3 Joint models for longitudinal and survival data

It is increasingly common for observational studies and trials to follow participants over time, recording abundant data on clinical features throughout the duration of the study. Moreover, routinely collected healthcare consumption data and population registries are being used more and more for research purposes, after being linked with other data sources. As a consequence, applied researchers often encounter longitudinally recorded covariates to account for when studying the clinical outcome of interest (e.g. time to an event, that is what I will focus on). Researchers then face two options: (1) select only one of the multiple values per individual and analyse as such, ignoring much of the available data, or (2) take into account the potential dependency and association between the repeatedly measured covariates and the outcome interest. The latter is usually the most sensible choice, as the longitudinal data can contain important predictors or surrogates of the time to event outcome. A powerful tool to achieve so is given by joint models for longitudinal and time to event data, in which the longitudinal and survival processes are modelled jointly into a single model allowing to infer their association. The development of such models was motivated by HIV/AIDS clinical trials, in which immune response was recorded over the duration of the trial and the association with survival was of interest. Seminal works on the topic are the papers by Wulfsohn and Tsiatis (1997), Tsiatis and Davidian (2004), Henderson, Diggle, and Dobson (2000), Pawitan and Self (1993); a more recent tractation of the topic is in Ibrahim, Chu, and Chen (2010), Rizopoulos (2012), Gould et al. (2015).

Previous attempts to tackle this problem consisted in (1) fitting a time-dependent Cox model (Cox 1972) by splitting individual rows every time a new observation from the longitudinal covariate becomes available, and (2) by using two-stages methods in which the longitudinal and survival data were modelled separately (Tsiatis, Degruttola, and Wulfsohn 1995). Nevertheless, it has been showed that joint modelling increases efficiency and reduces bias (Hogan and Laird 1998), while improving predictions at the same time (Rizopoulos et al. 2014).

Applications of joint models for longitudinal data to answer complex study questions using complex clinical data are increasingly common in medical literature, in a variety of settings: among others, cardiology (Sweeting and Thompson 2011), nephrology (Asar et al. 2015), and intensive care medicine (Andrinopoulou et al. 2017).

In this Chapter, I will focus on the basic joint model for longitudinal and survival data, with a single longitudinal process. I will present its formulation in Section 3.1, and the estimation process in Section 3.2. However, several extensions of the basic joint model presented in this Chapter have been proposed during the years, as the topic has received considerable attention. A review of the state of the art in joint models with a single longitudinal process is given by Gould et al. (2015). Furthermore, the joint model has been extended to allow incorporating multiple longitudinal processes at once, measured intermittently and not necessarily at the same time or with the same association structure with the survival component; a recent review on the topic is given by Hickey et al. (2016).

References

Wulfsohn, Michael S, and Anastasios A Tsiatis. 1997. “A Joint Model for Survival and Longitudinal Data Measured with Error.” Biometrics 53 (1): 330–39. doi:10.2307/2533118.

Tsiatis, Anastasios A, and Marie Davidian. 2004. “Joint Modeling of Longitudinal and Time-to-Event Data: An Overview.” Statistica Sinica 14: 809–34. http://www.jstor.org/stable/24307417.

Henderson, Robin, Peter Diggle, and Angela Dobson. 2000. “Joint Modelling of Longitudinal Measurements and Event Time Data.” Biostatistics 1 (4): 465–80. doi:10.1093/biostatistics/1.4.465.

Pawitan, Yudi, and Steve Self. 1993. “Modeling Disease Marker Processes in Aids.” Journal of the American Statistical Association 88 (423): 719–26. doi:10.2307/2290756.

Ibrahim, Joseph G, Haitao Chu, and Liddy M Chen. 2010. “Basic Concepts and Methods for Joint Models of Longitudinal and Survival Data.” Journal of Clinical Oncology 28 (16): 2796–2801. doi:10.1200/jco.2009.25.0654.

Rizopoulos, Dimitris. 2012. Joint Models for Longitudinal and Time-to-Event Data: With Applications in R. Biostatistics. Chapman & Hall / CRC.

Gould, A Lawrence, Mark E Boye, Michael J Crowther, Joseph G Ibrahim, George Quartey, Sandrine Micallef, and Frederic Y Bois. 2015. “Joint Modeling of Survival and Longitudinal Non-Survival Data: Current Methods and Issues. Report of the DIA Bayesian Joint Modeling Working Group.” Statistics in Medicine 34 (14): 2181–95. doi:10.1002/sim.6141.

Cox, David R. 1972. “Regression Models and Life-Tables.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 34 (2): 187–220. http://www.jstor.org/stable/2985181.

Tsiatis, Anastasios A, Victor Degruttola, and Michael S Wulfsohn. 1995. “Modeling the Relationship of Survival to Longitudinal Data Measured with Error. Applications to Survival and Cd4 Counts in Patients with AIDS.” Journal of the American Statistical Association 90 (429): 27–37. doi:10.1080/01621459.1995.10476485.

Hogan, Joseph W, and Nan M Laird. 1998. “Increasing Efficiency from Censored Survival Data by Using Random Effects to Model Longitudinal Covariates.” Statistical Methods in Medical Research 7 (1): 28–48. doi:10.1177/096228029800700104.

Rizopoulos, Dimitris, Laura A Hatfield, Bradley P Carlin, and Johanna JM Takkenberg. 2014. “Combining Dynamic Predictions from Joint Models for Longitudinal and Time-to-Event Data Using Bayesian Model Averaging.” Journal of the American Statistical Association 109 (508): 1385–97. doi:10.1080/01621459.2014.931236.

Sweeting, Michael J, and Simon G Thompson. 2011. “Joint Modelling of Longitudinal and Time-to-Event Data with Application to Predicting Abdominal Aortic Aneurysm Growth and Rupture.” Biometrical Journal 53 (5): 750–63. doi:10.1002/bimj.201100052.

Asar, Özgür, James Ritchies, Philip A Kalra, and Peter J Diggle. 2015. “Joint Modelling of Repeated Measurement and Time-to-Event Data: An Introductory Tutorial.” International Journal of Epidemiology 44 (1): 334–44. doi:10.1093/ije/dyu262.

Andrinopoulou, Montserrat Rué And Eleni-Rosalina, Danilo Alvares, Carmer Armero, Anabel Forte, and Lluis Blanch. 2017. “Bayesian Joint Modeling of Bivariate Longitudinal and Competing Risks Data: An Application to Study Patient-Ventilator Asynchronies in Critical Care Patients.” Biometrical Journal. doi:10.1002/bimj.201600221.

Hickey, Graeme L, Pete Philipson, Andrea Jorgensen, and Ruwanthi Kolamunnage-Dona. 2016. “Joint Modelling of Time-to-Event and Multivariate Longitudinal Outcomes: Recent Developments and Issues.” BMC Medical Research Methodology 16 (1). doi:10.1186/s12874-016-0212-5.