Chapter 16 Regression Analysis- Model Building PDF

Statistics for Business and Economics (13e) Statistics for Business and Economics (13e) Anderson, Sweeney, Williams, Camm, Cochran © 2017 Cengage Learning Slides by John Loucks St. Edwards University © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly access...

Statistics for Business and Economics (13e) Statistics for Business and Economics (13e) Anderson, Sweeney, Williams, Camm, Cochran © 2017 Cengage Learning Slides by John Loucks St. Edwards University © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 1 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Chapter 16 Regression Analysis: Model Building General Linear Model Determining When to Add or Delete Variables Variable Selection Procedures Multiple Regression Approach to Experimental Design Autocorrelation and the Durbin-Watson Test © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 2 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) General Linear Model Models in which the parameters (0, 1,... , p ) all have exponents of one are called linear models. A general linear model involving p independent variables is y = 0 + 1z1 + 2z2 + … pzp + e Each of the independent variables z is a function of x1, x2,..., xk (the variables for which data have been collected). © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 3 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) General Linear Model The simplest case is when we have collected data for just one variable x1 and want to estimate y by using a straight-line relationship. In this case z1 = x1. This model is called a simple first-order model with one predictor variable. y = 0 + 1x1 + e © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 4 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Modeling Curvilinear Relationships To account for a curvilinear relationship, we might set z1 = x1 and z2 = 𝑥1 2. This model is called a second-order model with one predictor variable. 𝑦 = 𝛽0 + 𝛽1 𝑥1 + 𝛽2 𝑥1 2 + 𝜀 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 5 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Interaction If the original data set consists of observations for y and two independent variables x1 and x2 we might develop a second-order model with two predictor variables. 𝑦 = 𝛽0 + 𝛽1 𝑥1 + 𝛽2 𝑥2 + 𝛽3 𝑥1 2 + 𝛽4 𝑥2 2 + 𝛽5 𝑥1 𝑥2 + 𝜀 In this model, the variable z5 = x1x2 is added to account for the potential effects of the two variables acting together. This type of effect is called interaction. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 6 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Transformations Involving the Dependent Variable Often the problem of nonconstant variance can be corrected by transforming the dependent variable to a different scale. Most statistical packages provide the ability to apply logarithmic transformations using either the base-10 (common log) or the base e = 2.71828... (natural log). Another approach, called a reciprocal transformation, is to use 1/y as the dependent variable instead of y. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 7 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Nonlinear Models That Are Intrinsically Linear Models in which the parameters (0, 1,... , p ) have exponents other than one are called nonlinear models. In some cases we can perform a transformation of variables that will enable us to use regression analysis with the general linear model. The exponential model involves the regression equation: 𝐸 𝑦 = 𝛽0 𝛽1 𝑥 We can transform this nonlinear model to a linear model by taking the logarithm of both sides. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 8 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Determining When to Add or Delete Variables To test whether the addition of x2 to a model involving x1 (or the deletion of x2 from a model involving x1 and x2) is statistically significant we can perform an F Test. The F Test is based on a determination of the amount of reduction in the error sum of squares resulting from adding one or more independent variables to the model. (SSE reduced − SSE full ) 𝐹= number of extra terms MSE(full) (SSE 𝑥1 − SSE 𝑥1 , 𝑥2 )/1 𝐹= SSE 𝑥1 , 𝑥2 /(𝑛 − 𝑝 − 1) © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 9 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Determining When to Add or Delete Variables The p–value criterion can also be used to determine whether it is advantageous to add one or more dependent variables to a multiple regression model. The p–value associated with the computed F statistic can be compared to the level of significance a. It is difficult to determine the p–value directly from the tables of the F distribution, but computer software packages, such as Minitab or Excel, provide the p-value. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 10 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection Procedures Stepwise Regression Iterative; one independent Forward Selection variable at a time is added or Backward Elimination deleted based on the F statistic Different subsets of the Best-Subsets Regression independent variables are evaluated The first 3 procedures are heuristics and therefore offer no guarantee that the best model will be found. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 11 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Stepwise Regression At each iteration, the first consideration is to see whether the least significant variable currently in the model can be removed because its F value is less than the user-specified or default Alpha to remove. If no variable can be removed, the procedure checks to see whether the most significant variable not in the model can be added because its F value is greater than the user-specified or default Alpha to enter. If no variable can be removed and no variable can be added, the procedure stops. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 12 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Stepwise Regression Compute F stat. and Any p-value for each indep. p-value < alpha variable not in model to enter No ? No Indep. variable Yes Any Yes with largest p-value > alpha Stop p-value is to remove removed ? from model Compute F stat. and next Indep. variable with p-value for each indep. smallest p-value is variable in model iteration entered into model Start with no indep. variables in model © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 13 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Forward Selection This procedure is similar to stepwise regression, but does not permit a variable to be deleted. This forward-selection procedure starts with no independent variables. It adds variables one at a time as long as a significant reduction in the error sum of squares (SSE) can be achieved. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 14 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Forward Selection Start with no indep. variables in model Compute F stat. and p-value for each indep. variable not in model Any Yes Indep. variable with p-value < alpha smallest p-value is to enter entered into model ? No Stop © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 15 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Backward Elimination This procedure begins with a model that includes all the independent variables the modeler wants considered. It then attempts to delete one variable at a time by determining whether the least significant variable currently in the model can be removed because its p-value is less than the user-specified or default value. Once a variable has been removed from the model it cannot reenter at a subsequent step. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 16 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Backward Elimination Start with all indep. variables in model Compute F stat. and p-value for each indep. variable in model Any Yes Indep. variable with p-value > alpha largest p-value is to remove removed from model ? No Stop © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 17 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Backward Elimination Example: Clarksville Homes Tony Zamora, a real estate investor, has just moved to Clarksville and wants to learn about the city’s residential real estate market. Tony has randomly selected 25 house-for-sale listings from the Sunday newspaper and collected the data partially listed on the next slide. Develop, using the backward elimination procedure, a multiple regression model to predict the selling price of a house in Clarksville. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 18 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Backward Elimination Partial Data Selling House Number Number Garage Segment Price Size of of Size of City ($000) (100s sq. ft.) Bedrms. Bathrms. (cars) Northwest 290 21 4 2 2 South 95 11 2 1 0 Northeast 170 19 3 2 2 Northwest 375 38 5 4 3 West 350 24 4 3 2 South 125 10 2 2 0 West 310 31 4 4 2 West 275 25 3 2 2 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 19 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Backward Elimination Regression Output Predictor Coef SE Coef T p Intercept -59.416 54.6072 -1.0881 0.28951 House Size 6.50587 3.24687 2.0037 0.05883 Bedrooms 29.1013 26.2148 1.1101 0.28012 Bathrooms 26.4004 18.8077 1.4037 0.17574 Cars -10.803 27.329 -0.3953 0.69680 Variable to be removed Greatest p-value >.05 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 20 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Backward Elimination Cars (garage size) is the independent variable with the highest p-value (.697) >.05. Cars variable is removed from the model. Multiple regression is performed again on the remaining independent variables. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 21 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Backward Elimination Regression Output Predictor Coef SE Coef T p Intercept -47.342 44.3467 -1.0675 0.29785 House Size 6.02021 2.94446 2.0446 0.05363 Bedrooms 23.0353 20.8229 1.1062 0.28113 Bathrooms 27.0286 18.3601 1.4721 0.15581 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 22 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Backward Elimination Bedrooms is the independent variable with the highest p-value (.281) >.05. Bedrooms variable is removed from the model. Multiple regression is performed again on the remaining independent variables. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 23 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Backward Elimination Regression Output Predictor Coef SE Coef T p Intercept -12.349 31.2392 -0.3953 0.69642 House Size 7.94652 2.38644 3.3299 0.00304 Bathrooms 30.3444 18.2056 1.6668 0.10974 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 24 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Backward Elimination Bathrooms is the independent variable with the highest p-value (.110) >.05. Bathrooms variable is removed from the model. Multiple regression is performed again on the remaining independent variable. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 25 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Backward Elimination Regression Output Predictor Coef SE Coef T p Intercept -9.8669 32.3874 -0.3047 0.76337 House Size 11.3383 1.29384 8.7633 8.7E-09 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 26 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Backward Elimination House size is the only independent variable remaining in the model. The estimated regression equation is: 𝑦 = −9.8669 + 11.3383(House Size) © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 27 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Best-Subsets Regression The three preceding procedures are one-variable-at-a-time methods offering no guarantee that the best model for a given number of variables will be found. Some software packages include best-subsets regression that enables the user to find, given a specified number of independent variables, the best regression model. Minitab output identifies the two best one-variable estimated regression equations, the two best two-variable equation, and so on. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 28 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable Selection: Best-Subsets Regression Example: PGA Tour Data The Professional Golfers Association keeps a variety of statistics regarding performance measures. Data include the average driving distance, percentage of drives that land in the fairway, percentage of greens hit in regulation, average number of putts, percentage of sand saves, and average score. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 29 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable-Selection Procedures Variable Names and Definitions Drive: average length of a drive in yards Fair: percentage of drives that land in the fairway Green: percentage of greens hit in regulation (a par-3 green is “hit in regulation” if the player’s first shot lands on the green) Putt: average number of putts for greens that have been hit in regulation Sand: percentage of sand saves (landing in a sand trap and still scoring par or better) Score: average score for an 18-hole round © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 30 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable-Selection Procedures Sample Data (Part 1) Drive Fair Green Putt Sand Score 277.6.681.667 1.768.550 69.10 259.6.691.665 1.810.536 71.09 269.1.657.649 1.747.472 70.12 267.0.689.673 1.763.672 69.88 267.3.581.637 1.781.521 70.71 255.6.778.674 1.791.455 69.76 272.9.615.667 1.780.476 70.19 265.4.718.699 1.790.551 69.73 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 31 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable-Selection Procedures Sample Data (Part 2) Drive Fair Green Putt Sand Score 272.6.660.672 1.803.431 69.97 263.9.668.669 1.774.493 70.33 267.0.686.687 1.809.492 70.32 266.0.681.670 1.765.599 70.09 258.1.695.641 1.784.500 70.46 255.6.792.672 1.752.603 69.49 261.3.740.702 1.813.529 69.88 262.2.721.662 1.754.576 70.27 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 32 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable-Selection Procedures Sample Data (Part 3) Drive Fair Green Putt Sand Score 260.5.703.623 1.782.567 70.72 271.3.671.666 1.783.492 70.30 263.3.714.687 1.796.468 69.91 276.6.634.643 1.776.541 70.69 252.1.726.639 1.788.493 70.59 263.0.687.675 1.786.486 70.20 263.0.639.647 1.760.374 70.81 253.5.732.693 1.797.518 70.26 266.2.681.657 1.812.472 70.96 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 33 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable-Selection Procedures Sample Correlation Coefficients Score Drive Fair Green Putt Drive -.154 Fair -.427 -.679 Green -.556 -.045.421 Putt.258 -.139.101.354 Sand -.278 -.024.265.083 -.296 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 34 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable-Selection Procedures Best Subsets Regression of SCORE Vars R-sq R-sq(a) C-p s D F G P S 1 30.9 27.9 26.9.39685 X 1 18.2 14.6 35.7.43183 X 2 54.7 50.5 12.4.32872 X X 2 54.6 50.5 12.5.32891 X X 3 60.7 55.1 10.2.31318 X X X 3 59.1 53.3 11.4.31957 X X X 4 72.2 66.8 4.2.26913 X X X X 4 60.9 53.1 12.1.32011 X X X X 5 72.6 65.4 6.0.27499 X X X X X © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 35 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable-Selection Procedures Minitab Output The regression equation Score = 74.678 -.0398(Drive) - 6.686(Fair) - 10.342(Green) + 9.858(Putt) Predictor Coef Stdev t-ratio p Constant 74.678 6.952 10.74.000 Drive -.0398.01235 -3.22.004 Fair -6.686 1.939 -3.45.003 Green -10.342 3.561 -2.90.009 Putt 9.858 3.180 3.10.006 s =.2691 R-sq = 72.4% R-sq(adj) = 66.8% © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 36 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Variable-Selection Procedures Minitab Output Analysis of Variance SOURCE DF SS MS F P Regression 4 3.79469.94867 13.10.000 Error 20 1.44865.07243 Total 24 5.24334 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 37 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Multiple Regression Approach to Experimental Design The use of dummy variables in a multiple regression equation can provide another approach to solving analysis of variance and experimental design problems. We will use the results of multiple regression to perform the ANOVA test on the difference in the means of three populations. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 38 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Multiple Regression Approach to Experimental Design Example: Reed Manufacturing Janet Reed would like to know if there is any significant difference in the mean number of hours worked per week for the department managers at her three manufacturing plants (in Buffalo, Pittsburgh, and Detroit). A simple random sample of five managers from each of the three plants was taken and the number of hours worked by each manager for the previous week is shown on the next slide. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 39 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Multiple Regression Approach to Experimental Design Plant 1 Plant 2 Plant 3 Observation Buffalo Pittsburgh Detroit 1 48 73 51 2 54 63 63 3 57 66 61 4 54 64 54 5 62 74 56 Sample Mean 55 68 57 Sample Variance 26.0 26.5 24.5 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 40 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Multiple Regression Approach to Experimental Design We begin by defining two dummy variables, A and B, that will indicate the plant from which each sample observation was selected. In general, if there are k populations, we need to define k – 1 dummy variables. A = 0, B = 0 if observation is from Buffalo plant A = 1, B = 0 if observation is from Pittsburgh plant A = 0, B = 1 if observation is from Detroit plant © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 41 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Multiple Regression Approach to Experimental Design Input Data Plant 1 Plant 2 Plant 3 Buffalo Pittsburgh Detroit A B y A B y A B y 0 0 48 1 0 73 0 1 51 0 0 54 1 0 63 0 1 63 0 0 57 1 0 66 0 1 61 0 0 54 1 0 64 0 1 54 0 0 62 1 0 74 0 1 56 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 42 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Multiple Regression Approach to Experimental Design E(y) = expected number of hours worked = 0 + 1A + 2B For Buffalo: E(y) = 0 + 1(0) + 2(0) = 0 For Pittsburgh: E(y) = 0 + 1(1) + 2(0) = 0 + 1 For Detroit: E(y) = 0 + 1(0) + 2(1) = 0 + 2 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 43 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Multiple Regression Approach to Experimental Design Excel produced the regression equation: y = 55 +13A + 2B Plant Estimate of E(y) Buffalo b0 = 55 Pittsburgh b0 + b1 = 55 + 13 = 68 Detroit b0 + b2 = 55 + 2 = 57 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 44 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Multiple Regression Approach to Experimental Design Next, we observe that if there is no difference in the means: E(y) for the Pittsburgh plant – E(y) for the Buffalo plant = 0 E(y) for the Detroit plant – E(y) for the Buffalo plant = 0 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 45 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Multiple Regression Approach to Experimental Design Because 0 equals E(y) for the Buffalo plant and 0 + 1 equals E(y) for the Pittsburgh plant, the first difference is equal to (0 + 1) - 0 = 1. Because 0 + 2 equals E(y) for the Detroit plant, the second difference is equal to (0 + 2) - 0 = 2. We would conclude that there is no difference in the three means if 1 = 0 and 2 = 0. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 46 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Multiple Regression Approach to Experimental Design The null hypothesis for a test of the difference of means is H0: 1 = 2 = 0 To test this null hypothesis, we must compare the value of MSR/MSE to the critical value from an F distribution with the appropriate numerator and denominator degrees of freedom. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 47 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Multiple Regression Approach to Experimental Design ANOVA Table Produced by Excel Source of Sum of Degrees of Mean Variation Squares Freedom Squares F p Regression 490 2 245 9.55.003 Error 308 12 25.667 Total 798 14 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 48 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Multiple Regression Approach to Experimental Design At a.05 level of significance, the critical value of F with k – 1 = 3 – 1 = 2 numerator d.f. and nT – k = 15 – 3 = 12 denominator d.f. is 3.89. Because the observed value of F (9.55) is greater than the critical value of 3.89, we reject the null hypothesis. Alternatively, we reject the null hypothesis because the p-value of.003 < a =.05. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 49 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Autocorrelation and the Durbin-Watson Test Often, the data used for regression studies in business and economics are collected over time. It is not uncommon for the value of y at one time period to be related to the value of y at previous time periods. In this case, we say autocorrelation (or serial correlation) is present in the data. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 50 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Autocorrelation and the Durbin-Watson Test With positive autocorrelation, we expect a positive residual in one period to be followed by a positive residual in the next period. With positive autocorrelation, we expect a negative residual in one period to be followed by a negative residual in the next period. With negative autocorrelation, we expect a positive residual in one period to be followed by a negative residual in the next period, then a positive residual, and so on. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 51 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Autocorrelation and the Durbin-Watson Test When autocorrelation is present, one of the regression assumptions is violated: the error terms are not independent. When autocorrelation is present, serious errors can be made in performing tests of significance based upon the assumed regression model. The Durbin-Watson statistic can be used to detect first-order autocorrelation. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 52 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Autocorrelation and the Durbin-Watson Test Durbin-Watson Test Statistic 𝑛 2 𝑡=2 𝑒𝑡 − 𝑒𝑡−1 𝑑= 𝑛 2 𝑡=1 𝑒𝑡 The ith residual is denoted 𝑒𝑖 = 𝑦𝑖 − 𝑦𝑖 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 53 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Autocorrelation and the Durbin-Watson Test Durbin-Watson Test Statistic The statistic ranges in value from zero to four. If successive values of the residuals are close together (positive autocorrelation is present), the statistic will be small. If successive values are far apart (negative autocorrelation is present), the statistic will be large. A value of two indicates no autocorrelation. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 54 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Autocorrelation and the Durbin-Watson Test Suppose the values of e (residuals) are not independent but are related in the following manner: et = 𝜌et-1 + zt where r is a parameter with an absolute value less than one and zt is a normally and independently distributed random variable with a mean of zero and variance of s 2. We see that if r = 0, the error terms are not related. The Durbin-Watson test uses the residuals to determine whether r = 0. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 55 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Autocorrelation and the Durbin-Watson Test The null hypothesis always is: 𝐻0 : 𝜌 = 0 there is no autocorrelation The alternative hypothesis is: 𝐻𝑎 : 𝜌 > 0 to test for positive autocorrelation 𝐻𝑎 : 𝜌 < 0 to test for negative autocorrelation 𝐻𝑎 : 𝜌 ≠ 0 to test for positive or negative autocorrelation © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 56 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Autocorrelation and the Durbin-Watson Test A Sample Of Critical Values For The Durbin-Watson Test For Autocorrelation Significance Points of dL and dU: a =.05 Number of Independent Variables 1 2 3 4 5 n dL dU dL dU dL dU dU dU dU dU 15 1.08 1.36 0.95 1.54 0.82 1.75 0.69 1.97 0.56 2.21 16 1.10 1.37 0.98 1.54 0.86 1.73 0.74 1.93 0.62 2.15 17 1.13 1.38 1.02 1.54 0.90 1.71 0.78 1.90 0.67 2.10 18 1.16 1.39 1.05 1.53 0.93 1.69 0.82 1.87 0.71 2.06 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 57 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) Autocorrelation and the Durbin-Watson Test Positive Incon- No evidence of autocor- clusive positive autocorrelation relation 0 dL dU 2 4-dU 4-dL 4 No evidence of Incon- Negative negative autocorrelation clusive autocor- relation 0 dL dU 2 4-dU 4-dL 4 Positive Incon- No evidence of Incon- Negative autocor- clusive autocorrelation clusive autocor- relation relation 0 dL dU 2 4-dU 4-dL 4 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 58 otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (13e) End of Chapter 16 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 59 otherwise on a password-protected website or school-approved learning management system for classroom use.