Accommodating informative dropout and death: a joint modelling - PowerPoint PPT Presentation

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions Accommodating informative dropout and death: a joint modelling approach for longitudinal and semi-competing risks data Qiuju Li MRC Biostatistics Unit, Cambridge, UK qiuju.li@mrc-bsu.cam.ac.uk Joint work with Dr. Li Su Workshop on Flexible Models for Longitudinal and Survival Data with Applications in Biostatistics (Warwick, 27th-29th July, 2015) Qiuju Li Joint modelling of longitudinal and semi-competing risks data

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions Outline 1 Introduction 2 Joint modelling of longitudinal and semi-competing risks data 3 Application: HERS data analysis 4 Conclusions Qiuju Li Joint modelling of longitudinal and semi-competing risks data

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions Introduction longitudinal and semi-competing risks data, e.g., CD4 counts, dropout and HIV-related death in the HIV epidemiology research study (HERS). complete data dropout 30 30 25 25 squart root of CD4 counts squart root of CD4 counts 20 20 15 15 10 10 5 5 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12 visit (subject=100058) visit (subject=101059) death dropout & death 30 30 25 25 squart root of CD4 counts squart root of CD4 counts Measurements 20 dropout 20 death 15 15 10 10 5 5 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12 visit (subject=100729) visit (subject=100241) Qiuju Li Joint modelling of longitudinal and semi-competing risks data

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions Some concepts mortal cohort; immortal cohort; (Aalen and Gunnes, 2010) longitudinal profile models: unconditional models, e.g., random-effects models f ( Y i ( t )); fully conditional models, e.g., f ( Y i ( t ) | S i = s ), s > t ; partly conditional models, e,g., f ( Y i ( t ) | S i > t ) (Kurland and Heagerty, 2005; Kurland et al. , 2009); GEE approaches; a likelihood-based joint modelling approach proposed subsequently. Qiuju Li Joint modelling of longitudinal and semi-competing risks data

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions Notation scheduled repeated measurements of a longitudinal outcome Y i = ( Y i 1 , . . . , Y iM ) ′ , taken at visits 1 , . . . , M , e.g., M = 12 for the HERS data; informative dropout and death dropout time denoted by D i , observed data i = min ( D i , S i , C i ), δ D D ∗ i = I ( D i ≤ S i , D i ≤ C i ); death time denoted by S i , observed data S ∗ i = min ( S i , C i ), δ S i = I ( S i ≤ C i ); C i denotes non-informative censoring, e.g., end of study; covariates X i , e.g., sex, treatment arm; Qiuju Li Joint modelling of longitudinal and semi-competing risks data

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions Time-to-event processes Time-to-event data, time to dropout: last visit of follow-up; time to death: T i time to death: τ 0 τ : the end of study. Qiuju Li Joint modelling of longitudinal and semi-competing risks data

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions Time-to-event processes Discrete time-to-event data ( mathematical attractiveness ), time to dropout: last visit of follow-up; time to death: T i time to death: τ t 1 t r − 1 t r t ( M − 1) 0 τ : the end of study. the discrete death time S i = r (Barrett et al, 2015, JRSSB) . Qiuju Li Joint modelling of longitudinal and semi-competing risks data

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions Joint models Joint models for the longitudinal and semi-competing risks data,  Y ij = x T ij β + z T ij b i + ǫ ij   Pr ( D i = r | D i ≥ r ) = 1 − Φ( x T D , ir α D + γ T D , r W D , ir b i ) ,  Pr ( S i = r | S i ≥ r ) = 1 − Φ( x T S , ir α S + γ T S , r W S , ir b i )  Φ( · ) standard normal cdf; β , α D , α S regression coefficients; γ D , r , γ S , r association effects; random effects b i ∼ N (0 , Σ b ); iid ∼ N (0 , σ 2 ); ǫ ij W D , ir b i , W S , ir b i vectors of linear combinations of random effects, e.g., W D , ir b i = ( b i 0 , b i 1 ) T ; conditional independence assumption given random effects b i ; Qiuju Li Joint modelling of longitudinal and semi-competing risks data

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions Likelihood function, � i , δ D i , δ S L i ( θ ; y i , D ∗ i , S ∗ i ) i � ∞ � = f (longitudinal data | θ, b i ) × −∞ i Pr (dropout data | θ, b i ) × Pr (death data | θ, b i ) × f ( b i | θ ) db i . Qiuju Li Joint modelling of longitudinal and semi-competing risks data

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions Semi-competing risks data Four possible scenarios of the observed time-to-event data, (1) neither dropout nor death: i = s , ( δ D i , δ S D ∗ i = d , S ∗ i ) = (0 , 0); (2) dropout only: i = s , ( δ D i , δ S D ∗ i = d , S ∗ i ) = (1 , 0); (3) death only: i = s , ( δ D i , δ S D ∗ i = d , S ∗ i ) = (0 , 1); (4) both dropout and death: i = s , ( δ D i , δ S D ∗ i = d , S ∗ i ) = (1 , 1); Qiuju Li Joint modelling of longitudinal and semi-competing risks data

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions For the scenario (1), the likelihood function of observed data i = d , δ D i = s , δ S { y i = ( y i 1 , . . . , y in i ) ′ , D ∗ i = 0 , S ∗ i = 0 } , i , δ D i , δ S L i ( θ ; y i , D ∗ i , S ∗ i ) � ∞ d � φ ( y i ; x i β + z i b i , σ 2 I n i ) Φ( x T D , ir α D + γ T = D , r W D , ir b i ) −∞ k =1 s � Φ( x T S , ir α S + γ T S , r W S , ir b i ) φ ( b i ; 0 , Σ b ) db i ℓ =1 = L i 1 ( ·\ b i )Φ ( d + s ) ( A ds + B ds h i ; 0 , I d + s + B ds H − 1 B T ds ) i closed-form likelihood (skewed normal distribution, Arnold 2009); L i 1 ( ·\ b i ), h i , H i , A ds , B ds function/vectors/matrices free of b i ; φ ( · ; µ, Σ) and Φ ( d + s ) ( · ; µ, Σ) denote multivariate normal pdf/cdf. Qiuju Li Joint modelling of longitudinal and semi-competing risks data

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions Marginal mean profile conditional on being alive Unconditional population mean profile for an immortal cohort E ( Y ij | x ij ) = x T ij β ; Conditional mean profile given being alive for a mortal cohort, we can compute E ( Y ij | x ij , S i ≥ j ) = x T ij β + z T ij E ( b i | S i ≥ j ) . Analogously, f ( b i | S i ≥ j ) is a multivariate skew-normal distribution, � � = Pr S j > ( j − 1) | b i f ( b i ) � � f ( b i | S i ≥ j ) = f b i | S i > ( j − 1) , Pr ( S i > ( j − 1)) closed form of its expectation can be obtained. Qiuju Li Joint modelling of longitudinal and semi-competing risks data

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions Statistical inference Maximum likelihood-based approach ( exact likelihood ); 1 R software utilising nlminb or optim . Bayesian approach; 2 implemented using WinBUGS. Qiuju Li Joint modelling of longitudinal and semi-competing risks data

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions HERS data description HIV epidemiology research study (HERS) (Smith et al ., 1997) # of subjects: 850 (HIV positive at baseline) CD4 counts reviewed every 6 months up to 12 visits time-to-event data # of subjects scenario (1): ( δ D i , δ S i ) = (0 , 0) 374 scenario (2): ( δ D i , δ S i ) = (1 , 0) 352 scenario (3): ( δ D i , δ S i ) = (0 , 1) 23 scenario (4): ( δ D i , δ S i ) = (1 , 1) 78 Objective: study the role of baseline patient characteristics (i.e., viral load, antiretroviral therapy (art), # of symptoms) on variation in longitudinal CD4 counts. Qiuju Li Joint modelling of longitudinal and semi-competing risks data

Introduction Joint modelling of longitudinal and semi-competing risks data Application: HERS data analysis Conclusions Models proposed for the HERS data  Y ij = β 0 + β 1 visit + β 2 ∼ 4 viral load + β 5 symptoms + β 6 art    + β 7 ∼ 9 visit*viral load + β 10 visit ∗ symptoms + β 11 visit ∗ art + b i 0 + b i 1 + ǫ ij           Pr ( D i = r | D i ≥ r ) = 1 − Φ( α D , i 0 + α D , i 1 ∼ 3 viral load + α D , i 4 symptoms  + α D , i 5 art + α D , i 6 r + α D , i 7 r 2 + γ D , 0 b i 0 + γ D , 1 b i 1 )         Pr ( S i = r | S i ≥ r ) = 1 − Φ( α S , i 0 + α S , i 1 ∼ 3 viral load + α S , i 4 symptoms     + α S , i 5 art + α S , i 6 r + α S , i 7 r 2 + γ S , 0 b i 0 + γ S , 1 b i 1 )   Qiuju Li Joint modelling of longitudinal and semi-competing risks data