Application of a Measure of Information to the Design and Comparison of Regression Experiments

A normal regression experiment can be represented by equation*1. 1 Yᵢ = ₉=₁ᵏ X₈₉ ⱼ + ᵢ (i = 1, , n) equation* where \ᵢ/i = 1, , n\ is a set of normally distributed random variables with zero means and non-singular dispersion matrix C, = (₁, , ₖ) is the parameter-vector of interest and X = (X₈₉) is a known n k matrix which will be called the allocation matrix. The rows of X will be called the allocation vectors. We denote the experiment by (X, C). We assume that C is known; generally it will be a function of X, C (X). The particular realisation of Y will be denoted y. The matrix F = X'C^-1X is the Fisher-information-matrix of (X, C). When F is non-singular, one answer to the question "What information does y give about? " is to quote F^-1, the dispersion matrix of the maximum-likelihood-estimates of. A strong argument in favour of this is that F^-1 is independent of both and y. The fact that it is independent of means that the answer is not "local"; the fact that it is independent of y leads to simplicity. This approach is taken by Box and Hunter 1 in their work on rotatable designs. However, we must accept the fact that many experimenters wish to have a one-dimensional answer to the question i. e. we must associate with (X, C) a single number which we call the "information". For instance Elfving 5 has developed the use of trace F^-1. In this paper we adopt the measure of information introduced by Lindley 7. In Section 2 we generalise Lindley's treatment of the regression situation to include the singular case, explain the uses of the measure and compare it with that of Elfving. Section 3 deals with the analogue of Elfving's main theorem. Theorems 4. 1 and 4. 2 of Section 4 provide links with the traditional variance approach. In Section 5 we derive the asymptotic form of the measure as the n of (1. 1) increases and show that this form can be derived also from Neyman-Pearsonian theory. In Section 6 the influence of nuisance parameters is discussed and an analogue of a theorem of Chernoff 2 is established.