1. Meta-analysis of joint longitudinal and event-time outcomes
Maria Sudell1*, Catrin Tudur-Smith1 and Ruwanthi Kolamunnage-Dona1
1Department of Biostatistics, University of Liverpool
Joint models for longitudinal and time-to-event data are preferred in the single study case to separate longitudinal or time-to-event analyses if the longitudinal and time-to-event outcomes are correlated, if time varying covariates to be included in time-to-event models are measured with error, or if study dropout issues exist.
The majority of the literature focuses on single study case methods, rather than multi-study cases such as meta-analyses
This investigation aimed to assess use of joint models for longitudinal and time-to-event data when compared to separate longitudinal or time-to-event analyses in two stage Individual Participant Data (IPD) meta-analyses (MA) through a simulation study.
Background
Example of results for 5 scenarios (low event rate with homogenous treatment effect all association values).*
*Full results available from first author
Overall observations from scenarios conducted
joineR software was long running due to bootstrapping for standard errors which could be an issue for a meta-analysis involving many studies
JM software was faster running but showed issues with coverage for piecewise constant methods
Joint methods preferred over separate methods for estimation of time-to-event treatment effect, with larger underestimation from separate effects observed with increasing association between longitudinal and time-to-event outcomes
Association estimates overall were good as were longitudinal treatment effect estimates (although coverage from JM unspecified baseline hazard decreasing with increasing association)
Results
Data was simulated in R using jointsim function (linked to joineR package) under proportional association assumption 1,2
Each scenario consisted of data from 5 simulated studies. Each study contained 500 individuals simulations generated for each scenario.
Longitudinal data were simulated under linear mixed effects model and event times were generated under Gompertz distribution3
A maximum of 5 longitudinal observations recorded at times 0, 1, 2, 3, 4 up to individual’s event time, and a treatment assignment term is included in both models
Data Simulation
20 scenarios considered (all possible combinations of homogenous/heterogeneous treatment effect, low or high event rate, and association level)
Homogenous data longitudinal treatment effect 2, time-to-event treatment effect 3
Heterogeneous treatment effect longitudinal treatment effect 𝑁 2,0.5 and time-to-event treatment effect 𝑁 3,0.5
Homogenous or Heterogeneous treatment Effect
Varied between simulations
Controlled through censoring process
Greater variation in event rates between simulations for high event rate data than low
Low (~25%) or High (~75%) event rate
Negative association similar behaviour in opposite direction
Association (0, 0.25, 0.5, 0.75, 1)
Choice of methods motivated by methods available in R software4
All longitudinal models (separate or joint) were linear containing fixed intercept, time and treatment, and random intercept and time terms. Random intercept not included in JM models due to high error rate
Time-to-event models were proportional hazards with varying baseline hazard modelling techniques and contained a fixed effect for the treatment assignment.
Models fitted to data from each study within each simulation separately
Study level results meta-analysed within simulations using both fixed and random9 methods
Separate MA conducted for longitudinal treatment effect, time-to-event treatment effect and association parameter. The overall (direct + indirect) time-to-event treatment effect was used from JM models
Within each scenario, mean estimates of parameters, their standard errors and coverages were recorded.
Two level methods
Separate Models
(R Package)
Longitudinal linear mixed effects model (nlme5)
Time-to-event Proportional Hazard models (survival6)
Joint Models
Unspecified baseline hazard model sharing only random effects between sub-models (joineR7)
Unspecified baseline hazard model sharing current longitudinal trajectory between sub-models (JM8)
Piecewise baseline hazard model sharing current longitudinal trajectory between sub-models (JM8)
Spline baseline hazard model sharing current longitudinal trajectory between sub-models (JM8)
For JM spline baseline hazard and piecewise baseline hazard, knots or cut points were not specified in function call – in a real data application this would be done to improve fit
JM unspecified baseline hazard could underestimate standard errors due to non-bootstrapping
Re-simulation of some scenarios in progress due to possible favouring of joineR methods over JM as data simulated under joineR model
Where association not equal to zero, separate methods underestimated time-to-event treatment effect compared to joint methods
General agreement between separate methods and joint methods for longitudinal treatment effect regardless of association
Joint methods agreed well.
General recommendation: Given differing model structures, use of JM with spline baseline hazard or joineR model dependent on aims of the investigation
Discussion
[1] Henderson, R., P. Diggle, and A. Dobson, Joint modelling of longitudinal measurements and event time data. Biostatistics (Oxford, England), (4): p , [2] Wulfsohn, M.S. and A.A. Tsiatis, A Joint Model for Survival and Longitudinal Data Measured with Error. 1997, International Biometric Society. p [3] Bender, R., T. Augustin, and M. Blettner, Generating survival times to simulate Cox proportional hazards models. STATISTICS IN MEDICINE, : p , [4] R Core Team, R: A Language and Environment for Statistical Computing. 2015, R Foundation for Statistical Computing: Vienna, Austria, [5] Pinheiro, J., et al., {nlme}: Linear and Nonlinear Mixed Effects Models. 2016, [6] Therneau, T.M., A Package for Survival Analysis in S. 2015, [7] Philipson, P., et al., joineR: Joint modelling of repeated measurements and time-to-event data. 2012, R package version , [8] Rizopoulos, D., JM: An R package for the joint modelling of longitudinal and time-to-event data. Journal of Statistical Software, (9): p , [9] DerSimonian, R. and N. Laird, Meta-analysis in clinical trials. Controlled Clinical Trials, (3): p
References