Correlation and regression
- 1. Correlation and Regression 10/13/14 1
- 2. Product Moment Correlation • The product moment correlation, r, summarizes the strength of association between two metric (interval or ratio scaled) variables, say X and Y. • It is an index used to determine whether a linear or straight-line relationship exists between X and Y. • As it was originally proposed by Karl Pearson, it is also known as the Pearson correlation coefficient. It is also referred to as simple correlation, bivariate correlation, or merely the correlation coefficient. • A correlation is a single number that describes the degree of relationship between two variables. (http:// www.socialresearchmethods.net/kb/statcorr.php ) 10/13/14 2
- 3. SPSS • Is there any correlation/association between attitude toward the city and duration of residence? H0 : There is no association/ correlation between duration of residence and attitude toward the city H1 : There is an association/ correlation between between duration of residence and attitude toward the city 10/13/14 3
- 4. Tab le 17.1 Respondent No Attitude Toward the City Duration of Residence Importance Attached to Weather 1 6 10 3 2 9 12 11 3 8 12 4 4 3 4 1 5 10 12 11 6 4 6 1 7 5 8 7 8 2 2 4 9 11 18 8 10 9 9 10 11 10 17 8 12 2 2 5 10/13/14 4
- 5. Descriptive Statistics Mean Std. Deviation N 6,58 3,32 12 9,33 5,26 12 Attitude toward the city Duration of residence Correlations 1,000 ,936** 10/13/14 5 , ,000 120,917 179,667 10,992 16,333 12 12 ,936** 1,000 ,000 , 179,667 304,667 16,333 27,697 12 12 Pearson Correlation Sig. (2-tailed) Sum of Squares and Cross-products Covariance N Pearson Correlation Sig. (2-tailed) Sum of Squares and Cross-products Covariance N Attitude toward the city Duration of residence Attitude toward the city Duration of residence Correlation is significant at the 0.01 **. level (2-tailed).
- 6. Correlation Coefficient n S (Xi – X) (Yi – Y) i=1 r = = COVxy / SxSy n n S (Xi – X)2 S (Yi – Y)2 i=1 i=1 10/13/14 6
- 7. Y X1 X2 r = (179.6668) / [(304/6669)(120.9168) = 0.9361 The respondent’s duration of residence in the city is strongly associated with attitude toward the city 10/13/14 7
- 8. city the toward Attitude N = 2 1 1 1 1 1 3 1 1 2 4 6 8 9 10 12 17 18 10/13/14 8 Duration of residence 12 10 8 6 4 2 0
- 9. • Analyze > correlate > bivariate – Options : - means and standar deviations - cross product deviations and covariances • Graphs > box plot > simple > define >ok 10/13/14 9
- 10. Regression Analysis • Regression analysis is a statistical procedure for analyzing associative relationship between a metric dependent variable and one or more metric independent variables. • Independent variable affect dependent variable 10/13/14 10
- 11. Regression Analysis can be used in the following ways: 1. Determine whether the independent variables explain a significant variation in the dependent variable: whether a relationship exists. 2. Determine how much of the variation in the dependent variable can be explained by the independent variables: strength of the relationship. 3. Determine the structure or form of the relationship: the mathematical equation relating the independent and dependent variables. 4. Predict the values of the dependent variables. 5. Control for other independent variables when evaluating the contributions of specific variable or set of variables. 10/13/14 11
- 12. Bivariate Regression • Bivariate regression is a procedure for deriving a mathematical relationship, in the form of an equation, between a single metric dependent variable and a single metric independent variable. • Bivariate regression example in marketing: – Can variation in sales be explained in terms of variation in advertising expenditures? What is the structure and form of this relationship, can it be modeled mathematically by an equation describing a straight line. 10/13/14 12
- 13. Multiple Regression The general form of the multiple regression model is as follows: Y = b0 + b1X1 + b2X2 + b3X3+ . . . + bkXk + e which is estimated by the following equation: Y = a + b1X1 + b2X2 + b3X3+ . . . + bXkk As before, the coefficient a represents the intercept, but the b's are now the partial regression coefficients. 10/13/14 13
- 14. Statistics Associated with Multiple Regression • Adjusted R2. R2, coefficient of multiple determination, is adjusted for the number of independent variables and the sample size to account for the diminishing returns. After the first few variables, the additional independent variables do not make much contribution. • Coefficient of multiple determination. The strength of association in multiple regression is measured by the square of the multiple correlation coefficient, R2, which is also called the coefficient of multiple determination. • F test. The F test is used to test the null hypothesis that the coefficient of multiple determination in the population, R2 pop, is zero. This is equivalent to testing the null hypothesis. The test statistic has an F distribution with k and (n - k - 1) degrees of freedom. 10/13/14 14
- 15. Model Summary R R Square Adjusted R Square Std. Error of the Estimate Change Statistics R Square Change F Change df1 df2 Sig. F Change ,975a ,951 ,942 ,762 ,951 105,826 2 11 ,000 Model 1 Predictors: (Constant), a. PRICE, QUALITY ANOVAb Sum of Squares df Mean Square F Sig. 122,831 2 61,415 105,826 ,000a 6,384 11 ,580 129,214 13 Regression Residual Total Model 1 a. Predictors: (Constant), PRICE, QUALITY b. Dependent Variable: PREFEREN Coefficientsa Unstandardized Coefficients B Std. Error Standardized Coefficients Beta t Sig. ,535 ,471 1,136 ,280 ,976 ,097 ,798 10,096 ,000 ,251 ,071 ,278 3,522 ,005 (Constant) QUALITY PRICE Model 1 a. Dependent Variable: PREFEREN 10/13/14 15