Advertisement

Indirect Comparison of Interaction Graphs

  • Ulrich Mansmann
  • Markus Schmidberger
  • Ralf Strobl
  • Vindi Jurinovic
Chapter

Abstract

Astrategy for testing differential conditional independence structures (CIS) between two graphs is introduced. The graphs have the same set of nodes and are estimated from data sampled under two different conditions. The test uses the entire pathplot in a Lasso regression as the information on how a node connects with the remaining nodes in the graph.

The interpretation of the paths as random processes allows defining stopping times which make the statistical properties of the test statistic accessible to analytic reasoning. A resampling approach is proposed to calculated p-values simultaneously for a hierarchical testing procedure. The hierarchical testing steps through a given hierarchy of clusters. First, collective effects are measured at the coarsest level possible (the global null hypothesis that no node in the graph shows a differential CIS). If the global null hypothesis can be rejected, finer resolution levels are tested for an effect until the level of individual nodes is reached.

The strategy is applied to association patterns of categories from the ICF in patients under post-acute rehabilitation. The patients are characterized by two different conditions. Acomprehensive understanding of differences in the conditional independence structures between the patient groups is pivotal for evidence-based intervention design on the policy, the service and the clinical level related to their treatment. Due to extensive computation, parallel computing offers an effective approach to implement our explorative tool and to locate nodes in a graph which show differential CIS between two conditions.

Keywords

Indirect Comparison Interaction Graph Lasso Regression Family Wise Error Rate Global Null Hypothesis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

This work is supported by the LMUinnovativ project Analysis and Modelling of Complex Systems in Biology and Medicine (Cluster B, Expression Analyses).

References

  1. Ashburner, M., Ball, C. A., Blake, J.A., Botstein, D., Butler, H., Cherry, J.M., Davis, A. P., Dolinski, K., Dwight, S. S., Eppig, J. T., Harris, M. A., Hill, D. P., Issel-Tarver, L., Kasarskis, A., Lewis, S., Matese, J. C., Richardson, J. E., Ringwald, M., Rubin, G. M. & Sherlock, G. (2000). Gene ontology:a tool for the unification of biology., Nature Genetics 25: 25–29.CrossRefGoogle Scholar
  2. Balasubramanian, R., LaFramboise, T., Scholtens, D. & Gentleman, R. (2004). A graph-theoretic approach to testing associations between disparate sources of functional genomics data., Bioinformatics 20(18): 3353–3362.CrossRefGoogle Scholar
  3. Banerjee, O., Ghaoui, L. E. & d’Aspremont, A. (2008). Model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data, Journal of Machine Learning Research pp. 485–516.Google Scholar
  4. Friedman, J., Hastie, T. & Tibshirani, R. (2007). Sparse inverse covariance estimation with the graphical lasso, Biostatistics .Google Scholar
  5. Gneiting, T. (2008). Editorial: Probabilistic forecasting, Journal of the Royal Statistical Society: Series A 17: 319–321.Google Scholar
  6. Goeman, J. (2008). penalized: L1 (lasso) and L2 (ridge) penalized estimation in GLMs and in the Cox model. http://www.msbi.nl/goeman. R package version 0.9-22.
  7. Goeman, J. (2009a). L1 and l2 penalized regression models. http://www.msbi.nl/goeman. R package version 0.9-24.
  8. Goeman, J. (2009b). L1 penalized estimation in the cox proportional hazards model., Biometrical Journal .Google Scholar
  9. Goeman, J. J. & Mansmann, U. (2008). Multiple testing on the directed acyclic graph of gene ontology., Bioinformatics 24(4): 537–544.CrossRefGoogle Scholar
  10. Kalisch, M. & Bühlmann, P. (2007). Estimating high dimensional acyclic graphs with the pcalgorithm, Journal of Machine Learning Research 8: 613–636.Google Scholar
  11. Meinshausen, N. (2008). Hierarchical testing of variable importance, Biometrika 95: 265–276.MATHCrossRefMathSciNetGoogle Scholar
  12. Meinshausen, N. & Bühlmann, P. (2006). High dimensional graphs and variable selection with the Lasso, The Annals of Statistics 34: 1436–1462.MATHCrossRefMathSciNetGoogle Scholar
  13. R Development Core Team (2009). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, http://www.R-project.org.
  14. Ruschhaupt, M. (2008). Erzeugung von positiv definiten Matrizen mit Nebenbedingungen zur Validierung von Netzwerkalgorithmen für Microarray-Daten, PhD thesis, LMU München, Fakultät für Mathematik, Informatik und Statistik, München, Germany. http://edoc.ub.uni-muenchen.de/view/subjects/fak16.html.
  15. Schäfer, J. & Strimmer, K. (2005). Ashrinkage approach to large-scale covariancematrix estimation and implications for functional genomics., Statistical Applications in Genetics and Molecular Biology 4: Article32.Google Scholar
  16. Schmidberger, M., Morgan, M., Eddelbuettel, D., Yu, H., Tierney, L. & Mansmann, U. (2009). State of the art in parallel computing with R, Journal of Statistical Software 31(1). http://www.jstatsoft.org/v31/i01/.
  17. Wainwright, M. J., Ravikumar, P. & Lafferty, J. D. (2006). High dimensional graphical model selection using l1-regularized logistic regression, Proceedings of Advances in neural information processing systems 9: 1465–1472.Google Scholar
  18. WHO(2001). International classification of functioning, disability and health (icf), Nature Genetics Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Ulrich Mansmann
    • 1
  • Markus Schmidberger
    • 1
  • Ralf Strobl
    • 2
  • Vindi Jurinovic
    • 1
  1. 1.IBELMU MunichMunichGermany
  2. 2.Institute for Health and Rehabilitation SciencesLMU MunichMunichGermany

Personalised recommendations