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Row-column interaction models, with an R implementation. (English) Zbl 1306.65153

Summary: We propose a family of models called row-column interaction models (RCIMs) for two-way table responses. RCIMs apply some link function to a parameter (such as the cell mean) to equal a row effect plus a column effect plus an optional interaction modelled as a reduced-rank regression. What sets this work apart from others is that our framework incorporates a very wide range of statistical models, e.g., (1) log-link with Poisson counts is Goodman’s RC model, (2) identity-link with a double exponential distribution is median polish, (3) logit-link with Bernoulli responses is a Rasch model, (4) identity-link with normal errors is two-way ANOVA with one observation per cell but allowing semi-complex modelling of interactions of the form \(\mathbf{AC}^T\), (5) exponential-link with normal responses are quasi-variances. Proposed here also is a least significant difference plot augmentation of quasi-variances. Being a special case of RCIMs, quasi-variances are naturally extended from the \(M=1\) linear/additive predictor \(\eta \) case (within the exponential family) to the \(M>1\) case (vector generalized linear model families). A rank-1 Goodman’s RC model is also shown to estimate the site scores and optimums of an equal-tolerances Poisson unconstrained quadratic ordination. New functions within the VGAM R package are described with examples. Altogether, RCIMs facilitate the analysis of matrix responses of many data types, therefore are potentially useful to many areas of applied statistics.

MSC:

62-08 Computational methods for problems pertaining to statistics
62J12 Generalized linear models (logistic models)
62-04 Software, source code, etc. for problems pertaining to statistics
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[1] Andrews HP, Snee RD, Sarner MH (1980) Graphical display of means. Am Stat 34:195-199
[2] de Rooij M (2007) The distance perspective of generalized biadditive models: scalings and transformations. J Comput Graph Stat 16:210-227 · doi:10.1198/106186007X180101
[3] Easton DF, Peto J, Babiker AGAG (1991) Floating absolute risk: an alternative to relative risk in survival and case-control analysis avoiding an arbitrary reference group. Stat Med 10:1025-1035 · doi:10.1002/sim.4780100703
[4] Firth D (2000) Quasi-variances in Xlisp-Stat and on the web. J Stat Softw 5:1-13, http://www.jstatsoft.org/v05/i04 · Zbl 1195.62123
[5] Firth, D.; Menezes, RX, No article title, Quasi-variances. Biometrika, 91, 65-80 (2004) · Zbl 1132.62340 · doi:10.1093/biomet/91.1.65
[6] Goodman LA (1981) Association models and canonical correlation in the analysis of cross-classifications having ordered categories. J Am Stat Assoc 76:320-334
[7] Gower JC, Lubbe SG, Le Roux NJ (2011) Understanding biplots. Wiley, Chichester · doi:10.1002/9780470973196
[8] Kotz S, Kozubowski TJ, Podgórski K (2001) The laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Birkhäauser, Boston · Zbl 0977.62003 · doi:10.1007/978-1-4612-0173-1
[9] McCullagh P, Nelder JA (1989) Generalized linear models, 2nd edn. Chapman & Hall, London · Zbl 0744.62098 · doi:10.1007/978-1-4899-3242-6
[10] Mosteller F, Tukey JW (1977) Data analysis and regression. Addison-Wesley, Reading, MA
[11] Powers DA, Xie Y (2008) Statistical methods for categorical data analysis, 2nd edn. Bingley, Emerald
[12] Schenker N, Gentleman JF (2001) On judging the significance of differences by examining the overlap between confidence intervals. Am Stat 55:182-186 · doi:10.1198/000313001317097960
[13] Scott D (2012) GeneralizedHyperbolic: the generalized hyperbolic distribution. http://CRAN.R-project.org/package=GeneralizedHyperbolic, R package version 0.8-1
[14] Turner H, Firth D (2007) g \[{ nm}\] nm: a package for generalized nonlinear models. R News 7:8-12, http://CRAN.R-project.org/doc/Rnews/
[15] Yee TW (2008) The \[{ VGAM}\] VGAM package. R News 8:28-39, http://CRAN.R-project.org/doc/Rnews/
[16] Yee TW (2010) The \[{ VGAM}\] VGAM package for categorical data analysis. J Stat Softw 32:1-34, http://www.jstatsoft.org/v32/i10/
[17] Yee TW (2014) Reduced-rank vector generalized linear models with two linear predictors. Comput Stat Data Anal 71:889-902 · Zbl 1471.62228 · doi:10.1016/j.csda.2013.01.012
[18] Yee TW (2015) Vector generalized linear and additive models. Springer, NY (in preparation)
[19] Yee TW, Hastie TJ (2003) Reduced-rank vector generalized linear models. Stat Model 3:15-41 · Zbl 1195.62123 · doi:10.1191/1471082X03st045oa
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