CHAPTER 2 - Biplots in Practice, by Michael Greenacre
In this video we see the three-dimensional scatterplot of species d versus pollution (y) and depth (x) and the regression plane defined by the model d = 6.135-1.388 y+0.148 x [see equation (2.1) and the schematic version on the left of Exhibit 2.2].
In this video we see the same three-dimensional scatterplot of species d versus pollution (y) and depth (x) , but with all variables standardized (mean 0, variance 1). The regression plane in these standardized variables is now d* = -0.446y*+0.347x*. [see schematic version on the right of Exhibit 2.2]. The plane can de uniquely defined by the white arrow pointing up the plane in the direction of steepest ascent, called a biplot axis or biplot direction. Lines in white are drawn at right angles to this direction according to units of standard deviation in the response variable d above and below the mean, as predicted by the regression model. On the x-y plane there is a shadow of the plane as well as the direction of the biplot arrow in red and the parallel lines in red which are now the corresponding contour lines of the plane. These contours can be seen in Exhibit 2.3, illustrating how the projections of the points on the biplot direction onto these contours are just the regression plane estimates.