Example of Simple and Multiple RegressionPrediction without an Independent VariableComputing the Mean and Variance with SPSSRequest the Mean and VarianceThe Descriptives OutputPrediction with One Independent VariableSimple Linear Regression OutputSlide 8Prediction with Two Independent VariablesRequesting a Correlation MatrixMultiple Regression OutputSlide 12Slide 13Prediction with Three Independent VariablesSlide 15Slide 16Slide 1 Example of Simple and Multiple RegressionThe purpose of this introductory example from pages 149-158 of the text is to demonstrate the basic concepts of regression analysis as one attempts to develop a predictive equation containing several independent variables. The dependent variable is the number of credit cards held by a family. The independent variables are family size and family income.The data for this problem is in the SPSS data set CreditCardData.Sav.Example of Simple and Multiple RegressionSlide 2 Prediction without an Independent VariableWith no information other than the number of credit cards per family, i.e. we only know the values for the dependent variable, Number of Credit Cards, our best estimate of the number of cards in a family is the mean. The cumulative amount of error in our guesses for all subjects in the data set is the sum of squared errors (squares of deviations from the mean).Recall that variance equals the sum of squared errors divided by (the number of cases minus 1 degree of freedom). While we cannot obtain the sum of squared errors directly, we can compute the variance and multiply it by the number of cases in our sample minus 1.Example of Simple and Multiple RegressionSlide 3 Computing the Mean and Variance with SPSS First, choose 'Descriptive Statistics | Descriptives...' from the Analyze menu. Second, in the Descriptives dialog box, move the variable 'Number of Credit Cards (ncards)' to the 'Variable(s)' list. Third, click on the 'Options...' button to request specific statistics. Example of Simple and Multiple RegressionSlide 4 Request the Mean and VarianceFirst, mark the check boxes for 'Mean','Std. Deviation', and 'Variance.'Second, click onthe 'Continue'button to closethe 'Descriptives:Options' dialogbox.Third, click onthe OK buttonto completeour request.Example of Simple and Multiple RegressionSlide 5 The Descriptives OutputIn the SPSS Output Navigator, we see that the variance is 3.143. If we multiply the variance by 7 (the number of cases in the study, 8 - 1 = 7), we compute the sum of squared errors to be equal to 22, which agrees with the text on page 151.7 If we use the mean for our best guess for each case, our measure of error is 22 units.7 The goal of regression is to use information from independent variables to reduce the amount of error associated with our guesses for the value of the dependent variable.Example of Simple and Multiple RegressionSlide 6 Prediction with One Independent Variable To use a single independent variable, family size, to predict the number of credit cards in a family, we first choose 'Regression | Linear...' from the Analyze menu. For this analysis, we accept all of the other defaults specified by SPSS. Fourth, click on the OK button to produce the output. Second, in the 'Linear Regression' dialog box, move the variable 'Number of Credit Cards (ncards)' to the 'Dependent:' variable list. Third, move the variable 'Family Size (famsize)' to the 'Independent(s)' list box. Example of Simple and Multiple RegressionA regression with a single independent variable and an independent variable is referred to as simple linear regression.Slide 7 Simple Linear Regression OutputThe regression coefficients are shown in the section of output shown to the right in the column titled 'B' of the coefficients table. Example of Simple and Multiple RegressionThe coefficient for the independent variable Family Size is .971. The intercept is labeled as the (Constant) which is 2.871. If we were to write the regression equation, it would be:Number of Credit Cards = 2.871 + 0.971 x Family SizeThe ANOVA table provides the information on the sum of squared errors.Slide 8 Simple Linear Regression OutputExample of Simple and Multiple RegressionThe 'Total' error or sum of squares (22) agrees with the calculation above for variance about the mean. When we use the information about the family size variable in estimating number of credit cards, we reduce the error in predicting number of credit cards to the 'Residual' sum of squares of 5.486 units. The difference between 22 and 5.486, 16.514, is the sum of squares attributed to the 'Regression' relationship between family size and number of credit cards.The ratio of the sum of squares attributed to the regression relationship (16.514) to the total sum of squares (22.0) is equal to the value of R Square in the Model Summary Table, i.e. 16.514 / 22.0 = 0.751. We would say that the pattern of variance in the independent variable, Family Size, explains 75.1% of the variance in the dependent variable, Number of Credit Cards.Slide 9 Prediction with Two Independent Variables First, click on the 'Dialog Recall' tool. Second, select 'Linear Regression' from the drop down menu. Third, move the variable 'Family Income (famincom)' to the list box of 'I ndependent(s): ' variables. ('Number of Credit Cards (ncards)' should still be in the 'Dependent:' variable text box and 'Family Size (famsize)' should still be listed in the 'Independent(s)' list box.) Fourth, click on the 'Statistics...' button to request a correlation matrix be added to the output. Example of Simple and Multiple RegressionExtending our analysis, we add another independent variable, Family Income, to the analysis. When we have more than one independent variable in the analysis, we refer to it as multiple regression.Slide 10 Requesting a Correlation MatrixFirst, mark the 'Descriptives'check box to add a correlationmatrix to the output.Second, click on the'Continue' button toclose the 'Statistics'dialog box.Third, click onthe OK buttonto produce theoutput.Example of Simple and Multiple RegressionSlide 11 Multiple Regression OutputExample of Simple and Multiple RegressionWe will examine the correlation matrix before we review the regression output:The ability of an independent variable to predict the dependent variable is based on the correlation between the independent and the dependent variable. When we add another independent variable, we must also concern ourselves with the