Learning Objectives Copyright © 2002 South-Western/Thomson Learning Data Analysis: Bivariate Correlation and Regression CHAPTER sixteen

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

Learning Objectives Bivariate Analysis Defined The degree of association between two variables Bivariate techniques Statistical methods of analyzing the relationship between two variables. Independent variable Affects the value of the dependent variable Dependent variable explained or caused by the independent variable To understand bivariate regression analysis. Bivariate Analysis of Association

Learning Objectives 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 To understand bivariate regression analysis. Bivariate Analysis of Association

Learning Objectives 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 Bivariate Regression To understand bivariate regression analysis.

Learning Objectives Y X A - Strong Positive Linear Relationship To understand bivariate regression analysis. Figure 16.1 Types of Relationships Found in Scatter Diagrams Bivariate Regression Example Bivariate Regression

Learning Objectives Y X B - Positive Linear Relationship Figure 16.1 Types of Relationships Found in Scatter Diagrams To understand bivariate regression analysis. Bivariate Regression

Learning Objectives Y X C - Perfect Negative Linear Relationship Figure 16.1 Types of Relationships Found in Scatter Diagrams To understand bivariate regression analysis. Bivariate Regression

Learning Objectives X D - Perfect Parabolic Relationship Figure 16.1 Types of Relationships Found in Scatter Diagrams To understand bivariate regression analysis. Bivariate Regression

Learning Objectives Y X E - Negative Curvilinear Relationship Figure 16.1 Types of Relationships Found in Scatter Diagrams To understand bivariate regression analysis. Bivariate Regression

Learning Objectives Y X F - No Relationship between X and Y Figure 16.1 Types of Relationships Found in Scatter Diagrams To understand bivariate regression analysis. Bivariate Regression

Learning Objectives 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. where Y = dependent variable X = independent variable e = error b = estimated slope of the regression line a = estimated Y intercept Bivariate Regression Y = a + bX + e

Learning Objectives Values for a and b can be calculated as follows: To understand bivariate regression analysis.  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 Bivariate Regression

Learning Objectives Strength of Association: R 2 Coefficient of Determination, R 2 : The measure of the strength of the linear relationship between X and Y. To become aware of the coefficient of determination, R 2. The Regression Line Predicted values for Y, based on calculated values. Bivariate Regression

Learning Objectives 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 To become aware of the coefficient of determination, R 2. Bivariate Regression

Learning Objectives R 2 = 1 - unexplained variance total variance =1 -  (Y i - Y i ) 2 n I = 1  (Y i - Y) 2 n I = 1 To become aware of the coefficient of determination, R 2. Bivariate Regression

Learning Objectives Statistical Significance of Regression Results Total variation = Explained variation + Unexplained variation To become aware of the coefficient of determination, R 2. 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. Bivariate Regression

Learning Objectives Total variation: Sum of squares (SST) To become aware of the coefficient of determination, R 2. SST =  (Y i - Y) 2 n i = 1  Y i 2 n i = 1 =  Y i 2 n i = 1 n Bivariate Regression

Learning Objectives Sum of squares due to regression (SSR) To become aware of the coefficient of determination, R 2. 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

Learning Objectives Error sums of squares (SSE) To become aware of the coefficient of determination, R 2. 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

Learning Objectives 0 X XiXi X (X, Y) a Y Total Variation Explained variation Y Unexplained variation Figure 16.4 Measures of Variation in a Regression Y i =a + bX i

Learning Objectives 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. To become aware of the coefficient of determination, R 2. Bivariate Regression

Learning Objectives Hypotheses about the Regression Coefficient Null Hypothesis H o : b = 0 Alternative Hypothesis H a : b  0 The appropriate test is the t-test. To become aware of the coefficient of determination, R 2. Bivariate Regression

Learning Objectives Correlation for Metric Data - Pearson’s Product Moment Correlation 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 Correlation Analysis To become aware of the coefficient of determination, R 2.

Learning Objectives 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 ] √ To become aware of the coefficient of determination, R 2. Correlation Analysis

Learning Objectives Correlation Using Ordinal Data: Spearman’s Rank- Order Correlation To analyze the degree of association between two ordinally scaled variables. Correlation analysis technique for use with ordinal data. Conclusions regarding rankings: 1. Positively correlated 2. Negatively correlated 3. Independent To understand Spearman Rank Order correlation. Correlation Analysis

Learning Objectives SUMMARY Bivariate Analysis of Association Bivariate Regression Correlation Analysis

Learning Objectives The End Copyright © 2002 South-Western/Thomson Learning