Regression Analysis

1. Regression Analysis

2. Regression Analysis 1. To comprehend the nature of correlation analysis. 2. To understand bivariate regression analysis. 3. To become aware of the coefficient of determination

3. Bivariate Analysis of Association • 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

4. Bivariate Analysis of Association • 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

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 • Nonlinear relationship

6. Bivariate Regression Bivariate Regression Example Types of Relationships Found in Scatter Diagrams Y X A - Strong Positive Linear Relationship

7. Bivariate Regression Types of Relationships Found in Scatter Diagrams Y X B - Positive Linear Relationship

8. Bivariate Regression Types of Relationships Found in Scatter Diagrams Y X C - Perfect Negative Linear Relationship

9. Bivariate Regression Types of Relationships Found in Scatter Diagrams X D - Perfect Parabolic Relationship

10. Bivariate Regression Types of Relationships Found in Scatter Diagrams Y X E - Negative Curvilinear Relationship

11. Bivariate Regression Types of Relationships Found in Scatter Diagrams Y X F - No Relationship between X and Y

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

13.  XiYi - nXY b =  X2i - n(X)2 a = Y - bX X = mean of value X Y = mean of value y Bivariate Regression Values for a and b can be calculated as follows: n = sample size

14. Bivariate Regression y= β0 +β1 + Є β1 = Sxy /Sxx β0 = y - β1x

15. Bivariate Regression • 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.

16. explained variance R2 = total variance explained variance = total variance - unexplained variance total variance - unexplained variance R2 = total variance unexplained variance 1 - = total variance Bivariate Regression

17. unexplained variance 1 - R2 = total variance n  (Yi - Yi)2 I = 1 = 1 - n  (Yi - Y)2 I = 1 Bivariate Regression Predicted response

18. Total variation = Explained variation + Unexplained variation Bivariate Regression To become aware of the coefficient of determination, R2. 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.

19. n SST =  (Yi - Y)2 i = 1 n  Yi2 n i = 1 =  Yi2 i = 1 n Bivariate Regression Total variation: Sum of squares (SST)

20. n SSR =  (Yi - Y)2 i = 1 2 n  Yi n n i = 1 = a  Yi b Xi Yi + i = 1 i = 1 n Bivariate Regression Sum of squares due to regression (SSR)

21. n SSE =  (Yi - Y)2 i = 1 n n n =  Y2i a Yi b XiYi i = 1 i = 1 i = 1 Bivariate Regression Error sums of squares (SSE)

22. Bivariate Regression • 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.

23. Bivariate Regression • Hypotheses about the Regression Coefficient • Null Hypothesis Ho: • b = 0 • Alternative Hypothesis Ha: • b  0 • The appropriate test is the t-test.

24. Measures of Variation in a Regression Unexplained variation Total Variation Explained variation (X, Y) Y Xi X Y Yi =a + bXi a 0 X

25. 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

26. √ + R = R2 - n  XY - ( X) - ( Y) R = √ [n  X2 - ( X) 2] [n  Y2 -  Y)2] Correlation Analysis To become aware of the coefficient of determination, R2. R can be computed directly from the data:

27. SUMMARY • Bivariate Analysis of Association • Bivariate Regression • Correlation Analysis