Chapter 14 Regression Summarized by Lana Hesler

Learning Objectives Understand the nature and purposes of regression analysis Recognize geographic problems where regression analysis is useful Understand how strength of relationship between variables is measured Determine the importance of residual analysis in regression Learn the basic concepts and purposes of multivariate regression

Regression Analysis Regression can be applied successfully in all areas of geography The most common application of regression is the identification of linear relationships between variables Bivariate regression ◦ Examines the influence of one variable or another Independent Variable ◦ Variable creating the influence or effect Dependent Variable ◦ Variable receiving the influence or effect

Regression Analysis Examples Medical Geographer ◦ Examine the relationship between the number of physicians located in the counties of a state and the income level of persons residing in these areas Environmental Geographer ◦ Examine the form of relationship between the acidity level in various locations in a chain of lakes and distance from a point pollution source Political Geographer ◦ Compare the strength of votes for a political party and the educational, financial, or racial composition of voters in the wards of a city

Strength in Bivariate Regression Determining the strength of relationship between two variables in regression is the same as measuring the relative ability of the independent variable to account for variation in the dependent variable.

Strength of variables Explained variation ◦ The amount of variation that can be accounted for by the independent variable X. Unexplained variation ◦ The portion of total variation in the dependent variable that cannot be accounted for by the independent variable Total sum of squares ◦ Summing the squared deviations from the mean in order to calculate the total variation TSS = ∑y 2 = ∑y 2 – (∑Y) 2 n

Residual Analysis Residual analysis is the amount of deviation of each point from the regression line RES = Y - ˆY RES = residual Y = actual value of the dependent variable ˆY = predicted regression line value of Y

Multivariate Regression Multivariate Regression is where a set of independent variables is used to explain a single dependent variable, offers a logical extension to the simple, two-variable regression procedure. Y = a+b 1 X 1 + b 2 X 2 + …+ b n X n Y = dependent variable X 1 …X n = independent variables a = Y-intercept or constant b 1 ….b n = regression coefficients

Lesson Review Regression analysis is a useful statistical procedure that supplements correlation Regression can be applied successfully in all areas of geography The strength of variable relationships must be determined as a basis for evaluating the explanatory ability of the regression model Residual analysis provides additional spatial and non-spatial information about the variation in the dependent variable that cannot be explained by the independent variable Although bivariate regression is usually effective, most real-world problems require multiple regression