zbMATH — the first resource for mathematics

Reduced-rank vector generalized linear models with two linear predictors. (English) Zbl 06975433
Summary: Vector generalized linear models (VGLMs) as implemented in the vgam R package permit multiple parameters to depend (via inverse link functions) on linear predictors. However it is often the case that one wishes different parameters to be related to each other in some way (i.e., to jointly satisfy certain constraints). Prominent and important examples of such cases include the normal or Gaussian family where one wishes to model the variance as a function of the mean, e.g., variance proportional to the mean raised to some power. Another example is the negative binomial family whose variance is approximately proportional to the mean raised to some power. It is shown that such constraints can be implemented in a straightforward manner via reduced rank regression (RRR) and easily used via the rrvglm() function. To this end RRR is briefly described and applied so as to impose parameter constraints in VGLMs with two parameters. The result is a rank-1 RR-VGLM. Numerous examples are given, some new, of the use of this technique. The implication here is that RRR offers hitherto undiscovered potential usefulness to many statistical distributions.

62 Statistics
Full Text: DOI
[1] Ahn, S. K.; Reinsel, G. C., Nested reduced-rank autoregressive models for multiple time series, Journal of the American Statistical Association, 83, 403, 849-856, (1988) · Zbl 0663.62097
[2] Anderson, T. W., Estimating linear restrictions on regression coefficients for multivariate normal distributions, Annals of Mathematical Statistics, 22, 3, 327-351, (1951) · Zbl 0043.13902
[3] Cameron, A. C.; Trivedi, P. K., Econometric models based on count data: comparisons and applications of some estimators and tests, Journal of Applied Econometrics, 1, 1, 29-53, (1986)
[4] Cameron, A. C.; Trivedi, P. K., Regression analysis of count data, (1998), Cambridge University Press Cambridge · Zbl 0924.62004
[5] Carroll, R. J.; Ruppert, D., Transformation and weighting in regression, (1988), Chapman and Hall New York, NY, USA · Zbl 0666.62062
[6] Dunn, P.K., 2012. tweedie: Tweedie exponential family models. R package version 2.1.5.
[7] Dunn, P.K., Smyth, G.K., 2012. dglm: double generalized linear models. R package version 1.6.2. URL: http://CRAN.R-project.org/package=dglm.
[8] Fiocco, M.; Putter, H.; van Houwelingen, J. C., Reduced rank proportional hazards model for competing risks, Biostatistics, 6, 3, 465-478, (2005) · Zbl 1070.62100
[9] Fletcher, D. J., Estimating overdispersion when Fitting a generalized linear model to sparse data, Biometrika, 99, 1, 230-237, (2012) · Zbl 1234.62109
[10] Fullerton, A. S., A conceptual framework for ordered logistic regression models, Sociological Methods & Research, 38, 2, 306-347, (2009)
[11] Greene, W., Functional forms for the negative binomial model for count data, Economics Letters, 99, 3, 585-590, (2008) · Zbl 1255.62010
[12] Heinen, A.; Rengifo, E., Multivariate reduced rank regression in non-Gaussian contexts, using copulas, Computational Statistics & Data Analysis, 52, 6, 2931-2944, (2008) · Zbl 05564680
[13] Hilbe, J. M., Negative binomial regression, (2011), Cambridge University Press Cambridge, UK, New York, USA · Zbl 1269.62063
[14] Hilbe, J.M., 2012. COUNT: functions, data and code for count data. R package version 1.2.3. URL: http://CRAN.R-project.org/package=COUNT.
[15] Izenman, A. J., Reduced-rank regression for the multivariate linear model, Journal of Multivariate Analysis, 5, 2, 248-264, (1975) · Zbl 0313.62042
[16] Jackman, S., 2012. pscl: classes and methods for R Developed in the Political Science Computational Laboratory, Stanford University. Department of Political Science, Stanford University, Stanford, California, USA, R package version 1.04.4. URL: http://pscl.stanford.edu/.
[17] Jørgensen, B., Exponential dispersion models, Journal of the Royal Statistical Society: Series B, 49, 2, 127-162, (1987) · Zbl 0662.62078
[18] Kleiber, C.; Zeileis, A., Applied econometrics with R, (2008), Springer New York, NY, USA · Zbl 1155.91004
[19] Kneib, T., Heinzl, F., Brezger, A., Bove, D.S., 2011. BayesX: R utilities accompanying the software package BayesX. R package version 0.2-5. URL: http://CRAN.R-project.org/package=BayesX.
[20] Lawless, J. F., Negative binomial and mixed Poisson regression, The Canadian Journal of Statistics, 15, 3, 209-225, (1987) · Zbl 0632.62060
[21] Li, S.; Yang, F.; Famoye, F.; Lee, C.; Black, D., Quasi-negative binomial distribution: properties and applications, Computational Statistics & Data Analysis, 55, 7, 2363-2371, (2011) · Zbl 1328.62175
[22] Liu, H., Chan, K.-S., 2009. Constrained generalized additive models for zero-inflated data. Tech. Rep. 388, University of Iowa, IA, USA. URL: http://www.stat.uiowa.edu/sites/default/files/techrep/tr388-version2.pdf.
[23] Liu, H.; Chan, K.-S., Introducing COZIGAM: an R package for unconstrained and constrained zero-inflated generalized additive model analysis, Journal of Statistical Software, 35, 11, 1-26, (2010), URL: http://www.jstatsoft.org/v35/i11/
[24] Neykov, N. M.; Filzmoser, P.; Neytchev, P. N., Robust joint modeling of mean and dispersion through trimming, Computational Statistics & Data Analysis, 56, 1, 34-48, (2012) · Zbl 1239.62018
[25] Richards, F. S.G., A method of maximum-likelihood estimation, Journal of the Royal Statistical Society. Series B. Methodological, 23, 2, 469-475, (1961) · Zbl 0104.13003
[26] Rigby, R. A.; Stasinopoulos, D. M., Generalized additive models for location, scale and shape, (with discussion), Applied Statistics, 54, 507-554, (2005) · Zbl 05188697
[27] Smyth, G. K.; Huele, A. F.; Verbyla, A. P., Exact and approximate REML for heteroscedastic regression, Statistical Modelling, 1, 3, 161-175, (2001) · Zbl 1104.62080
[28] Taylor, L. R., Aggregation, variance and the mean, Nature, 189, 4766, 732-735, (1961)
[29] Turner, H.; Firth, D., Gnm: a package for generalized nonlinear models, R News, 7, 2, 8-12, (2007), URL: http://CRAN.R-project.org/doc/Rnews/
[30] Venables, W. N.; Ripley, B. D., Modern applied statistics with S, (2002), Springer New York, URL: http://www.stats.ox.ac.uk/pub/MASS4 · Zbl 1006.62003
[31] Ver Hoef, J. M.; Boveng, P. L., Quasi-Poisson vs. negative binomial regression: how should we model overdispersed count data?, Ecology, 88, 11, 2766-2772, (2007)
[32] Winkelmann, R.; Zimmermann, K., Recent developments in count data modeling: theory and application, Journal of Economic Surveys, 9, 1, 1-36, (1995)
[33] Yee, T. W., A new technique for maximum-likelihood canonical Gaussian ordination, Ecological Monographs, 74, 4, 685-701, (2004)
[34] Yee, T. W., Constrained additive ordination, Ecology, 87, 1, 203-213, (2006)
[35] Yee, T. W., The VGAM package, R News, 8, 2, 28-39, (2008), URL: http://CRAN.R-project.org/doc/Rnews/
[36] Yee, T. W., The VGAM package for categorical data analysis, Journal of Statistical Software, 32, 10, 1-34, (2010), URL: http://www.jstatsoft.org/v32/i10/
[37] Yee, T. W.; Hastie, T. J., Reduced-rank vector generalized linear models, Statistical Modelling, 3, 1, 15-41, (2003) · Zbl 1195.62123
[38] Zeileis, A.; Kleiber, C.; Jackman, S., Regression models for count data in R, Journal of Statistical Software, 27, 8, 1-25, (2008), URL: http://www.jstatsoft.org/v27/i08
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.