Slides by JOHN LOUCKS St Edward s University

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Chapter 16 Regression Analysis: Model Building n n n General Linear Model Determining When to Add or Delete Variables Variable-Selection Procedures n Multiple Regression Approach to Experimental Design n Autocorrelation and the Durbin-Watson Test © 2008 Thomson South-Western. All Rights Reserved Slide 2

General Linear Model n Models in which the parameters ( 0, 1, . . . , p ) all have exponents of one are called linear models. n A general linear model involving p independent variables is n Each of the independent variables z is a function of x 1, x 2, . . . , xk (the variables for which data have been collected). © 2008 Thomson South-Western. All Rights Reserved Slide 3

General Linear Model n The simplest case is when we have collected data for just one variable x 1 and want to estimate y by using a straight-line relationship. In this case z 1 = x 1. n This model is called a simple first-order model with one predictor variable. © 2008 Thomson South-Western. All Rights Reserved Slide 4

Modeling Curvilinear Relationships n To account for a curvilinear relationship, we might set z 1 = x 1 and z 2 =. n This model is called a second-order model with one predictor variable. © 2008 Thomson South-Western. All Rights Reserved Slide 5

Interaction n If the original data set consists of observations for y and two independent variables x 1 and x 2 we might develop a second-order model with two predictor variables. n In this model, the variable z 5 = x 1 x 2 is added to account for the potential effects of the two variables acting together. n This type of effect is called interaction. © 2008 Thomson South-Western. All Rights Reserved Slide 6

Transformations Involving the Dependent Variable n Often the problem of nonconstant variance can be corrected by transforming the dependent variable to a different scale. n 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). n Another approach, called a reciprocal transformation, is to use 1/y as the dependent variable instead of y. © 2008 Thomson South-Western. All Rights Reserved Slide 7

Nonlinear Models That Are Intrinsically Linear n Models in which the parameters ( 0, 1, . . . , p ) have exponents other than one are called nonlinear models. n In some cases we can perform a transformation of variables that will enable us to use regression analysis with the general linear model. n The exponential model involves the regression equation: n We can transform this nonlinear model to a linear model by taking the logarithm of both sides. © 2008 Thomson South-Western. All Rights Reserved Slide 8

Determining When to Add or Delete Variables n To test whether the addition of x 2 to a model involving x 1 (or the deletion of x 2 from a model involving x 1 and x 2) is statistically significant we can perform an F Test. n 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. © 2008 Thomson South-Western. All Rights Reserved Slide 9

Determining When to Add or Delete Variables n 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. n The p –value associated with the computed F statistic can be compared to the level of significance a. n 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. © 2008 Thomson South-Western. All Rights Reserved Slide 10

Variable Selection Procedures n Stepwise Regression n Forward Selection n Backward Elimination n Best-Subsets Regression Iterative; one independent variable at a time is added or deleted based on the F statistic Different subsets of the independent variables are evaluated The first 3 procedures are heuristics. There is no guarantee that the best model will be found. © 2008 Thomson South-Western. All Rights Reserved Slide 11

Variable Selection: Stepwise Regression n 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. n 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. n If no variable can be removed and no variable can be added, the procedure stops. © 2008 Thomson South-Western. All Rights Reserved Slide 12

Variable Selection: Forward Selection n This procedure is similar to stepwise regression, but does not permit a variable to be deleted. n This forward-selection procedure starts with no independent variables. n It adds variables one at a time as long as a significant reduction in the error sum of squares (SSE) can be achieved. © 2008 Thomson South-Western. All Rights Reserved Slide 13

Variable Selection: Backward Elimination n This procedure begins with a model that includes all the independent variables the modeler wants considered. n 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 pvalue is less than the user-specified or default value. n Once a variable has been removed from the model it cannot reenter at a subsequent step. © 2008 Thomson South-Western. All Rights Reserved Slide 14

Variable Selection: Backward Elimination n 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 an upcoming slide. © 2008 Thomson South-Western. All Rights Reserved Slide 15

Variable Selection: Backward Elimination n Example: Clarksville Homes Develop, using the backward elimination procedure, a multiple regression model to predict the selling price of a house in Clarksville. © 2008 Thomson South-Western. All Rights Reserved Slide 16

Variable Selection: Backward Elimination n Cars (garage size) is the independent variable with the highest p-value (. 697) >. 05. n Cars variable is removed from the model. n Multiple regression is performed again on the remaining independent variables. © 2008 Thomson South-Western. All Rights Reserved Slide 19

Variable Selection: Backward Elimination n Bedrooms is the independent variable with the highest p-value (. 281) >. 05. n Bedrooms variable is removed from the model. n Multiple regression is performed again on the remaining independent variables. © 2008 Thomson South-Western. All Rights Reserved Slide 21

Variable Selection: Backward Elimination n Bathrooms is the independent variable with the highest p-value (. 110) >. 05. n Bathrooms variable is removed from the model. n Multiple regression is performed again on the remaining independent variable. © 2008 Thomson South-Western. All Rights Reserved Slide 23

Variable Selection: Backward Elimination n House size is the only independent variable remaining in the model. n The estimated regression equation is: © 2008 Thomson South-Western. All Rights Reserved Slide 25

Variable Selection: Best-Subsets Regression n The three preceding procedures are one-variable-at-atime methods offering no guarantee that the best model for a given number of variables will be found. n Some software packages include best-subsets regression that enables the user to find, given a specified number of independent variables, the best regression model. n Minitab output identifies the two best one-variable estimated regression equations, the two best twovariable equation, and so on. © 2008 Thomson South-Western. All Rights Reserved Slide 26

Variable Selection: Best-Subsets Regression n 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. © 2008 Thomson South-Western. All Rights Reserved Slide 27

Variable-Selection Procedures n 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 © 2008 Thomson South-Western. All Rights Reserved Slide 28

Variable-Selection Procedures n Sample Data Drive 277. 6 259. 6 269. 1 267. 0 267. 3 255. 6 272. 9 265. 4 Fair. 681. 691. 657. 689. 581. 778. 615. 718 Green. 667. 665. 649. 673. 637. 674. 667. 699 Putt 1. 768 1. 810 1. 747 1. 763 1. 781 1. 791 1. 780 1. 790 Sand. 550. 536. 472. 672. 521. 455. 476. 551 © 2008 Thomson South-Western. All Rights Reserved Score 69. 10 71. 09 70. 12 69. 88 70. 71 69. 76 70. 19 69. 73 Slide 29

Variable-Selection Procedures n Sample Correlation Coefficients Drive Fair Green Putt Sand Score -. 154 -. 427 -. 556. 258 -. 278 Drive Fair Green Putt -. 679 -. 045 -. 139 -. 024 . 421. 101. 265 . 354. 083 -. 296 © 2008 Thomson South-Western. All Rights Reserved Slide 30

Multiple Regression Approach to Experimental Design n The use of dummy variables in a multiple regression equation can provide another approach to solving analysis of variance and experimental design problems. n We will use the results of multiple regression to perform the ANOVA test on the difference in the means of three populations. © 2008 Thomson South-Western. All Rights Reserved Slide 31

Multiple Regression Approach to Experimental Design n 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). © 2008 Thomson South-Western. All Rights Reserved Slide 32

Multiple Regression Approach to Experimental Design n Example: Reed Manufacturing 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. © 2008 Thomson South-Western. All Rights Reserved Slide 33

Multiple Regression Approach to Experimental Design Observation 1 2 3 4 5 Sample Mean Sample Variance Plant 1 Buffalo 48 54 57 54 62 Plant 2 Pittsburgh 73 63 66 64 74 Plant 3 Detroit 51 63 61 54 56 55 26. 0 68 26. 5 57 24. 5 © 2008 Thomson South-Western. All Rights Reserved Slide 34

Multiple Regression Approach to Experimental Design n n 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 A = 1, B = 0 A = 0, B = 1 if observation is from Buffalo plant if observation is from Pittsburgh plant if observation is from Detroit plant © 2008 Thomson South-Western. All Rights Reserved Slide 35

Multiple Regression Approach to Experimental Design n Input Data Plant 1 Buffalo A B y Plant 2 Pittsburgh A B y Plant 3 Detroit A B y 0 0 0 1 1 1 0 0 0 0 0 48 54 57 54 62 0 0 0 73 63 66 64 74 © 2008 Thomson South-Western. All Rights Reserved 1 1 1 51 63 61 54 56 Slide 36

Multiple Regression Approach to Experimental Design E(y) = expected number of hours worked = 0 + 1 A + 2 B For Buffalo: For Pittsburgh: For Detroit: E(y) = 0 + 1(0) + 2(0) = 0 E(y) = 0 + 1(1) + 2(0) = 0 + 1 E(y) = 0 + 1(0) + 2(1) = 0 + 2 © 2008 Thomson South-Western. All Rights Reserved Slide 37

Multiple Regression Approach to Experimental Design Excel produced the regression equation: y = 55 +13 A + 2 B Plant Buffalo Pittsburgh Detroit Estimate of E(y) b 0 = 55 b 0 + b 1 = 55 + 13 = 68 b 0 + b 2 = 55 + 2 = 57 © 2008 Thomson South-Western. All Rights Reserved Slide 38

Multiple Regression Approach to Experimental Design n 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 © 2008 Thomson South-Western. All Rights Reserved Slide 39

Multiple Regression Approach to Experimental Design n 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. n Because 0 + 2 equals E(y) for the Detroit plant, the second difference is equal to ( 0 + 2) - 0 = 2. n We would conclude that there is no difference in the three means if 1 = 0 and 2 = 0. © 2008 Thomson South-Western. All Rights Reserved Slide 40

Multiple Regression Approach to Experimental Design n The null hypothesis for a test of the difference of means is H 0: 1 = 2 = 0 n 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. © 2008 Thomson South-Western. All Rights Reserved Slide 41

Multiple Regression Approach to Experimental Design n ANOVA Table Produced by Excel Source of Variation Regression Error Total Sum of Degrees of Squares Freedom 490 308 798 2 12 14 Mean Squares 245 25. 667 © 2008 Thomson South-Western. All Rights Reserved F p 9. 55 . 003 Slide 42

Multiple Regression Approach to Experimental Design n At a. 05 level of significance, the critical value of F with k – 1 = 3 – 1 = 2 numerator d. f. and n. T – k = 15 – 3 = 12 denominator d. f. is 3. 89. n Because the observed value of F (9. 55) is greater than the critical value of 3. 89, we reject the null hypothesis. n Alternatively, we reject the null hypothesis because the p-value of. 003 < a =. 05. © 2008 Thomson South-Western. All Rights Reserved Slide 43

Autocorrelation and the Durbin-Watson Test n Often, the data used for regression studies in business and economics are collected over time. n 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. n In this case, we say autocorrelation (or serial correlation) is present in the data. © 2008 Thomson South-Western. All Rights Reserved Slide 44

Autocorrelation and the Durbin-Watson Test n With positive autocorrelation, we expect a positive residual in one period to be followed by a positive residual in the next period. n With positive autocorrelation, we expect a negative residual in one period to be followed by a negative residual in the next period. n 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. © 2008 Thomson South-Western. All Rights Reserved Slide 45

Autocorrelation and the Durbin-Watson Test n When autocorrelation is present, one of the regression assumptions is violated: the error terms are not independent. n When autocorrelation is present, serious errors can be made in performing tests of significance based upon the assumed regression model. n The Durbin-Watson statistic can be used to detect first-order autocorrelation. © 2008 Thomson South-Western. All Rights Reserved Slide 46

Autocorrelation and the Durbin-Watson Test n 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. © 2008 Thomson South-Western. All Rights Reserved Slide 47

Autocorrelation and the Durbin-Watson Test n Suppose the values of e (residuals) are not independent but are related in the following manner: et = r et-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. n We see that if r = 0, the error terms are not related. n The Durbin-Watson test uses the residuals to determine whether r = 0. © 2008 Thomson South-Western. All Rights Reserved Slide 48

Autocorrelation and the Durbin-Watson Test n The null hypothesis always is: there is no autocorrelation n The alternative hypothesis is: to test for positive autocorrelation to test for negative autocorrelation to test for positive or negative autocorrelation © 2008 Thomson South-Western. All Rights Reserved Slide 49

Autocorrelation and the Durbin-Watson Test A Sample Of Critical Values For The Durbin-Watson Test For Autocorrelation Significance Points of d. L and d. U: a =. 05 n Number of Independent Variables 1 2 3 4 5 d. L d. U d. U 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 © 2008 Thomson South-Western. All Rights Reserved Slide 50

Autocorrelation and the Durbin-Watson Test Positive autocorrelation 0 Inconclusive d. L d. U No evidence of positive autocorrelation 2 No evidence of negative autocorrelation 0 d. L Positive autocorrelation 0 d. L d. U 2 No evidence of autocorrelation Inconclusive d. U 2 4 -d. U Inconclusive 4 -d. U © 2008 Thomson South-Western. All Rights Reserved 4 -d. L 4 Negative autocorrelation 4 -d. L 4 Slide 51