Simple Linear Regression - Introduction

$Often\; the\; relationship\; between\; two\; variables,\; Y\; and\; X,\; can\; be\; adequately\; represented\; by\; a$simple linear equation of the form

Y = b₀ + b₁X + e

In regression terminology Y is the response or dependent variable and X is the regressor or independent variable. The error term, e, suggests that the relationship is not a perfect one. To understand the relationship between Y and X, or to summarize the relationship, or to predict Y for a given X it is necessary to estimate the coefficients or parameters b₀ and b₁. One way to estimate the coefficients is to collect data where Y and X are observed directly and let the data indicate the appropriate values for the coefficients. For example, suppose observations of diameter at breast height (DBH) and stump diameter (at 1-foot) inside bark (DS) of 243 Virginia cove-grown white oak were made (fig. 1).

From inspection of the graph a linear equation would seem appropriate for describing the relationship between DBH and DS for cove-grown white oak in Virginia. Hand sketching a line through the data and recalling some basic algebra leads to b₀ (intercept) = 0.0 and b₁ (slope) = 0.75 so that, approximately, for these data

DBH = 0.75 DS