Chapter 15 - Multiple Regression

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1 Chapter - Multple Regresson Chapter - Multple Regresson Multple Regresson Model The equaton that descrbes how the dependent varable y s related to the ndependent varables x, x,... x p and an error term s: y = b + b x + b x b p x p + e where: b, b, b,..., b p are the parameters, and e s a random varable called the error term Multple Regresson Equaton and Estmated MRE Multple Regresson Equaton The equaton that descrbes how the mean value of y s related to x, x,... x p s: E(y) = b + b x + b x b p x p Estmated Multple Regresson Equaton y = b + b x + b x b p x p A smple random sample s used to compute sample statstcs b, b, b,..., b p that are used as the pont estmators of the parameters b, b, b,..., b p. Slde Slde Estmaton Process Multple Regresson Equaton Multple Regresson Model E(y) = b + b x + b x b p x p + e Multple Regresson Equaton E(y) = b + b x + b x b p x p Unknown parameters are b, b, b,..., b p Sample Data: x x... x p y Two varable model Y Ŷ b b X b X b, b, b,..., b p provde estmates of b, b, b,..., b p Estmated Multple Regresson Equaton yˆ b b x b x... b px p Sample statstcs are b, b, b,..., b p X Slde X Slde Least Squares Method Multple Regresson Model Least Squares Crteron mn ( y yˆ ) Computaton of Coeffcent Values The formulas for the regresson coeffcents b, b, b,... b p nvolve the use of matrx algebra. We wll rely on computer software packages to perform the calculatons. Example: Programmer Survey A software frm collected data for a sample of computer programmers. A suggeston was made that regresson analyss could be used to determne f salary was related to the years of experence and the score on the frm s programmer apttude test. The years of experence, score on the apttude test test, and correspondng annual salary ($s) for a sample of programmers s shown on the next slde. Slde Slde

2 Chapter - Multple Regresson Multple Regresson Model Multple Regresson Model ($s) ($s) Suppose we beleve that salary (y) s related to the years of experence (x ) and the score on the programmer apttude test (x ) by the followng regresson model: where y = annual salary ($) x = years of experence = score on programmer apttude test x y = b + b x + b x + e Slde 7 Slde Solvng for the Estmates of b, b, b Solvng for the Estmates of b, b, b Excel s Regresson Equaton Output Input Data x x y Computer Package for Solvng Multple Regresson Problems Least Squares Output b = b = b = R = etc. A B C D E 9 Coeffc. Std. Err. t Stat P-value Intercept Experence E te: Columns F-I are not shown. SALARY =.7 +.(EXPER) +.(SCORE) te: Predcted salary wll be n thousands of dollars. Slde 9 Slde Interpretng the Coeffcents Multple Coeffcent of Determnaton In multple regresson analyss, we nterpret each regresson coeffcent as follows: b represents an estmate of the change n y correspondng to a -unt ncrease n x when all other ndependent varables are held constant. b =. b =. s expected to ncrease by $, for each addtonal year of experence (when the varable score on programmer atttude test s held constant). s expected to ncrease by $ for each addtonal pont scored on the programmer apttude test (when the varable years of experence s held constant). Slde Relatonshp Among SST, SSR, SSE where: ( y y) SST = SSR + SSE ˆ = ( y y) + SST = total sum of squares SSR = sum of squares due to regresson SSE = sum of squares due to error ( y yˆ ) Slde

3 Chapter - Multple Regresson Multple Coeffcent of Determnaton Excel s ANOVA Output A B C D E F ANOVA df SS MS F Sgnfcance F Regresson E-7 Resdual Total SSR SST R = SSR/SST R =./99.7 =. Slde Adjusted Multple Coeffcent of Determnaton n a ( ) R R n p (.79).7 R a The coeffcent of determnaton R s the proporton of varablty n a data set that s accounted for by a statstcal model. In ths defnton, the term "varablty" s defned as the sum of squares. Adjusted R-square s a modfcaton of R-square that adjusts for the number of terms n a model. R-square always ncreases when a new term s added to a model, but adjusted R-square ncreases only f the new term mproves the model more than would be expected by chance. Slde Multple Regresson ng for Sgnfcance (F and t-test) In smple lnear regresson, the F and t tests provde the same concluson. In multple regresson, the F and t tests have dfferent purposes. The F test s used to determne whether a sgnfcant relatonshp exsts between the dependent varable and the set of all the ndependent varables. Hypotheses Statstcs Rejecton Rule ng for Sgnfcance: F H : b = b =... = b p = H a : One or more of the parameters s not equal to zero. F = MSR/MSE Reject H f p-value < a or f F > F a, where F a s based on an F dstrbuton wth p d.f. n the numerator and n - p - d.f. n the denomnator. The F test s referred to as the test for overall sgnfcance. Slde Slde F for Overall Sgnfcance ng for Sgnfcance: t Hypotheses H : b = b = H a : One or both of the parameters s not equal to zero. Rejecton Rule For a =. and d.f. =, 7; F. =.9 Reject H f p-value <. or F >.9 If the F test shows an overall sgnfcance, the t test s used to determne whether each of the ndvdual ndependent varables s sgnfcant. A separate t test s conducted for each of the ndependent varables n the model. Statstcs F = MSR/MSE =./. =.7 We refer to each of these t tests as a test for ndvdual sgnfcance. Concluson p-value <., so we can reject H. (Also, F =.7 >.9) Slde 7 Slde

4 Chapter - Multple Regresson ng for Sgnfcance: t Hypotheses Statstcs Rejecton Rule H : b H a : b b t s b Reject H f p-value < a or f t < -t a or t > t a where t a s based on a t dstrbuton wth n - p - degrees of freedom. Hypotheses Rejecton Rule t for Sgnfcance of Indvdual Parameters H : b H : b a For a =. and d.f. = 7, t. =. Reject H f p-value <., or f t < -. or t >. Slde 9 Slde t for Sgnfcance of Indvdual Parameters Excel s Regresson Equaton Output A B C D E 9 Coeffc. Std. Err. t Stat P-value Intercept Experence E te: Columns F-I are not shown. t statstc and p-value used to test for the ndvdual sgnfcance of Experence Statstcs Conclusons t for Sgnfcance of Indvdual Parameters b s.9 b b.9. s.77 b Reject both H : b = and H : b =. Both ndependent varables are sgnfcant. Slde Slde ng for Sgnfcance: Multcollnearty Categorcal Independent Varables The term multcollnearty refers to the correlaton among the ndependent varables. When the ndependent varables are hghly correlated (say, r >.7), t s not possble to determne the separate effect of any partcular ndependent varable on the dependent varable. If the estmated regresson equaton s to be used only for predctve purposes, multcollnearty s usually not a serous problem. In many stuatons we must work wth categorcal ndependent varables such as gender (male, female), method of payment (cash, check, credt card), etc. For example, x mght represent gender where x = ndcates male and x = ndcates female. In ths case, x s called a dummy or ndcator varable. Every attempt should be made to avod ncludng ndependent varables that are hghly correlated. Slde Slde

5 Chapter - Multple Regresson Categorcal Independent Varables Categorcal Independent Varables Example: Programmer Survey As an extenson of the problem nvolvng the computer programmer salary survey, suppose that management also beleves that the annual salary s related to whether the ndvdual has a graduate degree n computer scence or nformaton systems. The years of experence, the score on the programmer apttude test, whether the ndvdual has a relevant graduate degree, and the annual salary ($) for each of the sampled programmers are shown on the next slde Degr. ($s) Degr. ($s) Slde Slde Estmated Regresson Equaton Categorcal Independent Varables where: ^ y = b + b x + b x + b x y ^ = annual salary ($) x = years of experence x = score on programmer apttude test x = f ndvdual does not have a graduate degree f ndvdual does have a graduate degree x s a dummy varable Excel s Regresson Statstcs A B C SUMMARY OUTPUT Regresson Statstcs 7 Multple R.99 R Square.79 9 Adjusted R Square.7 Standard Error.97 Observatons Slde 7 Slde Excel s ANOVA Output Categorcal Independent Varables A B C D E F ANOVA df SS MS F Sgnfcance F Regresson E-7 Resdual Total Slde 9 Categorcal Independent Varables Excel s Regresson Equaton Output A B C D E 9 Coeffc. Std. Err. t Stat P-value Intercept Experence Grad. Degr t sgnfcant A B F G H I 9 Coeffc. Low. 9% Up. 9% Low. 9.% Up. 9.% Intercept Experence Grad. Degr Slde

6 Chapter - Multple Regresson More Complex Categorcal Varables If a categorcal varable has k levels, k - dummy varables are requred, wth each dummy varable beng coded as or. For example, a varable wth levels A, B, and C could be represented by x and x values of (, ) for A, (, ) for B, and (,) for C. Care must be taken n defnng and nterpretng the dummy varables. More Complex Categorcal Varables For example, a varable ndcatng level of educaton could be represented by x and x values as follows: Hghest Degree x x Bachelor s Master s Ph.D. Slde Slde

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