A predictive joint shared parameter model is proposed for discrete time-to-event and longitudinal data. A discrete survival model with frailty and a generalized linear mixed model for the longitudinal data are joined to predict the probability of events. This joint model focuses on predicting discrete time-to-event outcome, taking advantage of repeated measurements. We show that the probability of an event in a time window can be more precisely predicted by incorporating the longitudinal measurements. The model was investigated by comparison with a two-step model and a discrete-time survival model. Results from both a study on the occurrence of tuberculosis and simulated data show that the joint model is superior to the other models in discrimination ability, especially as the latent variables related to both survival times and the longitudinal measurements depart from 0.
AllisonPD. Discrete-time methods for the analysis of event histories. Sociol Methodol1982; 13: 61–98.
2.
LindseyJCRyanLM. Methods for interval-censored data. Stat Med1998; 17: 219–238.
3.
SingerJDWillettJB. It's about time: using discrete-time survival analysis to study duration and the timing of events. J Educat Stat1993; 18: 155–195.
4.
CoxDR. Regression models and life-tables. J Royal Stat Soc B (Methodological)1972; 34: 187–220.
5.
BrownCC. On the use of indicator variables for studying the time-dependence of parameters in a response-time model. Biometrics1975; 31: 863–872.
6.
EfronB. Logistic regression, survival analysis, and the Kaplan-Meier curve. J Am Stat Assoc1988; 83: 414–425.
7.
LairdNOlivierD. Covariance analysis of censored survival data using log-linear analysis techniques. J Am Stat Assoc1981; 76: 231–240.
8.
ThompsonWAJr. On the treatment of grouped observations in life studies. Biometrics1977; 33: 463–470.
9.
IbrahimJGChuHChenLM. Basic concepts and methods for joint models of longitudinal and survival data. J Clin Oncol2010; 28: 2796–2801.
10.
TsiatisAADeGruttolaVWulfsohnMS. Modeling the relationship of survival to longitudinal data measured with error. Applications to survival and CD4 counts in patients with AIDS. J Am Stat Assoc1995; 90: 27–37.
11.
WulfsohnMSTsiatisAA. A joint model for survival and longitudinal data measured with error. Biometrics1997; 53: 330–339.
12.
FollmannDWuM. An approximate generalized linear model with random effects for informative missing data. Biometrics1995; 51: 151–168.
13.
VoneshEF. Greene T and Schluchter MD. Shared parameter models for the joint analysis of longitudinal data and event times. Stat Med2006; 25: 143–163.
14.
Lang Wu, Wei Liu, Grace Y. Yi, and Yangxin Huang. Analysis of Longitudinal and Survival Data: Joint Modeling, Inference Methods, and Issues. J Probab Stat 2012. doi:10.1155/2012/640153, 17 pages.
15.
McCullagh P and Nelder JA. Generalized Linear Models, Second Edition (1989). Chapman and Hall, London.
16.
WangJ. EM algorithms for nonlinear mixed effects models. Computat Stat Data Anal2007; 51: 3244–3256.
17.
OehlertGW. A note on the delta method. Am Stat1992; 46: 27–29.
18.
BoothJGHobertJP. Standard errors of prediction in generalized linear mixed models. J Am Stat Assoc1998; 93: 262–272.
19.
CarlinBPGelfandAE. Approaches for empirical bayes confidence intervals. J Am Stat Assoc1990; 85: 105–114.
20.
CarlinBPGelfandAE. A sample reuse method for accurate parametric empirical bayes confidence intervals. J Royal Stat Soc B (Methodological)1991; 53: 189–200.
21.
LairdNMLouisTA. Empirical Bayes Confidence intervals based on bootstrap samples. J Am Stat Assoc1987; 82: 739–750.
22.
MorrisCN. Parametric empirical Bayes Inference: Theory and applications. J Am Stat Assoc1983; 78: 47–55.
23.
PiegorschWWCasellaG. Empirical Bayes estimation for logistic regression and extended parametric regression models. J Agri Biologic Environ Stat1996; 1: 231–249.
24.
GuwatuddeD. Tuberculosis in household contacts of infectious cases in Kampala, Uganda. Am J Epidemiol2003; 158: 887–898.
25.
SteinCM. Heritability analysis of cytokines as intermediate phenotypes of tuberculosis. J Infect Dis2003; 187: 1679–1685.
26.
MahaCS. Innate and adaptive immune responses during acute M. tuberculosis infection in adult household contacts in Kampala, Uganda. Am J Tropical Med Hygiene2012; 86: 690–697.
27.
SteinCM. Genome scan of M. tuberculosis infection and disease in Ugandans. PLoS One2008; 3(12): e4094–e4094.
28.
Mahan CS, Zalwango S, Thiel BA, et al. Innate and adaptive immune responses during acute M. tuberculosis infection in adult household contacts in Kampala, Uganda. 2012; 86: 690–697.
29.
PrenticeRLGloecklerLA. Regression analysis of grouped survival data with application to breast cancer data. Biometrics1978; 34: 57–67.
30.
Efron B and Tibshirani RJ. An introduction to the bootstrap. Monographs on Statistics and Applied Probability, 57. Chapman and Hall, New York, 1993. xvi+436 pp. ISBN: 0-412-04231-2.
31.
AbramowitzMStegunIA. Handbook of mathematical functions with formulas, graphs, and mathematical tables, US: Govt. Print. Off, 1964.
32.
CookNR. Use and misuse of the receiver operating characteristic curve in risk prediction. Circulation2007; 115: 928–935.
