Applications of Information Theory to Psychology: A Summary of Basic Concepts, Methods, and Results

By Fred Attneave | Go to book overview

Appendix I
The Calculation of Information Measures from Variance Statistics

The correlation coefficient, r, the correlation ratio, η (eta), and the transmission function, T, are alike in that all three are measures of relatedness. T is the most general of the three, since it may be calculated for any transmission matrix of the sort shown in Table 3 (p. 45), whether the variables are ordered --that is, whether adjacent categories represent successive steps on a continuum--or not. When the variables are ordered, but not related in a linear manner, either T or η may be calculated, and when a linear relationship holds, either T, η, or r may be. In the last case, η and r will be equal, or very nearly so: the square of either may be interpreted as the proportion of the variance of y which is predicted, or accounted for, by x. But how is T related to r and η? This relationship has a practical importance as well as a theoretical interest. The stability of a sample value of r (or of η, though to a lesser degree) is much greater than that of T + ̂, because T + ̂ involves many more degrees of freedom; therefore, if we had a matrix containing many cells and relatively few observations, and a linear relationship held between the variables, an estimate of T calculated from r might be considerably more accurate than one calculated by the use of the Shannon-Wiener measure.

Now, it may be shown (see Shannon [79] pp. 54-56) that the uncertainty associated with a normal distribution is

(54)

Here H has the meaning with which we are familiar if the unit in which σ is expressed is the "width" of one category, or alternative, along the continuum. Formula (54) may be re-

-95-

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