Christiana Charalambous

Transcript

1. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Christiana Charalambous School of Mathematics The University of Manchester, UK christiana.charalambous@manchester.ac.uk
2. Joint models for multi-outcome data and covariance structures via MCMC Outline Outline 1 Introduction Joint models for longitudinal and survival outcomes Joint models for mean and variance structures 2 Joint models for multi-outcome data and covariance structures via MCMC Model specification Bayesian estimation using MCMC
3. Joint models for multi-outcome data and covariance structures via MCMC Introduction Joint models for longitudinal and survival outcomes Benefits of joint modelling approach Joint models are popular in clinical trials because they Provide more efficient estimates of the treatment effects on both the longitudinal marker and the time to event Reduce bias in the estimates of the overall treatment effect (Ibrahim, Chu and Chen, 2010)
4. Joint models for multi-outcome data and covariance structures via MCMC Introduction Joint models for longitudinal and survival outcomes Benefits of joint modelling approach Joint models are popular in clinical trials because they Provide more efficient estimates of the treatment effects on both the longitudinal marker and the time to event Reduce bias in the estimates of the overall treatment effect (Ibrahim, Chu and Chen, 2010) In some cases, joint models can identify significant differences between treatment groups not identified by separate analyses (see, e.g. Powell et al, 2013), while in other instances (e.g. considering multiple longitudinal markers) they can increase prediction accuracy (Fieuws et al, 2008). The potential benefits of using meta-analysis to pool estimates accross studies where joint modelling was employed, have also recently been exlored (Sudell, Kolamunnage-Dona and Tudur-Smith, 2016).
5. Joint models for multi-outcome data and covariance structures via MCMC Introduction Joint models for longitudinal and survival outcomes Joint models for longitudinal and survival data These models could be useful in different situations: Survival models with measurement errors or missing data in time-dependent covariates.
6. Joint models for multi-outcome data and covariance structures via MCMC Introduction Joint models for longitudinal and survival outcomes Joint models for longitudinal and survival data These models could be useful in different situations: Survival models with measurement errors or missing data in time-dependent covariates. Longitudinal models with informative dropouts.
7. Joint models for multi-outcome data and covariance structures via MCMC Introduction Joint models for longitudinal and survival outcomes Joint models for longitudinal and survival data These models could be useful in different situations: Survival models with measurement errors or missing data in time-dependent covariates. Longitudinal models with informative dropouts. Longitudinal and survival processes which are associated via latent variables. (Henderson, Diggle and Dobson, 2000).
8. Joint models for multi-outcome data and covariance structures via MCMC Introduction Joint models for longitudinal and survival outcomes Joint models for longitudinal and survival data These models could be useful in different situations: Survival models with measurement errors or missing data in time-dependent covariates. Longitudinal models with informative dropouts. Longitudinal and survival processes which are associated via latent variables. (Henderson, Diggle and Dobson, 2000).
9. Joint models for multi-outcome data and covariance structures via MCMC Introduction Joint models for longitudinal and survival outcomes Joint models for longitudinal and survival data Rizopoulos and Ghosh (2011) noted that additional to the current value of a longitudinal outcome, other features, such as the slope or curvature of the true longitudinal trajectory, could be related to the risk of an event and more recently, Andrinopoulou and Rizopoulos (2016) proposed a bayesian approach to select which features of the longitudinal outcome have an influence on survival.
10. Joint models for multi-outcome data and covariance structures via MCMC Introduction Joint models for longitudinal and survival outcomes Joint models for longitudinal and survival data Rizopoulos and Ghosh (2011) noted that additional to the current value of a longitudinal outcome, other features, such as the slope or curvature of the true longitudinal trajectory, could be related to the risk of an event and more recently, Andrinopoulou and Rizopoulos (2016) proposed a bayesian approach to select which features of the longitudinal outcome have an influence on survival. We think of the problem from a different perspective; could the variance of the longitudinal process be linked to the survival process?
11. Joint models for multi-outcome data and covariance structures via MCMC Introduction Joint models for mean and variance structures Joint models for mean and variance structures Joint mean and variance modelling has a long history dating back to the late 1960s.
12. Joint models for multi-outcome data and covariance structures via MCMC Introduction Joint models for mean and variance structures Joint models for mean and variance structures Joint mean and variance modelling has a long history dating back to the late 1960s. Much work has been done since (McCullagh & Nelder, 1989, Pourahmadi, 1999; Lee and Nelder, 2001; Pan & MacKenzie, 2006, 2007; Charalambous et al, 2014, 2015).
13. Joint models for multi-outcome data and covariance structures via MCMC Introduction Joint models for mean and variance structures Joint models for mean and variance structures Joint mean and variance modelling has a long history dating back to the late 1960s. Much work has been done since (McCullagh & Nelder, 1989, Pourahmadi, 1999; Lee and Nelder, 2001; Pan & MacKenzie, 2006, 2007; Charalambous et al, 2014, 2015). Model both mean and variance on sets of covariates.
14. Joint models for multi-outcome data and covariance structures via MCMC Introduction Joint models for mean and variance structures Joint models for mean and variance structures Joint mean and variance modelling has a long history dating back to the late 1960s. Much work has been done since (McCullagh & Nelder, 1989, Pourahmadi, 1999; Lee and Nelder, 2001; Pan & MacKenzie, 2006, 2007; Charalambous et al, 2014, 2015). Model both mean and variance on sets of covariates. We propose to additionally model the within-subject variance of the longitudinal response and assume it’s associated (through random effects) with both the longitudinal and survival processes.
15. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Longitudinal model Let yi(t) denote the longitudinal outcome for the ith subject (i = 1, . . . , m) at time point t and yij = {yi(tij), j = 1, . . . , ni}.
16. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Longitudinal model Let yi(t) denote the longitudinal outcome for the ith subject (i = 1, . . . , m) at time point t and yij = {yi(tij), j = 1, . . . , ni}. yi(t) = xT 1i (t)β + zT 1i (t)b1i + εi(t) = µi(t) + εi(t)
17. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Longitudinal model Let yi(t) denote the longitudinal outcome for the ith subject (i = 1, . . . , m) at time point t and yij = {yi(tij), j = 1, . . . , ni}. yi(t) = xT 1i (t)β + zT 1i (t)b1i + εi(t) = µi(t) + εi(t) where x1i(t) and z1i(t) - p1 × 1 and q1 × 1 vectors of the design matrices for the fixed and random effects
18. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Longitudinal model Let yi(t) denote the longitudinal outcome for the ith subject (i = 1, . . . , m) at time point t and yij = {yi(tij), j = 1, . . . , ni}. yi(t) = xT 1i (t)β + zT 1i (t)b1i + εi(t) = µi(t) + εi(t) where x1i(t) and z1i(t) - p1 × 1 and q1 × 1 vectors of the design matrices for the fixed and random effects β - p1 × 1 vector of unknown fixed effects parameters
19. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Longitudinal model Let yi(t) denote the longitudinal outcome for the ith subject (i = 1, . . . , m) at time point t and yij = {yi(tij), j = 1, . . . , ni}. yi(t) = xT 1i (t)β + zT 1i (t)b1i + εi(t) = µi(t) + εi(t) where x1i(t) and z1i(t) - p1 × 1 and q1 × 1 vectors of the design matrices for the fixed and random effects β - p1 × 1 vector of unknown fixed effects parameters b1i - q1 × 1 vector of random effects for subject i
20. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Longitudinal model Let yi(t) denote the longitudinal outcome for the ith subject (i = 1, . . . , m) at time point t and yij = {yi(tij), j = 1, . . . , ni}. yi(t) = xT 1i (t)β + zT 1i (t)b1i + εi(t) = µi(t) + εi(t) where x1i(t) and z1i(t) - p1 × 1 and q1 × 1 vectors of the design matrices for the fixed and random effects β - p1 × 1 vector of unknown fixed effects parameters b1i - q1 × 1 vector of random effects for subject i ε1, . . . , εm are mutually independent, with εi ∼ Nni (0, Σi) and εi = (εi1, . . . , εini )
21. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Survival model Let Ti denote the observed failure time for the ith subject; then Ti = min(T∗ i , Ci), where T∗ i is the true event time and Ci is the censoring time. Also denote δi = I(T∗ i ≤ Ci), which indicates the occurrence of event or censoring.
22. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Survival model Let Ti denote the observed failure time for the ith subject; then Ti = min(T∗ i , Ci), where T∗ i is the true event time and Ci is the censoring time. Also denote δi = I(T∗ i ≤ Ci), which indicates the occurrence of event or censoring. λi(t) = lim dt→0 dt−1Pr(t ≤ T∗ i < t + dt|T∗ i ≥ t, x2i(t), z2i(t), b2i, ¯ ti(t), Ci) = λ0(t) exp(xT 2i (t)γ + zT 2i (t)b2i)
23. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Survival model Let Ti denote the observed failure time for the ith subject; then Ti = min(T∗ i , Ci), where T∗ i is the true event time and Ci is the censoring time. Also denote δi = I(T∗ i ≤ Ci), which indicates the occurrence of event or censoring. λi(t) = lim dt→0 dt−1Pr(t ≤ T∗ i < t + dt|T∗ i ≥ t, x2i(t), z2i(t), b2i, ¯ ti(t), Ci) = λ0(t) exp(xT 2i (t)γ + zT 2i (t)b2i) where x2i(t) and z2i(t) - p2 × 1 and q2 × 1 vectors of the design matrices for the fixed and random effects
24. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Survival model Let Ti denote the observed failure time for the ith subject; then Ti = min(T∗ i , Ci), where T∗ i is the true event time and Ci is the censoring time. Also denote δi = I(T∗ i ≤ Ci), which indicates the occurrence of event or censoring. λi(t) = lim dt→0 dt−1Pr(t ≤ T∗ i < t + dt|T∗ i ≥ t, x2i(t), z2i(t), b2i, ¯ ti(t), Ci) = λ0(t) exp(xT 2i (t)γ + zT 2i (t)b2i) where x2i(t) and z2i(t) - p2 × 1 and q2 × 1 vectors of the design matrices for the fixed and random effects γ - p2 × 1 vector of unknown fixed effects parameters
25. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Survival model Let Ti denote the observed failure time for the ith subject; then Ti = min(T∗ i , Ci), where T∗ i is the true event time and Ci is the censoring time. Also denote δi = I(T∗ i ≤ Ci), which indicates the occurrence of event or censoring. λi(t) = lim dt→0 dt−1Pr(t ≤ T∗ i < t + dt|T∗ i ≥ t, x2i(t), z2i(t), b2i, ¯ ti(t), Ci) = λ0(t) exp(xT 2i (t)γ + zT 2i (t)b2i) where x2i(t) and z2i(t) - p2 × 1 and q2 × 1 vectors of the design matrices for the fixed and random effects γ - p2 × 1 vector of unknown fixed effects parameters b2i - q2 × 1 vector of random effects for subject i
26. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Survival model Let Ti denote the observed failure time for the ith subject; then Ti = min(T∗ i , Ci), where T∗ i is the true event time and Ci is the censoring time. Also denote δi = I(T∗ i ≤ Ci), which indicates the occurrence of event or censoring. λi(t) = lim dt→0 dt−1Pr(t ≤ T∗ i < t + dt|T∗ i ≥ t, x2i(t), z2i(t), b2i, ¯ ti(t), Ci) = λ0(t) exp(xT 2i (t)γ + zT 2i (t)b2i) where x2i(t) and z2i(t) - p2 × 1 and q2 × 1 vectors of the design matrices for the fixed and random effects γ - p2 × 1 vector of unknown fixed effects parameters b2i - q2 × 1 vector of random effects for subject i ¯ ti(t) = {tij : tij ≤ t}
27. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Covariance model We can decompose Σi using the Modified Cholesky Decomposition (Pourahmadi, 1999), such that Σ−1 i = LT i D−1 i Li
28. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Covariance model We can decompose Σi using the Modified Cholesky Decomposition (Pourahmadi, 1999), such that Σ−1 i = LT i D−1 i Li where Li =      1 0 · · · 0 −φi21 1 · · · 0 . . . . . . ... . . . −φini1 −φini2 · · · 1      ,
29. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Covariance model We can decompose Σi using the Modified Cholesky Decomposition (Pourahmadi, 1999), such that Σ−1 i = LT i D−1 i Li where Li =      1 0 · · · 0 −φi21 1 · · · 0 . . . . . . ... . . . −φini1 −φini2 · · · 1      , φijk are the autoregressive parameters in ˆ yij = µij + j−1 k=1 φijk(yik − µik), with µij = E(yij|b1i)
30. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Covariance model We can decompose Σi using the Modified Cholesky Decomposition (Pourahmadi, 1999), such that Σ−1 i = LT i D−1 i Li where Li =      1 0 · · · 0 −φi21 1 · · · 0 . . . . . . ... . . . −φini1 −φini2 · · · 1      , φijk are the autoregressive parameters in ˆ yij = µij + j−1 k=1 φijk(yik − µik), with µij = E(yij|b1i) and Di = diag(σ2 i1 , . . . , σ2 ini ), with σ2 ij = var(yij − ˆ yij|b1i).
31. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Covariance model (continued) We can model the unconstrained parameters σ2 ij and φijk by log(σ2 ij ) = wT ij ζ + zT 3ij b3i and φijk = hT ijk ξ
32. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Covariance model (continued) We can model the unconstrained parameters σ2 ij and φijk by log(σ2 ij ) = wT ij ζ + zT 3ij b3i and φijk = hT ijk ξ where wij and hijk - p3 × 1 and p4 × 1 vectors of the design matrices for the unknown fixed effects ζ and ξ
33. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Covariance model (continued) We can model the unconstrained parameters σ2 ij and φijk by log(σ2 ij ) = wT ij ζ + zT 3ij b3i and φijk = hT ijk ξ where wij and hijk - p3 × 1 and p4 × 1 vectors of the design matrices for the unknown fixed effects ζ and ξ z3ij - q3 × 1 vector of the design matrix for the random effects b2i
34. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Assumptions We assume that bi = (b1i, b2i, b3i)i.i.d. ∼ Nq(0, G) with q = q1 + q2 + q3
35. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Assumptions We assume that bi = (b1i, b2i, b3i)i.i.d. ∼ Nq(0, G) with q = q1 + q2 + q3 and λ0(t) = λ0 for t −1 ≤ t < t ( = 1, . . . , nλ).
36. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Assumptions We assume that bi = (b1i, b2i, b3i)i.i.d. ∼ Nq(0, G) with q = q1 + q2 + q3 and λ0(t) = λ0 for t −1 ≤ t < t ( = 1, . . . , nλ). Thus, the parameters to be estimated are θ = (β, γ, λ0, ζ, ξ, G).
37. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Model specification Assumptions We assume that bi = (b1i, b2i, b3i)i.i.d. ∼ Nq(0, G) with q = q1 + q2 + q3 and λ0(t) = λ0 for t −1 ≤ t < t ( = 1, . . . , nλ). Thus, the parameters to be estimated are θ = (β, γ, λ0, ζ, ξ, G). We also assume apriori independence for the parameters, such that: π(θ) = π(β)π(γ)π(λ0)π(ζ)π(ξ)π(G)
38. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Bayesian estimation using MCMC Joint posterior distribution of θ and b π(θ, b|T, δ, Y ) ∝ L(θ; T, δ, Y, b)π(θ) where L(θ; T, δ, Y, b) = m i=1 π(Ti, δi|bi, θ)π(yi|bi, θ)π(bi|θ)
39. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Bayesian estimation using MCMC Joint posterior distribution of θ and b (continued) π(Ti, δi|bi, θ) = λ(Ti)δi S(Ti) = λ0(Ti) exp(xT 2i (Ti)γ + zT 2i (Ti)b2i) δi × exp − Ti 0 λi(u)du ,
40. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Bayesian estimation using MCMC Joint posterior distribution of θ and b (continued) π(Ti, δi|bi, θ) = λ(Ti)δi S(Ti) = λ0(Ti) exp(xT 2i (Ti)γ + zT 2i (Ti)b2i) δi × exp − Ti 0 λi(u)du , π(Yi|bi, θ) ∝ |Σi|−1/2 exp 1 2 (yi − x1iβ − z1ibi)T Σ−1 i (yi − x1iβ − z1ibi)
41. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Bayesian estimation using MCMC Joint posterior distribution of θ and b (continued) π(Ti, δi|bi, θ) = λ(Ti)δi S(Ti) = λ0(Ti) exp(xT 2i (Ti)γ + zT 2i (Ti)b2i) δi × exp − Ti 0 λi(u)du , π(Yi|bi, θ) ∝ |Σi|−1/2 exp 1 2 (yi − x1iβ − z1ibi)T Σ−1 i (yi − x1iβ − z1ibi) and π(bi|θ) ∝ |G|−1/2 exp 1 2 bT i G−1bi
42. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Bayesian estimation using MCMC Priors chosen for θ β ∼ Np1 (µβ, Sβ) γ ∼ Np2 (µγ, Sγ) ζ ∼ Np3 (µζ, Sζ) ξ ∼ Np4 (µξ, Sξ) G−1 ∼ Wq(V, df) λ0 ∼ Gamma(α1, α2) ∀
43. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Bayesian estimation using MCMC Hybrid RWM within Gibbs sampler We can obtain closed forms for the conditionals π(β|θ−β, T, δ, Y, b) ∝ π(Y |b, θ)π(β), π(G−1|θ−G, T, δ, Y, b) ∝ π(b|θ)π(G−1) and π(λ0 |θ−λ0 , T, δ, Y, b) ∝ π(T, δ|b, θ)π(λ0 ) which are densities proportional to the Normal, Wishart and Gamma distributions, respectively.
44. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Bayesian estimation using MCMC Hybrid RWM within Gibbs sampler We can obtain closed forms for the conditionals π(β|θ−β, T, δ, Y, b) ∝ π(Y |b, θ)π(β), π(G−1|θ−G, T, δ, Y, b) ∝ π(b|θ)π(G−1) and π(λ0 |θ−λ0 , T, δ, Y, b) ∝ π(T, δ|b, θ)π(λ0 ) which are densities proportional to the Normal, Wishart and Gamma distributions, respectively. None of the other conditional posteriors belong to known distributions, so we propose to sample those parameters using a random walk Metropolis algorithm.
45. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Bayesian estimation using MCMC Hybrid RWM within Gibbs sampler (continued) 1 Initialise algorithm by selecting θ0 and b0. For s = 1, . . . , N 2 Update βs, Gs and λ0 through their conditional posteriors. 3 Propose new values γ∗, ζ∗, ξ∗ and b∗, using Normal proposals, centered around the previous values γs−1, ζs−1, ξs−1 and bs−1. 4 Accept these new values with probability α = min π(γ∗, ζ∗, ξ∗, b∗|βs, Gs, λs 0 , T, δ, Y ) π(γs−1, ζs−1, ξs−1, bs−1|βs, Gs, λs 0 , T, δ, Y ) , 1 and set (γ, ζ, ξ, b)s = (γ, ζ, ξ, b)∗, otherwise set (γ, ζ, ξ, b)s = (γ, ζ, ξ, b)s−1. 5 Discarding the first K samples as burn-in, θK+1, . . . , θN will be (approximately) a sample from the posterior π(θ|T, δ, Y, b).
46. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Bayesian estimation using MCMC References Andrinopoulou, E.R. and Rizopoulos, D (2016) - Bayesian shrinkage approach for a joint model of longitudinal and survival outcomes assuming different association structures. Statistics in Medicine, 35(26), 4813–4823 Charalambous, C., Pan, J. and Tranmer, M. (2014) - Variable Selection in Joint Mean and Dispersion Models via Double Penalized Likelihood. Sankhya Series B, 76(2): 276–304. Charalambous, C., Pan, J. and Tranmer, M. (2015) - Variable selection in joint modelling of the mean and variance for hierarchical data. Statistical Modelling, 15(1): 24–50. Fieuws, S., Verbeke, G., Maes, B. and Vanrenterghem, Y. (2008) Predicting renal graft failure using multivariate longitudinal profiles. Biostatistics 9: 419–431.
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49. Joint models for multi-outcome data and covariance structures via MCMC Joint models for multi-outcome data and covariance structures via MCMC Bayesian estimation using MCMC References Rizopoulos, D. and Ghosh, P. (2011) - A Bayesian semiparametric multivariate joint model for multiple longitudinal outcomes and a time-to-event. Statistics in Medicine, 30:1366–1380 Sudell, M., Kolamunnage-Dona, R. and Tudur-Smith, C. (2016) - Joint models for longitudinal and time-to event data: a review of reporting quality with a view to meta-analysis. BMC Medical Research Methodology, 16:168