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Joint modelling of multivariate longitudinal and time-to-event data

Graeme Hickey
November 13, 2017

Joint modelling of multivariate longitudinal and time-to-event data

Presented at Durham University, Department of Mathematical Sciences

Graeme Hickey

November 13, 2017
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  1. Introduction Model Estimation Software & Example revisited Prediction References
    Joint modelling of multivariate longitudinal and
    time-to-event data
    Graeme L. Hickey1, Pete Philipson2, Andrea Jorgensen1,
    Ruwanthi-Kolamunnage-Dona1
    1Department of Biostatistics, University of Liverpool, UK
    2Mathematics, Physics and Electrical Engineering, University of Northumbria, UK
    [email protected]
    Funded by Grant MR/M013227/1
    13th November 2017
    GL. Hickey Joint modelling of multivariate data 1 / 48

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  2. Introduction Model Estimation Software & Example revisited Prediction References
    Joint modelling
    Covariate data
    Longitudinal
    outcomes data
    Random
    effects

    GL. Hickey Joint modelling of multivariate data 2 / 48

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  3. Introduction Model Estimation Software & Example revisited Prediction References
    Joint modelling
    Covariate data
    Time-to-event
    outcomes data
    Frailties

    GL. Hickey Joint modelling of multivariate data 2 / 48

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  4. Introduction Model Estimation Software & Example revisited Prediction References
    Joint modelling
    Covariate data
    Time-to-event
    outcomes data
    Frailties

    Longitudinal
    outcomes data
    Random
    effects

    GL. Hickey Joint modelling of multivariate data 2 / 48

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  5. Introduction Model Estimation Software & Example revisited Prediction References
    Joint modelling
    Covariate data
    Time-to-event
    outcomes data
    Frailties

    Longitudinal
    outcomes data
    Random
    effects


    GL. Hickey Joint modelling of multivariate data 2 / 48

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  6. Introduction Model Estimation Software & Example revisited Prediction References
    Why use a joint model?
    Interest lies with
    adjustment of inferences about longitudinal measurements for
    possibly outcome-dependent drop-out
    adjustment of inferences about the time-to-event distribution
    conditional on intermediate and/or error prone longitudinal
    measurements
    the joint evolution of the measurement and event time processes
    biomarker surrogacy
    dynamic prediction
    GL. Hickey Joint modelling of multivariate data 3 / 48

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  7. Introduction Model Estimation Software & Example revisited Prediction References
    Motivation for multivariate joint models
    Clinical studies often repeatedly measure multiple biomarkers or
    other measurements and an event time
    Research has predominantly focused on a single event time and
    single measurement outcome
    Ignoring correlation leads to bias and reduced efficiency in
    estimation
    Harnessing all available information in a single model is
    advantageous and should lead to improved model predictions
    GL. Hickey Joint modelling of multivariate data 4 / 48

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  8. Introduction Model Estimation Software & Example revisited Prediction References
    Clinical example
    Figure source: https://www.medgadget.com
    Primary biliary cirrhosis (PBC)
    is a chronic liver disease char-
    acterized by inflammatory de-
    struction of the small bile ducts,
    which eventually leads to cirrho-
    sis of the liver and death
    GL. Hickey Joint modelling of multivariate data 5 / 48

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  9. Introduction Model Estimation Software & Example revisited Prediction References
    Clinical example
    Consider a subset of 154 patients randomized to placebo
    treatment from Mayo Clinic trial (Murtaugh et al. 1994)
    Multiple biomarkers repeatedly measured at intermittent times,
    of which we consider 3 clinically relevant ones:
    1 serum bilirunbin (mg/dl)
    2 serum albumin (mg/dl)
    3 prothrombin time (seconds)
    GL. Hickey Joint modelling of multivariate data 6 / 48

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  10. Introduction Model Estimation Software & Example revisited Prediction References
    Objective 1
    1 Determine if longitudinal biomarker trajectories are associated
    with death
    GL. Hickey Joint modelling of multivariate data 7 / 48

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  11. Introduction Model Estimation Software & Example revisited Prediction References
    Objective 1
    1 Determine if longitudinal biomarker trajectories are associated
    with death
    Objective 2
    1 Dynamically predict the biomarker trajectories and time to death
    for a new patient
    GL. Hickey Joint modelling of multivariate data 7 / 48

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  12. Introduction Model Estimation Software & Example revisited Prediction References
    Objective 1
    1 Determine if longitudinal biomarker trajectories are associated
    with death
    Objective 2
    1 Dynamically predict the biomarker trajectories and time to death
    for a new patient
    Objective 3
    1 Wrap it all up into a freely available software package
    GL. Hickey Joint modelling of multivariate data 7 / 48

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  13. Introduction Model Estimation Software & Example revisited Prediction References
    Prothrombin time (0.1 x seconds)−4
    Albumin (mg dL)
    Serum bilirubin (loge
    mg dL)
    0 5 10 0 5 10 0 5 10
    0.0
    0.5
    1.0
    1.5
    2
    4
    6
    8
    −2
    0
    2
    4
    Time from registration (years)
    Alive Dead
    GL. Hickey Joint modelling of multivariate data 8 / 48

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  14. Introduction Model Estimation Software & Example revisited Prediction References
    0 2 4 6 8 10 12 14
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    Time from registration (years)
    Survival probability
    nevents = 69, 44.8%
    GL. Hickey Joint modelling of multivariate data 9 / 48

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  15. Introduction Model Estimation Software & Example revisited Prediction References
    Data
    For each subject i = 1, . . . , n, we observe
    yi = (yi1
    , . . . , yiK
    ) is a K-variate continuous outcome vector,
    where each yik denotes an (nik × 1)-vector of observed
    longitudinal measurements for the k-th outcome type:
    yik = (yi1k, . . . , yinik k)
    Observation times tijk for j = 1, . . . , nik, which can differ
    between subjects and outcomes
    (Ti , δi ), where Ti = min(T∗
    i
    , Ci ), where T∗
    i
    is the true event
    time, Ci corresponds to a potential right-censoring time, and δi
    is the failure indicator equal to 1 if the failure is observed
    (T∗
    i
    ≤ Ci ) and 0 otherwise
    GL. Hickey Joint modelling of multivariate data 10 / 48

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  16. Introduction Model Estimation Software & Example revisited Prediction References
    Longitudinal sub-model
    Following Henderson et al. (2000) for the univariate case
    yi (t) = µi (t) + W1i (t) + εi (t),
    where
    εi (t) is the model error term, which is i.i.d. N(0, σ2) and
    independent of W1i (t)
    µi (t) = xi
    (t)β is the mean response
    xi (t) is a p-vector of (possibly) time-varying covariates with
    corresponding fixed effect terms β
    W1i (t) is a zero-mean latent Gaussian process
    GL. Hickey Joint modelling of multivariate data 11 / 48

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  17. Introduction Model Estimation Software & Example revisited Prediction References
    Longitudinal sub-model
    We can extend it to K-separate sub-models (with k = 1, . . . , K)
    yik(t) = µik(t) + W (k)
    1i
    (t) + εik(t),
    where
    εik(t) is the model error term, which is i.i.d. N(0, σ2
    k
    ) and
    independent of W (k)
    1i
    (t)
    µik(t) = xik
    (t)βk is the mean response
    xik(t) is a pk-vector of (possibly) time-varying covariates with
    corresponding fixed effect terms βk
    W (k)
    1i
    (t) is a zero-mean latent Gaussian process
    GL. Hickey Joint modelling of multivariate data 11 / 48

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  18. Introduction Model Estimation Software & Example revisited Prediction References
    Time-to-event sub-model
    λi (t) = lim
    dt→0
    P(t ≤ Ti < t + dt | Ti ≥ t)
    dt
    = λ0(t) exp vi
    (t)γv + W2i (t) ,
    where
    λ0(·) is an unspecified baseline hazard function
    vi (t) is a q-vector of (possibly) time-varying covariates with
    corresponding fixed effect terms γv
    W2i (t) is a zero-mean latent Gaussian process, independent of
    the censoring process
    GL. Hickey Joint modelling of multivariate data 12 / 48

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  19. Introduction Model Estimation Software & Example revisited Prediction References
    Correlation
    Following Laird and Ware (1982):
    W (k)
    1i
    (t) = zik
    (t)bik for k = 1, . . . , K
    GL. Hickey Joint modelling of multivariate data 13 / 48

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  20. Introduction Model Estimation Software & Example revisited Prediction References
    Correlation
    Following Laird and Ware (1982):
    W (k)
    1i
    (t) = zik
    (t)bik for k = 1, . . . , K
    Three sources of correlation:
    GL. Hickey Joint modelling of multivariate data 13 / 48

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  21. Introduction Model Estimation Software & Example revisited Prediction References
    Correlation
    Following Laird and Ware (1982):
    W (k)
    1i
    (t) = zik
    (t)bik for k = 1, . . . , K
    Three sources of correlation:
    1 Within-subject correlation between longitudinal measurements:
    bik ∼ N(0, Dkk)
    GL. Hickey Joint modelling of multivariate data 13 / 48

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  22. Introduction Model Estimation Software & Example revisited Prediction References
    Correlation
    Following Laird and Ware (1982):
    W (k)
    1i
    (t) = zik
    (t)bik for k = 1, . . . , K
    Three sources of correlation:
    1 Within-subject correlation between longitudinal measurements:
    bik ∼ N(0, Dkk)
    2 Between longitudinal outcomes correlation: cov(bik, bil ) = Dkl
    for k = l
    GL. Hickey Joint modelling of multivariate data 13 / 48

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  23. Introduction Model Estimation Software & Example revisited Prediction References
    Correlation
    Following Laird and Ware (1982):
    W (k)
    1i
    (t) = zik
    (t)bik for k = 1, . . . , K
    Three sources of correlation:
    1 Within-subject correlation between longitudinal measurements:
    bik ∼ N(0, Dkk)
    2 Between longitudinal outcomes correlation: cov(bik, bil ) = Dkl
    for k = l
    3 Correlation between sub-models1: W2i (t) = K
    k=1
    γykW (k)
    1i
    (t)
    1Extends model proposed Henderson et al. (2000)
    GL. Hickey Joint modelling of multivariate data 13 / 48

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  24. Introduction Model Estimation Software & Example revisited Prediction References
    Association structure: alternatives
    Many other proposals for association structures in the literature:
    Current value parameterisation: W2i (t) = γy {µi (t) + W1i (t)}
    Random effects parameterisation: W2i (t) = γy1
    bi
    Bivariate distribution: (W1i , W2i ) ∼ N(0, Ω)
    Random-slopes parameterisation:
    W2i (t) = γy1 {µi (t) + W1i (t)} + γy2

    ∂t
    {µi (t) + W1i (t)}
    . . .
    GL. Hickey Joint modelling of multivariate data 14 / 48

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  25. Introduction Model Estimation Software & Example revisited Prediction References
    Likelihood
    We can re-write the longitudinal sub-model as
    yi | bi , β, Σi ∼ N(Xi β + Zi bi , Σi ), with bi | D ∼ N(0, D),
    where β = (β1
    , . . . , βK
    ), bi = (bi1
    , . . . , biK
    ) , and
    Xi =





    Xi1 · · · 0
    .
    .
    .
    ...
    .
    .
    .
    0 · · · XiK





    ,
    Zi =





    Zi1 · · · 0
    .
    .
    .
    ...
    .
    .
    .
    0 · · · ZiK





    ,
    D =





    D11 · · · D1K
    .
    .
    .
    ...
    .
    .
    .
    D1K
    · · · DKK





    Σi =





    σ2
    1
    Ini1
    · · · 0
    .
    .
    .
    ...
    .
    .
    .
    0 · · · σ2
    K
    IniK





    GL. Hickey Joint modelling of multivariate data 15 / 48

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  26. Introduction Model Estimation Software & Example revisited Prediction References
    Likelihood
    The observed data likelihood is given by
    n
    i=1

    −∞
    f (yi | bi , θ)f (Ti , δi | bi , θ)f (bi | θ)dbi
    where θ = (β , vech(D), σ2
    1
    , . . . , σ2
    K
    , λ0(t), γv
    , γy
    )
    GL. Hickey Joint modelling of multivariate data 16 / 48

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  27. Introduction Model Estimation Software & Example revisited Prediction References
    Likelihood
    The observed data likelihood is given by
    n
    i=1

    −∞
    f (yi | bi , θ)f (Ti , δi | bi , θ)f (bi | θ)dbi
    where θ = (β , vech(D), σ2
    1
    , . . . , σ2
    K
    , λ0(t), γv
    , γy
    ), and
    f (yi | bi , θ) =
    K
    k=1
    (2π)−nik
    2 |Σi |−1
    2
    exp −
    1
    2
    (yi − Xi β − Zi bi ) Σ−1
    i
    (yi − Xi β − Zi bi )
    GL. Hickey Joint modelling of multivariate data 16 / 48

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  28. Introduction Model Estimation Software & Example revisited Prediction References
    Likelihood
    The observed data likelihood is given by
    n
    i=1

    −∞
    f (yi | bi , θ)f (Ti , δi | bi , θ)f (bi | θ)dbi
    where θ = (β , vech(D), σ2
    1
    , . . . , σ2
    K
    , λ0(t), γv
    , γy
    ), and
    f (Ti , δi | bi ; θ) = λ0(Ti ) exp vi
    γv + W2i (Ti , bi )
    δi
    exp −
    Ti
    0
    λ0(u) exp vi
    γv + W2i (u, bi ) du
    GL. Hickey Joint modelling of multivariate data 16 / 48

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  29. Introduction Model Estimation Software & Example revisited Prediction References
    Likelihood
    The observed data likelihood is given by
    n
    i=1

    −∞
    f (yi | bi , θ)f (Ti , δi | bi , θ)f (bi | θ)dbi
    where θ = (β , vech(D), σ2
    1
    , . . . , σ2
    K
    , λ0(t), γv
    , γy
    ), and
    f (bi | θ) = (2π)− r
    2 |D|−1
    2 exp −
    1
    2
    bi
    D−1bi ,
    with r = dim(bi )
    GL. Hickey Joint modelling of multivariate data 16 / 48

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  30. Introduction Model Estimation Software & Example revisited Prediction References
    Estimation
    Multiple approaches have been considered over the years:
    Markov chain Monte Carlo (MCMC)
    Direct likelihood maximisation (e.g. Newton-methods)
    Generalised estimating equations
    EM algorithm (treating the random effects as missing data)
    . . .
    GL. Hickey Joint modelling of multivariate data 17 / 48

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  31. Introduction Model Estimation Software & Example revisited Prediction References
    EM algorithm (Dempster et al. 1977)
    E-step. At the m-th iteration, we compute the expected
    log-likelihood of the complete data conditional on the observed data
    and the current estimate of the parameters.
    Q(θ | ˆ
    θ(m)) =
    n
    i=1
    E log f (yi , Ti , δi , bi | θ) ,
    =
    n
    i=1

    −∞
    log f (yi , Ti , δi , bi | θ) f (bi | Ti , δi , yi ; ˆ
    θ(m))dbi
    GL. Hickey Joint modelling of multivariate data 18 / 48

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  32. Introduction Model Estimation Software & Example revisited Prediction References
    EM algorithm (Dempster et al. 1977)
    E-step. At the m-th iteration, we compute the expected
    log-likelihood of the complete data conditional on the observed data
    and the current estimate of the parameters.
    Q(θ | ˆ
    θ(m)) =
    n
    i=1
    E log f (yi , Ti , δi , bi | θ) ,
    =
    n
    i=1

    −∞
    log f (yi , Ti , δi , bi | θ) f (bi | Ti , δi , yi ; ˆ
    θ(m))dbi
    M-step. We maximise Q(θ | ˆ
    θ(m)) with respect to θ. namely,
    ˆ
    θ(m+1) = arg max
    θ
    Q(θ | ˆ
    θ(m))
    GL. Hickey Joint modelling of multivariate data 18 / 48

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  33. Introduction Model Estimation Software & Example revisited Prediction References
    M-step: closed form estimators
    ˆ
    λ0(t) =
    n
    i=1
    δi I(Ti = t)
    n
    i=1 E exp vi
    γv + W2i (t, bi ) I(Ti ≥ t)
    ˆ
    β =
    n
    i=1
    Xi
    Xi
    −1 n
    i=1
    Xi
    (yi − Zi E[bi ])
    ˆ
    σ2
    k
    =
    1
    n
    i=1
    nik
    n
    i=1
    (yik − Xikβk) (yik − Xikβk − 2ZikE[bik])
    +trace Zik
    ZikE[bikbik
    ]
    ˆ
    D =
    1
    n
    n
    i=1
    E bi bi
    GL. Hickey Joint modelling of multivariate data 19 / 48

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  34. Introduction Model Estimation Software & Example revisited Prediction References
    M-step: non-closed form estimators
    There is no closed form update for γ = (γv
    , γy
    ), so use a one-step
    Newton-Raphson iteration
    ˆ
    γ(m+1) = ˆ
    γ(m) + I ˆ
    γ(m) −1
    S ˆ
    γ(m) ,
    where
    S(γ) =
    n
    i=1
    δi E [˜
    vi (Ti )] −
    Ti
    0
    λ0(u)E ˜
    vi (u) exp{˜
    vi
    (u)γ} du
    I(γ) = −

    ∂γ
    S(γ)
    with ˜
    vi (t) = vi
    , zi1
    (t)bi1, . . . , ziK
    (t)biK a (q + K)–vector
    GL. Hickey Joint modelling of multivariate data 20 / 48

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  35. Introduction Model Estimation Software & Example revisited Prediction References
    MCEM algorithm
    E-step requires calculating several multidimensional integrals of
    form E h(bi ) | Ti , δi , yi ; ˆ
    θ
    Gauss-quadrature can be slow if dim(bi ) is large ⇒ might not
    scale well as K increases
    Instead, we use the Monte Carlo Expectation-Maximization
    (MCEM; Wei and Tanner 1990)
    M-step updates remain the same
    GL. Hickey Joint modelling of multivariate data 21 / 48

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  36. Introduction Model Estimation Software & Example revisited Prediction References
    Monte Carlo E-step
    Conventional EM algorithm: use quadrature to compute
    E h(bi ) | Ti , δi , yi ; ˆ
    θ =

    −∞
    h(bi )f (bi | yi ; ˆ
    θ)f (Ti , δi | bi ; ˆ
    θ)dbi

    −∞
    f (bi | yi ; ˆ
    θ)f (Ti , δi | bi ; ˆ
    θ)dbi
    ,
    where
    h(·) = any known fuction,
    bi | yi , θ ∼ N Ai Zi
    Σ−1
    i
    (yi − Xi β) , Ai , and
    Ai = Zi
    Σ−1
    i
    Zi + D−1 −1
    GL. Hickey Joint modelling of multivariate data 22 / 48

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  37. Introduction Model Estimation Software & Example revisited Prediction References
    Monte Carlo E-step
    MCEM algorithm E-step: use Monte Carlo integration to compute
    E h(bi ) | Ti , δi , yi ; ˆ
    θ ≈
    1
    N
    N
    d=1
    h b(d)
    i
    f Ti , δi | b(d)
    i
    ; ˆ
    θ
    1
    N
    N
    d=1
    f Ti , δi | b(d)
    i
    ; ˆ
    θ
    where
    h(·) = any known fuction,
    bi | yi , θ ∼ N Ai Zi
    Σ−1
    i
    (yi − Xi β) , Ai , and
    Ai = Zi
    Σ−1
    i
    Zi + D−1 −1
    b(1)
    i
    , b(2)
    i
    , . . . , b(N)
    i
    ∼ bi | yi , θ a Monte Carlo draw
    GL. Hickey Joint modelling of multivariate data 22 / 48

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  38. Introduction Model Estimation Software & Example revisited Prediction References
    Speeding up convergence
    Monte Carlo integration converges at a rate of O(N−1/2), which
    is independent of K and r = dim(bi )
    EM algorithm convergences linearly
    Can we speed this up?
    GL. Hickey Joint modelling of multivariate data 23 / 48

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  39. Introduction Model Estimation Software & Example revisited Prediction References
    Speeding up convergence
    Monte Carlo integration converges at a rate of O(N−1/2), which
    is independent of K and r = dim(bi )
    EM algorithm convergences linearly
    Can we speed this up?
    1 Antithetic variates
    2 Quasi-Monte Carlo
    GL. Hickey Joint modelling of multivariate data 23 / 48

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  40. Introduction Model Estimation Software & Example revisited Prediction References
    Variance reduction
    Instead of directly sampling from the MVN distribution for bi | yi ; θ,
    we apply a variance reduction technique
    Antithetic simulation
    Sample Ω ∼ N(0, Ir ) and obtain the pairs
    Ai Zi
    Σ−1
    i
    (yi − Xi β) ± Ci Ω,
    where Ci is the Cholesky decomposition of Ai such that Ci Ci
    = Ai
    Negative correlation between the N/2 pairs ⇒ smaller variance in the
    sample means than would be obtained from N independent
    simulations
    GL. Hickey Joint modelling of multivariate data 24 / 48

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  41. Introduction Model Estimation Software & Example revisited Prediction References
    Convergence
    In standard EM, convergence usually declared at (m + 1)-th iteration
    if one of the following criteria satisfied
    Relative change: ∆(m+1)
    rel
    = max |ˆ
    θ(m+1)−ˆ
    θ(m)|

    θ(m)|+ 1
    < 0
    Absolute change: ∆(m+1)
    abs
    = max |ˆ
    θ(m+1) − ˆ
    θ(m)| < 2
    for some choice of 0, 1, and 2
    GL. Hickey Joint modelling of multivariate data 25 / 48

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  42. Introduction Model Estimation Software & Example revisited Prediction References
    Convergence
    In MCEM framework, there are 2 complications to account for
    GL. Hickey Joint modelling of multivariate data 26 / 48

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  43. Introduction Model Estimation Software & Example revisited Prediction References
    Convergence
    In MCEM framework, there are 2 complications to account for
    1 spurious convergence declared due to random chance
    GL. Hickey Joint modelling of multivariate data 26 / 48

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  44. Introduction Model Estimation Software & Example revisited Prediction References
    Convergence
    In MCEM framework, there are 2 complications to account for
    1 spurious convergence declared due to random chance
    ⇒ Solution: require convergence for 3 iterations in succession
    GL. Hickey Joint modelling of multivariate data 26 / 48

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  45. Introduction Model Estimation Software & Example revisited Prediction References
    Convergence
    In MCEM framework, there are 2 complications to account for
    1 spurious convergence declared due to random chance
    ⇒ Solution: require convergence for 3 iterations in succession
    2 estimators swamped by Monte Carlo error, thus precluding
    convergence
    GL. Hickey Joint modelling of multivariate data 26 / 48

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  46. Introduction Model Estimation Software & Example revisited Prediction References
    Convergence
    In MCEM framework, there are 2 complications to account for
    1 spurious convergence declared due to random chance
    ⇒ Solution: require convergence for 3 iterations in succession
    2 estimators swamped by Monte Carlo error, thus precluding
    convergence
    ⇒ Solution: increase Monte Carlo size N as algorithm moves
    closer towards maximizer
    GL. Hickey Joint modelling of multivariate data 26 / 48

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  47. Introduction Model Estimation Software & Example revisited Prediction References
    Dynamic MC size
    Using large N when far from maximizer = computationally
    inefficient
    Using small N when close to maximizer = unlikely to detect
    convergence
    Solution (proposed by Ripatti et al. 2002): after a ‘burn-in’ phase,
    calculate the coefficient of variation statistic
    cv(∆(m+1)
    rel
    ) =
    sd(∆(m−1)
    rel
    , ∆(m)
    rel
    , ∆(m+1)
    rel
    )
    mean(∆(m−1)
    rel
    , ∆(m)
    rel
    , ∆(m+1)
    rel
    )
    ,
    and increase N to N + N/δ if cv(∆(m+1)
    rel
    ) > cv(∆(m)
    rel
    ) for some
    small positive integer δ
    GL. Hickey Joint modelling of multivariate data 27 / 48

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  48. Introduction Model Estimation Software & Example revisited Prediction References
    Quasi-Monte Carlo
    Replaces the (pseudo-)random sequence by a deterministic one
    Quasi-random sequences yield smaller errors than standard
    Monte Carlo integration methods
    Convergence is O (logN)r
    N
    Research on-going. . .
    GL. Hickey Joint modelling of multivariate data 28 / 48

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  49. Introduction Model Estimation Software & Example revisited Prediction References
    Quasi-Monte Carlo
    q
    q
    q
    q
    q
    q
    q
    q
    q
    q
    q
    q
    q
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    OMC AMC QMC
    Uniform
    0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00
    0.00
    0.25
    0.50
    0.75
    1.00
    U1
    U2
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    OMC AMC QMC
    Normal
    −4 −2 0 2 −4 −2 0 2 −4 −2 0 2
    −2
    0
    2
    4
    Z1
    Z2
    Key: OMC = ordinary Monte Carlo; AMC = antithetic Monte Carlo; QMC = quasi-Monte Carlo
    GL. Hickey Joint modelling of multivariate data 29 / 48

    View Slide

  50. Introduction Model Estimation Software & Example revisited Prediction References
    Standard error estimation
    Method 1: Bootstrap
    Conceptually simple + theoretically superior (Hsieh et al. 2006). . .
    but computationally slow!
    GL. Hickey Joint modelling of multivariate data 30 / 48

    View Slide

  51. Introduction Model Estimation Software & Example revisited Prediction References
    Standard error estimation
    Method 1: Bootstrap
    Conceptually simple + theoretically superior (Hsieh et al. 2006). . .
    but computationally slow!
    Method 2: Empirical information matrix approximation
    Following McLachlan and Krishnan (2008), SE(θ) ≈ I−1/2
    e (ˆ
    θ), where
    Ie(θ) =
    n
    i=1
    si (θ)si
    (θ) −
    1
    n
    S(θ)S (θ),
    S(θ) = n
    i=1
    si (θ) is the score vector for θ−λ0(t)
    (baseline hazards a
    profiled out of the likelihood)
    GL. Hickey Joint modelling of multivariate data 30 / 48

    View Slide

  52. Introduction Model Estimation Software & Example revisited Prediction References
    joineRML
    +
    getVarCov()
    vcov()
    fixef()
    ranef()*
    AIC()
    BIC()
    confint()
    formula()
    sampleData()
    dynSurv()*
    dynLong()*
    print()
    summary()
    plot()
    sigma()
    coef()
    update()
    baseHaz()
    residuals()
    fitted()
    logLik()
    bootSE()
    Rich collection of associated methods
    * associated with additional plot methods
    mjoint()
    Version 0.3.0 available on CRAN
    https://cran.r-project.org/web/packages/joineRML/
    Developmental version available on
    GitHub
    https://github.com/graemeleehickey/joineRML
    Parallel
    Computing
    +
    GL. Hickey Joint modelling of multivariate data 31 / 48

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  53. Introduction Model Estimation Software & Example revisited Prediction References
    Alternative options
    Pre-2017: none!
    2017-onwards:
    joineRML: discussed today
    stjm: a new extension to the Stata package2 written by Michael
    Crowther
    megenreg: similar to stjm, but can handle other models
    rstanarm: development branch that absorbs package written by
    Sam Brilleman3
    JMbayes: a new extension4 to the R package written by Dimitris
    Rizopoulos
    2Crowther MJ. Joint Statistical Meeting. Seattle; 2015.
    3
    github.com/sambrilleman/rstanjm
    4
    github.com/drizopoulos/JMbayes
    GL. Hickey Joint modelling of multivariate data 32 / 48

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  54. Introduction Model Estimation Software & Example revisited Prediction References
    Proposed model for PBC data
    Longitudinal sub-model
    log(serBilir) = (β0,1
    + b0i,1
    ) + (β1,1
    + b1i,1
    )year + εij1,
    albumin = (β0,2
    + b0i,2
    ) + (β1,2
    + b1i,2
    )year + εij2,
    (0.1 × prothrombin)−4 = (β0,3
    + b0i,3
    ) + (β1,3
    + b1i,3
    )year + εij3,
    bi ∼ N6
    (0, D), and εijk
    ∼ N(0, σ2
    k
    ) for k = 1, 2, 3;
    Time-to-event sub-model
    λi
    (t) = λ0
    (t) exp {γv age + W2i
    (t)} ,
    W2i
    (t) = γbil
    (b0i,1
    + b1i,1
    t) + γalb
    (b0i,2
    + b1i,2
    t) + γpro
    (b0i,3
    + b1i,3
    t).
    GL. Hickey Joint modelling of multivariate data 33 / 48

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  55. Introduction Model Estimation Software & Example revisited Prediction References
    Example code
    data(pbc2)
    placebo <- subset(pbc2, drug == "placebo")
    fit.pbc <- mjoint(
    formLongFixed = list(
    "bil" = log(serBilir) ˜ year,
    "alb" = albumin ˜ year,
    "pro" = (0.1 * prothrombin)ˆ-4 ˜ year),
    formLongRandom = list(
    "bil" = ˜ year | id,
    "alb" = ˜ year | id,
    "pro" = ˜ year | id),
    formSurv = Surv(years, status2) ˜ age,
    data = placebo,
    timeVar = "year",
    control = list(tol0 = 0.001, burin = 400))
    GL. Hickey Joint modelling of multivariate data 34 / 48

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  56. Introduction Model Estimation Software & Example revisited Prediction References
    Results
    Parameter Estimate SE 95% CI
    β0,1 0.5541 0.0858 (0.3859, 0.7223)
    β1,1 0.2009 0.0201 (0.1616, 0.2402)
    β0,2 3.5549 0.0356 (3.4850, 3.6248)
    β1,2 -0.1245 0.0101 (-0.1444, -0.1047)
    β0,3 0.8304 0.0212 (0.7888, 0.8719)
    β1,3 -0.0577 0.0062 (-0.0699, -0.0456)
    γv 0.0462 0.0151 (0.0166, 0.0759)
    γbil
    0.8181 0.2046 (0.4171, 1.2191)
    γalb
    -1.7060 0.6181 (-2.9173, -0.4946)
    γpro
    -2.2085 1.6070 (-5.3582, 0.9412)
    GL. Hickey Joint modelling of multivariate data 35 / 48

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  57. Introduction Model Estimation Software & Example revisited Prediction References
    Results
    Parameter Estimate SE 95% CI
    β0,1 0.5541 0.0858 (0.3859, 0.7223)
    β1,1 0.2009 0.0201 (0.1616, 0.2402)
    β0,2 3.5549 0.0356 (3.4850, 3.6248)
    β1,2 -0.1245 0.0101 (-0.1444, -0.1047)
    β0,3 0.8304 0.0212 (0.7888, 0.8719)
    β1,3 -0.0577 0.0062 (-0.0699, -0.0456)
    γv 0.0462 0.0151 (0.0166, 0.0759)
    γbil
    0.8181 0.2046 (0.4171, 1.2191)
    γalb
    -1.7060 0.6181 (-2.9173, -0.4946)
    γpro
    -2.2085 1.6070 (-5.3582, 0.9412)
    GL. Hickey Joint modelling of multivariate data 35 / 48

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  58. Introduction Model Estimation Software & Example revisited Prediction References
    Results
    Effect of multivariate inference over univariate joint model:
    Parameter Model Estimate 95% CI
    γbil
    UV 1.2182 (0.9789, 1.6130)
    γbil
    MV 0.8181 (0.4171, 1.2191)
    γalb
    UV -3.0770 (-4.4865, -2.3466)
    γalb
    MV -1.7060 (-2.9173, -0.4946)
    γpro
    UV -7.2078 (-10.5410, -5.3917)
    γpro
    MV -2.2085 (-5.3582, 0.9412)
    UV = univariate joint model (fitted with joineR package); MV =
    multivariate joint model
    GL. Hickey Joint modelling of multivariate data 36 / 48

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  59. Introduction Model Estimation Software & Example revisited Prediction References
    Dynamic prediction
    So far we have only discussed inference from joint models
    How we can use them for prediction?
    Predict what?
    1 Failure probability at time u > t given longitudinal data observed
    up until time t
    2 Longitudinal trajectories at time u > t given longitudinal data
    observed up until time t
    GL. Hickey Joint modelling of multivariate data 37 / 48

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  60. Introduction Model Estimation Software & Example revisited Prediction References
    Dynamic prediction: example
    Bivariate joint model
    We will consider the PBC data again (as above) with K = 2
    biomarkers only: serurm bilirubin (log-transformed) and albumin
    (untransformed), since prothrombin time was non-significant in the
    trivariate model
    GL. Hickey Joint modelling of multivariate data 38 / 48

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  61. Introduction Model Estimation Software & Example revisited Prediction References
    Dynamic prediction: survival
    For a new subject i = n + 1, we want to calculate
    P[T∗
    n+1
    ≥ u | T∗
    n+1
    > t, yn+1; θ] = E
    Sn+1 (u | W2,n+1(u, bn+1; θ); θ)
    Sn+1 (t | W2,n+1(t, bn+1; θ); θ)
    ,
    where W2i (t, bi ; θ) = {W2i (s, vi ; θ); 0 ≤ s < t} and the expectation is
    taken with respect to the distribution
    p(bn+1 | T∗
    n+1
    > t, yn+1; θ)
    GL. Hickey Joint modelling of multivariate data 39 / 48

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  62. Introduction Model Estimation Software & Example revisited Prediction References
    Dynamic prediction: survival
    Rizopoulos (2011) proposed two estimators for this:
    1 A first-order approximation
    P[T∗
    n+1
    ≥ u | T∗
    n+1
    > t, yn+1
    ; θ] ≈
    Sn+1
    u | W2,n+1
    (u, ˆ
    bn+1
    ; ˆ
    θmle
    ); ˆ
    θmle
    Sn+1
    t | W2,n+1
    (t, ˆ
    bn+1
    ; ˆ
    θmle
    ); ˆ
    θmle
    ,
    where ˆ
    bn+1
    is the mode of p(bn+1 | T∗
    n+1
    > t, yn+1
    ; θ)
    2 A simulated scheme
    1 Draw θ(l) ∼ N(ˆ
    θmle
    , V (ˆ
    θmle
    ))
    2 Draw b(l)
    n+1
    ∼ p(bn+1 | T∗
    n+1
    > t, yn+1
    ; θ) [Metropolis-Hastings]
    3 Calculate
    Sn+1 u | W2,n+1(u,b(l)
    n+1
    ;θ(l));θ(l)
    Sn+1 t | W2,n+1(t,b(l)
    n+1
    ;θ(l));θ(l)
    4 Repeat Steps 1–3 l = 2, . . . , L times
    GL. Hickey Joint modelling of multivariate data 40 / 48

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  63. Introduction Model Estimation Software & Example revisited Prediction References
    Example code
    # New patient
    nd <- subset(placebo, id == "11") # patient 11
    # First-order prediction (default)
    pred1 <- dynSurv(fit.pbc, nd[1:5, ])
    pred1
    plot(pred1)
    # Simulated prediction
    pred2 <- dynSurv(fit.pbc, nd[1:5, ], type = "simulated", scale = 2)
    pred2
    plot(pred2)
    GL. Hickey Joint modelling of multivariate data 41 / 48

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  64. Introduction Model Estimation Software & Example revisited Prediction References
    Dynamic predicton: survival
    GL. Hickey Joint modelling of multivariate data 42 / 48
    0.20
    0.25
    0.30
    0.35
    0.40
    0.45
    data.t$tk[[k]]
    log(bilirubin)
    2.5
    3.0
    3.5
    4.0
    4.5
    5.0
    5.5
    data.t$tk[[k]]
    albumin
    0 0 2 4 6 8 10 12 14
    xpts
    ypts
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    Time
    Event−free probability

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  65. Introduction Model Estimation Software & Example revisited Prediction References
    Dynamic prediction: longitudinal
    For a new subject i = n + 1, we want to calculate
    E yn+1(u) | T∗
    n+1
    > t, yn+1; θ = Xn+1
    (u)β + Zn+1
    (u)E[bn+1],
    GL. Hickey Joint modelling of multivariate data 43 / 48

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  66. Introduction Model Estimation Software & Example revisited Prediction References
    Dynamic prediction: longitudinal
    Again, we can use the same estimation proposals:
    1 A first-order approximation
    E [yn+1
    (u) | T∗
    n+1
    > t, yn+1
    ; θ] ≈ Xn+1
    (u)ˆ
    β + Zn+1
    (u)ˆ
    bn+1,
    where ˆ
    bn+1
    is the mode of p(bn+1 | T∗
    n+1
    > t, yn+1
    ; θ)
    2 A simulated scheme
    1 Draw θ(l) ∼ N(ˆ
    θmle
    , V (ˆ
    θmle
    ))
    2 Draw b(l)
    n+1
    ∼ p(bn+1 | T∗
    n+1
    > t, yn+1
    ; θ) [Metropolis-Hastings]
    3 Calculate Xn+1
    (u)β(l) + Zn+1
    (u)b(l)
    n+1
    4 Repeat Steps 1–3 l = 2, . . . , L times
    GL. Hickey Joint modelling of multivariate data 44 / 48

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  67. Introduction Model Estimation Software & Example revisited Prediction References
    Example code
    # First-order prediction (default)
    pred1 <- dynLong(fit.pbc, nd[1:5, ])
    pred1
    plot(pred1)
    # Simulated prediction
    pred2 <- dynLong(fit.pbc, nd[1:5, ], type = "simulated", scale = 2)
    pred2
    plot(pred2)
    GL. Hickey Joint modelling of multivariate data 45 / 48

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  68. Introduction Model Estimation Software & Example revisited Prediction References
    Dynamic predicton: longitudinal
    GL. Hickey Joint modelling of multivariate data 46 / 48
    0
    1
    2
    3
    4
    5
    xpts
    log(bilirubin)
    1.5
    2.0
    2.5
    3.0
    3.5
    4.0
    xpts
    albumin
    0 2 4 6 8 10 12 14
    Time

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  69. Introduction Model Estimation Software & Example revisited Prediction References
    Open challenges
    How can we incorporate high-dimensional K? E.g. K = 10?
    Data reduction techniques: can we project high-dimensional K
    onto a lower order plane?
    Speed-up calculations using approximations (e.g. Laplace
    approximations)
    GL. Hickey Joint modelling of multivariate data 47 / 48

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  70. Introduction Model Estimation Software & Example revisited Prediction References
    References
    Murtaugh, Paul A et al. (1994). Primary biliary cirrhosis: prediction of short-term survival based on repeated patient visits.
    Hepatology 20(1), pp. 126–134.
    Henderson, R et al. (2000). Joint modelling of longitudinal measurements and event time data. Biostatistics 1(4), pp. 465–480.
    Laird, NM and Ware, JH (1982). Random-effects models for longitudinal data. Biometrics 38(4), pp. 963–74.
    Dempster, AP et al. (1977). Maximum likelihood from incomplete data via the EM algorithm. J Roy Stat Soc B 39(1), pp. 1–38.
    Wei, GC and Tanner, MA (1990). A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation
    algorithms. J Am Stat Assoc 85(411), pp. 699–704.
    Ripatti, S et al. (2002). Maximum likelihood inference for multivariate frailty models using an automated Monte Carlo EM algorithm.
    Life Dat Anal 8(2002), pp. 349–360.
    Hsieh, F et al. (2006). Joint modeling of survival and longitudinal data: Likelihood approach revisited. Biometrics 62(4),
    pp. 1037–1043.
    McLachlan, GJ and Krishnan, T (2008). The EM Algorithm and Extensions. Second. Wiley-Interscience.
    Rizopoulos, Dimitris (2011). Dynamic predictions and prospective accuracy in joint models for longitudinal and time-to-event data.
    Biometrics 67(3), pp. 819–829.
    GL. Hickey Joint modelling of multivariate data 48 / 48

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