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).

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

Aircraft dataAircraft SPR RGF PLF SLF

a FH-1 1.468 3.30 0.166 0.10

b FJ-1 1.605 3.64 0.154 0.10

c F-86A 2.168 4.87 0.177 2.90

d F9F-2 2.054 4.72 0.275 1.10

e F-94A 2.467 4.11 0.298 1.00

f F3D-1 1.294 3.75 0.150 0.90

g F-89A 2.183 3.97 0.000 2.40

h XF10F-1 2.426 4.65 0.117 1.80

i F9F-6 2.607 3.84 0.155 2.30

j F100-A 4.567 4.92 0.138 3.20

k F4D-1 4.588 3.82 0.249 3.50

m F11F-1 3.618 4.32 0.143 2.80

n F-101A 5.855 4.53 0.172 2.50

p F3H-2 2.898 4.48 0.178 3.00

q F102-A 3.880 5.39 0.101 3.00

r F-8A 0.455 4.99 0.008 2.64

s F-104A 8.088 4.50 0.251 2.70

t F-105B 6.502 5.20 0.366 2.90

u YF-107A 6.081 5.56 1.06 2.90

v F-106A 7.105 5.40 0.089 3.20

w F-4B 8.548 4.20 0.222 2.90

Scatterplot of RGF and SPR

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Usual linear biplot display (Gabriel)

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Uses inner-product interpretation rirj cosθij

Same biplot but with calibrated axes interpretation

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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.

Same biplot but with better choice of origin and axes rotated to correspond closely to a

conventional x-y plot

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Nonlinear PCA biplot (cases omitted)

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The variables are of ordered categorical type, with four categories for each variable, as shown on the next slide.

Origin shifted, cases shown

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G eo g r ap h icAttr ib u tes

S o c ia lAttr ib u tes

I n d iv d u alsVer yS o m ew h atN o t v er yN o t a t a ll

G r ea t Br ita in : Bip lo t o f 4 - p o in t s c a les q u an tif ied b y n o n lin ear p r in c ip a l c o m p o n en ts

Interaction

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

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

Inner-product biplot with equal scaling

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Distance plot with equal scaling

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Distance plot with asymmetric scaling (XD versus Y)

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All column points

Possibilities and Difficulties

• Different scalings have a major influence on the distances

• Utilize this to 1. Optimize the correlation between data and

distances

2. 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 axesF u

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Key : Var ie ty

1 0 g m 2Hig h N en v ir o n m en tsL o w N en v ir o n m en ts

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Symmetric approachesA problem with these biplots is that they do not

treat rows and columns symmetrically.

rirjcosij = 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

2222

12112

1***

2211

)()( jijiji

jijijiij

dcdcbam

dcdcbamy

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 throughHo parallel to the opposite sides.

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PLAN

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) rirjcosij = 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

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e u rope

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Some References

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.