Regression, correlation and structure

The paper sets out the basic statistical theory of the association between two or more measured quantities. The distinction is made between a mathematical variable and a random variable or variate, and by a simple example the concept of the “power" of a significance test is explained. The analysis of association is necessarily carried out by the use of models and the theory involved in each of four such models is explained. The models are: (i) The Regression model which contains only one random variable (yi)whose mean is linearly dependent on a mathematical variable (xi), and the power of the associated significance test is increased by examining the means of the Yi s at widely separated values of x. (ii) The Bivariate Normal Distribution model which contains two interdependent variates. Distinction is made between the regression line considered as a property of the bivariate distribution and alternatively as part of the definition of the regression model. (iii) The Errors in Variables model in which two random variables are measurements involving random errors, and the model is used to arrive at an estimate of the underlying linear relationship between the unobserved structural variables which may be random or not. (iv) Berkson’s model in which analysis is made of the relationship between two quantities one of which is an observed random variable whose means are dependent linearly on unobserved random variables whose means, in turn, are known specified constants.