Chapter Thirteen Copyright © 2006 John Wiley & Sons, Inc. Bivariate Correlation and Regression

John Wiley & Son, Inc 2 1.To comprehend the nature of correlation analysis. 2.To understand bivariate regression analysis. 3.To become aware of the coefficient of determination, R 2. 4.To understand Spearman Rank Order correlation. Learning Objectives

John Wiley & Son, Inc 3 To understand bivariate regression analysis. Bivariate Analysis of Association Bivariate Techniques –Statistical methods of analyzing the relationship between two variables. Multivariate Techniques –When more than two variables are involved Independent Variable (Predictor) –Affects the value of the dependent variable Dependent Variable (Criterion) –explained or caused by the independent variable

John Wiley & Son, Inc 4 Types of Bivariate Procedures –Bivariate regression –Pearson product moment correlation –Spearman rank-order correlation –Two group t-tests –chi-square analysis of cross-tabulation or contingency tables –ANOVA (analysis of variance) for two groups Bivariate Analysis of Association To understand bivariate regression analysis.

John Wiley & Son, Inc 5 Bivariate Regression Bivariate Regression Defined –Analyzing the strength of the linear relationship between the dependent variable and the independent variable. Nature of the Relationship –Plot in a scatter diagram Dependent variable –Y is plotted on the vertical axis Independent variable –X is plotted on the horizontal axis Linear Relationship Nonlinear Relationship To understand bivariate regression analysis.

John Wiley & Son, Inc 6 Y X A - Strong Positive Linear Relationship Exhibit 13.1 Types of Relationships Found in Scatter Diagrams Bivariate Regression Example To understand bivariate regression analysis. Bivariate Regression

John Wiley & Son, Inc 7 Y X B - Positive Linear Relationship Exhibit 13.1 Types of Relationships Found in Scatter Diagrams To understand bivariate regression analysis. Bivariate Regression

John Wiley & Son, Inc 8 Y X C - Perfect Negative Linear Relationship Exhibit 13.1 Types of Relationships Found in Scatter Diagrams To understand bivariate regression analysis. Bivariate Regression

John Wiley & Son, Inc 9 X D - Perfect Parabolic Relationship Exhibit 13.1 Types of Relationships Found in Scatter Diagrams Y To understand bivariate regression analysis. Bivariate Regression

John Wiley & Son, Inc 10 Y X E - Negative Curvilinear Relationship Exhibit 13.1 Types of Relationships Found in Scatter Diagrams To understand bivariate regression analysis. Bivariate Regression

John Wiley & Son, Inc 11 Y X F - No Relationship between X and Y Exhibit 13.1 Types of Relationships Found in Scatter Diagrams To understand bivariate regression analysis. Bivariate Regression

John Wiley & Son, Inc 12 where Y = dependent variable X = independent variable e = error b = estimated slope of the regression line a = estimated Y intercept Y = a + bX + e Least Squares Estimation Procedure –Results in a straight line that fits the actual observations better than any other line that could be fitted to the observations. To understand bivariate regression analysis. Bivariate Regression

John Wiley & Son, Inc 13 Values for a and b can be calculated as follows:  X i Y i - nXY b =  X 2 i - n(X) 2 n = sample size a = Y - bX X = mean of value X Y = mean of value y To understand bivariate regression analysis. Bivariate Regression

John Wiley & Son, Inc 14 To become aware of the coefficient of determination, R 2. The Regression Line –Predicted values for Y, based on calculated values. Strength of Association: R 2 –Coefficient of Determination, R 2 The measure of the strength of the linear relationship between X and Y. Coefficient of determination measures the percentage of the total variation in Y that is explained by the variation in X The R 2 statistic ranges from 0 to 1. Bivariate Regression

John Wiley & Son, Inc 15 R 2 = explained variance total variance explained variance = total variance - unexplained variance R 2 = total variance - unexplained variance total variance = 1 - unexplained variance total variance Bivariate Regression To become aware of the coefficient of determination, R 2.

John Wiley & Son, Inc 16 R 2 = 1 - unexplained variance total variance =1 -  (Y i - Y i ) 2 n I = 1  (Y i - Y) 2 n I = 1 Bivariate Regression To become aware of the coefficient of determination, R 2.

John Wiley & Son, Inc 17 Statistical Significance of Regression Results The total variation is a measure of variation of the observed Y values around their mean. It measures the variation of the Y values without any consideration of the X values. Total variation = Explained variation + Unexplained variation Bivariate Regression To become aware of the coefficient of determination, R 2.

John Wiley & Son, Inc 18 Total variation: Sum of squares (SST) SST =  (Y i - Y) 2 n i = 1  Y i 2 n i = 1 =  Y i 2 n i = 1 n Bivariate Regression To become aware of the coefficient of determination, R 2.

John Wiley & Son, Inc 19 Sum of squares due to regression (SSR) SSR =  (Y i - Y) 2 n i = 1  Y i n i = 1 = a  Y i n i = 1 n b  X i Y i n i = Bivariate Regression To become aware of the coefficient of determination, R 2.

John Wiley & Son, Inc 20 Error sums of squares (SSE) SSE =  (Y i - Y) 2 n i = 1  Y 2 i n i = 1 = a  Y i n i = 1 b  X i Y i n i = 1 Bivariate Regression To become aware of the coefficient of determination, R 2.

John Wiley & Son, Inc 21 0 X XiXi X (X, Y) a Y Total Variation Explained variation Y Unexplained variation Exhibit 13.7 Measures of Variation in a Regression Y i =a + bX i

John Wiley & Son, Inc 22 Hypotheses Concerning the Overall Regression –Null Hypothesis H o There is no linear relationship between X and Y. –Alternative Hypothesis H a : There is a linear relationship between X and Y. Bivariate Regression To become aware of the coefficient of determination, R 2.

John Wiley & Son, Inc 23 Hypotheses about the Regression Coefficient b –Null Hypothesis H o b = 0 –Alternative Hypothesis H a : b  0 –The appropriate test is the t-test. Bivariate Regression To become aware of the coefficient of determination, R 2.

John Wiley & Son, Inc 24 Correlation Analysis To comprehend the nature of correlation analysis Correlation for Metric Data - Pearson’s Product Moment Correlation –Correlation The degree to which changes in one variable (the dependent variable) are associated with the changes in another –Correlation analysis Analysis of the degree to which changes in one variable are associated with changes in another variable. –Pearson’s product moment correlation Correlation analysis technique for use with metric data

John Wiley & Son, Inc 25 R = + - R2R2 √ R can be computed directly from the data: R = n  XY - (  X) - (  Y) [n  X 2 - (  X) 2 ] [n  Y 2 -  Y) 2 ] √ Correlation Analysis To comprehend the nature of correlation analysis

John Wiley & Son, Inc 26 To understand Spearman Rank Order correlation. Correlation Using Ordinal Data: Spearman’s Rank- Order Correlation –To analyze the degree of association between two ordinal scaled variables. –Spearman’s Coefficient of Rank-Order Coefficient R—the appropriate measure for analyzing ordinal data and like the coefficient of correlation R, has a lower limit of -1 and an upper limit of +1. Conclusions regarding rankings: –1. Positively correlated –2. Negatively correlated –3. Independent Correlation Analysis

John Wiley & Son, Inc 27 Bivariate Analysis of Association Bivariate Regression Correlation Analysis SUMMARY

John Wiley & Son, Inc 28 The End Copyright © 2006 John Wiley & Son, Inc