6.3 Regression and the bivariate normal model
6.3.1 The model
The problem we will consider in this section is that of using the values of one variable to explain or predict values of another. We shall refer to an explanatory and a dependent variable, although it is conventional to refer to an independent and a dependent variable. An important reason for preferring the phrase explanatory variable is that the word ‘independent’ if used in this context has nothing to do with the use of the word in the phrase ‘independent random variable’. Some authors, for example, Novick and Jackson (1974, Section 9.1), refer to the dependent variable as the criterion variable. The theory can be applied, for example, to finding a way of predicting the weight (the dependent variable) of typical individuals in terms of their height (the explanatory variable). It should be noted that the relationship which best predicts weight in terms of height will not necessarily be the best relationship for predicting height in terms of weight.
The basic situation and notation are the same as in the last two sections, although in this case there is not the symmetry between the two variables that there was there. We shall suppose that the xs represent the explanatory variable and the ys the dependent variables.
There are two slightly different situations. In the first, the experimenters are free to set the values of xi, whereas in the second both values are random, although one is thought of as having a causal or ...