Basic & Clinical Biostatistics: Fourth Edition (LANGE Basic Science) by Beth Dawson & Robert G. Trapp

Author:Beth Dawson & Robert G. Trapp [Dawson, Beth]
Language: eng
Format: azw3
Publisher: McGraw-Hill Education
Published: 2004-04-15T04:00:00+00:00

Least Squares Method

Several times previously in this text, we mentioned the linear nature of the pattern of points in a scatterplot. For example, in Figure 8–2, a straight line can be drawn through the points representing the values of BMI and percent body fat to indicate the direction of the relationship. The least squares method is a way to determine the equation of the line that provides a good fit to the points.

To illustrate the method, consider the straight line in Figure 8–6. Elementary geometry can be used to determine the equation for any straight line. If the point where the line crosses, or intercepts, the Y-axis is denoted by a and the slope of the line by b, then the equation is

Y = a + bX

The slope of the line measures the amount Y changes each time X changes by 1 unit. If the slope is positive, Y increases as X increases; if the slope is negative, Y decreases as X increases; and vice versa. In the regression model, the slope in the population is generally symbolized by β1, called the regression coefficient; and β0 denotes the intercept of the regression line, that is, β1 and β0 are the population parameters in regression. In most applications, the points do not fall exactly along a straight line. For this reason, the regression model contains an error term, e, which is the distance the actual values of Y depart from the regression line. Putting all this together, the regression equation is given by

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