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MULTIPLE REGRESSION. is a method used to examine the relationship between one dependent variable Y and one or more independent variables X i . The regression parameters or coefficients b i in the regression equation
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is a method used to examine the relationship between one dependent variable Y and one or more independent variables Xi. The regression parameters or coefficients bi in the regression equation • In multiple regression, more than one variable is used to predict the criterion.
-deal with interval and ratio level variables • -assess causal linkages • -forecast future outcomes
Regression line A straight drawn through the scatter plot which represents the best possible “fit” for making predictions of Y from X. • Slope in regression, the change regression line for a unit increase in X.The slope is interpreted as the change in the Y variable associated with a unit change in X variable.
General form for Multiple Regression Actual (true Model) • y is the dependent variable • xi = independent (explanatory) variables • Β0 is the actual constant • Є is the error term which models the unsystematic error of the Y terms from the predicted Y terms • xi = independent (explanatory) variables • Β= is the actual coefficient of the independent variable.
Prediction (estimated) Modelў=bo+b1x1+b2x2+b3x3+…bnxn • y is the dependent variable • Βo is the actual constant • Є is the error term which models the unsystematic error of the Y terms from the predicted Y terms • Ў is the predicted value using the mulitple regression model • Bo is the estimated constant • xi = independent (explanatory) variables • b1= is the predicted coefficient of the independent variable,called the partial regression coefficients,which represent the change in the predicted value of y per unit change int the x value provided that the other xis are held constant
Example Problem The ABC corporation is opening new retail sales outlets and they want to staff these stores with employees most likely to be successful at selling the products. To meet this goal, ABC decides to study the sales staff at existing stores to determine if intelligence and extroversion (i.e., a friendly and outgoing personality) predict sales performance of current employees. ABC's logic is that if intelligence and extroversion predicts sales performance, then a good strategy for new stores is to hire intelligent extroverts for the sales positions. To conduct the study, all current retail sales employees at existing stores take psychological tests designed to measure intelligence and extroversion. Also, past sales performance data is checked for each employee. In the end, there are three scores for each sales person: an intelligence score (on a scale of 50-low intelligence to 150-high intelligence), an extroversion score (on a scale of 15-low extroversion to 30-high extroversion), and sales performance expressed as the average dollar amount sold per week.
In these types of studies, the variables used to forecast (intelligence and extroversion) are called "predictors" and the variable being forecast (sales performance) is called the "criterion". The predictor and criterion data are presented below for the 20 current sales employees of the ABC corporation.
To analyze these data, one option is to examine the bivariate (i.e., two variable)correlationand the bivariate regressionequation of the intelligence vs. sales performance relationship and the extroversion vs. sales performance relationship. For intelligence vs. sales performance, the bivariate correlation r = .33 for the above data. For the extroversion vs. sales relationship, r = .55. Both of these relationships are positive, and are moderately strong relative to what is often observed in "real world" studies similar to this. The interpretation is that sales performance increases as ABC sales people become more intelligent and more extroverted. Thescatterplots and associated bivariate regression equations shown below are another way to examine these data.
Summary • Multiple regression analysis is a powerful tool when a researcher wants to predict the future. • Multiple regression involves using two or more variables (predictors) to predict a third variable (criterion). • Multiple regression equations with two predictor variables can be illustrated graphically using a three-dimensional scatterplot. • The plane of best fit is the plane which minimizes the magnitude of errors when predicting the criterion variable from values on the predictors variables. • The multiple regression formula can be used to predict an individual observation's most likely score on the criterion variable.
