Y = β 0 + β 1 X 1 + β 2 X β k X k + ε

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1 Chapter 3 Secton 3.1 Model Assumptons: Multple Regresson Model Predcton Equaton Std. Devaton of Error Correlaton Matrx Smple Lnear Regresson: 1.) Lnearty.) Constant Varance 3.) Independent Errors 4.) Normalty of the Errors Multple Regresson: 1.) Lnearty.) Constant Varance 3.) Independent Errors 4.) Normalty of the Errors Notce that the assumptons are the same for both smple and multple lnear regresson. Smple Lnear Regresson Model Y = β 0 + β 1 X + ε Data Model Error where ε ~ N(0, σ ε ) and ndependent Multple Regresson Model Y = β 0 + β 1 X 1 + β X β k X k + ε Data Model Error Model: Conssts of k predctors for a total of k+1 parameters. Error: Each error s Independent and dstrbuted normally wth constant varance,.e. ε ~ N(0, σ ε ) Data: For each of the 1,,, n cases we need a value for Y and for all of X 1,..., X k The 4 Step Process for Multple Regresson: Collect data for Y and all predctors. CHOOSE a form of the model. Select predctors; possbly transform Y. Choose any functons of predctors. FIT Estmate the coeffcents ˆ, ˆ 1, k ˆ Estmate the resdual standard error: σ^ ε. ASSESS the ft. Test ndvdual predctors: t-tests. Test the overall ft: ANOVA, R. Examne resduals. USE Predctons, CI s, and PI s. Multple Regresson Model Y = β 0 + β 1 X 1 + β X β k X k + ε k predctors Recall n smple lnear regresson we ft the model usng least squares, that s, we found the β ^ that mnmzed Y Ŷ). We wll do the same thng n multple regresson. The predcton model wll be: ^ ^ ^ ^ ^ Y = β 0 + β 1 X 1 + β X β k X k 6 1

2 Example: Multple Predctors Response Varable: Y = Actve pulse (n bpm) after walkng up and down 3 flghts of stars Predctors: X 1 = Restng pulse (n bpm) X = Heght (n nches) X 3 = Gender (0 = M, 1 = F) Sample sze n = 3, k = 3 Data: Pulse.txt (has other varables too) Correlaton Matrx Actve Rest Gender Hgt Actve Rest Gender Hgt Notce: Correlatons of X s wth Y = Actve Correlatons of X s wth each other In partcular, Gender & Hgt have hgh r Predcton Equaton where the coeffcents are chosen to mnmze: SSE Y Y ˆ Example: Y = Actve pulse rate Ŷ Rest 0.685Hgt 4.46Gender 9 10 Multple Regresson n R mymodel=lm(actve~rest+hgt+gender) Usual commands stll work. summary(mymodel) anova(mymodel) plot(mymodel)... Regresson Output > mymodel=lm(actve~rest+hgt+gender) Coeffcents: (Intercept) Rest <e-16 *** Hgt Gender Resdual standard error: on 8 degrees of freedom Multple R-squared: 0.374, Adjusted R-squared: F-statstc: 45.1 on 3 and 8 DF, p-value: <.e-16 1

3 Std. Devaton of Error Term = Resdual standard error (n R) Recall: ~ N(0, ) S MSE Gven by R SSE n k 1 df n k R Regresson Output (Intercept) Rest <e-16 *** Hgt Gender Resdual standard error: on 8 degrees of freedom Multple R-squared: 0.374, Adjusted R-squared: F-statstc: 45.1 on 3 and 8 DF, p-value: <.e-16 > anova(mymodel) Response: Actve Df Sum Sq Mean Sq F value Pr(>F) Rest <e-16 *** Hgt Gender Resduals SSE MSE 14 Correlaton Matrx > newpulse=pulse.df[,c(1,,4,7)] #extract columns 1,,4, and 7 > cor(newpulse) Actve Rest Gender Hgt Actve Rest Gender Hgt Some R Lnear Model Commands (some for later n the course) Once you have ft, e.g., model=lm(y~x1+x+x3) summary(model) t-tests for coeffcents, etc. anova(model) (sequental) sums of squares plot(model) modelng checkng plots rstandard(model) standardzed resduals rstudent(model) studentzed resduals hatvalues(model) leverage (h ) Chapter 3 Secton 3. Multple Regresson Inference n Multple Regresson Parttonng Varablty Adjusted R CI, PI for Multple Regresson t-test for Correlaton H 0 : ρ = 0 H 1 : ρ 0 t. s. No change! r n 1 r Compare to t n Use ths to: (1) Identfy potental good predctors of Y. () Look for relatonshps among predctors. 3

4 t-test for Slope Note: We now have several slopes to test. H 0 : β = 0 H 1 : β 0 ˆ t. s. S ˆ Compare to t n (k + 1) Reject H 0 The th predctor s useful n ths model, gven others already n the model. All gven by R wth a p-value Lose 1 d.f. for each coeffcent Example: Hgt and Actve Test #1: Compute and test the correlaton between Hgt and Actve pulse rates. Test #: Compute and test the coeffcent of Hgt n a multple regresson model (along wth Rest and Gender) to predct Actve pulse rates. We wll see that we get dfferent results. What s gong on? Correlaton Matrx Actve Rest Gender Hgt H 0 : ρ = 0 r t. s. Actve Rest Gender H 1 : ρ 0 1 r Hgt n Coeffcents: (Intercept) e-11 *** Hgt ** Resdual standard error: on 30 degrees of freedom Multple R-squared: , Adjusted R-squared: F-statstc: on 1 and 30 DF, p-value: Negatve Coeffcent/Correlaton when only Hgt s n model DF = 30, p-value = > cor.test(actve,hgt) data: Actve and Hgt t = , df = 30, p-value = alternatve hypothess: true correlaton s not equal to 0 95 percent confdence nterval: Regresson Output > mymodel=lm(actve~rest+hgt+gender) Coeffcents: (Intercept) Rest <e-16 *** Hgt Gender Resdual standard error: on 8 degrees of freedom Multple R-squared: 0.374, Adjusted R-squared: F-statstc: 45.1 on 3 and 8 DF, p-value: <.e-16 t-test for Correlaton versus t-test for Slope t-test for correlaton: Assesses the lnear assocaton between two varables by themselves. t-test for slope: Assesses the lnear assocaton after accountng for the other predctors n the model. In ths example, heght and gender are correlated. So t-test s for slope of heght, once gender (and rest) already n model. 4

5 SSModel ( Yˆ Y ) + SSE ( Y Y ) Parttonng Varablty = 0 1X1... kx k + ε SSModel SSE = Y ˆ Y ) ( + Error after regresson Total varablty n Y ANOVA F-test for Overall Ft H 0 : β 1 = β =... = β k = 0 H 1 : Some β 0 k n-k-1 k (n-k-1) Null model (no X s used) Effectve model F k,n-k-1 5 Multple Regresson Model Populaton model: Y = β 0 + β 1 X 1 + β X β k X k + ε Ftted model (from sample): ^ ^ ^ ^ ^ Y = β 0 + β 1 X 1 + β X β k X k We can test: Indvdual terms (t-test) and overall ft (F-test from ANOVA table) R Regresson Output (Intercept) Rest <e-16 *** Hgt Gender Resdual standard error: on 8 degrees of freedom Multple R-squared: 0.374, Adjusted R-squared: F-statstc: 45.1 on 3 and 8 DF, p-value: <.e-16 > anova(mymodel) Response: Actve Df Sum Sq Mean Sq F value Pr(>F) Rest <e-16 *** Hgt Gender Resduals Test ndvdual terms (gven other terms) Test for Overall model Wll learn next what these test. Note that R does not provde SSModel, and overall F test s not gven by anova command anymore. 8 R Multple Regresson Output, so far we have covered these: Resduals: Mn 1Q Medan 3Q Max Coeffcents: (Intercept) Rest <e-16 *** Hgt Gender Resdual standard error: on 8 degrees of freedom Multple R-squared: 0.374, Adjusted R-squared: F-statstc: 45.1 on 3 and 8 DF, p-value: <.e-16 > anova(mymodel) Analyss of Varance Table R Multple Regresson Output Usual F test and p-value s n summary( )(last slde), not n anova output. Response: Actve Df Sum Sq Mean Sq F value Pr(>F) Rest <e-16 *** Hgt Gender Resduals Sequental sum of squares: New varablty explaned as each predctor s added. SSModel = = wth 3 d.f. = =

6 Coeffcent of Multple Determnaton R SSModel Now nterpreted as the % of varablty n the response varable (Y) that s explaned by a lnear combnaton of these predctors. R SSModel The % of varablty n the response varable (actve pulse) that s explaned by a lnear combnaton of the predctors (restng pulse, heght, gender). Why Do We Call It R? SSModel R For a smple lnear model: If r s the correlaton between X and Y, then r = R. Does ths make sense for multple regresson? Each predctor has a dfferent correlaton wth Y. Why Do We Call It R? Another way to get R : Compute the correlaton r between the Y values and the predcted Y^ values: r = R. For a smple model: Yˆ ˆ ˆ 1X Corr( X, Y ) Corr( Yˆ, Y ) 0 What Makes a Good Model? Hgh R Small SSE Large ANOVA test statstcs Put n predctors Trade-off Strong t-tests Good predctors Parsmony Take out predctors Two purposes for regresson: (1) to model and understand; () to predct. (1) parsmony, construct a smple model () ncrease R, construct a complex model But can we beleve that a model wll yeld good predctons for ponts that weren t used to ft the model n the frst place? 6

7 Addng addtonal predctors wll: Increase SSModel Decrease SSE Increase R But s the ncrease n R worth t? R Recall: adj R SSModel 1 Adjusted R SSE (n 1 (n SSE k 1) 1) 1 ˆ S Y (Adjusts for the number of predctors n the model) R Multple Regresson Output >summary(mymodel) Coeffcents: (Intercept) Gender Hgt Rest <e-16 *** Resdual standard error: on 8 degrees of freedom Multple R-squared: 0.374, Adjusted R-squared: F-statstc: 45.1 on 3 and 8 DF, p-value: <.e-16 Compare Models usng Adjusted R-Squared (Intercept) Hgt Rest <e-16 *** Resdual standard error: on 9 degrees of freedom Multple R-squared: , Adjusted R-squared: F-statstc: on and 9 DF, p-value: <.e-16 (Intercept) Rest <e-16 *** Resdual standard error: on 30 degrees of freedom Multple R-squared: 0.365, Adjusted R-squared: F-statstc: 13. on 1 and 30 DF, p-value: <.e CI s and PI s for Y Recall: For a smple lnear model, when we predct Y for a partcular value of X = x p ˆ 1 Y t (1) CI for μ / S Y n x p X SSX Where s the average Y for all wth X = x p? () PI for ndvdual Y Y ˆ t / S 1 1 n x X p SSX Where are most Y s when X = x p? What about predctng Y wth multple X s? CI for μ Y CI s and PI s for Multple Regresson For a partcular set of predctor values: (x 1, x,, x k ) yˆ ˆ ˆ x ˆ x... ˆ x PI for Indvdual Y k k y ˆ t * ˆ y ˆ t * ˆ d.f.=n k 1 Stuff 1 Stuff SE Ft 7

8 R: CI and PI for Multple Regresson Read the fle Pulse > model<-lm(actve~rest+hgt+gender, data=pulse) > newx=data.frame(rest=63,hgt=65,gender=1) > predct(model,newx,nterval="confdence") ft lwr upr > predct(model,newx,nterval="predcton") ft lwr upr All cases n the Pulse dataset > predct(model,pulse,nterval="predcton") ft lwr upr Etc 8

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