Copyright © 2010 Pearson Education, Inc Chapter Seventeen Correlation and Regression

Copyright © 2010 Pearson Education, Inc ) Chapter Outline 1) Overview 2) Correlation 3) Regression Analysis

Copyright © 2010 Pearson Education, Inc ) Correlation Coefficient The correlation coefficient, r, summarizes the strength of association between two metric (interval or ratio scaled) variables, say X and Y. In other words, you can have a correlation coefficient for Likert scale items, not dichotomous items. It is an index used to determine whether a linear (straight-line) relationship exists between X and Y. As it was originally proposed by Karl Pearson, it is known as the Pearson correlation coefficient.

Copyright © 2010 Pearson Education, Inc Linear relationships For example: How much does weight (Y) go up as height (X) goes up by one unit? Height and weight are positively correlated.

Copyright © 2010 Pearson Education, Inc Product Moment Correlation R can be calculated from a sample of n observations: sum X = average of all x’s Y = average of all y’s Don’t worry, we can do this in SPSS…

Copyright © 2010 Pearson Education, Inc Product Moment Correlation r varies between -1.0 and The correlation coefficient between two variables will be the same regardless of their underlying units of measurement. For example, comparing a 5 point scale to a 7 point scale is okay. Note: correlation does not equal causation! e.g. ice cream does not cause a sunburn.

Copyright © 2010 Pearson Education, Inc Explaining Attitude Toward the City of Residence

Copyright © 2010 Pearson Education, Inc Interpretation of the Correlation Coefficient The correlation coefficient ranges from −1 to 1. A value of 1 implies that all data points lie on a line for which Y increases as X increases. A value of −1 implies that all data points lie on a line for which Y decreases as X increases. A value of 0 implies that there is no linear correlation between the variables.

Copyright © 2010 Pearson Education, Inc Positive and Negative Correlation

Copyright © 2010 Pearson Education, Inc CorrelationNegativePositive None−0.09 to to 0.09 Small−0.3 to − to 0.3 Medium−0.5 to − to 0.5 Strong−1.0 to − to 1.0 Interpretation of the Correlation Coefficient As a rule of thumb, correlation values can be interpreted in the following manner:

Copyright © 2010 Pearson Education, Inc SPSS Windows: Correlations 1. Select ANALYZE from the SPSS menu bar. 2. Click CORRELATE and then BIVARIATE. 3. Move “variable x” into the VARIABLES box. Then move “variable y” into the VARIABLES box. 4. Check PEARSON under CORRELATION COEFFICIENTS. 5. Check ONE-TAILED under TEST OF SIGNIFICANCE. 6. Check FLAG SIGNIFICANT CORRELATIONS. 7. Click OK.

Copyright © 2010 Pearson Education, Inc SPSS Example: Correlation Correlations AgeInternetUsage InternetShoppi ng AgePearson Correlation Sig. (1- tailed) N20 InternetUsagePearson Correlation Sig. (1- tailed).000 N20 InternetShoppingPearson Correlation Sig. (1- tailed) N20

Copyright © 2010 Pearson Education, Inc Linear Relationships and Regression Analysis Regression analysis is a predictive analysis technique in which one or more variables are used to predict the level of another by use of the straight-line formula, y=a+bx.

Copyright © 2010 Pearson Education, Inc Bivariate Linear Regression Analysis Bivariate regression analysis is a type of regression in which only two variables are used in the regression, predictive model. One variable is termed the dependent variable (y), the other is termed the independent variable (x). The independent variable is used to predict the dependent variable, and it is the x in the regression formula.

Copyright © 2010 Pearson Education, Inc Bivariate Linear Regression Analysis With bivariate analysis, one variable is used to predict another variable. The straight-line equation is the basis of regression analysis.

Copyright © 2010 Pearson Education, Inc Bivariate Linear Regression Analysis

Copyright © 2010 Pearson Education, Inc Bivariate Linear Regression Analysis: Basic Procedure Independent variable: used to predict the independent variable (x in the regression straight-line equation) Dependent variable: that which is predicted (y in the regression straight-line equation) Least squares criterion: used in regression analysis; guarantees that the “best” straight-line slope and intercept will be calculated

Copyright © 2010 Pearson Education, Inc Bivariate Linear Regression Analysis: Basic Procedure The regression model, intercept, and slope must always be tested for statistical significance Regression analysis predictions are estimates that have some amount of error in them Standard error of the estimate: used to calculate a range of the prediction made with a regression equation

Copyright © 2010 Pearson Education, Inc Testing for Statistical Significance of the Intercept and the Slope The t test is used to determine whether the intercepts and slope are significantly different from 0 (the null hypothesis). If the computed t value is greater than the table t value, the null hypothesis is not supported.

Copyright © 2010 Pearson Education, Inc Making a Prediction

Copyright © 2010 Pearson Education, Inc Ch 1921 Bivariate Linear Regression Analysis: Basic Procedure Regression predictions are made with confidence intervals.

Copyright © 2010 Pearson Education, Inc Ch 1922 Multiple Regression Analysis Multiple regression analysis uses the same concepts as bivariate regression analysis, but uses more than one independent variable. General conceptual model identifies independent and dependent variables and shows their basic relationships to one another.

Copyright © 2010 Pearson Education, Inc Ch 1923 Multiple Regression Analysis: A Conceptual Model

Copyright © 2010 Pearson Education, Inc Ch 1924 Multiple Regression Analysis Multiple regression means that you have more than one independent variable to predict a single dependent variable

Copyright © 2010 Pearson Education, Inc Ch 1925 Example of Multiple Regression

Copyright © 2010 Pearson Education, Inc Ch 1926 Example of Multiple Regression We wish to predict customers’ intentions to purchase a Lexus automobile. We performed a survey that included an attitude- toward-Lexus variable, a word-of-mouth variable, and an income variable. Here is the result:

Copyright © 2010 Pearson Education, Inc Ch 1927 Example of Multiple Regression This multiple regression equation means that we can predict a consumer’s intention to buy a Lexus level if you know three variables: Attitude toward Lexus, Friends’ negative comments about Lexus, and Income level using a scale with 10 income grades.

Copyright © 2010 Pearson Education, Inc Ch 1928 Example of Multiple Regression Calculation of Lexus purchase intention using the multiple regression equation: Multiple regression is a powerful tool because it tells us which factors predict the dependent variable, which way (the sign) each factor influences the dependent variable, and even how much (the size of b) each factor influences it.

Copyright © 2010 Pearson Education, Inc Thank you! Questions??