Chapter 13: Multiple Regression

Transcription

1 Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to estmate the populaton parameter β 0, β 1, and β usng the followng regresson equaton: 1

2 Usng Data Analyss - Regresson for the frst 3 columns of Potato.xls: The ft equaton allows us now to predct estmated values for the dfferent expermental condtons. For example ph 4 and pressure 15 leads to: ^ Y Our textbook now refers to Mntab for nformaton about a confdence nterval estmate of the average as well as a predcton nterval estmate for a future ndvdual value. The coeffcent of Multple Determnaton (R ) SSR SST R Y Ths essentally means that 41.58% of the varaton can be explaned by the effect of ph and pressure on the sold content. The remanng 58.4% s due to random scatterng of the data. Sometmes an adjusted R s suggested, whch takes the number of data ponts (n) and the number of explanatory varables (k ) nto account. R adj 1 n ( 1 R ) 1 ( ) Y.1 n k

3 13. Resdual Analyss By plottng the dfference between the actual data pont value and the predcted value (resdual), one can check the data for possble trends. The resduals are plotted aganst the varous parameters, X 1 and X, as well as aganst the data pont values, Y, or the tme Testng of Sgnfcance of the Multple Regresson Model Essentally testng: H 0 : β 1 β 0 (s there a slope wth respect to any of the parameters?) SST n ( Y Y ) 1 SSR n ( Y Y ) 1 ˆ SSE n ( Y Yˆ ) Confdence Interval Estmate for each Slope The standard error for each regresson coeffcent s provded n the PhStat analyss. Testng for the sgnfcance of each (slope) regresson coeffcent s: or the Confdence Interval Estmate for the slope s: b j ± tn k 1 s b j 3

4 13.5 Testng Portons of the Multple Regresson Model Whch parameters are really mportant?? One runs the data analyss frst wth all parameters ncluded and then a second tme wth all parameters except the one beng tested. The SS-regresson then allows us to defne the dfference between the two stuatons as beng the contrbuton of that specfc (excluded) parameter. The followng example demonstrates ths for the ph-parameter n potato.xls Analyss ncludng ph and Lower Pressure: Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 54 ANOVA df SS MS F Sgnfcance F Regresson E-06 Resdual Total Coeffcents Standard Error t Stat P-value Intercept PH E-06 Lower Pressure Result wth ph beng excluded: Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 54 ANOVA df SS MS F Sgnfcance F Regresson Resdual Total Coeffcents Standard Error t Stat P-value Intercept E-15 Lower Pressure The sgnfcance of ph s now determned through the F-rato of MSR(X ph ) / MSR(error) ( ) / (58.41 / 51) 6.31 The F-cut-off s 4.04 for 1 and 51 degrees of freedom at the α 5% level. Thus, ph s sgnfcantly mprovng the regresson ft. 4

5 Coeffcent of Partal Determnaton Smlar to the complete analyss of R as descrbed n 13.1, one can also analyze how much varaton n the data can be explaned by a specfc parameter as follows: In our current example: R R Y1. Y ( ph ) ( lower pressure) For a fxed lower pressure 34.03% of the varaton can be explaned by the varaton n ph. And alternatvely 1.7% of the varaton can be explaned by the varaton of lower pressure. 5

6 13.6 The Quadratc Curvlnear Regresson Model Essentally the testng procedure and nterpretaton of the results s the same as for the lnear regresson analyss. However, the second parameter now uses the square value of the frst parameter. In a very smlar way the regresson models can be expanded to cubc or any other hgher order relatons. The predcted Y-values are: Y ˆ b + b X + b 0 The sgnfcance of the Quadratc Curvlnear Model s gven by the F-rato between the mean square regresson and the mean square error. F MSR / MSE As well as the Coeffcent of Multple Determnaton : R Y.1 SSR / SST Estmaton ntervals for ndvdual regresson coeffcents are descrbed usng the t-test wth t b / s b s b beng the standard error of the correspondng parameter n the Excel data analyss output. Detals about the calculaton of the standard error of a parameter are outsde the scope of ths class, but the systematc calculaton s descrbed on page 417 n Appled Statstcs and Probablty for Engneers, 3 rd edton by Montgomery and Runger. The t-dstrbuton s to be used for n {# of ft-parameters} for the correspondng degrees of freedom Dummy-Varable Models Categorcal varables can be substtuted wth dummy varables. For example, operator A s assgned a value of 0 and operator B s assgned a value of 1 (or wet 0 and dry 1). However, we should use ths only n cases where the dummy varable has only two values. We can the further evaluate f the slope of the lnear regresson for our other parameters s affected by the dummy varable by addng an nteracton factor. 1 1 X 1 6

7 13.8 Usng Transformatons n Regresson Models Smlarly to what was already descrbed n secton 13.6, we can use all other knds of transformatons and regresson models. E.g.: Square root transformaton Y β + β X + β X + ε Multplcatve model β β Y β0 X1 1 X ε transforms to lny ln β0 + β1 ln X1 + β ln X + ln ε Exponental model β X X Y e 0 + β1 1 β + ε transforms to lny β0 + β1 X1 + β X + ln ε In short, we can use any possble mathematcal relaton as long as we can descrbe the model as a combnaton of lnear factors Collnearty Collnearty descrbes a stuaton, where factors are hghly correlated. It then becomes dffcult to separate the ndvdual effects from the cross-correlated effects and the effectveness of a specfc model can hghly fluctuate dependng on whch parameters are beng ncluded. One way to evaluate collnearty s to use the varance nflatonary factor: 1 VIFj wth R 1 R j beng the coeffcent of determnaton when usng j all other X-varables except X j. Values close to 1 ndcate uncorrelated varables, whereas values of 5 or hgher are consdered sgnfcant. In other words, a large VIF value ndcates that two (or more) varables are actually closely related. The two varables are not ndependent of each other but they are rather two measurements of the same effect. For example, measurng the bouncng heght of a ball and correlatng that to the speed of a fallng ball 5 cm before httng the ground (varable 1) and the heght of where the ball was released ntally (varable ) are hghly correlated varables. There s no need to nclude both varables n a model. 7

8 13.10 Model Buldng We lke to acheve a model that ncludes the fewest number of varables. 8

9 Frst we elmnate hghly correlated varables. We then could use varous combnatons of varables and see f all of them show sgnfcance above a certan threshold n our regresson model. In a more systematc way, we can use all possble combnaton of varables. The followng fgure shows the result for the potato-processng data. We can now evaluate the adjusted R values of the dfferent models. The larger the R value s, the more of the data scatterng s beng explaned by the model. An alternatve approach use the so-called C p statstc for evaluaton f a certan model should be consdered. Here models where C p s less than the number of varables beng used n the model + 1 are beng consdered for further evaluaton. Thereafter we agan evaluate each model n detal and see f all varables show a sgnfcant effect (P-values below a gven cut-off value). Nevertheless, choosng an approprate model s a hghly subjectve process! 9

10 13.11 Ptfalls n Multple Regresson The regresson coeffcent for one partcular varable s nterpreted for the case that all other varables are held constant. We need resdual plots for each ndependent varable. Interacton plots are needed for each parameter when usng dummy varables. VIF evaluaton s needed to decde on parameters to be ncluded n a model. Examne several alternatve subsets for models. Sample sze 10 x larger than the # of varables n the model. Evaluate a suffcently wde range for each varable. Stablty over tme s a key necessty to use a fttng model for predctve purposes. 10