Tutorial I: Motivation for Joint Modeling & Joint …

1 Tutorial I: Motivation for Joint Modeling & Joint Models forLongitudinal and Survival DataDimitris RizopoulosDepartment of Biostatistics, Erasmus University Medical Modeling and BeyondMeeting and Tutorials on Joint Modeling With Survival, longitudinal , and Missing DataApril 14, 2016, DiepenbeekContents1 Motivating longitudinal Studies.. Research Questions.. Recent Developments.. Joint Models..152 Linear Mixed-Effects Features of longitudinal Data.. The Linear Mixed Model..20 Tutorial I: Joint Models for longitudinal and Survival Data: April 14, Missing Data in longitudinal Studies.. Missing Data Mechanisms..323 Relative Risk Features of Survival Data.. Relative Risk Models.. Time Dependent Covariates.. Extended Cox Model..494 The Basic Joint Joint Modeling Framework..55 Tutorial I: Joint Models for longitudinal and Survival Data: April 14, Estimation.. A Comparison with the TD Cox.. Joint Models in R.. Connection with Missing Data..78 Tutorial I: Joint Models for longitudinal and Survival Data: April 14, 2016ivWhat are these Tutorials About Often in follow-up studies different types of outcomes are collected Explicitoutcomes multiple longitudinal responses ( , markers, blood values) time-to-event(s) of particular interest ( , death, relapse) Implicitoutcomes missing data ( , dropout, intermittent missingness) random visit timesTutorial I: Joint Models for longitudinal and Survival Data: April 14, 2016vWhat are these Tutorials About (cont'd) Methods for the separate analysis of such outcomes are well established in theliterature Survival data: Cox model, accelerated failure time models.

2 longitudinal data mixed effects models, GEE, marginal models, .. Tutorial I: Joint Models for longitudinal and Survival Data: April 14, 2016viWhat are these Tutorials About (cont'd)Purpose of these tutorials is to introduce the basics of popularJoint Modelings TechniquesTutorial I: Joint Models for longitudinal and Survival Data: April 14, 2016viiChapter 1 IntroductionTutorial I: Joint Models for longitudinal and Survival Data: April 14, Motivating longitudinal Studies AIDS:467 HIV infected patients who had failed or were intolerant to zidovudinetherapy (AZT)(Abrams et al., NEJM, 1994) The aim of this study was to compare the efficacy and safety of two alternativeantiretroviral drugs, didanosine (ddI) and zalcitabine (ddC) Outcomes of interest: time to death randomized treatment: 230 patients ddI and 237 ddC CD4 cell count measurements at baseline, 2, 6, 12 and 18 months prevOI: previous opportunistic infectionsTutorial I: Joint Models for longitudinal and Survival Data: April 14, Motivating longitudinal Studies (cont'd)Time (months)CD4 cell count0510152025051015ddC051015ddITutoria l I: Joint Models for longitudinal and Survival Data: April 14, Motivating longitudinal Studies (cont'd) Meier EstimateTime (months)Survival ProbabilityddCddITutorial I: Joint Models for longitudinal and Survival Data: April 14, Motivating longitudinal Studies (cont'd) Research Questions: How strong is the association between CD4 cell count and the risk for death?

3 Is CD4 cell count a good biomarker?*if treatment improves CD4 cell count, does it also improve survival? Tutorial I: Joint Models for longitudinal and Survival Data: April 14, Motivating longitudinal Studies (cont'd) PBC:Primary Biliary Cirrhosis: a chronic, fatal but rare liver disease characterized by in ammatory destruction of the small bile ducts within the liver Data collected by Mayo Clinic from 1974 to 1984(Murtaugh et al., Hepatology, 1994) Outcomes of interest: time to death and/or time to liver transplantation randomized treatment: 158 patients received D-penicillamine and 154 placebo longitudinal serum bilirubin levelsTutorial I: Joint Models for longitudinal and Survival Data: April 14, Motivating longitudinal Studies (cont'd)Time (years)log serum Bilirubin 101233805103951051068708290 1012393 1012313414817320005102162420510269 10123290 Tutorial I: Joint Models for longitudinal and Survival Data: April 14, Motivating longitudinal Studies (cont'd) Meier EstimateTime (years)Survival ProbabilityplaceboD penicilTutorial I: Joint Models for longitudinal and Survival Data: April 14, Motivating longitudinal Studies (cont'd) Research Questions: How strong is the association between bilirubin and the risk for death?

4 How the observed serum bilirubin levels could be utilized to provide predictions ofsurvival probabilities? Can bilirubin discriminate between patients of low and high risk? Tutorial I: Joint Models for longitudinal and Survival Data: April 14, Research Questions Depending on the questions of interest, different types of statistical analysis arerequired We will distinguish between two general types of analysis separate analysis per outcome Joint analysis of outcomes Focus on each outcome separately does treatment affect survival? are the average longitudinal evolutions different between males and females? .. Tutorial I: Joint Models for longitudinal and Survival Data: April 14, Research Questions (cont'd) Focus on multiple outcomes Complex hypothesis testing: does treatment improve the average longitudinalpro les in all markers? Complex effect estimation: how strong is the association between the longitudinalevolution of CD4 cell counts and the hazard rate for death?

5 Association structure among outcomes:*how the association between markers evolves over time (evolution of theassociation)*how marker-speci c evolutions are related to each other (association of theevolutions) Tutorial I: Joint Models for longitudinal and Survival Data: April 14, Research Questions (cont'd) Prediction: can we improve prediction for the time to death by considering allmarkers simultaneously? Handling implicit outcomes: focus on a single longitudinal outcome but withdropout or random visit timesTutorial I: Joint Models for longitudinal and Survival Data: April 14, Recent Developments Up to now emphasis has been restrictedorcoercedto separate analysis per outcome or given to naive types of Joint analysis ( , last observation carried forward) Main reasons lack of appropriate statistical methodology lack of efficient computational approaches & softwareTutorial I: Joint Models for longitudinal and Survival Data: April 14, Recent Developments (cont'd) However, recently there has been an explosion in the statistics and biostatisticsliterature of Joint Modeling approaches Many different approaches have been proposed that can handle different types of outcomes can be utilized in pragmatic computing time can be rather exible most importantly:can answer the questions of interestTutorial I: Joint Models for longitudinal and Survival Data.

6 April 14, Joint Models LetY1andY2two outcomes of interest measured on a number of subjects for whichjoint Modeling is of scienti c interest both can be measured longitudinally one longitudinal and one survival We have various possible approaches to construct a Joint densityp(y1; y2)offY1; Y2g Conditional models:p(y1; y2) =p(y1)p(y2jy1) Copulas:p(y1; y2) =cfF(y1);F(y2)gp(y1)p(y2)ButRandom Effects Modelshave (more or less) prevailedTutorial I: Joint Models for longitudinal and Survival Data: April 14, Joint Models (cont'd) Random Effects Models specifyp(y1; y2) = p(y1; y2jb)p(b)db= p(y1jb)p(y2jb)p(b)db Unobserved random effectsbexplain the association betweenY1andY2 Conditional Independence assumptionY1??Y2jbTutorial I: Joint Models for longitudinal and Survival Data: April 14, Joint Models (cont'd) Features: Y1andY2can be of different type*one continuous and one categorical*one continuous and one survival*.. Extensions to more than two outcomes straightforward Speci c association structure betweenY1andY2is assumed Computationally intensive (especially in high dimensions) Tutorial I: Joint Models for longitudinal and Survival Data: April 14, 201617 Chapter 2 Linear Mixed-Effects ModelsTutorial I: Joint Models for longitudinal and Survival Data: April 14, Features of longitudinal Data Repeated evaluations of the same outcome in each subject in time CD4 cell count in HIV-infected patients serum bilirubin in PBC patientsMeasurements on the same subject are expected tobe (positively) correlated This implies that standard statistical tools, such as thet-test and simple linearregression that assume independent observations, are not optimal for longitudinaldata I: Joint Models for longitudinal and Survival Data: April 14, The Linear Mixed Model The direct approach to model correlated data)multivariate regressionyi=Xi +"i; "i N(0; Vi).

7 Where yithe vector of responses for theith subject Xidesign matrix describing structural component Vicovariance matrix describing the correlation structure There are several options for modelingVi, , compound symmetry, autoregressiveprocess, exponential spatial correlation, Gaussian spatial correlation, .. Tutorial I: Joint Models for longitudinal and Survival Data: April 14, The Linear Mixed Model (cont'd) Alternative intuitive approach:Each subject in the population has her ownsubject-speci c mean response pro le over timeTutorial I: Joint Models for longitudinal and Survival Data: April 14, The Linear Mixed Model (cont'd)012345020406080 TimeLongitudinal OutcomeSubject 1 Subject 2 Tutorial I: Joint Models for longitudinal and Survival Data: April 14, The Linear Mixed Model (cont'd) The evolution of each subject in time can be described by a linear modelyij=~ i0+~ i1tij+"ij; "ij N(0; 2);where yijthejth response of theith subject ~ i0is the intercept and~ i1the slope for subjecti Assumption:Subjects are randomly sampled from a population)subject-speci cregression coefficients are also sampled from a population of regression coefficients~ i N(.

8 D) Tutorial I: Joint Models for longitudinal and Survival Data: April 14, The Linear Mixed Model (cont'd) We can reformulate the model asyij= ( 0+bi0) + ( 1+bi1)tij+"ij;where s are known as the xed effects bis are known as therandom effects In accordance for the random effects we assumebi=24bi0bi135 N(0; D) Tutorial I: Joint Models for longitudinal and Survival Data: April 14, The Linear Mixed Model (cont'd) Put in a general form8>>> <>>>:yi=Xi +Zibi+"i;bi N(0; D); "i N(0; 2 Ini);with Xdesign matrix for the xed effects Zdesign matrix for the random effectsbi bi??"iTutorial I: Joint Models for longitudinal and Survival Data: April 14, The Linear Mixed Model (cont'd) Interpretation: jdenotes the change in the averageyiwhenxjis increased by one unit biare interpreted in terms of how a subset of the regression parameters for theithsubject deviates from those in the population Advantageous feature: population+subject-speci c predictions describes mean response changes in the population +bidescribes individual response trajectoriesTutorial I: Joint Models for longitudinal and Survival Data: April 14, The Linear Mixed Model (cont'd) Example:We t a linear mixed model for the AIDS dataset assuming different average longitudinal evolutions per treatment group ( xed part) random intercepts & random slopes (random part)8>>> <>>>:yij= 0+ 1tij+ 2fddIi tijg+bi0+bi1tij+"ij;bi N(0; D); "ij N(0.

9 2) Note:We did not include a main effect for treatment due to randomizationTutorial I: Joint Models for longitudinal and Survival Data: April 14, The Linear Mixed Model (cont'd)Value 07:189 0:222 32:359<0:001 1 0:163 0:021 7:855<0:001 20:0280:0300:9520:342 No evidence of differences in the average longitudinal evolutions between the twotreatmentsTutorial I: Joint Models for longitudinal and Survival Data: April 14, Missing Data in longitudinal Studies A major challenge for the analysis of longitudinal data is the problem of missing data studies are designed to collect data on every subject at a set of prespeci edfollow-up times often subjects miss some of their planned measurements for a variety of reasonsTutorial I: Joint Models for longitudinal and Survival Data: April 14, Missing Data in longitudinal Studies (cont'd) Implications of missingness: we collect less data than originally planned)loss of efficiency not all subjects have the same number of measurements)unbalanced datasets missingness may depend on outcome)potential bias For the handling of missing data, we introduce the missing data indicatorrij=8<:1ifyijis observed0otherwiseTutorial I: Joint Models for longitudinal and Survival Data: April 14, Missing Data in longitudinal Studies (cont'd) We obtain a partition of the complete response vectoryi observed datayoi, containing thoseyijfor whichrij= 1 missing dataymi, containing thoseyijfor whichrij= 0 For the remaining we will focus on dropout)notation can be simpli ed Discrete dropout time:rdi= 1 +ni j=1rij(ordinal variable) Continuous time:T idenotes the time to dropoutTutorial I: Joint Models for longitudinal and Survival Data.

10 April 14, Missing Data Mechanisms To describe the probabilistic relation between the measurement and missingnessprocesses Rubin (1976, Biometrika) has introduced three mechanisms Missing Completely At Random (MCAR): The probability that responses are missingis unrelated to bothyoiandymip(rijyoi;ymi) =p(ri) Examples subjects go out of the study after providing a pre-determined number ofmeasurements laboratory measurements are lost due to equipment malfunctionTutorial I: Joint Models for longitudinal and Survival Data: April 14, Missing Data Mechanisms (cont'd) Missing At Random (MAR): The probability that responses are missing is related toyoi, but is unrelated toymip(rijyoi;ymi) =p(rijyoi) Examples study protocol requires patients whose response value exceeds a threshold to beremoved from the study physicians give rescue medication to patients who do not respond to treatmentTutorial I: Joint Models for longitudinal and Survival Data: April 14, Missing Data Mechanisms (cont'd) Missing Not At Random (MNAR): The probability that responses are missing isrelated toymi, and possibly also toyoip(rijymi)orp(rijyoi;ymi) Examples in studies on drug addicts, people who return to drugs are less likely than othersto report their status in longitudinal studies for quality-of-life, patients may fail to complete thequestionnaire at occasions when their quality-of-life is compromisedTutorial I: Joint Models for longitudinal and Survival Data: April 14, Missing Data Mechanisms (cont'd) Features of MNAR The observed data cannot be considered a random sample from the targetpopulation Only procedures that explicitly model the Joint distributionfyoi.