Abstract
This paper presents a novel method to test mean differences of geometric object properties (GOPs). The method is designed for data whose representations include both Euclidean and non-Euclidean elements. It is based on advanced statistical analysis methods such as backward means on spheres. We develop a suitable permutation test to find global and simultaneously individual morphological differences between two populations based on the GOPs. To demonstrate the sensitivity of the method, an analysis exploring differences between hippocampi of first-episode schizophrenics and controls is presented. Each hippocampus is represented by a discrete skeletal representation (s-rep). We investigate important model properties using the statistics of populations. These properties are highlighted by the s-rep model that allows accurate capture of the object interior and boundary while, by design, being suitable for statistical analysis of populations of objects. By supporting non-Euclidean GOPs such as direction vectors, the proposed hypothesis test is novel in the study of morphological shape differences. Suitable difference measures are proposed for each GOP. Both global and simultaneous GOP analyses showed statistically significant differences between the first-episode schizophrenics and controls.
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Abbreviations
- CDF:
-
Cumulative distribution function
- CG:
-
Control group
- CPNG:
-
Composite principal nested great spheres
- CPNS:
-
Composite principal nested spheres
- DiProPerm:
-
Direction projection permutation
- DWD:
-
Distance-weighted discrimination
- FDR:
-
False discovery rate
- FWER:
-
Familywise error rate
- GOP:
-
Geometric object property
- KDE:
-
Kernel density estimate
- MD:
-
Mean difference
- MRI:
-
Magnetic resonance imaging
- PCA:
-
Principal component analysis
- PDM:
-
Point distribution model
- PGA:
-
Principal geodesic analysis
- PNG:
-
Principal nested great spheres
- PNS:
-
Principal nested spheres
- PP1:
-
Pre-processing step 1
- PP2:
-
Pre-processing step 2
- RFT:
-
Random field theory
- ROC:
-
Receiver operating characteristic
- S-rep:
-
Skeletal representation
- SG:
-
Schizophrenia group
References
Abramovich, F., Benjamini, Y.: Adaptive thresholding of wavelet coefficients. Comput. Stat. Data Anal. 22(4), 351–361 (1996)
Albertson, R.C., Streelman, J.T., Kocher, T.D.: Genetic basis of adaptive shape differences in the cichlid head. J. Hered. 94(4), 291–301 (2003)
Benjamini, Y., Hochberg, Y.: Controlling the false discovery rate: a practical and powerful approach to multiple testing. J. R. Stat. Soc. Ser. B Stat. Methodol 57(1), 289–300 (1995)
Bookstein, F.L.: Landmark methods for forms without landmarks: morphometrics of group differences in outline shape. Med. Image Anal. 10(3), 225–243 (1996)
Bullmore, E., Fadili, J., Breakspear, M., Salvador, R., Suckling, J., Brammer, M.: Wavelets and statistical analysis of functional magnetic resonance images of the human brain. Stat. Methods Med. Res. 12(5), 375–399 (2003)
Chumbley, J.R., Friston, K.J.: False discovery rate revisited: FDR and topological inference using Gaussian random fields. NeuroImage 44(1), 62–70 (2009)
Cootes, T.F., Taylor, C., Cooper, D., Graham, J.: Training models of shape from sets of examples. In: Hogg, D., Boyle, R. (eds.) Proceedings of the British Machine Vision Conference, pp. 9–18. Springer, Berlin (1992)
Cramér, H., Wold, H.: Some theorems on distribution functions. J. Lond. Math. Soc. 1(11), 4 (1936)
Damon, J.: Smoothness and geometry of boundaries associated to skeletal structures: sufficient conditions for smoothness. Ann. Inst. Fourier 53, 1001–1045 (2003)
Damon, J.: Swept regions and surfaces: modeling and volumetric properties. Conf. Comput. Alg. Geom. 2006(392), 66–91 (2008)
Damon, J., Marron, J.S.: Backwards principal component analysis and principal nested relations. J. Math. Imaging Vis. 50, 107–114 (2014)
Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley, Chichester (1998)
Edgington, E.: Randomization Tests, 3rd edn. Dekker, New York (1995)
Ferrarini, L., Palm, W.M., Olofsen, H., van Buchem, M.A., Reiber, J.H., Admiraal-Behloul, F., et al.: Shape differences of the brain ventricles in Alzheimer’s disease. NeuroImage 32(3), 1060–1069 (2006)
Fletcher, P.T., Lu, C., Pizer, S.M., Joshi, S.: Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Trans. Med. Imaging 23, 995–1005 (2004)
Fréchet, M.: Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. Henri Poincaré 10, 215–310 (1948)
Gerig, G., Styner, M., Shenton, M.E., Lieberman, J.A.: Shape versus size: improved understanding of the morphology of brain structures. In: MICCAI pp. 24–32 (2001)
Goodall, C.: Procrustes methods in the statistical analysis of shape. J. R. Stat. Soc. Ser. B Stat. Methodol. 53(2B), 285–339 (1991)
Gouttard, S., Styner, M., Joshi, S., Gerig, G.: Subcortical structure segmentation using probabilistic atlas prior. In: Proceedings of the SPIE Medical Imaging, vol. 65122, pp. J1–J11 (2007)
Hong, J., Vicory, J., Schulz, J., Styner, M., Marron, J., Pizer, S.M.: Classification of medically imaged objects via s-rep statistics. Med. Image Anal. (to appear)
Huckemann, S., Hotz, T., Munk, A.: Intrinsic shape analysis: geodesic PCA for Riemannian manifolds modulo isometric Lie group actions. Stat. Sin. 20(1), 1–58 (2010)
Jung, S., Dryden, I.L., Marron, J.S.: Analysis of principal nested spheres. Biometrika 99(3), 551–568 (2012)
Jung, S., Foskey, M., Marron, J.S.: Principal arc analysis on direct product manifolds. Ann. Appl. Stat. 5(1), 578–603 (2011)
Jung, S., Liu, X., Marron, J.S., Pizer, S.M.: Generalized PCA via the backward stepwise approach in image analysis. In: Angeles, J. et al. (eds.) Brain, Body and Machine: Proceedings of an International Symposium on the 25th Anniversary of McGill University Centre for Intelligent Machines, Advances in Intelligent and Soft Computing, vol. 83, pp. 111–123 (2010)
Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30(5), 509–541 (1977)
Kendall, D.G., Barden, D., Carne, T.K., Le, H.: Shape and Shape Theory. Wiley, Chichester (1999)
Kilner, J.M., Kiebel, S.J., Friston, K.J.: Applications of random field theory to electrophysiology. Neurosci. Lett. 374, 174–178 (2005)
Kurtek, S., Ding, Z., Klassen, E., Srivastava, A.: Parameterization-invariant shape statistics and probabilistic classification of anatomical surfaces. Inf. Process. Med. Imaging 22, 147–158 (2011)
Mamah, D., Harms, M.P., Barch, D.M., Styner, M.A., Lieberman, J., Wang, L.: Hippocampal shape and volume changes with antipsychotics in early stage psychotic illness. Front. Psychiatry 3(96), 1–10 (2012)
Marozzi, M.: Some remarks about the number of permutations one should consider to perform a permutation test. Statistica 64(1), 193–202 (2004)
Marron, J.S., Todd, M.J., Ahn, J.: Distance weighted discrimination. J. Am. Stat. Assoc. 102(480), 1267–1271 (2007)
McClure, R.K., Styner, M., Maltbie, E., Liebermann, J.A., Gouttard, S., Gerig, G., Shi, X., Zhu, H., et al.: Localized differences in caudate and hippocampal shape are associated with schizophrenia but not antipsychotic type. Psychiatry Res. Neuroimaging 211(1), 1–10 (2013)
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., Bilder, R.M.: Regional specificity of hippocampal volume reductions in first-episode schizophrenia. NeuroImage 21(4), 1563–1575 (2004)
Nichols, T.E., Hayasaka, S.: Controlling the familywise error rate in functional neuroimaging: a comparative review. Stat. Methods Med. Res. 12(5), 419–446 (2003)
Nitrc: S-rep fitting, statistics, and segmentation. http://www.nitrc.org/projects/sreps (2013)
Pantazis, D., Nichols, T.E., Baillet, S., Leahy, R.M.: A comparison of random field theory and permutation methods for the statistical analysis of MEG data. NeuroImage 25(2B), 383–394 (2005)
Pennec, X.: Statistical computing on manifolds: from Riemannian geometry to computational anatomy. Emerg. Trends Vis. Comput. 5416, 347–386 (2008)
Pesarin, F.: Multivariate Permutation Tests with Applications to Biostatistics. Wiley, Chichester (2001)
Pizer, S.M., Hong, J., Jung, S., Marron, J.S., Schulz, J., Vicory, J.: Relative statistical performance of s-reps with principal nested spheres vs. PDMs. In: Proceedings of Shape 2014, Symposium of statistical shape models and applications. SICAS (2014)
Pizer, S.M., Jung, S., Goswami, D., Zhao, X., Chaudhuri, R., Damon, J.N., Huckemann, S., Marron, J.S.: Nested sphere statistics of skeletal models. Innovations for Shape Analysis: Models and Algorithms, Lecture Notes in Computer Science, pp. 93–115. Springer, Berlin (2013)
Qiao, X., Zhang, H.H., Liu, Y., Todd, M.J., Marron, J.S.: Weighted distance weighted discrimination and its asymptotic properties. J. Am. Stat. Assoc. 105(489), 401–414 (2010)
Rohde, G.K., Ribeiro, A.J.S., Dahl, K.N., Murphy, R.F.: Deformation-based nuclear morphometry: capturing nuclear shape variation in HeLa cells. Cytom. A 73(4), 341–350 (2008)
Schulz, J., Jung, S., Huckemann, S., Pierrynowski, M., Marron, J.S., Pizer, S.M.: Analysis of rotational deformations from directional data. J. Comput. Graph. Stat. 24(2), 539–560 (2015)
Shi, X., Ibrahim, J.G., Lieberman, J., Styner, M., Li, Y., Zhu, H.: Two-stage empirical likelihood for longitudinal neuroimaging data. Ann. Appl. Stat. 5(2B), 1132–1158 (2011)
Siddiqi, K., Pizer, S.: Medial Representations: Mathematics, Algorithms and Applications. Computational Imaging and Vision, vol. 37, 1st edn. Springer, Dordrecht (2008)
Silverman, B.: Monographs on Statistics and Applied Probability, 1st edn. Springer, New York (1986)
Styner, M., Lieberman, J., Pantazis, D., Gerig, G.: Boundary and medial shape analysis of the hippocampus in schizophrenia. Med. Image Anal. 8(3), 197–203 (2004)
Terriberry, T., Joshi, S., Gerig, G.: Hypothesis Testing with Nonlinear Shape Models. In: Christensen, G., Sonka, M. (eds.) Information Processing in Medical Imaging, Lecture Notes in Computer Science, vol. 3565, pp. 15–26. Springer, Berlin (2005)
Van De Ville, D., Blu, T., Unser, M.: Integrated wavelet processing and spatial statistical testing of fMRI data. NeuroImage 23(4), 1472–1485 (2004)
Vicory, J., Saboo, R., Juttokonda, M.R., Rosenman, J.G., Niethammer, M., Pizer, S.M.: Constrained smooth interpolation of slice-segmented medical images via Laplacian of curvature flow (2015, in preparation)
Wang, L., Joshi, S.C., Miller, M.I., Csernansky, J.G.: Statistical analysis of hippocampal asymmetry in schizophrenia. NeuroImage 14(3), 531–545 (2001)
Wei, S., Lee, C., Wichers, L., Marron, J.S.: Direction-projection-permutation for high dimensional hypothesis tests. J. Comput. Graph. Stat. (2015). doi: 10.1080/10618600.2015.1027773
Acknowledgments
The following researchers have also contributed to this work: Jared Vicory (UNC) gave advice on running Pablo and provided earlier fits of 62 hippocampi, Juan Carlos Prieto (CREATIS-INSA, France) provided the implementation of a crest interpolation term in Pablo and removed bugs from the program, Sungkyu Jung (University of Pittsburgh, USA) provided Fig. 2, program code and additional discussions about CPNS, Martin Styner (UNC) provided the hippocampus dataset and answered questions. The first author acknowledges support from the Norwegian Research Council through grant 176872/V30 in the eVita program and additional support from the Tromsø Telemedicine Laboratory and the Department of Electrical Engineering and Computer Science at the University of Stavanger, Norway.
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Schulz, J., Pizer, S.M., Marron, J.S. et al. Non-linear Hypothesis Testing of Geometric Object Properties of Shapes Applied to Hippocampi. J Math Imaging Vis 54, 15–34 (2016). https://doi.org/10.1007/s10851-015-0587-7
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DOI: https://doi.org/10.1007/s10851-015-0587-7