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Decomposition and model selection for large contingency tables. (English) Zbl 1207.62126
Summary: Large contingency tables summarizing categorical variables arise in many areas. One example is in biology, where large numbers of biomarkers are cross-tabulated according to their discrete expression level. Interactions of the variables are of great interest and are generally studied with log-linear models. The structure of a log-linear model can be visually represented by a graph from which the conditional independence structure can then be easily read off. However, since the number of parameters in a saturated model grows exponentially in the number of variables, this generally comes with a heavy computational burden. Even if we restrict ourselves to models of lower-order interactions or other sparse structures, we are faced with the problem of a large number of cells which play the role of sample size. This is in sharp contrast to high-dimensional regression or classification procedures because, in addition to a high-dimensional parameter, we also have to deal with the analogue of a huge sample size. Furthermore, high-dimensional tables naturally feature a large number of sampling zeros which often leads to the nonexistence of the maximum likelihood estimate. We therefore present a decomposition approach, where we first divide the problem into several lower-dimensional problems and then combine these to form a global solution. Our methodology is computationally feasible for log-linear interaction models with many categorical variables each or some of them having many levels. We demonstrate the proposed method on simulated data and apply it to a bio-medical problem in cancer research.

MSC:
62H17 Contingency tables
62J12 Generalized linear models (logistic models)
62P10 Applications of statistics to biology and medical sciences; meta analysis
05C90 Applications of graph theory
65C60 Computational problems in statistics (MSC2010)
62-04 Software, source code, etc. for problems pertaining to statistics
Software:
MICE
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References:
[1] Bishop, Discrete Multivariate Analysis (1975)
[2] Breiman, Random forests, Machine Learning 45 pp 5– (2001) · Zbl 1007.68152
[3] Christensen, Linear Models for Multivariate, Time Series, and Spatial Data (1991) · Zbl 0717.62079
[4] Dahinden, Mining tissue microarray data to uncover combinations of biomarker expression patterns that improve intermediate staging and grading of clear cell renal cell cancer, Clinical Cancer Research 16 pp 88– (2010)
[5] Dahinden, Penalized likelihood for sparse contingency tables with an application to full-length cDNA libraries, BMC Bioinformatics 8 pp 476– (2007)
[6] Darroch, Markov fields and log-linear interaction models for contingency tables, Annals of Statistics 8 pp 522– (1980) · Zbl 0444.62064
[7] Imai, Hypoxia attenuates the expression of E-cadherin via up-regulation of SNAIL in ovarian carcinoma cells, The American Journal of Pathology 163 pp 1437– (2003)
[8] Jackson, L., Gray, A. and Fienberg, S. ( 2007). Sequential category aggregation and partitioning approach for multi-way contingency tables based on survey and census data, preprint. · Zbl 1149.62049
[9] Kallioniemi, Tissue microarray technology for high-throughput molecular profiling of cancer, Human Molecular Genetics 10 pp 657– (2001)
[10] Kim, S. ( 2005). Log-linear modelling for contingency tables by using marginal model structures. Research Report 05, Division of Applied Mathematics, Korea Advanced Institute of Science and Technology.
[11] Kim, In vitro transcriptional activation of p21 promoter by p53, Biochemical and Biophysical Research Communication 234 pp 300– (1997)
[12] Lauritzen (1996)
[13] Mazal, Expression of aquaporins and PAX-2 compared to CD10 and cytokeratin 7 in renal neoplasms: a tissue microarray study, Modern Pathology 18 pp 535– (2005)
[14] Olesen, Maximal prime subgraph decomposition of Bayesian networks, IEEE Transactions on Systems, Man, and Cybernetics, Part B 32 pp 21– (2002)
[15] Osipov, Expression of p27 and VHL in renal tumors, Applied Immunohistochemistry & Molecular Morphology 10 pp 344– (2002)
[16] Ravikumar, High-dimensional graphical model selection using l1-regularized logistic regression, Annals of Statistics (2009)
[17] Roe, p53 stabilization and transactivation by a von Hippel-Lindau protein, Molecular Cell 22 pp 395– (2006)
[18] Strobl, Bias in random forest variable importance measures: illustrations, sources and a solution, BMC Bioinformatics 8 pp 25– (2007)
[19] Tibshirani, Regression shrinkage and selection via the lasso, Journal of the Royal Statistical Society B 58 pp 267– (1996) · Zbl 0850.62538
[20] van Buuren, S. and Oudshoorn, C. ( 2007). Mice: multivariate imputation by chained equations. R package version 1.16. http://web.inter.nl.net/users/S.van.Buuren/mi/html/mice.htm.
[21] Wainwright, Advances in Neural Information Processing Systems 19 pp 1465– (2007)
[22] Wenger, R., Stiehl, D. and Camenisch, G. ( 2005). Integration of oxygen signaling at the consensus HRE. Science Signaling: Signal Transduction Knowledge Environment (STKE) 2005, re12.
[23] Yuan, Model selection and estimation in regression with grouped variables, Journal of the Royal Statistical Society B 68 pp 49– (2006) · Zbl 1141.62030
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