ( u, t ; θ ) p ( θ | D n ) dθ BAYES May 21-23th, 2013 17

Dynamic predictions - Heart Valve data (cont’d) • We estimate the conditional probabilities using a MCMC algorithm. Specifically, ◃ Randomly select θ ∗ from the MCMC of the joint model l > t, ˜ Y 1 l ( t ) , ˜ ◃ Draw b ∗ l ∼ { b l |∪ K k =1 T ∗ Y 2 l ( t ) , θ ∗ } ◃ Compute π lk ( u, t, b ∗ l ; θ ∗ ) = CIF ( u, t, b ∗ l ; θ ∗ ) /S ( t, b ∗ l ; θ ∗ ) • Repeat for a patient N times • Then, estimation of the conditional probabilities: mean/median { π lk ( u, t, b ∗ l ; θ ∗ ) } BAYES May 21-23th, 2013 18

Dynamic predictions - Heart Valve data (cont’d) Patient 80 Aortic Gradient (mmHg) 10 8 6 4 2 0 4 Aortic Regurgitation 3 2 1 0 2 4 6 8 10 12 Time (years) BAYES May 21-23th, 2013 19

Dynamic predictions - Heart Valve data (cont’d) 12 Death 1.0 Reoperation Aortic Gradient (mmHg) 10 8 0.8 6 4 Conditional CIF 0.6 2 0 6 0.4 Aortic Regurgitation 5 4 3 0.2 2 1 0.0 0 0 5 10 15 20 0 5 10 15 20 Follow−up time Time (years) BAYES May 21-23th, 2013 20

Dynamic predictions - Heart Valve data (cont’d) 12 Death 1.0 Reoperation Aortic Gradient (mmHg) 10 8 0.8 6 4 Conditional CIF 0.6 2 0 6 0.4 Aortic Regurgitation 5 4 3 0.2 2 1 0.0 0 0 5 10 15 20 0 5 10 15 20 Follow−up time Time (years) BAYES May 21-23th, 2013 20

Dynamic predictions - Heart Valve data (cont’d) 12 Death 1.0 Reoperation Aortic Gradient (mmHg) 10 8 0.8 6 4 Conditional CIF 0.6 2 0 6 0.4 Aortic Regurgitation 5 4 3 0.2 2 1 0.0 0 0 5 10 15 20 0 5 10 15 20 Follow−up time Time (years) BAYES May 21-23th, 2013 20

Dynamic predictions - Heart Valve data (cont’d) 12 Death 1.0 Reoperation Aortic Gradient (mmHg) 10 8 0.8 6 4 Conditional CIF 0.6 2 0 6 0.4 Aortic Regurgitation 5 4 3 0.2 2 1 0.0 0 0 5 10 15 20 0 5 10 15 20 Follow−up time Time (years) BAYES May 21-23th, 2013 20