LESSON 23: MULTIPLE REGRESSION Outline Multiple Regression Analysis

MULTIPLE REGRESSION Lessons 21 and 22 discuss regression analysis using only one independent variable. However, often a variable depends on more than one independent variable. For example, ground/surface water level, rainfall, market demand of a product, share prices, etc. are often more of a mystery than well-understood. Researchers attempt to explain variations in each of these with many independent variables.

MULTIPLE REGRESSION For two independent variables, the estimated multiple regression equation: The scatter diagram for such a case is three dimensional. The method of least squares finds Y-intercept a, and estimated partial regression coefficients b1, b2 by minimizing the sum of the squares of deviations

MULTIPLE REGRESSION Normal equations for multiple regression: The regression coefficients are found by solving these equations simultaneously.

MULTIPLE REGRESSION Usually, multiple regression explains variations more than the simple regression. Consequently, multiple regression gives a higher coefficient of correlation. For the case of two independent variable, the standard error of estimate The cases of more than two independent variables are similar.

MULTIPLE REGRESSION Example 1: Using watershed rainfall and upstream dam releases as independent variables and seasonal mean river flow as the dependent variable, determine the equation for the estimated regression plane from the following sample data: Seasonal Mean Watershed Upstream Dam River Flow (cfps) Rainfall (inches) Releases (cfps) 240,000 25 90,000 210,000 19 100,000 220,000 23 65,000 175,000 16 120,000

MULTIPLE REGRESSION Example 2: Consider Example 1. Compute the standard error of the estimate for Y.

MULTIPLE REGRESSION Example 3: Predict the seasonal mean river flow when watershed rainfall = 20 inches and upstream dam releases = 100,000 cubic feet per second.

READING AND EXERCISES Lesson 23 Reading: Section 4-3 pp. 107-118 4-16, 4-17