Shared random-effect models for the joint analysis of longitudinal and time-to-event data: application to the prediction of prostate cancer recurrence

Sène, Mbéry; Bellera, Carine A.; Proust-Lima, Cécile

Journal de la société française de statistique, Tome 155 (2014) no. 1, pp. 134-155.

Dans la dernière décennie, la recherche en modélisation conjointe s’est développée très rapidement dans le domaine des biostatistiques et de la recherche médicale. Ce type de modèles permet d’étudier simultanément un marqueur longitudinal et un temps d’événement corrélés. Parmi eux, les modèles à effets aléatoires partagés, qui définissent un modèle mixte pour le marqueur longitudinal et un modèle de survie pour le temps d’événement incluant les caractéristiques du modèle mixte comme variables explicatives, ont reçu le plus d’attention. En effet, ces modèles étendent naturellement le modèle de survie avec variables explicatives dépendantes du temps et offrent un cadre flexible pour explorer le lien entre le biomarqueur longitudinal et le risque d’événement.

L’objectif de cet article est de passer brièvement en revue la méthodologie du modèle à effets aléatoires partagés et de détailler son implémentation et son évaluation à travers un exemple réel d’étude de progression de cancer de la prostate après une radiothérapie. En particulier, différentes spécifications de la dépendance entre le biomarqueur longitudinal, l’antigène spécifique de la prostate (PSA), et le risque de rechute clinique sont investiguées pour bien comprendre le lien entre la dynamique du PSA et le risque de rechute clinique. Ces différents modèles conjoints sont comparés en termes de qualité d’ajustement et d’adéquation aux hypothèses du modèle conjoint mais aussi en termes de pouvoir prédictif en utilisant la cross-entropie pronostique. En effet, en plus de mieux comprendre le lien entre la dynamique de PSA et le risque de rechute clinique, la perspective dans les études sur le cancer de la prostate est de fournir des outils pronostiques dynamiques de rechute clinique basés sur toute l’histoire du biomarqueur.

In the last decade, joint modeling research has expanded very rapidly in biostatistics and medical research. This type of models enables the simultaneous study of a longitudinal marker and a correlated time-to-event. Among them, the shared random-effect models that define a mixed model for the longitudinal marker and a survival model for the time-to-event including characteristics of the mixed model as covariates received the main interest. Indeed, they extend naturally the survival model with time-dependent covariates and offer a flexible framework to explore the link between a longitudinal biomarker and a risk of event.

The objective of this paper is to briefly review the shared random-effect model methodology and detail its implementation and evaluation through a real example from the study of prostate cancer progression after a radiation therapy. In particular, different specifications of the dependency between the longitudinal biomarker, the prostate-specific antigen (PSA), and the risk of clinical recurrence are investigated to better understand the link between the PSA dynamics and the risk of clinical recurrence. These different joint models are compared in terms of goodness-of-fit and adequation to the joint model assumptions but also in terms of predictive accuracy using the expected prognostic cross-entropy. Indeed, in addition to better understand the link between the PSA dynamics and the risk of clinical recurrence, the perspective in prostate cancer studies is to provide dynamic prognostic tools of clinical recurrence based on the biomarker history.

[1] Crowther, Michael J.; Abrams, Keith R.; Lambert, Paul C. Flexible parametric joint modelling of longitudinal and survival data, Statistics in Medicine, Volume 31 (2012) no. 30, pp. 4456-4471

[2] Chi, Yueh-Yun; Ibrahim, Joseph G. Joint models for multivariate longitudinal and multivariate survival data., Biometrics, Volume 62 (2006) no. 2, p. 432-45 | Zbl

[3] Commenges, Daniel; Liquet, Benoit; Proust-Lima, Cécile Choice of prognostic estimators in joint models by estimating differences of expected conditional Kullback-Leibler risks., Biometrics, Volume 68 (2012) no. 2, p. 380-7 | Zbl

[4] Cox, David R. Regression models and life tables, Journal of the Royal Statistical Society, Series B, Volume 34 (1972), pp. 187-220 | Zbl

[5] Faucett, Cheryl L.; Thomas, Duncan C. Simultaneously modelling censored survival data and repeatedly measured covariates: a Gibbs sampling approach., Statistics in medicine, Volume 15 (1996) no. 15, p. 1663-85

[6] Guo, Xu; Carlin, Bradley P. Separate and Joint Modeling of Longitudinal and Event Time Data Using Standard Computer Packages, The American Statistician, Volume 58 (2004) no. 1, pp. 16-24

[7] Gerds, Thomas A.; Schumacher, Martin Consistent Estimation of the Expected Brier Score in General Survival Models with Right-Censored Event Times, Biometrical Journal, Volume 48 (2006) no. 6, pp. 1029-1040 | Zbl

[8] Gerds, Thomas A.; Schumacher, Martin Efron-type measures of prediction error for survival analysis., Biometrics, Volume 63 (2007) no. 4, p. 1283-7 | Zbl

[9] Henderson, Robin; Diggle, Peter; Dobson, Angela Joint modelling of longitudinal measurements and event time data., Biostatistics (Oxford, England), Volume 1 (2000) no. 4, p. 465-80 | Zbl

[10] Henderson, Robin; Diggle, Peter; Dobson, Angela Identification and efficacy of longitudinal markers for survival., Biostatistics (Oxford, England), Volume 3 (2002) no. 1, pp. 33-50 | Zbl

[11] Huang, Xin; Li, Gang; Elashoff, Robert; Pan, Jianxin A general joint model for longitudinal measurements and competing risks survival data with heterogeneous random effects, Lifetime Data Analysis, Volume 17 (2011), pp. 80-100 | Zbl

[12] Ibrahim, Joseph G; Chu, Haitao; Chen, Liddy M Basic concepts and methods for joint models of longitudinal and survival data., Journal of clinical oncology : official journal of the American Society of Clinical Oncology, Volume 28 (2010) no. 16, p. 2796-801

[13] Ibrahim, Joseph G.; Chen, Ming-Hui; Sinha, Debajyoti Bayesian Survival Analysis., Springer (2001) | Zbl

[14] Jacqmin-Gadda, Héléne; Thiébaut, Rodolphe; Dartigues, Jean F. Modélisation conjointe de données longitudinales quantitatives et délais censurés., Revue d’épidémiologie et de santé publique, Volume 52 (2004) no. 6, p. 502-10

[15] Kalbfleisch, JD; Prentice, RL The statistical analysis of failure time data, Second Edition, John Wiley (2002)

[16] Lesaffre, Emmanuel; Spiessens, Bart On the effect of the number of quadrature points in a logistic random effects model: an example, Journal of the Royal Statistical Society: Series C (Applied Statistics), Volume 50 (2001) no. 3, pp. 325-335 | DOI | Zbl

[17] Lin, Haiqun; Turnbull, Bruce W.; McCulloch, Charles E.; Slate, Elizabeth H. Latent class models for joint analysis of longitudinal biomarker and event process data : Application to longitudinal prostate-specific antigen readings and prostate cancer., Journal of the American Statistical Association, Volume 97 (2002) no. 457 | Zbl

[18] Law, Ngayee J; Taylor, Jeremy M G; Sandler, Howard The joint modeling of a longitudinal disease progression marker and the failure time process in the presence of cure., Biostatistics (Oxford, England), Volume 3 (2002) no. 4, p. 547-63 | Zbl

[19] Laird, Nan M.; Ware, James H. Random-effects models for longitudinal data., Biometrics, Volume 38 (1982), pp. 963-974 | Zbl

[20] Proust-Lima, Cécile; Sène, Mbéry; Taylor, Jeremy M G.; Jacqmin-Gadda, Héléne Joint latent class models for longitudinal and time-to-event data: A review., Statistical methods in medical research (2012)

[21] Proust-Lima, Cécile; Taylor, Jeremy M G. Development and validation of a dynamic prognostic tool for prostate cancer recurrence using repeated measures of posttreatment PSA: a joint modeling approach., Biostatistics (Oxford, England), Volume 10 (2009) no. 3, p. 535-49 | Zbl

[22] Proust-Lima, Cécile; Taylor, Jeremy M G.; Scott, Willams; Ankerst, D.; Liu, N.; Kestin, L.; K, Bae; Howard, Sandler Determinants of change in prostate-specific antigen over time and its association with recurrence after external beam radiation therapy for prostate cancer in five large cohorts., International Journal of Radiation Oncology Biology Physics (2008)

[23] Rizopoulos, Dimitris; Ghosh, Pulak A Bayesian semiparametric multivariate joint model for multiple longitudinal outcomes and a time-to-event., Statistics in medicine, Volume 30 (2011) no. 12, p. 1366-80

[24] Ratcliffe, Sarah J.; Guo, Wensheng; Ten Have, Thomas R. Joint Modeling of Longitudinal and Survival Data via a Common Frailty., Biometrics, Volume 60 (2004), pp. 892-899 | Zbl

[25] Rizopoulos, Dimitris JM : An R Package for the Joint Modelling of longitudinal and time-to-event data., Journal Of Statistical Software, Volume 35 (2010) no. 9

[26] Rizopoulos, Dimitris Dynamic predictions and prospective accuracy in joint models for longitudinal and time-to-event data., Biometrics, Volume 67 (2011) no. 3, p. 819-29 | Zbl

[27] Rizopoulos, Dimitris Fast fitting of joint models for longitudinal and event time data using a pseudo-adaptive Gaussian quadrature rule, Computational Statistics and Data Analysis, Volume 56 (2012) no. 3, pp. 491 -501 | DOI | Zbl

[28] Rizopoulos, Dimitris; Verbeke, Geert; Lesaffre, Emmanuel Fully exponential Laplace approximations for the joint modelling of survival and longitudinal data., Journal of the Royal Statistical Society: Series B (Statistical Methodology), Volume 71 (2009) no. 3, pp. 637-654 | Zbl

[29] Schoop, Rotraut; Schumacher, Martin; Graf, Erika Measures of prediction error for survival data with longitudinal covariates., Biometrical journal. Biometrische Zeitschrift, Volume 53 (2011) no. 2, p. 275-93 | Zbl

[30] Sweeting, Michael J.; Thompson, Simon G. Joint modelling of longitudinal and time-to-event data with application to predicting abdominal aortic aneurysm growth and rupture, Biometrical Journal, Volume 53 (2011) no. 5, pp. 750-763 | Zbl

[31] Tsiatis, Anastasios A.; Davidian, Marie Joint modeling of longitudinal and time-to-event data : An overview., Statistica Sinica, Volume 14 (2004), pp. 809-834 | Zbl

[32] Tsiatis, Anastasios A.; Victor, DeGruttola; Wulfsohn, Michael S. 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 Stastical Association, Volume 90 (1995) no. 429 | Zbl

[33] Wu, Lang; Liu, Wei; Yi, G Y; Huang, Yangxin Analysis of Longitudinal and Survival Data: Joint Modeling, Inference Methods, and Issues., Journal of Probability and Statistics (2012) | Zbl

[34] Wang, Yan; Taylor, Jeremy M G. Jointly Modeling Longitudinal and Event Time Data With Application to Acquired Immunodeficiency Syndrome., Journal of the American Statistical Association, Volume 96 (2001) no. 455 | Zbl

[35] Wulfsohn, Michael S.; Tsiatis, Anastasios A. A joint model of survival and longitudinal data measured with error., Biometrics, Volume 53 (1997), pp. 330-339 | Zbl

[36] Yu, Menggang; Law, Ngayee J.; Taylor, Jeremy M G.; Sandler, Howard M. Joint Longitudinal-Survival-Cure Models and Their Application to Prostate Cancer., Statistica Sinica, Volume 14 (2004), pp. 835-862 | Zbl

[37] Yu, Menggang; Taylor, Jeremy M G.; Sandler, Howard M. Individual Prediction in Prostate Cancer Studies Using a Joint Longitudinal Survival-Cure Model, Journal of the American Statistical Association, Volume 103 (2008) no. 481, pp. 178-187 | Zbl

[38] Zeng, Donglin; Cai, Jianwen Asymptotic results for maximum likelihood estimators in joint analysis of repeated measurements and survival time, The Annals of Statistics, Volume 33 (2005) no. 5, pp. 2132-2163 | Zbl

[39] Zheng, Yingye; Heagerty, Patrick J. Prospective Accuracy for Longitudinal Markers, Biometrics, Volume 63 (2007) no. 2, pp. 332-341 | DOI | Zbl