Chapter 13 Multiple Regression

Chapter Outline Multiple Regression Model Least Squares Method Coefficient of Determination Model Assumptions Testing for Significance Estimation and Prediction Categorical Independent Variables

Multiple Regression Model The equation that describes how the dependent variable y is related to the independent variables x1, x2, . . . xp and an error term is: y = b0 + b1x1 + b2x2 + . . . + bpxp + e where: b0, b1, b2, . . . , bp are the parameters, and e is a random variable called the error term

Multiple Regression Equation Multiple regression equation is: The equation that describes how the mean value of y is related to x1, x2, . . . xp is: E(y) = 0 + 1x1 + 2x2 + . . . + pxp

Estimated Multiple Regression Equation ^ y = b0 + b1x1 + b2x2 + . . . + bpxp A simple random sample is used to compute sample statistics b0, b1, b2, . . . , bp that are used as the point estimates of the parameters b0, b1, b2, . . . , bp.

Estimation Process E(y) = 0 + 1x1 + 2x2 +. . .+ pxp + e Multiple Regression Model E(y) = 0 + 1x1 + 2x2 +. . .+ pxp + e Multiple Regression Equation E(y) = 0 + 1x1 + 2x2 +. . .+ pxp Unknown parameters are b0, b1, b2, . . . , bp Sample Data: x1 x2 . . . xp y . . . . Estimated Multiple Regression Equation Sample statistics are b0, b1, b2, . . . , bp b0, b1, b2, . . . , bp provide estimates of

Least Squares Method Least Squares Criterion Computation of Coefficient Values: The formulas for the regression coefficients b0, b1, b2, . . . bp involve the use of matrix algebra. We will rely on computer software packages to perform the calculations.

Example: Employee Salary Survey Multiple Regression Example: Employee Salary Survey A local firm collected data for a sample of 20 employees. A suggestion was made that regression analysis could be used to determine if salary was related to the years of experience and the score on the firm’s aptitude test. The gender of employees, years of experience, score on the aptitude test, and corresponding annual salary ($1000s) for a sample of 20 employees is shown on the next slide.

Multiple Regression Example: Employee Salary Survey (data)

Multiple Regression Model Suppose we believe that salary (y) is related to the years of experience (x1) and the score on the aptitude test (x2) by the following regression model: y = 0 + 1x1 + 2x2 +  where y = annual salary ($1000) x1 = years of experience x2 = score on aptitude test

Solving for the Estimates of 0, 1, 2 Least Squares Output Input Data x1 x2 y 4 78 24 7 100 43 . . . 3 89 30 Computer Package for Solving Multiple Regression Problems b0 = b1 = b2 = R2 = etc.

Solving for the Estimates of 0, 1, 2 Excel’s Regression Output – Parameter Estimates Note: All the numbers are rounded to the 4th decimal point.

Estimated Regression Equation SALARY = 3.174 + 1.404(YEARS) + 0.251(SCORE) Note: Predicted salary will be in thousands of dollars.

Interpreting the Coefficients In multiple regression analysis, we interpret each regression coefficient as follows: bi represents an estimate of the change in y corresponding to a 1-unit increase in xi when all other independent variables are held constant.

Interpreting the Coefficients b1 = 1.404 Salary is expected to increase by $1,404 for each additional year of experience (when the variable score on attitude test is held constant).

Interpreting the Coefficients b2 = 0.251 Salary is expected to increase by $251 for each additional point scored on the aptitude test (when the variable years of experience is held constant).

Multiple Coefficient of Determination Relationship Among SST, SSR, SSE SST = SSR + SSE where: SST = total sum of squares (i.e. total variability of y) SSR = sum of squares due to regression (i.e. the variability of y that is explained by regression) SSE = sum of squares due to error (i.e. the variability of y that cannot be explained by regression)

Multiple Coefficient of Determination Excel’s ANOVA Output SST SSR

Multiple Coefficient of Determination r2 = SSR/SST = 500.3285/599.7855 = .83418 The regression relationship is strong. About 83.4% of the variability in the salary of employees can be explained by the years of experience and the aptitude score.

Adjusted Multiple Coefficient of Determination Note: p is the number of slope coefficients.

Assumptions About the Error Term e y = b0 + b1x1 + b2x2 + … + bpxp +e 1. The error  is a random variable with mean of zero. 2. The variance of  , denoted by  2, is the same for all values of the independent variable. 3. The values of  are independent. 4. The error  is a normally distributed random variable reflecting the deviation between the y value and the expected value of y given by 0 + 1x1 + 2x2 + . . + pxp.

Testing for Significance In simple linear regression, the F and t tests provide the same conclusion. In multiple regression, the F and t tests have different purposes.

Testing for Significance The F test is used to test the overall significance of a regression model. The t test is used to test the individual significance, i.e. whether each of the individual independent variables is significant.

Testing for Significance: F Test H0: 1 = 2 = . . . = p = 0 Ha: One or more of the parameters is not equal to zero. Hypotheses Test Statistics F = MSR/MSE Rejection Rule Reject H0 if p-value < a or if F > F , where F is based on an F distribution with p d.f. in the numerator and n - p - 1 d.f. in the denominator.

F Test: Employee Salary Survey H0: 1 = 2 = 0 Ha: One or both of the parameters is not equal to zero. Hypotheses Test Statistics F = MSR/MSE = 250.16/5.85 = 42.76 For  = .05 and d.f. = (2, 17); F.05 = 3.59 Reject H0 if p-value < .05 or F > 3.59 Rejection Rule

F Test: Employee Salary Survey p-value Conclusion p-value < .05, so we can reject H0. (Also, F = 42.76 > 3.59)

Testing for Significance: t Test Hypotheses Test Statistics Rejection Rule Reject H0 if p-value < a or if t < -tor t > t where t is based on a t distribution with n - p - 1 degrees of freedom.

t Test: Employee Salary Survey Hypotheses Test Statistics Rejection Rule For  = .05 and d.f. = 17, t.025 = 2.11 Reject H0 if p-value < .05, or if t < -2.11 or t > 2.11

t Test: Employee Salary Survey Conclusions Reject both H0: 1 = 0 and H0: 2 = 0. Both independent variables are significant.

Test for Significance: Multicollinearity The term multicollinearity refers to the correlation among the independent variables. When the independent variables are highly correlated (say, |r | > .7), it is not possible to determine the separate effect of any particular independent variable on the dependent variable.

Using the Estimated Regression Equation for Estimation and Prediction The procedures for estimating the mean value of y and predicting an individual value of y in multiple regression are similar to those in simple regression. We substitute the given values of x1, x2, . . . , xp into the estimated regression equation and use the corresponding value of y as the point estimate.

Categorical Independent Variables In many situations we must work with categorical independent variables such as gender (male, female), method of payment (cash, check, credit card), etc. For example, xi might represent gender where xi = 0 indicates male and xi = 1 indicates female. In this case, xi is called a dummy or indicator variable.

Categorical Independent Variables Example: Employee Salary Survey As an extension of the problem involving the employee salary survey, suppose that management wants to find out if the annual salary is related to employees’ gender. The years of experience, the score on the aptitude test, employees’ gender, and the annual salary ($000) for each of the sampled 20 employees are shown on the next slide.

Estimated Regression Equation y = b0 + b1x1 + b2x2 + b3x3 ^ ^ where: y = annual salary ($1000) x1 = years of experience x2 = score on aptitude test x3 = 0 if an employee is female; 1 if an employee is male. x3 is a dummy variable

Categorical Independent Variables

Categorical Independent Variables Excel’s Regression Statistics

Categorical Independent Variables Excel’s ANOVA Output

Categorical Independent Variables Excel’s Regression Equation Output Not significant