Bivariate Correlation and Regression

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Learning Objectives 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, R2.
Learning Objectives Bivariate Analysis of To understand bivariate Association regression analysis. Bivariate Analysis Defined The degree of association between two variables Bivariate techniques Statistical methods of analyzing the relationship between two variables. Multivariate Techniques When more than two variables are involved Independent variable Affects the value of the dependent variable Dependent variable explained or caused by the independent variable
Learning Objectives Bivariate Analysis of To understand bivariate Association regression analysis. Types of Bivariate Procedures • Two group t-tests • chi-square analysis of cross-tabulation or contingency tables • ANOVA (analysis of variance) for two groups • Bivariate regression • Pearson product moment correlation
Learning Objectives Bivariate Regression To understand bivariate regression analysis. 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 • Nonlinear relationship
Learning Objectives Bivariate Regression To understand bivariate regression analysis. Bivariate Regression Example Figure 13.1 Types of Relationships Found in Scatter Diagrams Y X A - Strong Positive Linear Relationship
Learning Objectives Bivariate Regression To understand bivariate regression analysis. Figure 13.1 Types of Relationships Found in Scatter Diagrams Y X B - Positive Linear Relationship
Learning Objectives Bivariate Regression To understand bivariate regression analysis. Figure 13.1 Types of Relationships Found in Scatter Diagrams Y X C - Perfect Negative Linear Relationship
Learning Objectives Bivariate Regression To understand bivariate regression analysis. Figure 13.1 Types of Relationships Found in Scatter Diagrams X D - Perfect Parabolic Relationship
Learning Objectives Bivariate Regression To understand bivariate regression analysis. Figure 13.1 Types of Relationships Found in Scatter Diagrams Y X E - Negative Curvilinear Relationship
Learning Objectives Bivariate Regression To understand bivariate regression analysis. Figure 13.1 Types of Relationships Found in Scatter Diagrams Y X F - No Relationship between X and Y
Learning Objectives Bivariate Regression To understand bivariate regression analysis. 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. Y = a + bX + e where Y = dependent variable a = estimated Y intercept b = estimated slope of the regression line X = independent variable e = error
Learning Objectives Bivariate Regression To understand bivariate regression analysis. Values for a and b can be calculated as follows:  XiYi - nXY b = 2 2  X i - n(X) a = Y - bX X = mean of value X Y = mean of value y n = sample size
Learning Objectives Bivariate Regression To become aware of the coefficient of determination, R2. The Regression Line Predicted values for Y, based on calculated values. Strength of Association: R2 Coefficient of Determination, R2: The measure of the strength of the linear relationship between X and Y.
Learning Objectives Bivariate Regression To become aware of the coefficient of determination, R2. explained variance R2 = total variance explained variance = total variance - unexplained variance total variance - unexplained variance R2 = total variance unexplained variance = 1 - total variance
Learning Objectives Bivariate Regression To become aware of the coefficient of determination, R2. unexplained variance R2 = 1 - total variance n 2  (Yi - Yi) = 1 - I = 1 n 2  (Yi - Y) I = 1
Learning Objectives Bivariate Regression To become aware of the coefficient of determination, R2. Statistical Significance of Regression Results Total variation = Explained variation + Unexplained variation 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.
Learning Objectives Bivariate Regression To become aware of the coefficient of determination, R2. Total variation: Sum of squares (SST) n SST = 2  (Yi - Y) i = 1 n 2 n  Yi =  Y 2 i = 1 i = 1 i n
Learning Objectives Bivariate Regression To become aware of the coefficient of determination, R2. Sum of squares due to regression (SSR) n SSR = 2  (Yi - Y) i = 1 2 n n n  Yi = a  Y b X Y i = 1 i = 1 i + i = 1 i i n
Learning Objectives Bivariate Regression To become aware of the coefficient of determination, R2. Error sums of squares (SSE) n SSE = 2  (Yi - Y) i = 1 n n n =  Y2 a Y b X Y i = 1 i i = 1 i i = 1 i i
Learning Objectives Bivariate Regression To become aware of the coefficient of determination, R2. Hypotheses Concerning the Overall Regression Null Hypothesis Ho: There is no linear relationship between X and Y. Alternative Hypothesis Ha: There is a linear relationship between X and Y.
Learning Objectives Bivariate Regression To become aware of the coefficient of determination, R2. Hypotheses about the Regression Coefficient Null Hypothesis Ho: b = 0 Alternative Hypothesis Ha: b 0 The appropriate test is the t-test.
Figure 13.4 Measures of Variation in aLearning Regression Objectives Y Yi =a + bXi Unexplained Total variation Variation Explained variation Y (X, Y) a 0 X X Xi
Learning Objectives Correlation Analysis To become aware of the coefficient of determination, R2. 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
Learning Objectives Correlation Analysis To become aware of the coefficient of determination, R2. R = + 2 - √ R R can be computed directly from the data: n  XY - ( X) - ( Y) R = √ [n  X2 - ( X) 2] [n  Y2 -  Y)2]
Learning Objectives SUMMARY • Bivariate Analysis of Association • Bivariate Regression • Correlation Analysis