Two-stage empirical likelihood for longitudinal neuroimaging data. (English) Zbl 1223.62024

Summary: Longitudinal imaging studies are essential to understanding the neural development of neuropsychiatric disorders, substance use disorders, and the normal brain. The main objective of this paper is to develop a two-stage adjusted exponentially tilted empirical likelihood (TETEL) for the spatial analysis of neuroimaging data from longitudinal studies. The TETEL method as a frequentist approach allows us to efficiently analyze longitudinal data without modeling temporal correlation and to classify different time-dependent covariate types. To account for spatial dependence, the TETEL method developed here specifically combines all the data in the closest neighborhood of each voxel (or pixel) on a 3-dimensional (3D) volume (or 2D surface) with appropriate weights to calculate adaptive parameter estimates and adaptive test statistics. Simulation studies are used to examine the finite sample performance of the adjusted exponential tilted likelihood ratio statistic and TETEL. We demonstrate the application of our statistical methods to the detection of the difference in the morphological changes of the hippocampus across time between schizophrenia patients and healthy subjects in a longitudinal schizophrenia study.


62G05 Nonparametric estimation
92C50 Medical applications (general)
92C55 Biomedical imaging and signal processing
62P10 Applications of statistics to biology and medical sciences; meta analysis
65C60 Computational problems in statistics (MSC2010)
Full Text: DOI arXiv


[1] Almli, C. R., Rivkin, M. J., McKinstry, R. C. and Brain Development Cooperative Group (2007). The NIH MRI study of normal brain development (objective-2): Newborns, infants, toddlers, and preschoolers. NeuroImage 35 308-325.
[2] Arndt, S., Cohen, G., Alliger, R. J., Swayze, V. W. and Andreasen, N. C. (1991). Problems with ratio and proportion measures of imaged cerebral structures. Psychiatry Res. 40 79-89.
[3] Ashburner, J. and Friston, K. J. (2000). Voxel-based morphometry: The methods. NeuroImage 11 805-821.
[4] Beckmann, C. F., Jenkinson, M. and Smith, S. M. (2003). General multilevel linear modeling for group analysis in fMRI. NeuroImage 20 1052-1063.
[5] Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. R. Stat. Soc. Ser. B 57 289-300. · Zbl 0809.62014
[6] Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Ann. Statist. 29 1165-1188. · Zbl 1041.62061
[7] Bowman, F. D., Caffo, B., Bassett, S. S. and Kilts, C. (2008). A Bayesian hierarchical framework for spatial modeling of fMRI data. NeuroImage 39 146-156.
[8] Cao, J. and Worsley, K. J. (2001). Applications of random fields in human brain mapping. In Spatial Statistics: Methodological Aspects and Applications (M. Moore, ed.). Lecture Notes in Statistics 159 170-182. Springer, New York. · Zbl 1022.92021
[9] Chen, J., Variyath, A. M. and Abraham, B. (2008). Adjusted empirical likelihood and its properties. J. Comput. Graph. Statist. 17 426-443.
[10] Chung, M. K., Dalton, K. M. and Davidson, R. J. (2007). Tensor-based cortical surface morphometry via weighted spherical harmonic representation. IEEE Transactions on Medical Imaging 26 566-581.
[11] Diggle, P. J., Heagerty, P. J., Liang, K.-Y. and Zeger, S. L. (2002). Analysis of Longitudinal Data , 2nd ed. Oxford Statistical Science Series 25 . Oxford Univ. Press, Oxford. · Zbl 1031.62002
[12] Dryden, I. L. and Mardia, K. V. (1998). Statistical Shape Analysis . Wiley, Chichester. · Zbl 0901.62072
[13] Duvernoy, H. (2005). The Human Hippocampus . Springer, New York.
[14] Friston, K. J. (2007). Statistical Parametric Mapping: The Analysis of Functional Brain Images . Academic Press, London.
[15] Friston, K. J., Holmes, A. P., Poline, J. B., Price, C. J. and Frith, C. D. (1996). Detecting activations in PET and fMRI: Levels of inference and power. NeuroImage 4 223-235.
[16] Friston, K. J., Stephan, K. E., Lund, T. E., Morcom, A. and Kiebel, S. (2005). Mixed-effects and fMRI studies. NeuroImage 24 244-252.
[17] Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica 50 1029-1054. · Zbl 0502.62098
[18] Hayasaka, S., Phan, L. K., Liberzon, I., Worsley, K. J. and Nichols, T. E. (2004). Nonstationary cluster-size inference with random field and permutation methods. NeuroImage 22 676-687.
[19] Hecke, W. V., Sijbers, J., Backer, S. D., Poot, D., Parizel, P. M. and Leemans, A. (2009). On the construction of a ground truth framework for evaluating voxel-based diffusion tensor MRI analysis methods. NeuroImage 46 692-707.
[20] Huettel, S. A., Song, A. W. and McCarthy, G. (2004). Functional Magnetic Resonance Imaging . Sinauer, Sunderland, MA.
[21] Imbens, G. W., Spady, R. H. and Johnson, P. (1998). Information-theoretic approaches to inference in moment condition models. Econometrica 66 333-357. · Zbl 1055.62512
[22] Jones, D. K., Symms, D. K., Cercignani, M. and Howard, R. J. (2005). The effect of filter size on VBM analyses of DT-MRI data. NeuroImage 26 546-554.
[23] Lai, T. L. and Small, D. (2007). Marginal regression analysis of longitudinal data with time-dependent covariates: A generalized method-of-moments approach. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 79-99.
[24] Liang, K. Y. and Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika 73 13-22. · Zbl 0595.62110
[25] Lieberman, J. A., Tollefson, G. D., Charles, C., Zipursky, R., Sharma, T., Kahn, R. S., Keefe, R. S. E., Green, A. I., Gur, R. E., McEvoy, J., Perkins, D., Hamer, R. M., Gu, H. and Tohen, M. (2005). Antipsychotic drug effects on brain morphology in first-episode psychosis. Archives of General Psychiatry 62 361-370.
[26] Lindquist, M. A. and Wager, T. D. (2008). Spatial smoothing in fMRI using prolate spheroidal wave functions. Hum. Brain Mapp. 29 1276-1287.
[27] Liu, Y. and Chen, J. (2010). Adjusted empirical likelihood with high-order precision. Ann. Statist. 38 1341-1362. · Zbl 1189.62054
[28] Logan, B. R. and Rowe, D. B. (2004). An evalution of thresholding techniques in fMRI analysis. NeuroImage 22 95-108.
[29] Luo, W.-L. and Nichols, T. E. (2003). Diagnosis and exploration of massively univariate neuroimaging models. NeuroImage 19 1014-1032.
[30] Luo, H. and Puthusserypady, S. (2005). A sparse Bayesian method for determination of flexible design matrix for fMRI data analysis. IEEE Trans. Circuits Syst. I. Regul. Pap. 52 2699-2706.
[31] Narr, K. L., Thompson, P. M., Szeszko, P., Robinson, D., Jang, S., Woods, R. P., Kim, S., Hayashi, K. M., Asunction, D., Toga, A. W. and Bilder, R. M. (2004). Regional specificity of hippocampal volume reductions in first-episode schizophrenia. NeuroImage 21 1563-1575.
[32] Newey, W. K. and Smith, R. J. (2004). Higher order properties of GMM and generalized empirical likelihood estimators. Econometrica 72 219-255. · Zbl 1151.62313
[33] Owen, A. B. (2001). Empirical Likelihood . Chapman & Hall/CRC, New York. · Zbl 0989.62019
[34] Penny, W., Flandin, G. and Trujillo-Barreto, N. (2007). Bayesian comparison of spatially regularised general linear models. Hum. Brain Mapp. 28 275-293.
[35] Pepe, M. S. and Anderson, G. L. (1994). A cautionary note on inference for marginal regression models with longitudinal data and general correlated response data. Comm. Statist. Simul. Comput. 23 939-951.
[36] Pizer, S. M., Fletcher, P. T., Joshi, S., Thall, A., Chen, J. Z., Fridman, Y., Fritsch, D. S., Gash, A. G., Glotzer, J. M., Jiroutek, M. R., Lu, C., Muller, K. E., Tracton, G., Yushkevich, P. and Chaney, E. L. (2003). Deformable m-reps for 3D medical image segmentation. Int. J. Comput. Vis. 55 85-106.
[37] Poline, J. and Mazoyer, B. (1994). Analysis of individual brain activation maps using hierarchical description and multiscale detection. IEEE Transactions in Medical Imaging 4 702-710.
[38] Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations. Ann. Statist. 22 300-325. · Zbl 0799.62049
[39] Qu, A., Lindsay, B. G. and Li, B. (2000). Improving generalised estimating equations using quadratic inference functions. Biometrika 87 823-836. · Zbl 1028.62045
[40] Rogers, B. P., Morgan, V. L., Newton, A. T. and Gore, J. C. (2007). Assessing functional connectivity in the human brain by fMRI. Magn. Reson. Imaging 25 1347-1357.
[41] Rowe, D. B. (2005). Parameter estimation in the magnitude-only and complex-valued fMRI data models. NeuroImage 25 1124-1132.
[42] Salmond, C. H., Ashburner, J., Vargha-Khadem, F., Connelly, A., Gadian, D. G. and Friston, K. J. (2002). Distributional assumptions in voxel-based morphometry. NeuroImage 17 1027-1030.
[43] Schennach, S. M. (2007). Point estimation with exponentially tilted empirical likelihood. Ann. Statist. 35 634-672. · Zbl 1117.62024
[44] Shafie, K., Sigal, B., Siegmund, D. and Worsley, K. J. (2003). Rotation space random fields with an application to fMRI data. Ann. Statist. 31 1732-1771. · Zbl 1043.92019
[45] Shi, X., Ibrahim, J. G., Lieberman, J., Styner, M., Li, Y. and Zhu, H. (2011). Supplement to “Two-stage empirical likelihood for longitudinal neuroimaging data.” . · Zbl 1223.62024
[46] Snook, L., Plewes, C. and Beaulieu, C. (2007). Voxel based versus region of interest analysis in diffusion tensor imaging of neurodevelopment. NeuroImage 34 243-252.
[47] Styner, M. and Gerig, G. (2003). Automatic and robust computation of 3d medial models incorporating object variability. Int. J. Comput. Vis. 55 107-122.
[48] Styner, M., Lieberman, J. A., Pantazis, D. and Gerig, G. (2004). Boundary and medial shape analysis of the hippocampus in schizophrenia. Med. Image Anal. 8 197-203.
[49] Styner, M., Lieberman, J. A., McClure, R. K., Weinberger, D. R., Jones, D. W. and Gerig, G. (2005). Morphometric analysis of lateral ventricles in schizophrenia and healthy controls regarding genetic and disease-specific factors. Proc. Natl. Acad. Sci. USA 102 4872-4877.
[50] Thompson, P. M., Cannon, T. D. and Toga, A. W. (2002). Mapping genetic influences on human brain structure. Annals of Medicine 24 523-536.
[51] Thompson, P. M. and Toga, A. W. (2002). A framework for computational anatomy. Comput. Vis. Sci. 5 13-34. · Zbl 1001.92032
[52] Woolrich, M. W., Behrens, T. E. J., Beckmann, C. F., Jenkinson, M. and Smith, S. M. (2004). Multilevel linear modelling for fMRI group analysis using Bayesian inference. NeuroImage 21 1732-1747.
[53] Worsley, K. J., Taylor, J. E., Tomaiuolo, F. and Lerch, J. (2004). Unified univariate and multivariate random field theory. NeuroImage 23 189-195.
[54] Yue, Y., Loh, J. M. and Lindquist, M. A. (2010). Adaptive spatial smoothing of fMRI images. Stat. Interface 3 3-13. · Zbl 1245.62118
[55] Zhu, H., Li, Y., Tang, N., Bansal, R., Hao, X., Weissman, M. M. and Peterson, B. S. (2008a). Statistical modelling of brain morphological measures within family pedigrees. Statist. Sinica 18 1569-1591. · Zbl 1151.62094
[56] Zhu, H., Ibrahim, J. G., Tang, N. and Zhang, H. (2008b). Diagnostic measures for empirical likelihood of general estimating equations. Biometrika 95 489-507. · Zbl 1437.62678
[57] Zhu, H. T., Zhou, H., Chen, J., Li, Y., Styner, M. and Lieberman, J. (2009). Adjusted exponentially tilted likelihood with applications to brain morphology. Biometrics 65 919-927. · Zbl 1172.62072
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