Row–column interaction models, with an R implementation

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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\mathbf{A}\mathbf{C}^T$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="bold">A</mi> <msup> <mrow> <mi mathvariant="bold">C</mi> </mrow> <mi>T</mi> </msup> </mrow> </math> </EquationSource> </InlineEquation>, (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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$M=1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>M</mi> <mo>=</mo> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation> linear/additive predictor <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\eta $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">η</mi> </math> </EquationSource> </InlineEquation> case (within the exponential family) to the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$M>1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>M</mi> <mo>></mo> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation> 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 <Emphasis FontCategory="SansSerif">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. Copyright Springer-Verlag Berlin Heidelberg 2014