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The Visualisation of Multiplicative Interaction. John Gower, Open University, U.K. and Mark De Rooij, Leiden University, NL. 1. Asymmetric case (e.g. PCA). 2. Symmetric case (e.g. multiplicative interaction as in the biadditive model for genotype/environment interaction).

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The visualisation of multiplicative interaction

The Visualisation of Multiplicative Interaction

John Gower, Open University, U.K.

and

Mark De Rooij, Leiden University, NL


Biplot axes are coordinate axes

1. Asymmetric case (e.g. PCA).

2. Symmetric case (e.g. multiplicative interaction as in the biadditive model for genotype/environment interaction).

Biplot axes are coordinate axes

We shall first review the asymmetric case and then show how the ideas extend to the symmetric case and suggest some useful variants.




Usual linear biplot display gabriel
Usual linear biplot display (Gabriel)

Uses inner-product interpretation rirj cosθij



Improving the display

Once it is recognised that biplot axes behave as coordinate axes, the usual devices of choosing a convenient origin and orientation are available.

In particular, efforts can be made to disentangle the cases from the axes representing the variables.

There is no loss of information but the displays are more helpful.

Improving the display


Same biplot but with better choice of origin and axes rotated to correspond closely to a conventional x-y plot


Nonlinear pca biplot cases omitted
Nonlinear PCA biplot (cases omitted) rotated to correspond closely to a conventional x-y plot

The variables are of ordered categorical type, with four

categories for each variable, as shown on the next slide.


Origin shifted cases shown
Origin shifted, cases shown rotated to correspond closely to a conventional x-y plot


Interaction
Interaction rotated to correspond closely to a conventional x-y plot

The same methods may be used to display multiplicative interaction biplots for biadditive models but one has to choose either the rows (say genotypes) or columns (say environments) to play the role of variables plotted as calibrated axes; the others are then plotted as points.


Inner product biplot
Inner-product biplot rotated to correspond closely to a conventional x-y plot

Black points-varieties, Red points (unlabelled) indicate varieties.


Inner product biplot with equal scaling
Inner-product biplot with equal scaling rotated to correspond closely to a conventional x-y plot


Distance plot with equal scaling
Distance plot with equal scaling rotated to correspond closely to a conventional x-y plot


Distance plot with asymmetric scaling xd versus y
Distance plot with asymmetric scaling (XD versus Y) rotated to correspond closely to a conventional x-y plot


Possibilities and difficulties
Possibilities and Difficulties rotated to correspond closely to a conventional x-y plot

  • Different scalings have a major influence on the distances

  • Utilize this to

    • Optimize the correlation between data and distances

    • Minimize the constant in order to get optimal discriminability between the magnitude of distances

  • But never take a look at the plot without noticing the main effects


Biplot with scaled axes
Biplot with scaled axes rotated to correspond closely to a conventional x-y plot


Symmetric approaches
Symmetric approaches rotated to correspond closely to a conventional x-y plot

A problem with these biplots is that they do not treat rows and columns symmetrically.

rirjcosij = rirjsin(ij+90) so we may rotate the environment points through a rightangle and replace inner-products by areas of triangles.

Another way is to note that

So that inner-products may be replaced by Euclidean distance, provided we reparameterise the main effects.


Biplot – Area interpretation, The area of the triangle shows the interaction

between Variety Ho at “environment” H4. The sign depends on whether area is

determined clockwise or anticlockwise, so the line joining O and Ho separates

positive and negative interactions. Equal area loci are lines through H4 and through

Ho parallel to the opposite sides.


PLAN shows the interaction

1 Show Rob Kempton’s Gabriel type display, followed by its calibrated axis version (if possible with shifted origin and rotation). I hope to scan this from the original paper.

2 Say not symmetric as is desirable and offer (a) area display and (b) distance display.

3 Sketch algebra of (a) rirjcosij = rirjsin(ij+90) and (b) for equivalence of biadditive model parameterised in terms of innerproducts and distances. (b) needs something on tuning constants and handling of “main effects”.


Some other types of biplot
Some other types of biplot shows the interaction


Some references
Some References shows the interaction

Denis, J-B., & Gower, J.C. (1996) Asymptotic confidence regions for biadditive models: Interpreting genotype-environment interactions. Applied Statistics,45, 479-493.

De Rooij, M. and Heiser, W.J. (2005). Graphical representations and odds ratios in a distance association model for the analysis of cross-classified data, Psychometrika, 70, *-*.

Gower, J.C. and Hand, D. J. (1996) Biplots. London: Chapman and Hall, 277 + xvi pp.

Kempton, R. A. (1984) The use of biplots in interpreting variety by environment interactions. J. Agric. Sci. Camb., 103, 123-135.


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