The main goal of this paper is to employ longitudinal trajectories in a significant number of sub-regional brain volumetric MRI data as statistical predictors for Alzheimer's disease (AD) classification. We use logistic regression in a Bayesian framework that includes many functional predictors. The direct sampling of regression coefficients from the Bayesian logistic model is difficult due to its complicated likelihood function. In high-dimensional scenarios, the selection of predictors is paramount with the introduction of either spike-and-slab priors, non-local priors, or Horseshoe priors. We seek to avoid the complicated Metropolis-Hastings approach and to develop an easily implementable Gibbs sampler. In addition, the Bayesian estimation provides proper estimates of the model parameters, which are also useful for building inference. Another advantage of working with logistic regression is that it calculates the log of odds of relative risk for AD compared to normal control based on the selected longitudinal predictors, rather than simply classifying patients based on cross-sectional estimates. Ultimately, however, we combine approaches and use a probability threshold to classify individual patients. We employ 49 functional predictors consisting of volumetric estimates of brain sub-regions, chosen for their established clinical significance. Moreover, the use of spike-and-slab priors ensures that many redundant predictors are dropped from the model.

References

• Adaszewski, S., Dukart, J., Kherif, F., Frackowiak, R., & Draganski, B. (2013). How early can we predict Alzheimer's disease using computational anatomy? Neurobiology of Aging, 34(12), 2815–2826. https://doi.org/10.1016/j.neurobiolaging.2013.06.015 
• Arlt, S., Buchert, R., Spies, L., Eichenlaub, M., Lehmbeck, J. T., & Jahn, H. (2013). Association between fully automated MRI-based volumetry of different brain regions and neuropsychological test performance in patients with amnestic mild cognitive impairment and Alzheimer's disease. European Archives of Psychiatry and Clinical Neuroscience, 263(4), 335–344. https://doi.org/10.1007/s00406-012-0350-7 
• Casanova, R., Wagner, B., Whitlow, C. T., Williamson, J. D., S. A. Shumaker, Maldjian, J. A., & Espeland, M. A. (2011). High dimensional classification of structural MRI Alzheimer's disease data based on large scale regularization. Frontiers in Neuroinformatics, 5, 22. https://doi.org/10.3389/fninf.2011.00022 
• Casella, G., Ghosh, M., Gill, J., & Kyung, M. (2010). Penalized regression, standard errors, and Bayesian lassos. Bayesian Analysis, 5(2), 369–411. 
• Fan, Y., Batmanghelich, N., Clark, C. M., & Davatzikos, C., and Alzheimer's Disease Neuroimaging Initiative and others (2008). Spatial patterns of brain atrophy in MCI patients, identified via high-dimensional pattern classification, predict subsequent cognitive decline. NeuroImage, 39(4), 1731–1743. https://doi.org/10.1016/j.neuroimage.2007.10.031 
• Fan, Y., James, G. M., & Radchenko, P. (2015). Functional additive regression. The Annals of Statistics, 43(5), 2296–2325. https://doi.org/10.1214/15-AOS1346 
• George, E. I., & McCulloch, R. E. (1993). Variable selection via Gibbs sampling. Journal of the American Statistical Association, 88(423), 881–889. https://doi.org/10.1080/01621459.1993.10476353 
• George, E. I., & McCulloch, R. E. (1997). Approaches for Bayesian variable selection. Statistica Sinica, 7, 339–373. https://www.jstor.org/stable/24306083 
• Hobert, J. P., & Geyer, C. J. (1998). Geometric ergodicity of Gibbs and block Gibbs samplers for a hierarchical random effects model. Journal of Multivariate Analysis, 67(2), 414–430. https://doi.org/10.1006/jmva.1998.1778 
• Ishwaran, H., & Rao, J. S., et al. (2005). Spike and slab variable selection: Frequentist and bayesian strategies. The Annals of Statistics, 33(2), 730–773. https://doi.org/10.1214/009053604000001147 
• Jack, C. R., Petersen, R. C., Xu, Y., O'Brien, P. C., Smith, G. E., Ivnik, R. J., E. G. Tangalos, & Kokmen, E. (1998). Rate of medial temporal lobe atrophy in typical aging and Alzheimer's disease. Neurology, 51(4), 993–999. https://doi.org/10.1212/WNL.51.4.993 
• Jack, C. R., Petersen, R. C., Xu, Y. C., O'Brien, P. C., Smith, G. E., Ivnik, R. J., Boeve, B. F., Waring, S. C., Tangalos, E. G., & Kokmen, E. (1999). Prediction of ad with MRI-based hippocampal volume in mild cognitive impairment. Neurology, 52(7), 1397–1397. https://doi.org/10.1212/WNL.52.7.1397 
• James, G. M. (2002). Generalized linear models with functional predictors. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(3), 411–432. https://doi.org/10.1111/rssb.2002.64.issue-3 
• Jiang, W. (2006). On the consistency of bayesian variable selection for high dimensional binary regression and classification. Neural Computation, 18(11), 2762–2776. https://doi.org/10.1162/neco.2006.18.11.2762 
• Jiang, W. (2007). Bayesian variable selection for high dimensional generalized linear models: Convergence rates of the fitted densities. The Annals of Statistics, 35(4), 1487–1511. https://doi.org/10.1214/009053607000000019 
• Johnstone, I. M., & Silverman, B. W. (2004). Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences. The Annals of Statistics, 32(4), 1594–1649. https://doi.org/10.1214/009053604000000030 
• Lee, S. H., Bachman, A. H., Yu, D., Lim, J., & Ardekani, B. A., and Alzheimer's Disease Neuroimaging Initiative and others (2016). Predicting progression from mild cognitive impairment to Alzheimer's disease using longitudinal callosal atrophy. Alzheimer's & Dementia: Diagnosis, Assessment & Disease Monitoring, 2(1), 68–74. https://doi.org/10.1016/j.dadm.2016.01.003 
• Leng, C., Tran, M.-N., & Nott, D. (2014). Bayesian adaptive lasso. Annals of the Institute of Statistical Mathematics, 66(2), 221–244. https://doi.org/10.1007/s10463-013-0429-6 
• Li, Q., Lin, N. (2010). The bayesian elastic net. Bayesian Analysis, 5(1), 151–170. https://doi.org/10.1214/10-BA506 
• Majumder, A. (2017). Variable selection in high-dimensional setup: A detailed illustration through marketing and MRI data. Michigan State University. 
• Meier, L., S. Van De Geer, & Bühlmann, P. (2008). The group lasso for logistic regression. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 70(1), 53–71. https://doi.org/10.1111/j.1467-9868.2007.00627.x 
• Misra, C., Fan, Y., & Davatzikos, C. (2009). Baseline and longitudinal patterns of brain atrophy in mci patients, and their use in prediction of short-term conversion to ad: Results from adni. NeuroImage, 44(4), 1415–1422. https://doi.org/10.1016/j.neuroimage.2008.10.031 
• Müller, H.-G. (2005). Functional modelling and classification of longitudinal data. Scandinavian Journal of Statistics, 32(2), 223–240. https://doi.org/10.1111/sjos.2005.32.issue-2 
• Park, T., & Casella, G. (2008). The bayesian lasso. Journal of the American Statistical Association, 103(482), 681–686. https://doi.org/10.1198/016214508000000337 
• Polson, N. G., Scott, J. G., & Windle, J. (2013). Bayesian inference for logistic models using pólya–gamma latent variables. Journal of the American Statistical Association, 108(504), 1339–1349. https://doi.org/10.1080/01621459.2013.829001 
• Raz, N., Lindenberger, U., Rodrigue, K. M., Kennedy, K. M., Head, D., Williamson, A., Dahle, C., Gerstorf, D., & Acker, J. D. (2005). Regional brain changes in aging healthy adults: General trends, individual differences and modifiers. Cerebral Cortex, 15(11), 1676–1689. https://doi.org/10.1093/cercor/bhi044 
• Seixas, F. L., Zadrozny, B., Laks, J., Conci, A., & Saade, D. C. M. (2014). A bayesian network decision model for supporting the diagnosis of dementia, Alzheimer's disease and mild cognitive impairment. Computers in Biology and Medicine, 51(8), 140–158. https://doi.org/10.1016/j.compbiomed.2014.04.010 
• Shen, L., Qi, Y., Kim, S., Nho, K., Wan, J., Risacher, S. L., & Saykin, A. J., and others. (2010). Sparse bayesian learning for identifying imaging biomarkers in ad prediction. In International Conference on Medical Image Computing and Computer-Assisted Intervention (pp. 611–618). Springer. 
• Shi, G. (2017). Bayesian variable selection: Extensions of nonlocal priors. Michigan State University. Statistics.
• Smith, M., & Kohn, R. (1996). Nonparametric regression using Bayesian variable selection. Journal of Econometrics, 75(2), 317–343. https://doi.org/10.1016/0304-4076(95)01763-1 
• Wang, X., Nan, B., Zhu, J., & Koeppe, R. (2014). Regularized 3d functional regression for brain image data via haar wavelets. The Annals of Applied Statistics, 8(2), 1045. https://doi.org/10.1214/14-AOAS736 
• Windle, J., Polson, N. G., & Scott, J. G. (2014). Sampling polya-gamma random variates: Alternate and approximate techniques. arXiv preprint arXiv:1405.0506. 
• Xu, X., Ghosh, M. (2015). Bayesian variable selection and estimation for group lasso. Bayesian Analysis, 10(4), 909–936. https://doi.org/10.1214/14-BA929
• Yang, W., Chen, X., Xie, H., & Huang, X. (2010). Ica-based automatic classification of magnetic resonance images from adni data. In Life System Modeling and Intelligent Computing (pp. 340–347). Springer. 
• Zhang, D., & Shen, D., and Alzheimer's Disease Neuroimaging Initiative. (2012). Predicting future clinical changes of MCI patients using longitudinal and multimodal biomarkers. PloS One, 7(3), e33182. https://doi.org/10.1371/journal.pone.0033182 
• Zhu, H., Vannucci, M., & Cox, D. D. (2010). A Bayesian hierarchical model for classification with selection of functional predictors. Biometrics, 66(2), 463–473. https://doi.org/10.1111/j.1541-0420.2009.01283.x 

To cite this article: Asish Banik, Taps Maiti & Andrew Bender (2022) Bayesian penalized model for classification and selection of functional predictors using longitudinal MRI data from ADNI, Statistical Theory and Related Fields, 6:4, 327-343, DOI: 10.1080/24754269.2022.2064611 To link to this article: https://doi.org/10.1080/24754269.2022.2064611