The Annals of Applied Statistics

Biomarker change-point estimation with right censoring in longitudinal studies

Xiaoying Tang, Michael I. Miller, and Laurent Younes

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider in this paper a statistical two-phase regression model in which the change point of a disease biomarker is measured relative to another point in time, such as the manifestation of the disease, which is subject to right-censoring (i.e., possibly unobserved over the entire course of the study). We develop point estimation methods for this model, based on maximum likelihood, and bootstrap validation methods. The effectiveness of our approach is illustrated by numerical simulations, and by the estimation of a change point for amygdalar atrophy in the context of Alzheimer’s disease, wherein it is related to the cognitive manifestation of the disease.

Article information

Ann. Appl. Stat. Volume 11, Number 3 (2017), 1738-1762.

Received: February 2016
Revised: January 2017
First available in Project Euclid: 5 October 2017

Permanent link to this document

Digital Object Identifier

Change-point estimation right censoring medical imaging


Tang, Xiaoying; Miller, Michael I.; Younes, Laurent. Biomarker change-point estimation with right censoring in longitudinal studies. Ann. Appl. Stat. 11 (2017), no. 3, 1738--1762. doi:10.1214/17-AOAS1056.

Export citation


  • Alzheimer’s Association (2015). 2015 Alzheimer’s disease facts and figures. Alzheimer’s Dement. 11 332–384.
  • Atiya, M., Hyman, B. T., Albert, M. and Killiany, R. (2003). Structural magnetic resonance imaging in established and prodromal Alzheimer disease: A review. Alzheimer Dis. Assoc. Disord. 17 177–195.
  • Bauer, M., Bruveris, M. and Michor, P. W. (2014). Overview of the geometries of shape spaces and diffeomorphism groups. J. Math. Imaging Vision 50 60–97.
  • Cavedo, E., Boccardi, M., Ganzola, R., Canu, E., Beltramello, A., Caltagirone, C., Thompson, P. M. and Frisoni, G. B. (2011). Local amygdala structural differences with 3T MRI in patients with Alzheimer disease. Neurology 76 727–733.
  • Chen, J. and Gupta, A. K. (2000). Parametric Statistical Change Point Analysis. Birkhäuser, Boston, MA.
  • Csernansky, J. G., Wang, L., Swank, J., Miller, J. P., Gado, M., McKeel, D., Miller, M. I. and Morris, J. C. (2005). Preclinical detection of Alzheimer’s disease: Hippocampal shape and volume predict dementia onset in the elderly. NeuroImage 25 783–792.
  • den Heijer, T., Geerlings, M. I., Hoebeek, F. E., Hofman, A., Koudstaal, P. J. and Breteler, M. M. B. (2006). Use of hippocampal and amygdalar volumes on magnetic resonance imaging to predict dementia in cognitively intact elderly people. Arch. Gen. Psychiatry 63 57–62.
  • Dupuy, J.-F. (2006). Estimation in a change-point hazard regression model. Statist. Probab. Lett. 76 182–190.
  • Farley, J. U. and Hinich, M. J. (1970). A test for a shifting slope coefficient in a linear model. J. Amer. Statist. Assoc. 65 1320–1329.
  • Feder, P. I. (1975). On asymptotic distribution theory in segmented regression problems—Identified case. Ann. Statist. 3 49–83.
  • Fischl, B. (2012). FreeSurfer. NeuroImage 62 774–781.
  • Gombay, E. and Horváth, L. (1994a). Limit theorems for change in linear regression. J. Multivariate Anal. 48 43–69.
  • Gombay, E. and Horváth, L. (1994b). An application of the maximum likelihood test to the change-point problem. Stochastic Process. Appl. 50 161–171.
  • Hamann, S. (2001). Cognitive and neural mechanisms of emotional memory. Trends Cogn. Sci. 5 394–400.
  • Hebert, L. E., Weuve, J., Scherr, P. A. and Evans, D. A. (2013). Alzheimer disease in the United States (2010–2050) estimated using the 2010 census. Neurology 80 1778–1783.
  • Hinkley, D. V. (1969). Inference about the intersection in two-phase regression. Biometrika 56 495–504.
  • Hinkley, D. V. (1971). Inference in two-phase regression. J. Amer. Statist. Assoc. 66 736–743.
  • Hudson, D. J. (1966). Fitting segmented curves whose join points have to be estimated. J. Amer. Statist. Assoc. 61 1097–1129.
  • Jack, C. R., Petersen, R. C., O’Brien, P. C. and Tangalos, E. G. (1992). MR-based hippocampal volumetry in the diagnosis of Alzheimer’s disease. Neurology 42 183–188.
  • Jack, C. R., Petersen, R. C., Xu, Y. C., Waring, S. C., O’Brien, P. C., Tangalos, E. G., Smith, G. E., Ivnik, R. J. and Kokmen, E. (1997). Medial temporal atrophy on MRI in normal aging and very mild Alzheimer’s disease. Neurology 49 786–794.
  • Kantarci, K. K. and Jack, C. R. (2003). Neuroimaging in Alzheimer disease: An evidence-based review. Neuroimaging Clin. N. Am. 13 197–209.
  • Larrieu, S., Letenneur, L., Orgogozo, J. M., Fabrigoule, C., Amieva, H., Le Carret, N., Barberger-Gateau, P. and Dartigues, J. F. (2002). Incidence and outcome of mild cognitive impairment in a population-based prospective cohort. Neurology 59 1594–1599.
  • Li, Y., Qian, L. and Zhang, W. (2013). Estimation in a change-point hazard regression model with long-term survivors. Statist. Probab. Lett. 83 1683–1691.
  • Ma, J., Miller, M. I. and Younes, L. (2010). A Bayesian generative model for surface template estimation. Int. J. Biomed. Imaging 2010 974957.
  • Miller, M. I., Trouvé, A. and Younes, L. (2015). Hamiltonian systems in computational anatomy: 100 years since D’Arcy Thompson. Annu. Rev. Biomed. Eng. 17.
  • Miller, M. I., Younes, L. and Trouvé, A. (2014). Diffeomorphometry and geodesic positioning systems for human anatomy. Technology 2 36–43.
  • Miller, M. I., Younes, L., Ratnanather, J. T., Brown, T., Trinh, H., Lee, D. S., Tward, D., Mahon, P. B., Mori, S. and Albert, M. (2015). Amygdalar atrophy in symptomatic Alzheimer’s disease based on diffeomorphometry: The BIOCARD cohort. Neurobiol. Aging 36 S3–S10. Supplement 1: Novel Imaging Biomarkers for Alzheimer’s Disease and Related Disorders (NIBAD).
  • Mueller, S. G., Weiner, M. W., Thal, L. J., Petersen, R. C., Jack, C. R., Jagust, W., Trojanowski, J. Q., Toga, A. W. and Beckett, L. (2005). Ways toward an early diagnosis in Alzheimer’s disease: The Alzheimer’s Disease Neuroimaging Initiative (ADNI). Alzheimer’s Dement. 1 55–66.
  • Nguyen, H. T., Rogers, G. S. and Walker, E. A. (1984). Estimation in change-point hazard rate models. Biometrika 71 299–304.
  • Pierson, R., Johnson, H., Harris, G., Keefe, H., Paulsen, J. S., Andreasen, N. C. and Magnotta, V. A. (2011). Fully automated analysis using BRAINS: AutoWorkup. NeuroImage 54 328–336.
  • Pons, O. (2003). Estimation in a Cox regression model with a change-point according to a threshold in a covariate. Ann. Statist. 31 442–463. Dedicated to the memory of Herbert E. Robbins.
  • Poulin, S. P., Dautoff, R., Morris, J. C., Barrett, L. F., Dickerson, B. C., Alzheimer’s Disease Neuroimaging Initiative et al. (2011). Amygdala atrophy is prominent in early Alzheimer’s disease and relates to symptom severity. Psychiatry Research. Neuroimaging 194 7–13.
  • Price, J. L. (2003). Comparative aspects of amygdala connectivity. Ann. N.Y. Acad. Sci. 985 50–58.
  • Quandt, R. E. (1958). The estimation of the parameters of a linear regression system obeying two separate regimes. J. Amer. Statist. Assoc. 53 873–880.
  • Reuter, M. (2010). Hierarchical shape segmentation and registration via topological features of Laplace–Beltrami eigenfunctions. Int. J. Comput. Vis. 89 287–308.
  • Rusinek, H., De Santi, S., Frid, D., Tsui, W.-H., Tarshish, C. Y., Convit, A. and de Leon, M. J. (2003). Regional brain atrophy rate predicts future cognitive decline: 6-year longitudinal MR imaging study of normal aging. Radiology 229 691–696.
  • Scott, S. A., DeKosky, S. T. and Scheff, S. W. (1991). Volumetric atrophy of the amygdala in Alzheimer’s disease: Quantitative serial reconstruction. Neurology 41 351–356.
  • Scott, S. A., Sparks, D. L., Scheff, S. W., Dekosky, S. T. and Knox, C. A. (1992). Amygdala cell loss and atrophy in Alzheimer’s disease. Ann. Neurol. 32 555–563.
  • Sheline, Y. I., Gado, M. H. and Price, J. L. (1998). Amygdala core nuclei volumes are decreased in recurrent major depression. NeuroReport 9 2023–2028.
  • Sprent, P. (1961). Some hypotheses concerning two phase regression lines. Biometrics 17 634–645.
  • Tang, X., Oishi, K., Faria, A. V., Hillis, A. E., Albert, M. S., Mori, S. and Miller, M. I. (2013). Bayesian parameter estimation and segmentation in the multi-atlas random orbit model. PLoS ONE 8 e65591.
  • Tang, X., Holland, D., Dale, A. M., Younes, L., Miller, M. I. and Alzheimer’s Disease Neuroimaging Initiative (2014). Shape abnormalities of subcortical and ventricular structures in mild cognitive impairment and Alzheimer’s disease: Detecting, quantifying, and predicting. Hum. Brain Mapp. 35 3701–3725.
  • Tang, X., Holland, D., Dale, A. M., Younes, L., Miller, M. I. and Alzheimer’s Disease Neuroimaging Initiative (2015). The diffeomorphometry of regional shape change rates and its relevance to cognitive deterioration in mild cognitive impairment and Alzheimer’s disease. Hum. Brain Mapp. 36 2093–2117.
  • Thambisetty, M., Simmons, A., Velayudhan, L., Hye, A., Campbell, J., Zhang, Y., Wahlund, L.-O., Westman, E., Kinsey, A., Güntert, A. et al. (2010). Association of plasma clusterin concentration with severity, pathology, and progression in Alzheimer disease. Arch. Gen. Psychiatry 67 739–748.
  • Tsuchiya, K. and Kosaka, K. (1990). Neuropathological study of the amygdala in presenile Alzheimer’s disease. J. Neurol. Sci. 100 165–173.
  • Wu, C. Q., Zhao, L. C. and Wu, Y. H. (2003). Estimation in change-point hazard function models. Statist. Probab. Lett. 63 41–48.
  • Younes, L. (2010). Shapes and Diffeomorphisms. Applied Mathematical Sciences 171. Springer, Berlin.
  • Younes, L., Albert, M., Miller, M. I., BIOCARD Research Team et al. (2014). Inferring changepoint times of medial temporal lobe morphometric change in preclinical Alzheimer’s disease. NeuroImage Clin. 5 178–187.