Goodness of fit of biplots and correspondence analysis

The present paper examines proportional goodness of fit to variables recorded on individuals, the variances and covariances of the variables, and the form and distances between individuals.No single plot displays all three optimally in the sense of least squares. However, even aspects which are non-optimally fitted by biplots and Benzecri plots often closely preserve the optimal fit. This is shown by means of a preservation-of-fit function which depends on the type of display and on the ratio of the second to the first singular value of the data matrix. This function is never below 0·5, so at least half the fit is always preserved, and it is close to 1 unless the ratio of the singular values is small. That explains the frequently observed similarity of the various biplots and the Benzecri plot and the fact that they usually lead to the same conclusions. It follows that in many applications it is reasonable to use either the symmetric biplot or the Benzecri plot or a compromise maximin preservation plot, and that the difference between these three is usually unimportant.