Joint modelling of longitudinal data and survival data

Joint modelling of longitudinal and survival data in Stata Dr Michael J Crowther1,2, Prof Paul C Lambert1,2 & Prof Keith R Abrams1 Department of Health Sciences, University of Leicester, UK Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden Timetable – Day 11.00 11.30 Welcome and introduction 11.30 12.30 Introduction to survival analysis and longitudinal analysis 12.30 13.30 Lunch 13.30 14.30 Practical 14.30 15.30 Survival analysis with time-varying covariates and two-stage models 15.30 16.00 Tea/Coffee 16.00 17.00 Practical 19.00 Course Dinner Timetable – Day 9.00 9.15 Lab review of day 9.15 10.15 Joint models of longitudinal and survival data 10.15 11.00 Practical 11.00 11.30 Tea/Coffee 11.30 12.30 Practical continued 12.30 13.30 Lunch 13.30 14.30 Alternative association structures and prediction 14.30 15.30 Practical 15.30 16.00 Wrap-up session - further topics 16.00 Tea/Coffee and farewell Participant Introductions What is JM? & Terminology • Joint/Simultaneous Modelling • broad inter-linked processes: – Biomarker process (longitudinal) [mixed] model – Time to clinical outcome process (survival/time-toevent) model • Focus may be … – Estimating biomarker profile/trajectory allowing for informative dropout, e.g death – Estimating relationship between underlying [adjusting for measurement error] biomarker profile/trajectory and clinical outcome What does JM add? – What does JM add? – -.5 Biomarker 1.5 Panel 2 logb Follow-up time (years) Longitudinal prediction (including BLUPS) What does JM add? – Panel 1.5 1.0 0.4 Biomarker 0.6 -.5 0.2 0.0 10 12 Follow-up time Longitudinal response Predicted conditional survival 95% pointwise confidence interval Why the need for JM? – • Technology is very rapidly evolving … more biomarkers are being used/collected … • e-Health agenda means that more routinely collected biomarker data is being linked to outcome data • For example, CPRD now links primary care records with Hospital Episode Statistics (HES) data, cancer registry and mortality data Survival probability 0.8 Why the need for JM? – • Clinical (and health policy) decision making is not now just about who will (or will not) benefit from a particular treatment, e.g Cetuximab, bevacizumab and panitumumab for the treatment of metastatic colorectal cancer after first-line chemotherapy [NICE TA 242] – stratified medicine • Acknowledge uncertainty … personalised medicine – try patient on a treatment and see if they “respond” or not, BUT … Why the need for JM? – • requires quick, reliable and valid (i.e linked to clinical outcomes) surrogates that can be monitored repeatedly and routinely, e.g biomarkers – so patients can stop asap if not responding • For example, PSA-defined response in previously treated metastatic prostate cancer [NICE TA 259] Personalised Medicine/Health and JM in future • Will require (realistically complex) real-time dynamic risk prediction models • e.g digital health – bluetooth enabled Wearable Technologies (WT) – Obesity reduction/physical activity – CVD • e.g digital medicine – bluetooth enabled Digested Technology (IT) BUT … back to today (and tomorrow)! • • • • • • Survival models Longitudinal models Survival models with time-varying covariates 2-stage approaches to Joint Models Fully Joint Models How to link survival and longitudinal submodels using association structures • Prediction Introduction Survival Data Longitudinal Data Lecture 2: Introduction to survival analysis and longitudinal analysis 22nd April 2015 Paul C Lambert Department of Health Sciences, University of Leicester, UK, and Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden paul.lambert@le.ac.uk University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 / 49 Introduction Survival Data Longitudinal Data Outline Introduction Survival Data Longitudinal Data University of Leicester Joint Modelling in Stata Introduction 22nd-23rd April 2015 Survival Data / 49 Longitudinal Data Outline Introduction Survival Data Longitudinal Data University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 / 49 Introduction Survival Data Longitudinal Data Introduction This course is essentially about simultaneously fitting a survival model and a longitudinal model First, we will review key features of both of these types of model separately In particular we will discuss their use in Stata Michael will then discuss reasons why we may want to consider both outcomes simultaneously University of Leicester Joint Modelling in Stata Introduction 22nd-23rd April 2015 Survival Data / 49 Longitudinal Data Clinical example 312 patients with primary biliary cirrhosis Cirrhosis is a slowly progressing disease in which healthy liver tissue is replaced with scar tissue, eventually preventing the liver from functioning properly 1945 repeated measures of serum bilirubin, a measure of liver function Treated with D-penicillamine or a placebo Outcome of all-cause death, where 140 (44.8%) patients died Research question: How does serum bilirubin change over time, and are those changes associated with survival? In this session we will consider the survival model and longitudinal model separately University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 / 49 Introduction Further association structures Predicting survival conditional on a biomarker profile Discussion References Alternative association structures Both the current value and slope associations can be thought of as time-dependent association structures The risk of event at time t is linked to a value, or transformation, of the biomarker at time t Alternatively, we could pick out specific aspects of the profile, say the subject-specific intercept University of Leicester Introduction Joint Modelling in Stata Further association structures 22nd-23rd April 2015 18 / 35 Predicting survival conditional on a biomarker profile Discussion References Alternative association structures Random effects parametrisation Once again, we our longitudinal outcome yi (t) = ❳ T (t)β + ❩ T (t)❜ + ei (t) ✐ ✐ ✐ We can define time-independent association structures h(t|Mi (t), ✈ ) = h0 (t) exp{φT ✈ + αT (β + ❜ )} ✐ ✐ ✐ Which includes both the population level mean of the random effect, plus the subject specific deviation University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 19 / 35 Introduction Further association structures Predicting survival conditional on a biomarker profile Discussion References Let’s simplify, say we assumed a linear growth curve yi (t) = (β0 + b0i ) + (β1 + b1i )t + ei (t) and we wanted to link our subject-specific baseline values of the outcome h(t|Mi (t), ✈ ) = h0 (t) exp{φT ✈ + α1 (β0 + b0i )} ✐ ✐ (1) where exp(α1 ) is the hazard ratio for a one unit increase in the baseline value of the longitudinal outcome i.e the intercept University of Leicester Introduction Joint Modelling in Stata Further association structures 22nd-23rd April 2015 Predicting survival conditional on a biomarker profile 20 / 35 Discussion References The previous model allows us to investigate the association between the baseline value of the biomarker and our survival endpoint But in the model, we are still using all of the repeated measures to improve our estimates of the baseline value (Crowther et al., 2013) On a technical note, time-independent structures are generally more computationally efficient as we can write down our survival function directly University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 21 / 35 Introduction Further association structures Predicting survival conditional on a biomarker profile Discussion References Specifying the association using stjm stjm longdepvar [varlist ], panel(varname ) [gh(#) ] The intercept association can be specified using intassociation Random coefficients can be linked using association(#), where # must be specified in rfp() University of Leicester Introduction Joint Modelling in Stata Further association structures 22nd-23rd April 2015 22 / 35 Predicting survival conditional on a biomarker profile stjm logb, panel(id) survmodel(weibull) rfp(1) survcov(trt) /// > nocurrent intassociation -> gen double _time_1 = X^(1) (where X = _t0) Obtaining initial values: Fitting full model: Joint model estimates Number of obs = Panel variable: id Number of panels = Number of failures = Log-likelihood = -1955.7015 Coef Longitudinal _time_1 _cons Survival assoc:int _cons ln_lambda trt _cons ln_gamma _cons Std Err z Discussion 1945 312 140 P>|z| [95% Conf Interval] 1782451 5007463 012694 0581723 14.04 8.61 0.000 0.000 1533653 3867306 2031248 614762 1.333708 1138519 11.71 0.000 1.110562 1.556853 1564195 -4.551237 1763784 3061135 0.89 -14.87 0.375 0.000 -.1892759 -5.151209 5021149 -3.951266 4325337 0722281 5.99 0.000 2909692 5740982 Random effects Parameters References Estimate Std Err [95% Conf Interval] sd(_time_1) sd(_cons) corr(_time_1,_cons) 1653823 9999101 4816444 0113989 0426311 0743054 1444842 9197504 323452 1893031 1.087056 6136461 sd(Residual) 3516559 0068733 3384392 3653886 id: Unstructured University of Leicester Joint Modelling Stataeffects model 22nd-23rd April 2015 Longitudinal submodel: Linear inmixed Survival submodel: Weibull proportional hazards model Integration method: Adaptive Gauss-Hermite quadrature using nodes Cumulative hazard: Gauss-Kronrod quadrature using 15 nodes 23 / 35 Introduction Further association structures Predicting survival conditional on a biomarker profile Discussion References Choosing association structures You should have a clear clinical idea of which association structure(s) to use Model selection tools can be useful to guide choice, such as AIC/BIC Current methodological developments include partitioning AIC into longitudinal and survival components University of Leicester Introduction Joint Modelling in Stata Further association structures 22nd-23rd April 2015 Predicting survival conditional on a biomarker profile 24 / 35 Discussion References Outline Introduction Further association structures Predicting survival conditional on a biomarker profile Discussion University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 25 / 35 Introduction Further association structures Predicting survival conditional on a biomarker profile Discussion References Dynamic prediction from a joint model In this section we will focus on subject-specific, individualised predictions We are interested in predicting survival, conditional on a set of biomarker measurements Further info in (Rizopoulos, 2011) University of Leicester Introduction Joint Modelling in Stata Further association structures 22nd-23rd April 2015 Predicting survival conditional on a biomarker profile 26 / 35 Discussion References Let’s look at two patients in particular, from the cirrhosis dataset, patient and patient 102 Patient has observations with their final measurement at 8.83 years Patient 102 has 12 observations with their final measurement at 9.18 years What is the probability of dying conditional on each patient’s bilirubin profiles? University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 27 / 35 Introduction Further association structures Predicting survival conditional on a biomarker profile Patient 102 1.5 log(serum bilirubin) -.5 log(serum bilirubin) -.5 University of Leicester Introduction References 1.5 Patient Discussion Follow-up time (years) Joint Modelling in Stata Further association structures 10 Follow-up time (years) 10 22nd-23rd April 2015 Predicting survival conditional on a biomarker profile 28 / 35 Discussion References Some technical details We have a set of biomarker measurements, Yi (t) = {yi (s), ≤ s < t} And we are interested in, P{Ti∗ ≥ u|Ti∗ > t, Yi (t), Dn } where, u > t, and Dn is our sample which the joint model was fitted University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 29 / 35 Introduction Further association structures Predicting survival conditional on a biomarker profile Discussion References Given our fitted joint model, we have our estimated ˆ and variance-covariance matrix, Vˆ parameters, θ, To tailor our predictions to individual patients, we must estimate the subject-specific deviations of the random effects, bˆi This gives us two sources of variability which must be accounted for in predicting conditional survival University of Leicester Introduction Joint Modelling in Stata Further association structures 22nd-23rd April 2015 Predicting survival conditional on a biomarker profile 30 / 35 Discussion References We therefore proceed with a simulation approach: ˆ Vˆ ) Draw θ(k) ∼ N(θ, (k) Draw bi ∼ (bi |Ti∗ > t, Yi (t), θ(k) ) Calculate Pi (u|t) = Si (u|Mi (u), θ(k) )/Si (t|Mi (t), θ(k) ) Repeat k = 1, , K times University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 31 / 35 Introduction Further association structures Predicting survival conditional on a biomarker profile Discussion References Conditional survival predictions Panel Panel 102 1.0 1.5 1.5 1.0 0.8 Biomarker 0.4 0 0.6 0.0 10 15 Follow-up time 0.2 -.5 0.2 -.5 Survival probability 0.4 Survival probability 0.6 Biomarker 1 0.8 0.0 10 15 Follow-up time Longitudinal response Predicted conditional survival 95% pointwise confidence interval University of Leicester Introduction Joint Modelling in Stata Further association structures 22nd-23rd April 2015 Predicting survival conditional on a biomarker profile 32 / 35 Discussion References Outline Introduction Further association structures Predicting survival conditional on a biomarker profile Discussion University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 33 / 35 Introduction Further association structures Predicting survival conditional on a biomarker profile Discussion References Discussion Joint modelling allows you to link survival with practically any component of a longitudinal trajectory It is important to consider clinically plausible association structures How to choose association structures is still debated Dynamic risk prediction is a particularly appealing aspect and tool of the joint model framework, which could play an important role in clinical practice in the coming years, in a variety of disease areas University of Leicester Introduction Joint Modelling in Stata Further association structures 22nd-23rd April 2015 Predicting survival conditional on a biomarker profile 34 / 35 Discussion References References I M J Crowther, P C Lambert, and K R Abrams Adjusting for measurement error in baseline prognostic biomarkers included in a time-to-event analysis: A joint modelling approach BMC Med Res Methodol, 13(146), 2013 Dimitris Rizopoulos Dynamic predictions and prospective accuracy in joint models for longitudinal and time-to-event data Biometrics, 67(3):819–829, 2011 doi: 10.1111/j.1541-0420.2010.01546.x URL http://dx.doi.org/10.1111/j.1541-0420.2010.01546.x Marcel Wolbers, Abdel Babiker, Caroline Sabin, Jim Young, Maria Dorrucci, Genevi` eve Chˆ ene, Cristina Mussini, Kholoud Porter, Heiner C Bucher, and CASCADE Pretreatment CD4 cell slope and progression to AIDS or death in HIV-infected patients initiating antiretroviral therapy–the CASCADE collaboration: a collaboration of 23 cohort studies PLoS Med, 7(2):e1000239, 2010 doi: 10.1371/journal.pmed.1000239 URL http://dx.doi.org/10.1371/journal.pmed.1000239 University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 35 / 35 Introduction Summary References Lecture 6: Further topics 23rd April 2015 Dr Michael J Crowther1,2,∗ Department of Health Sciences, University of Leicester, UK, and Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden ∗ michael.crowther@le.ac.uk University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 / 10 Introduction Summary References Introduction We have covered a lot in the past two days There is also a lot we didn’t include! In this lecture, I’ll briefly cover some extensions to the joint model framework, which may be of interest This is also an opportunity for you to ask any final questions University of Leicester Joint Modelling in Stata Introduction 22nd-23rd April 2015 Summary / 10 References Choice of survival submodel In the lectures and practicals we predominantly assumed a Weibull survival model This may not be flexible enough in many practical situations In stjm there are a variety of options, but we tend to favour using splines (Crowther et al., 2012) University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 / 10 Introduction Summary 1.00 Marginal survival function − degree of freedom Marginal survival function − degrees of freedom 1.00 0.75 Survival probability 0.75 Survival probability References 0.50 0.25 0.50 0.25 0.00 0.00 Follow−up time Kaplan−Meier estimate Marginal survival function University of Leicester Joint Modelling in Stata Introduction Follow−up time 95% Confidence Interval 22nd-23rd April 2015 Summary / 10 References Extensions to the longitudinal submodel Most previous work in joint modelling has assumed a continuous longitudinal outcome The extension to binary, count, ordinal repeated measures all follow naturally within a generalised linear mixed effects model (Viviani et al., 2013) RCTs in critical care often measure daily binary indicators such as ventilation or infection free days, which can be subject to informative dropout due to death Recurrent non-fatal events and a terminal event can be modelled as a joint longitudinal Poisson process and survival, for example in heart failure This can then be extended to the multivariate generalised linear mixed effects model University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 / 10 Introduction Summary References Extensions to the survival submodel Competing risks/multi-state models (Dantan et al., 2011) Delayed entry Estimation is more complex due to selection effects, requires a second set of numerical integration This allows age to be used as the timescale It also allows us to utilise previous measurements University of Leicester Joint Modelling in Stata Introduction 22nd-23rd April 2015 / 10 Summary References -5 -3 Longitudinal response -1 Patient has a heart attacked at age 60 50 55 outcome University of Leicester Joint Modelling in Stata 60 Age (years) 65 70 lowess outcome age 22nd-23rd April 2015 / 10 Introduction Summary References -5 -3 Longitudinal response -1 Patient has a heart attacked at age 60 50 55 60 Age (years) outcome University of Leicester 70 lowess outcome age Joint Modelling in Stata Introduction 65 22nd-23rd April 2015 Summary / 10 References Summary Joint modelling is now a well established statistical technique for investigating the inter-relationships between a longitudinal outcome and a survival outcome It can provide us with more efficient estimates of covariate effects User friendly software is now available which allows the methods to be used in practice University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 / 10 Introduction Summary References Finally, Thank you for attending! Feedback forms are included in your course material, please complete and return to us (either a hard copy or via email) University of Leicester Joint Modelling in Stata Introduction 22nd-23rd April 2015 Summary / 10 References References I Michael J Crowther, Keith R Abrams, and Paul C Lambert Flexible parametric joint modelling of longitudinal and survival data Stat Med, 31(30):4456–4471, 2012 doi: 10.1002/sim.5644 URL http://dx.doi.org/10.1002/sim.5644 E Dantan, P Joly, J-F Dartigues, and H Jacqmin-Gadda Joint model with latent state for longitudinal and multistate data Biostatistics, 12(4):723–736, Oct 2011 doi: 10.1093/biostatistics/kxr003 URL http://dx.doi.org/10.1093/biostatistics/kxr003 Sara Viviani, Marco Alfo, and Dimitris Rizopoulos Generalized linear mixed joint model for longitudinal and survival outcomes Statistics and Computing, pages 1–11, 2013 ISSN 0960-3174 doi: 10.1007/s11222-013-9378-4 URL http://dx.doi.org/10.1007/s11222-013-9378-4 University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 10 / 10 ... Introduction Survival Data Longitudinal Data Outline Introduction Survival Data Longitudinal Data University of Leicester Joint Modelling in Stata Introduction 22nd-23rd April 2015 Survival Data / 49 Longitudinal. .. Introduction Survival Data Longitudinal Data Outline Introduction Survival Data Longitudinal Data University of Leicester Introduction Joint Modelling in Stata 22nd-23rd April 2015 Survival Data / 49 Longitudinal. .. University of Leicester Joint Modelling in Stata 22nd-23rd April 2015 29 / 49 Introduction Survival Data Longitudinal Data Outline Introduction Survival Data Longitudinal Data University of Leicester Joint