CHAPTER Smple Lear Regreo EXAMPLE A expermet volvg fve ubject coducted to determe the relatohp betwee the percetage of a certa drug the bloodtream ad the legth of tme t take the ubject to react to a tmulu. Reacto Tme VS. Drug Percetage Subject Amout of Drug Tme % Reacto Tme Secod Mary Joh 3 Carl 3 4 Sara 4 5 Wllam 5 4 Frt recogze that the depedet varable x Amout of Drug ad the depedet varable y Reacto Tme. 4.5 4 3.5 3.5.5.5 3 4 5 6 The above catter plot dcate that a model for th tuato the frt-order lear model E( y) β + βx probably adequate, we ca try to ue the ample data to etmate the mg parameter β & β of the leat quare le.. Prelmary computato for the drug reacto problem x y x x y Total y 4 3 9 4 6 4 6 4 8 5 4 5 6 x 5 y 55 y 6 xy 37
The leat quare le yˆ ˆ β ˆ + βx ha the followg properte:. The um of error (SE) equal zero.. The um of quared error (E) maller tha that for ay other traght le model. Formula for the Leat Square Etmate Slope: ˆ xy β xx y-tercept: ˆ β ˆ y βx Where ( x )( y ) xy x y ad xx x ( x ) Example: Fd the leat quare predcto le for our example above. Let E( y) β + βxbe our traght le model where y reacto tme ecod ad x amout of drug. Ug the umber from above we ca get: ( x )( y ) xy x y 37 (5)()/5 7 ( x ) xx x 55 ( 5 ) 55 45 5 the ˆ xy β 7/.7 xx ( y ) ( ) x ˆ 5 ad ˆ β ˆ y βx β The the leat quare le the gve by: yˆ.+.7x.7. 5 5 Example: What would the average reacto tme be whe the % of drug.5%? Sce we foud yˆ.+.7x, we plug.5 for x to get the predcted average reacto tme(y) whe the percetage.5%. yˆ.+.7x -. + (.7)(.5).65 Thu, whe the percet of drug.5%, we predct the average reacto tme to be.65 ecod.
Iterpretato of ˆ β ad ˆβ of the leat quare le ˆ ˆ ˆ β + ˆ β x y β : y tercept, the value of y whe x ˆβ : lope, the mea amout of creae (or decreae) of y for every ut creae x. Example: Gve practcal terpretato to ˆ β ad ˆβ of the example. We foud yˆ.+.7x, where ˆ β -. ad ˆβ.7 The reacto tme (y) -. ecod whe the amout of drug (x) % For every % creae o the amout of drug (x) the bloodtream, the mea reacto tme (y) etmated to creae.7 ecod. To fd E, we have followg formula: E ˆ β Where xy y ( y ) Example: Now let fd the um quare error of the leat quare predcto le ( y ) ( ) y 6 6 6 5 the E ˆ βxy 6-(.7)(7). Model Aumpto Aumpto : The mea of the probablty dtrbuto for ( ε ). why E( y) β + βx. Recall the orgal model wa y β + βx+ ε. That Aumpto : The varace for ( ε ) a cotat deoted byσ. Now matter what x value we ue the model the dtrbuto of the radom error ha the ame varace. ε ormal. Aumpto 3: The probablty dtrbuto of ( ) Aumpto 4: The value of ( ε ) aocated wth ay two oberved value of y are depedet.
A etmator of σ E E S Degree of Freedom for Error A etmator of σ E S S S kow a the etmated tadard error of the regreo model. Iterpretato of, the etmated tadard devato of( ε ) : We expect approxmately 95% of the oberved y value to le wth of ther repectve leat quare predcted y value, ŷ. Example: Fd ad terpret the etmate of σ E..367 o.367. 6 5 Iterpretato: We expect approxmately 95% of the oberved y value to le wth. of ther repectve leat quare le.. Makg Iferece about β our lope Recall β our lope for the lear model: y β + βx+ ε. If the true value of the lope equal to zero that mea y β + βx+ ε become y β + x+ ε β + ε, th mea that x ha o role predctg y. If that the cae, our model ot ueful. For th reao, we wll wat to tet the clam that the lope equal to zero. We would lke to reject that clam becaue f we are uable to reject t we have a uele model. To be able to perform a hypothe tet to make a ferece about β (the lope), we eed to kow the amplg dtrbuto of our etmator ˆ β. Samplg Dtrbuto of ˆ β If we make the four aumpto about ε (ee ecto.3), the amplg dtrbuto of the leat quare etmator ˆβ of the lope wll be ormal wth mea β (the true lope) σ ad tadard devato σ ˆ β xx
We etmate σ by βˆ ˆ β ad refer to th quatty a the etmated tadard xx error of the leat quare lope ˆβ E (recall S S ). A Tet of Model Utlty: Smple Lear Regreo Hypothe Tet Stattc Rejecto Rego H o; β t > t a H ; β > ˆ β ˆ A β t H o; β / xx t < -t ˆ β a H ; β < A H ; β df- t < - t or t H o ; β A α / > tα/ Example: At the % gfcace level, tet the clam that there a potve lear relatohp betwee Amout of Drug (x) ad the Reacto Tme (y). Ue α.5 Tet Stattc: t H o; β H ; β > A ˆ β Ad df - 5-3 XX.7.6 3.63 Rejecto Rego t >.535 α.5 Deco: Reject Ho at α.5.353 t Cocluo: There eough evdece to coclude that there a potve lear relato betwee Amout of Drug (x) ad the Reacto Tme (y) (two addtoal example about th ecto were gve cla)
A (-a)% Cofdece Iterval for the Sample Lear Regreo Slope β ˆ β ± t ˆ ± t α / ˆ β β α / Ad t a / baed o (-) degree of freedom. XX Example: Fd the 95% CI for the lope β, the expected chage reacto tme(y) for a % creae the amout of drug the bloodtream (x) ˆ β ± t α / (.9,.3) XX.6.7 ± 3.8.7 ±.6 Iterpretato: We are 95% cofdet that the true mea creae reacto tme(y) per addtoal % of drug betwee.9 ad.3 ecod. The Coeffcet of Correlato r The coeffcet of correlato, r relatohp betwee two varable x ad y. xy a meaure of the tregth of the lear xx See page 66 to look at catter plot ad the correpodg value for r. People ometme mterpret r. Pleae remember that f r t doe ot mea there o relatohp betwee x ad y t jut mea there doe ot eem to be a lear relatohp betwee them. Alo, f r cloe to oe, t doe ot mea x caue y or that y caue x. It oly mea there ome lear relatohp betwee the two varable, but the relatohp could be due to ome other ukow caue. Example: Fd the coeffcet of correlato of the Reacto Tme(y) VS Amout of drug (x) XY 7 r.94 XX YY 6 Sce r potve ad ear dcate that the reacto ted(y) to creae a the amout of drug(x) the bloodtream creae, trog potve lear relatohp.
r The coeffcet of determato Aother way to meaure the uefule of the model to meaure the cotrbuto of x predctg y. To do th, we calculate how much the error of predcto of y were reduced by ug the formato provded by x. YY total ample varato of the obervato aroud the ample mea for y, ad E the remag uexplaed ample varablty after fttg the le ŷ ˆ β ˆ + βx. The coeffcet of determato Explaed ample varablty YY E r Total ample varablty r E YY YY E Iterpretato of r ( r )% of the ample varato y ca be explaed by ug x to predct y a traght le model. Example: Fd ad terpret the coeffcet of determato for the drug reacto example. E. r.87 (ug the above formula) YY 6 r (.94).87 (ug the correlato coeffcet r) Iterpretato: 8.7% of the ample varato reacto tme(y) ca be explaed by ug the amout of drug(x) to predct reacto tme(y) a traght le model. YY
READING FROM MINITAB OUTPUT Regreo Aaly: Tme veru Drug The regreo equato Tme -. +.7 Drug Predctor Coef SE Coef T P Cotat -..635 -.6.885 Drug.7.95 3.66.35 S.6553 R-Sq 8.7% R-Sq(adj) 75.6% Aaly of Varace Source DF MS F P Regreo 4.9 4.9 3.36.35 Redual Error 3..3667 Total 4 6. Frt, we recogzed that the depedet varable x Amout of Drug ad the depedet varable y Reacto Tme. To fd ˆ β ad ˆβ, look uder the coeffcet colum, ˆ β Cotat coef -. ad ˆβ Drug coef.7. OR jut look at Tme -. +.7 Drug, ad recogze the ˆ β ad ˆβ. So the leat quare le ˆ ˆ β + ˆ β x for th example yˆ.+.7x. y We ca alo fgure out the E from the mtab uder Redual Error, o E.. The etmator of σ,, the MS of redual error, o.3667. The etmator of σ,, ca be foud S.6553 The tet tattc for makg ferece about lope β uder T of Drug ad t dfdf Redual Error o we get Tet Stattc: t 3.66 wth df3 Coeffcet of Determato r clearly gve already a 8.7% Coeffcet of Correlato r, potve ce we kow that the lope ˆβ potve, o we jut obta t by r ( g) r, o r +.87. 9388 To get the CI for β, we ue jut ue the formula ˆ β ± tα / ˆ, where β β Cotat Coef ad SE Coef DRUG. ˆβ