Regression analysis

INTRODUCTION

In almost all fields of study, the researcher is frequently faced with the problem of trying to describe the relation between a response variable and a set of one or more input variables. Given data on input (predictor, independent) variables labeled x ₁, x ₂,..., x _p and the associated response (output, dependent) variable y, the objective is to determine an equation relating output to input. The reasons for developing such an equation include the following:

• 1. to predict the response from a given set of inputs;

• 2. to determine the effect of an input on the response; and

• 3. to confirm, refute, or suggest theoretical or empirical relations.

To illustrate, the simplest situation is that of a single input for which a linear relation is assumed. Thus, if the relation is exact, it is given for appropriate values of β₀ and β₁ by

(1)

The determination of β₀ and β₁ in this case is easy, requiring only two distinct pairs of observations (x ₁, y ₁) and (x ₂, y ₂).

In general, the...