© The McGraw-Hill Companies, Inc., Chapter 10 Correlation and Regression

© The McGraw-Hill Companies, Inc., Objectives Correlation: Draw a scatter plot for a set of ordered pairs. Find the correlation coefficient, r. Test the hypothesis H 0 :  = 0.

© The McGraw-Hill Companies, Inc., Objectives Regression: Find the equation of the regression line. Find the coefficient of determination. Find the standard error of estimate.

© The McGraw-Hill Companies, Inc., Scatter Plots A scatter plot (x, y) x y A scatter plot is a graph of the ordered pairs (x, y) of numbers consisting of the independent variable, x, and the dependent variable, y.

© The McGraw-Hill Companies, Inc., Scatter Plots - Scatter Plots - Example Construct a scatter plot for the data obtained in a study of age and systolic blood pressure of six randomly selected subjects. The data is given on the next slide.

© The McGraw-Hill Companies, Inc., Scatter Plots - Scatter Plots - Example

© The McGraw-Hill Companies, Inc., Scatter Plots - Scatter Plots - Example Positive Relationship

© The McGraw-Hill Companies, Inc., Scatter Plots - Scatter Plots - Other Examples Negative Relationship

© The McGraw-Hill Companies, Inc., Scatter Plots - Scatter Plots - Other Examples No Relationship

© The McGraw-Hill Companies, Inc., Correlation Coefficient correlation coefficient The correlation coefficient computed from the sample data measures the strength and direction of a relationship between two variables. Sample correlation coefficient, r. Population correlation coefficient, 

© The McGraw-Hill Companies, Inc., Range of Values for the Correlation Coefficient  Strong negative relationship Strong positive relationship No linear relationship

© The McGraw-Hill Companies, Inc., Formula for the Correlation Coefficient r                  r nxyxy nxxnyy          Where n is the number of data pairs

© The McGraw-Hill Companies, Inc., Correlation Coefficient – Correlation Coefficient – Example (Verify) correlation coefficient Compute the correlation coefficient for the age and blood pressure data.

© The McGraw-Hill Companies, Inc., The Significance of the Correlation Coefficient population corelation coefficient The population corelation coefficient, , is the correlation between all possible pairs of data values (x, y) taken from a population.

© The McGraw-Hill Companies, Inc., The Significance of the Correlation Coefficient H 0 :  = 0 H 1 :   0 This tests for a significant correlation between the variables in the population.

© The McGraw-Hill Companies, Inc., Formula for the t tests for the Correlation Coefficient t n r with dfn    

© The McGraw-Hill Companies, Inc., Example Test the significance of the correlation coefficient for the age and blood pressure data. Use  = 0.05 and r = Step 1: Step 1: State the hypotheses. H 0 :  = 0 H 1 :  0

© The McGraw-Hill Companies, Inc., Step 2: Step 2: Find the critical values. Since  = 0.05 and there are 6 – 2 = 4 degrees of freedom, the critical values are t = and t = – Step 3: Step 3: Compute the test value. t = (verify). Example

© The McGraw-Hill Companies, Inc., Step 4: Step 4: Make the decision. Reject the null hypothesis, since the test value falls in the critical region (4.059 > 2.776). Step 5: Step 5: Summarize the results. There is a significant relationship between the variables of age and blood pressure. Example

© The McGraw-Hill Companies, Inc., 2000 Using SPSS Open SPSS and add the example data in side-by-side columns Click: Analyze | Correlate | Bivariate… Move both variables into Variables Check Pearson then OK box Output follows

© The McGraw-Hill Companies, Inc., 2000 SPSS Output Matrix shows all correlations but ones on principal diagonal are irrelevant (i.e., 1s) Actual r is 0.897; * means it was significant. Correlations AgePressure AgePearson Correlation1.897 * Sig. (2-tailed).015 N66 PressurePearson Correlation.897 * 1 Sig. (2-tailed).015 N66 *. Correlation is significant at the 0.05 level (2-tailed).

© The McGraw-Hill Companies, Inc., The scatter plot for the age and blood pressure data displays a linear pattern. We can model this relationship with a straight line. This regression line is called the line of best fit or the regression line. The equation of the line is y = a + bx. Regression

© The McGraw-Hill Companies, Inc., Formulas for the Regression Line Formulas for the Regression Line y = a + bx.                      a yxxxy nxx b n xy nxx               Where a is the y intercept and b is the slope of the line. 

© The McGraw-Hill Companies, Inc., Example Find the equation of the regression line for the age and the blood pressure data. Substituting into the formulas give a = and b = (verify). Hence, y = x. ainterceptb slope Note, a represents the intercept and b the slope of the line.

© The McGraw-Hill Companies, Inc., Example y = x

© The McGraw-Hill Companies, Inc., Using the Regression Line to Predict The regression line can be used to predict a value for the dependent variable (y) for a given value of the independent variable (x). Caution: Caution: Use x values within the experimental region when predicting y values.

© The McGraw-Hill Companies, Inc., Example Use the equation of the regression line to predict the blood pressure for a person who is 50 years old. Since y = x, then y = (50) =  129. Note that the value of 50 is within the range of x values. 

© The McGraw-Hill Companies, Inc., Coefficient of Determination and Standard Error of Estimate coefficient of determination The coefficient of determination, denoted by r 2, is a measure of the variation of the dependent variable that is explained by the regression line and the independent variable.

© The McGraw-Hill Companies, Inc., Coefficient of Determination and Standard Error of Estimate r 2 is the square of the correlation coefficient. coefficient of nondetermination The coefficient of nondetermination is (1 – r 2 ). Example: If r = 0.90, then r 2 = 0.81.

© The McGraw-Hill Companies, Inc., Coefficient of Determination and Standard Error of Estimate standard error of estimate The standard error of estimate, denoted by s est, is the standard deviation of the observed y values about the predicted y values. The formula is given on the next slide.

© The McGraw-Hill Companies, Inc., Formula for the Standard Error of Estimate   s yy n or s yaybxy n est         

© The McGraw-Hill Companies, Inc., Standard Error of Estimate - Standard Error of Estimate - Example From the regression equation, y = x and n = 6, find s est. Here, a = 55.57, b = 8.13, and n = 6. Substituting into the formula gives s est = 6.48 (verify).

© The McGraw-Hill Companies, Inc., 2000 Using SPSS Open SPSS and add the example data in side-by-side columns Click: Analyze | Regression | Linear… Move dependent (Pressure) variable to Dependent: box Move independent variable to Independent(s): box

© The McGraw-Hill Companies, Inc., 2000 SPSS Output Look in box labelled Coefficients. Slope of regression is in red circle (0.964). y-intercept is in green (81.048). Coefficients a ModelUnstandardized CoefficientsStandardized Coefficients BStd. ErrorBetatSig. 1(Constant) Age a. Dependent Variable: Pressure