- Количество слайдов: 26
Lecture 22 • Multiple Regression (Sections 19. 3 -19. 4)
19. 1 Introduction • In this chapter we extend the simple linear regression model, and allow for any number of independent variables. • We expect to build a model that fits the data better than the simple linear regression model.
Examples of Multiple Regression • Business decisionmaking: La Quinta Inns wants to decide where to locate new inns. It wants to predict operating margin based on variables related to competition, market awareness, demand generators, demographics and physical location. • College admissions: The admissions officer wants to predict which students will be most successful. She wants to predict college GPA based on GPA from high school, SAT score and amount of time participating in extracurricular activities.
More Examples • Improving operations: A parcel delivery service would like to increase the number of packages that are sorted in each of its hub locations. Three factors that the company can control and that influence sorting performance are the number of sorting lines, the number of sorting workers, and the number of truck drivers. What can the company do to improve sorting performance? • Understanding relationships: Executive compensation. Does it matter how long the executive has been at the firm controlling for other factors? Do CEOs pay themselves less if they have a large stake in the stock of the company controlling for other factors? Does having an MBA increase executive salary controlling for other factors?
Introduction • We shall use computer printout to – Assess the model • How well it fits the data • Is it useful • Are any required conditions violated? – Employ the model • Interpreting the coefficients • Predictions using the prediction equation • Estimating the expected value of the dependent variable
Example 19. 1 • Where to locate a new motor inn? – La Quinta Motor Inns is planning an expansion. – Management wishes to predict which sites are likely to be profitable. – Several areas where predictors of profitability can be identified are: • • • Competition Market awareness Demand generators Demographics Physical quality
Profitabil ity Competition Rooms Market awareness Nearest Customers Office space Distance to Number of hotels/motels the nearest rooms within La Quinta inn. 3 miles from the site. Margin Community College Income enrollment Physical Disttwn Median Distance to household downtown. income.
19. 2 Model and Required Conditions • We allow for k independent variables to potentially be related to the dependent variable: Coefficients Random error variable y = b 0 + b 1 x 1+ b 2 x 2 + …+ bkxk + e Dependent variable Independent variables
Multiple Regression for k = 2, Graphical Demonstration - I y The simple linear regression model allows for one independent variable, “x” y =b 0 + b 1 x + e b 1 x 1 y = b 0 + b 1 x Note how the straight line y = b 0 0 y= becomes a plane, and. . . + b 2 x 2 + b 1 x 12 x 2 b b 0 x 1 + b 2 x 2 y = b 1 x+ + 2 x 2 b + x 1 b x b+ + 1 b+b 122 x 22 0 x = bb x y =+b 0 1 b 11+1 x 2 X 1 y b b y b 00 b 0 x 1 + b 2 = = +1 1 y y =0 + y = b The multiple linear regression model allows for more than one independent variable. Y = b 0 + b 1 x 1 + b 2 x 2 + e X 2
Multiple Regression for k = 2, Graphical Demonstration - II y y= b 0+ b 1 x 2 Note how a parabola becomes a parabolic Surface. b 0 X 1 y = b 0 + b 1 x 12 + b 2 x 2 X 2
Required conditions for the error variable • The error e is normally distributed. • The mean of the error e is equal to zero for each combination of x’s, i. e. , . • The standard deviation is constant (se) for all values of x’s. • The errors are independent.
Estimating the Coefficients and Assessing the Model, Example • Data were collected from randomly selected 100 inns that belong to La Quinta, and ran for the following suggested model: Margin = b 0 + b 1 Rooms + b 2 Nearest + b 3 Office + b 4 College + b 5 Income + b 6 Disttwn Xm 19 -01
19. 3 Estimating the Coefficients and Assessing the Model • The procedure used to perform regression analysis: – Estimate the model coefficients and statistics using least squares using JMP. – Diagnose violations of required conditions. Try to remedy problems when identified. – Assess the model fit using statistics obtained from the sample. – If the model assessment indicates good fit to the data, use it to interpret the coefficients and generate predictions.
Model Assessment • The model is assessed using three tools: – The standard error of estimate – The coefficient of determination – The F-test of the analysis of variance • The standard error of estimates participates in building the other tools.
Standard Error of Estimate • The standard deviation of the error is estimated by the Standard Error of Estimate: • The magnitude of se is judged by comparing it to
Standard Error of Estimate • From the printout, se = 5. 51 • Calculating the mean value of y, • It seems se is not particularly small relative to y. • Question: Can we conclude the model does not fit the data well?
Coefficient of Determination • The definition is • From the printout, R 2 = 0. 5251 • 52. 51% of the variation in operating margin is explained by the six independent variables. 47. 49% remains unexplained. • When adjusted for degrees of freedom, Adjusted R 2 = 1 -[SSE/(n-k-1)] / [SS(Total)/(n-1)] = = 49. 44%
Testing the Validity of the Model • We pose the question: Is there at least one independent variable linearly related to the dependent variable? • To answer the question we test the hypothesis H 0: b 0 = b 1 = b 2 = … = b k H 1: At least one bi is not equal to zero. • If at least one bi is not equal to zero, the model has some validity.
Testing the Validity of the La Quinta Inns Regression Model • The hypotheses are tested by an ANOVA procedure.
Testing the Validity of the La Quinta Inns Regression Model [Variation in y] = SSR + SSE. If SSR is large relative to SSE, much of the variation in y is explained by the regression model; the model is useful and thus, the null hypothesis should be rejected. Thus, we reject for large F. Rejection region F>Fa, k, n-k-1
Testing the Validity of the La Quinta Inns Regression Model Conclusion: There is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. At least one of the bi is not equal to zero. Thus, at least one independent variable is linearly related to y. Fa, k, n-k-1 = F 0. 05, 6, 100 -6 -1=2. 17 This linear regression model is valid F = 17. 14 > 2. 17 Also, the p-value (Significance F) = 0. 0000 Reject the null hypothesis.
Interpreting the Coefficients • b 0 = 38. 14. This is the intercept, the value of y when all the variables take the value zero. Since the data range of all the independent variables do not cover the value zero, do not interpret the intercept. • b 1 = – 0. 0076. In this model, for each additional room within 3 mile of the La Quinta inn, the operating margin decreases on average by. 0076% (assuming the other variables are held constant).
Interpreting the Coefficients • b 2 = 1. 65. In this model, for each additional mile that the nearest competitor is to a La Quinta inn, the operating margin increases on average by 1. 65% when the other variables are held constant. • b 3 = 0. 020. For each additional 1000 sq-ft of office space, the operating margin will increase on average by. 02% when the other variables are held constant. • b 4 = 0. 21. For each additional thousand students the operating margin increases on average by. 21% when the other variables are held constant.
Interpreting the Coefficients • b 5 = 0. 41. For additional $1000 increase in median household income, the operating margin increases on average by. 41%, when the other variables remain constant. • b 6 = -0. 23. For each additional mile to the downtown center, the operating margin decreases on average by. 23% when the other variables are held constant.
Testing the Coefficients • The hypothesis for each bi is • JMP printout H 0: b i = 0 H 1: b i ¹ 0 Test statistic d. f. = n - k -1
La Quinta Inns, Predictions Xm 19 -01 • Predict the average operating margin of an inn at a site with the following characteristics: – – – 3815 rooms within 3 miles, Closet competitor. 9 miles away, 476, 000 sq-ft of office space, 24, 500 college students, $35, 000 median household income, 11. 2 miles distance to downtown center. MARGIN = 38. 14 - 0. 0076(3815) +1. 65(. 9) + 0. 020(476) +0. 21(24. 5) + 0. 41(35) - 0. 23(11. 2) = 37. 1%