Interaction: Fisher’s Optimal Scores Revisited

John C. Gower (), Sugnet Gardner-Lubbe () and Niel J. Le Roux ()
Additional contact information
John C. Gower: The Open University
Sugnet Gardner-Lubbe: Stellenbosch University
Niel J. Le Roux: Stellenbosch University

Journal of Agricultural, Biological and Environmental Statistics, 2018, vol. 23, issue 1, No 6, 92-112

Abstract: Abstract The normal way of fitting a bilinear model to values of a quantitative variable classified by rows and columns is first to fit the main effects of rows and columns and then fit a biadditive model to the residual table. This gives the ordinary least squares optimal fit, usually visualised by biplots and, among other applications areas, is routinely used in genotype $$\times $$ × environment trials. When the dependent variable is categorical, Fisher (1938) proposed scoring the categories by maximising the sum-of-squares of the row and column main effects relative to the residual sum-of-squares; then, the usual biadditive analysis again becomes available, although not used by Fisher. In this paper, it is shown that with categorical variables Fisher could have derived scores by maximising the sum-of-squares of the biadditive part of the model relative to the residual sum-of-squares. With Fisher’s data, this would have three advantages: (i) a better fitting model, (ii) a more parsimonious model and (iii) novel and more powerful visualisation techniques for expressing interactions. Other ways of partitioning the contributions from main effects and biadditivity are also discussed. The results show how small changes in model construction can have profound effects on interpretation. Potential applications extend far beyond the data used by Fisher to whenever the categorical nature of the dependent variable offers the freedom to generate quantifications with generalised interaction parameterisations. Supplementary materials accompanying this paper appear on-line.

Keywords: Biadditive models; Biplots; Categorical data; Modelling; Multiplicative interactions; Non-additive interactions; Optimal scores (search for similar items in EconPapers)
Date: 2018
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13253-017-0311-8 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jagbes:v:23:y:2018:i:1:d:10.1007_s13253-017-0311-8

Ordering information: This journal article can be ordered from
http://www.springer.com/journal/13253

DOI: 10.1007/s13253-017-0311-8

Access Statistics for this article

Journal of Agricultural, Biological and Environmental Statistics is currently edited by Stephen Buckland

More articles in Journal of Agricultural, Biological and Environmental Statistics from Springer, The International Biometric Society, American Statistical Association
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().