Bayesian joint models for multiple longitudinal biomarkers and a time-to-event outcome: software development and a melanoma case study Sam Brilleman 1,2 , Michael J. Crowther 3 , Margarita Moreno-Betancur 2,4,5 , Serigne Lo 6 , Jacqueline Buros Novik 7 , Rory Wolfe 1,2 38th Annual Conference of the International Society for Clinical Biostatistics Vigo, Spain 9-13 th July 2017 1 Monash University, Melbourne, Australia 4 Murdoch Childrens Research Institute, Melbourne, Australia 2 Victorian Centre for Biostatistics (ViCBiostat) 5 University of Melbourne, Melbourne, Australia 3 University of Leicester, Leicester, UK 6 Melanoma Institute Australia, Sydney, Australia 7 Icahn School of Medicine at Mount Sinai, New York, US
Collection of biomarker data from a melanoma patient 2
Collection of biomarker data from a melanoma patient 3
Collection of biomarker data from a Fitting a linear melanoma patient mixed model Analysing to lymphocyte changes in LDH Fitting a Cox counts model for survival The 1990’s analysis 4
Collection of biomarker data from a Fitting a linear melanoma patient mixed model Analysing to lymphocyte changes in LDH Fitting a Cox counts model for survival The 2017 analysis The 1990’s analysis Fitting a joint model for LDH, lymphocytes and survival 5
Background • What is joint modelling? • The joint estimation of regression models which, traditionally, we would have estimated separately: • A (multivariate) longitudinal mixed model for a longitudinal outcome(s) • A time-to-event model for the time to an event of interest • The “sub”models are linked through shared (subject-specific) parameters • Why use it? • We want to understand how (some function of) the longitudinal outcome is associated with risk of the event • can allow for measurement error in the biomarker • can allow for discrete-time measurement of the biomarker • “Dynamic” predictions of the risk of the event • Separating out “direct” and “indirect” effects of treatment • Adjusting for informative dropout 6
𝑧 𝑗𝑘𝑙 𝑢 is the value at time 𝑢 of the 𝑙 th longitudinal marker ( 𝑙 = 1, … , 𝐿 ) Joint model formulation for the 𝑗 th individual ( 𝑗 = 1, … , 𝑂 ) at the 𝑘 th time point ( 𝑘 = 1, … , 𝐾 𝑗𝑙 ) 𝑈 𝑗 is “true” event time, 𝐷 𝑗 is the censoring time ∗ = min 𝑈 𝑗 , 𝐷 𝑗 𝑈 𝑗 and 𝑒 𝑗 = 𝐽(𝑈 𝑗 ≤ 𝐷 𝑗 ) Longitudinal submodel 𝑧 𝑗𝑘𝑙 𝑢 follows a distribution in the exponential family with expected value 𝜈 𝑗𝑘𝑙 𝑢 and ′ ′ 𝜃 𝑗𝑘𝑙 𝑢 = 𝑙 𝜈 𝑗𝑘𝑙 𝑢 = 𝒚 𝒋𝒌𝒍 𝑢 𝜸 𝒍 + 𝒜 𝒋𝒌𝒍 𝑢 𝒄 𝒋𝒍 𝒄 𝒋𝟐 ⋮ = 𝒄 𝒋 ~ 𝑂 0, 𝚻 𝒄 𝒋𝑳 Event submodel 𝐿 𝑅 𝑙 ′ 𝑢 𝜹 + ℎ 𝑗 (𝑢) = ℎ 0 (𝑢) exp 𝒙 𝒋 𝛽 𝑙𝑟 𝑔 𝑙𝑟 (𝜸 𝒍 , 𝒄 𝒋𝒍 ; 𝑢) 𝑙=1 𝑟=1 7
𝑧 𝑗𝑘𝑙 𝑢 is the value at time 𝑢 of the 𝑙 th longitudinal marker ( 𝑙 = 1, … , 𝐿 ) Joint model formulation for the 𝑗 th individual ( 𝑗 = 1, … , 𝑂 ) at the 𝑘 th time point ( 𝑘 = 1, … , 𝐾 𝑗𝑙 ) 𝑈 𝑗 is “true” event time, 𝐷 𝑗 is the censoring time ∗ = min 𝑈 𝑗 , 𝐷 𝑗 𝑈 𝑗 and 𝑒 𝑗 = 𝐽(𝑈 𝑗 ≤ 𝐷 𝑗 ) Longitudinal submodel 𝑧 𝑗𝑘𝑙 𝑢 follows a distribution in the exponential family with expected value 𝜈 𝑗𝑘𝑙 𝑢 and ′ ′ 𝜃 𝑗𝑘𝑙 𝑢 = 𝑙 𝜈 𝑗𝑘𝑙 𝑢 = 𝒚 𝒋𝒌𝒍 𝑢 𝜸 𝒍 + 𝒜 𝒋𝒌𝒍 𝑢 𝒄 𝒋𝒍 𝒄 𝒋𝟐 ⋮ = 𝒄 𝒋 ~ 𝑂 0, 𝚻 association term 𝒄 𝒋𝑳 (some function of parameters from the longitudinal submodel) Event submodel 𝐿 𝑅 𝑙 ′ 𝑢 𝜹 + ℎ 𝑗 (𝑢) = ℎ 0 (𝑢) exp 𝒙 𝒋 𝛽 𝑙𝑟 𝑔 𝑙𝑟 (𝜸 𝒍 , 𝒄 𝒋𝒍 ; 𝑢) 𝑙=1 𝑟=1 association parameter 8
𝐿 𝑅 𝑙 ′ 𝑢 𝜹 + Association structures ℎ 0 (𝑢) exp 𝒙 𝒋 𝛽 𝑙𝑟 𝑔 𝑙𝑟 (𝜸 𝒍 , 𝒄 𝒋𝒍 ; 𝑢) 𝑙=1 𝑟=1 𝜃 𝑗𝑙 𝑢 Linear predictor (or expected value of the biomarker) at time 𝑢 𝑒𝜃 𝑗𝑙 𝑢 Rate of change in the linear predictor (or biomarker) at time 𝑢 𝑒𝑢 𝑢 න 𝜃 𝑗𝑙 𝑡 𝑒𝑡 Area under linear predictor (or biomarker trajectory), up to time 𝑢 0 𝜃 𝑗𝑙 𝑢 − 𝑣 Lagged value (for some lag time 𝑣 ) Interactions between values of the different biomarkers (for 𝑙 ≠ 𝑙 ′ ) 𝜃 𝑗𝑙 𝑢 × 𝜃 𝑗𝑙 ′ 𝑢 𝜃 𝑗𝑙 𝑢 × 𝑑 𝑗 𝑢 Interactions with observed data (e.g. for some observed covariate 𝑑 𝑗 𝑢 ) 9
Joint modelling software • An abundance of methodological developments in joint modelling • Not all methods have been translated into “user - friendly” software • Well established software for one longitudinal outcome • e.g. stjm (Stata); joineR, JM, JMbayes, frailtypack (R); JMFit (SAS) • Recent software developments for multiple longitudinal outcomes * • released packages: joineRML (R, available on CRAN) • development packages: survtd , rstanarm , JMbayes (R, available on GitHub); stjm • Each package has their strengths and limitations • e.g. (non-)normally distributed longitudinal outcomes, selected association structures, speed, etc. * Hickey et al. (2016) provide a nice “recent” review of multivariate joint model software 10
Joint modelling software • An abundance of methodological developments in joint modelling • Not all methods have been translated into “user - friendly” software • Well established software for one longitudinal outcome • e.g. stjm (Stata); joineR, JM, JMbayes, frailtypack (R); JMFit (SAS) • Recent software developments for multiple longitudinal outcomes * • released packages: joineRML (R, available on CRAN) • development packages: survtd , rstanarm , JMbayes (R, available on GitHub); stjm • Each package has their strengths and limitations • e.g. (non-)normally distributed longitudinal outcomes, selected association structures, speed, etc. * Hickey et al. (2016) provide a nice “recent” review of multivariate joint model software 11
Joint modelling software • An abundance of methodological developments in joint modelling • Not all methods have been translated into “user - friendly” software • Well established software for one longitudinal outcome • e.g. stjm (Stata); joineR, JM, JMbayes, frailtypack (R); JMFit (SAS) • Recent software developments for multiple longitudinal outcomes * • released packages: joineRML (R, available on CRAN) • development packages: survtd , rstanarm , JMbayes (R, available on GitHub); stjm • Each package has their strengths and limitations • e.g. (non-)normally distributed longitudinal outcomes, selected association structures, speed, etc. * Hickey et al. (2016) provide a nice “recent” review of multivariate joint model software 12
RStanArm Stan RStan Bayesian joint models via Stan R package C++ library R for for interface Applied full Bayesian for Regression • Development version available on GitHub Stan inference Modelling • https://github.com/sambrilleman/rstanarm (soon to be migrated to https://github.com/stan-dev/rstanarm then CRAN) • Can specify multiple longitudinal outcomes • Allows for multilevel clustering in longitudinal submodels (e.g. time < patients < clinics) • Variety of families (and link functions) for the longitudinal outcomes • e.g. normal, binomial, Poisson, negative binomial, Gamma, inverse Gaussian • Variety of association structures • Variety of prior distributions • Regression coefficients: normal, student t, Cauchy, shrinkage priors (horseshoe, lasso) • Novel decomposition of covariance matrix for the random effects • Posterior predictions – including “dynamic predictions” of event outcome • Baseline hazard • Weibull, piecewise constant, B-splines regression https://github.com/sambrilleman/rstanarm 13
RStanArm Stan RStan Bayesian joint models via Stan R package C++ library R for for interface Applied full Bayesian for Regression • Development version available on GitHub Stan inference Modelling • https://github.com/sambrilleman/rstanarm (soon to be migrated to https://github.com/stan-dev/rstanarm then CRAN) • Can specify multiple longitudinal outcomes • Allows for multilevel clustering in longitudinal submodels (e.g. time < patients < clinics) • Variety of families (and link functions) for the longitudinal outcomes • e.g. normal, binomial, Poisson, negative binomial, Gamma, inverse Gaussian • Variety of association structures • Variety of prior distributions • Regression coefficients: normal, student t, Cauchy, shrinkage priors (horseshoe, lasso) • Novel decomposition of covariance matrix for the random effects • Posterior predictions – including “dynamic predictions” of event outcome • Baseline hazard • Weibull, piecewise constant, B-splines regression https://github.com/sambrilleman/rstanarm 14
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