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A previous paper, Biplots: Quantitative data, dealt exclusively with biplots for quantitative data. This paper is mainly concerned with qualitative data or data in the form of counts. Qualitative data can be nominal or ordinal, and it is usually reported in a coded numerical form. In the analysis of qualitative data, many methods can be grouped as quantification methods (e.g., categorical principal component analysis, correspondence analysis, multiple correspondence analysis, homogeneity analysis): transforming qualities into quantitative values that may then be treated with quantitative methods. All the features of quantitative biplots are found in qualitative biplots, but calibrated interpolation axes become labeled category-level points and calibrated prediction axes become prediction regions. Interpretation remains in terms of distance, inner products, and sometimes area. WIREs Comput Stat 2016, 8:82-111. doi: 10.1002/wics.1377 For further resources related to this article, please visit the . [ABSTRACT FROM AUTHOR]

Biplots provide visualizations of two things, usually, but not necessarily, in two dimensions. This paper deals exclusively with biplots for quantitative data X; qualitative data or data in the form of counts will be addressed in a subsequent paper. Data X may represent either (1) a matrix with n rows representing samples/cases and columns representing p quantitative variables or (2) a two-way table whose rows and columns both represent classifying variables. Data sets of both types (1) and (2) are considered. Plotting symbols are usually points (typically for samples and distinguished by shape and/or color) and lines (typically for variables which may be calibrated or treated as arrowed vectors). Furthermore, variables may be nonlinear in both regularity of calibration and/or curvature. Interpretation is through distance, inner-products, and sometimes area. Biplots may be improved by judicious shifts of axes, by scaling and by rotation. Nearly always, biplots give approximations to X and measures, incorporated in the biplot, expressing the degree of approximation are discussed. These aspects are illustrated with reference to examples from principal component analysis, nonlinear biplots, biplots for biadditive models, canonical variate analysis and the analysis of distance between grouped samples. WIREs Comput Stat 2015, 7:42-62. doi: 10.1002/wics.1338 For further resources related to this article, please visit the . Conflict of interest: The authors have declared no conflicts of interest for this article. [ABSTRACT FROM AUTHOR]

Free-choice profiling is a technique that is used in sensory studies whereby assessors are allowed to describe products by using their own preferred descriptors and own number of descriptors. We show how a study of this type on eight varieties of port wine can be analysed and, in particular, how to associate the descriptors of the wines with each assessor and with a group average representation. We show how biplots combined with generalized orthogonal Procrustes analysis can be constructed and used to explore consistencies and idiosyncrasies among assessors’ uses of their descriptors and scales. [ABSTRACT FROM AUTHOR]

We examine the use of triadic distances as a basis for multidimensional scaling (MDS). The MDS of triadic distances (MDS3) and a conventional MDS of dyadic distances (MDS2) both give Euclidean representations. Our analysis suggests that MDS2 and MDS3 can be expected to give very similar results, and this is strongly supported by numerical examples. We have concentrated on the perimeter and generalized Euclidean models of triadic distances, both of which are linear transformations of dyadic distances and so might be suspected of explaining our findings; however an MDS3 of the nonlinear variance definition of triadic distance also closely approximated the MDS2 representation. An appendix gives some matrix results that we have found useful and also gives matrix representations and alternative derivations of some known properties of triadic distances. [ABSTRACT FROM AUTHOR]

An understanding of how genotypes of an agricultural crop interact with the environment in which they are grown is important for assessing plant production. A breeding trial for 21 genotypes of rye-grass grown at seven locations is used to illustrate the interpretation of genotype-environment interactions. Statisticians have proposed many ways of modelling these interactions, but a subclass of bilinear models, that we term biadditive, fits especially well. We emphasize assessing and interpreting the interaction parameters of biadditive models by constructing confidence regions in biplot representations. When a biadditive model is valid, this new development underpins better informed decisions on variety recommendation and genotype selection. [ABSTRACT FROM AUTHOR]