Advertisement

Reduced-Rank VGLMs

  • Thomas W. Yee
Chapter
Part of the Springer Series in Statistics book series (SSS)

Abstract

This chapter looks at a subclass of VGLMs called Reduced-Rank VGLMs (RR-VGLMs). They are built on the idea of latent variables, and are the same as VGLMs except some of their constraint matrices are estimated. RR-VGLMs have with interesting properties and applications. It is a dimension-reduction method, e.g., when applied to the multinomial logit model it leads to the stereotype model. Another related class of models is row–column interaction models (RCIMs), and two-parameter RR-VGLMs are described in this chapter. Some applications mentioned here and/or developed elsewhere include quasi-variances and indirect gradient analysis.

Keywords

Latent Variable Linear Discriminant Analysis Latent Trait Constraint Matrice Structural Zero 
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.

References

  1. Agresti, A. 2013. Categorical Data Analysis (Third ed.). Hoboken: Wiley.zbMATHGoogle Scholar
  2. Ahn, S. K. and G. C. Reinsel 1988. Nested reduced-rank autoregressive models for multiple time series. Journal of the American Statistical Association 83(403):849–856.zbMATHMathSciNetGoogle Scholar
  3. Anderson, J. A. 1984. Regression and ordered categorical variables. Journal of the Royal Statistical Society, Series B 46(1):1–30. With discussion.Google Scholar
  4. Anderson, T. W. 1951. Estimating linear restrictions on regression coefficients for multivariate normal distributions. Annals of Mathematical Statistics 22(3):327–351.zbMATHMathSciNetCrossRefGoogle Scholar
  5. Andrews, H. P., R. D. Snee, and M. H. Sarner 1980. Graphical display of means. American Statistician 34(4):195–199.Google Scholar
  6. Baker, F. B. and S.-H. Kim 2004. Item Response Theory: Parameter Estimation Techniques (Second ed.). New York: Marcel Dekker.Google Scholar
  7. Bock, R. D. and M. Leiberman 1970. Fitting a response model for n dichotomously scored items. Psychometrika 35(2):179–197.CrossRefGoogle Scholar
  8. Carroll, R. J. and D. Ruppert 1988. Transformation and Weighting in Regression. New York: Chapman and Hall.zbMATHCrossRefGoogle Scholar
  9. de Gruijter, D. N. M. and L. J. T. Van der Kamp 2008. Statistical Test Theory for the Behavioral Sciences. Boca Raton, FL, USA: Chapman & Hall/CRC.Google Scholar
  10. Firth, D. 2003. Overcoming the reference category problem in the presentation of statistical models. Sociological Methodology 33(1):1–18.CrossRefGoogle Scholar
  11. Firth, D. and R. X. de Menezes 2004. Quasi-variances. Biometrika 91(1):65–80.zbMATHMathSciNetCrossRefGoogle Scholar
  12. Gabriel, K. R. and S. Zamir 1979. Lower rank approximation of matrices by least squares with any choice of weights. Technometrics 21(4):489–498.zbMATHCrossRefGoogle Scholar
  13. Goodman, L. A. 1981. Association models and canonical correlation in the analysis of cross-classifications having ordered categories. Journal of the American Statistical Association 76(374):320–334.MathSciNetGoogle Scholar
  14. Hilbe, J. M. 2011. Negative Binomial Regression (Second ed.). Cambridge, UK; New York, USA: Cambridge University Press.Google Scholar
  15. Izenman, A. J. 1975. Reduced-rank regression for the multivariate linear model. Journal of Multivariate Analysis 5(2):248–264.zbMATHMathSciNetCrossRefGoogle Scholar
  16. Izenman, A. J. 2008. Modern Multivariate Statistical Techniques: Regression, Classification, and Manifold Learning. New York, USA: Springer.CrossRefGoogle Scholar
  17. Liu, H. and K. S. Chan 2010. Introducing COZIGAM: An R package for unconstrained and constrained zero-inflated generalized additive model analysis. Journal of Statistical Software 35(11):1–26.CrossRefGoogle Scholar
  18. McCullagh, P. and J. A. Nelder 1989. Generalized Linear Models (Second ed.). London: Chapman & Hall.zbMATHCrossRefGoogle Scholar
  19. Mosteller, F. and J. W. Tukey 1977. Data Analysis and Regression. Reading, MA, USA: Addison-Wesley.Google Scholar
  20. Rasch, G. 1961. On general laws and the meaning of measurement in psychology. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability 4:321–333.Google Scholar
  21. Reinsel, G. C. and R. P. Velu 1998. Multivariate Reduced-Rank Regression: Theory and Applications. New York, USA: Springer-Verlag.zbMATHCrossRefGoogle Scholar
  22. Reinsel, G. C. and R. P. Velu 2006. Partically reduced-rank multivariate regression models. Statistica Sinica 16(3):899–917.zbMATHMathSciNetGoogle Scholar
  23. Richards, F. S. G. 1961. A method of maximum-likelihood estimation. Journal of the Royal Statistical Society, Series B 23(2):469–475.zbMATHMathSciNetGoogle Scholar
  24. Schenker, N. and J. F. Gentleman 2001. On judging the significance of differences by examining the overlap between confidence intervals. American Statistician 55(3):182–186.MathSciNetCrossRefGoogle Scholar
  25. Seber, G. A. F. and C. J. Wild 1989. Nonlinear Regression. New York, USA: Wiley.zbMATHCrossRefGoogle Scholar
  26. Smyth, G. K. 1996. Partitioned algorithms for maximum likelihood and other nonlinear estimation. Statistics and Computing 6(3):201–216.CrossRefGoogle Scholar
  27. Smyth, G. K., A. F. Huele, and A. P. Verbyla 2001. Exact and approximate REML for heteroscedastic regression. Statistical Modelling 1(3):161–175.zbMATHCrossRefGoogle Scholar
  28. Taylor, L. R. 1961. Aggregation, variance and the mean. Nature 189(4766): 732–735.CrossRefGoogle Scholar
  29. Yee, T. W. 2014. Reduced-rank vector generalized linear models with two linear predictors. Computational Statistics & Data Analysis 71:889–902.MathSciNetCrossRefGoogle Scholar
  30. Yee, T. W. and A. F. Hadi 2014. Row-column interaction models, with an R implementation. Computational Statistics 29(6):1427–1445.zbMATHMathSciNetCrossRefGoogle Scholar
  31. Yee, T. W. and T. J. Hastie 2003. Reduced-rank vector generalized linear models. Statistical Modelling 3(1):15–41.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Thomas Yee 2015

Authors and Affiliations

  • Thomas W. Yee
    • 1
  1. 1.Department of StatisticsUniversity of AucklandAucklandNew Zealand

Personalised recommendations