Correlation: Examine Quantitative Bivariate Data
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1 Correlato ad Regreo Correlato: Eame Quattatve Bvarate Data The correlato, ρ, betwee two radom varable, X ad Y, defed a, ( X µ ρ average σx X ) ( Y µ Y σ Y ) product of the tadard devate of X ad Y, quatfe the tregth of lear relatohp. Regreo Aal I there a relato betwee umber of power boat the area ad umber of maatee klled? Year NPB( ) Nkll( ) Maatee Klled 6 3 Scatter Plot 6 7 Number of Power Boat 8 Eample: Calculato of Correlato Coeffcet Σ 7945 (Σ) 6335 Σ 4 (Σ) Σ Σ 56 Σ 475 S 989.5, S S 37 r , r A. Chag
2 Correlato ad Regreo A. Chag Formula for Pearo ample correlato coeffcet, r : Properte of correlato coeffcet: < r < It meaure the tregth ad drecto of the lear relato betwee two quattatve varable. r f all pot le eactl o a traght le. ρ he otato for populato correlato coeffcet. Correlato Doe Not Impl Cauato Eample: The umber of powerboat regtered ma ot be the drect caue for the death of Maatee..,,, where ) ( ) ( ) )( ( r, r cloe to r cloe to r cloe to zero r cloe to zero
3 Correlato ad Regreo t-tet for correlato Hpothe : Ho: ρ ρ, v.. Ha: ρ ρ Tet Stattc : t Deco rule : Reject Ho, f C.V. approach: t < t α/ or t > t α/ p-value approach: p-value < α Eample : (Maatee) I there a gfcat correlato betwee powerboat ad death of maatee? Ho: ρ, v.. Ha: ρ. 94 t (. 886)/( 4 ) d.f. 4 -, p-value <.5, reject Ho, there gfcat lear relato. Spearma Rak Correlato Coeffcet Spearma correlato coeffcet calculated baed o the rak of the obervato,.e., r rak of amog all, ad r rak of amog all, where d r r. If te ue average rak. Tet for correlato ug rak correlato coeffcet If ample ze le tha ue pecal table. If ample ze large ue t-dtrbuto table. The t-tet the ame a the tet procedure for o-rak data. It le etve to outler wth the dadvatage of lo of formato. Eample : (Maatee) r ρ ( r ) /( ) r ~ t-dtrbuto d.f. ( )( ) r r r r 6 ( ( ) ( ) r r r r d ) Year NPB( ) Rak Nkll( ) Rak Σ d (-) + (-5) + (3-6.5) + 57 r t ( 4 ) p-value <.5 <.5. There gfcat correlato. A. Chag 3
4 Correlato ad Regreo Eample : I a vetgato of the relato betwee the average aual temperature ad the mortalt de for a tpe of breat cacer wome certa rego of Europe, data were collected from 5 Europea coutre. 9 SPSS Output 8 7 Summar Mortalt Ide 6 3 Average Temperature 6 R R Square Adjuted R Square Std. Error of the Etmate.94 a a. Predctor: (Cotat), Average Temperature Sample correlato coeffcet: Eample : I a vetgato, coutre were cluded to tud the relato betwee female lfe epectac ad the brthrate Female lfe epectac Brth per populato, 99 6 R Summar R Square Adjuted R Square Std. Error of the Etmate.87 a a. Predctor: (Cotat), Brth per populato, 99 Sample correlato coeffcet: Predctg the erupto tme after oberved a erupto of 4 mute durato Iter-erupto Tme Durato A. Chag 4
5 Correlato ad Regreo Smple Lear Regreo the Lear Relato Betwee Two Quattatve varable, (repoe varable) ad (eplaator varable). µ + e (Smple Lear Regreo ) that to determe µ β +β. What are the value of β ad β? Leat-Square Etmate of Regreo Parameter Fd oluto of β ad β of a traght le that mmze the followg varablt meaure (total accumulated quared error). Σ [ (β +β )] 6 Maatee Klled 3? +? Redual (or Error) ˆ SS β ˆ ˆ, β β SS Leat-Square Etmate of Regreo Parameter 6 Number of Power Boat 7 8 Eample : (Maatee) ˆ 37 β Aother formula: β ˆ r, where ad are tadard devato of ad ˆ β Equato of the regreo le : ŷ βˆ ˆ + β ; ŷ If at a certa ear the umber of power boat regtered 7,, etmate how ma maatee o average would be klled. ŷ , at A. Chag 5
6 Correlato ad Regreo Problem of etrapolato Etrapolated reult for a value out of the cope of A poble tred Scope of data Regreo ad caualt Regreo telf provde o formato about caual patter ad mut upplemeted b addtoal aal (wth deged ad cotrolled epermet) to obta ght about caual relatohp. Frt Order Smple Lear Regreo aumpto: β +β + ε wth error depedet ad detcall dtrbuted a ε Ν (, σ ) ad mea of at µ β +β. Mea of at A. Chag 6
7 Correlato ad Regreo Redual: e - ˆ β ˆ + ˆ ŷ β ŷ Eample : Fd the redual at 46. Oberved, ad predcted The redual Source of varablt: Repoe varable, Eplaator varable, Error Redual Sum of Square (SSRed), or Sum of Square of Error (SSE) Total Sum of Square (SSTo) SSTo SSR + SSE ( - ) Sum of Square due to Regreo (SSR) SSTo SSE ANOVA Table Source of Var. S.S. d.f. M.S. F ( ˆ - ) Regreo SSR SSR/MSR MSR/MSE Error SSE SSE/( )MSE Total (corrected) SSTo e - ˆ ) ( Etmato of : MSE SSE / ( ) 8.87 (Degree of freedom ) Etmated Stadard Error of the regreo model: 4.8 A. Chag 7
8 Correlato ad Regreo R Summar R Square Adjuted R Square Std. Error of the Etmate.94 a a. Predctor: (Cotat), NPOWERBT ANOVA b Sum of Square df Mea Square F Sg. Regreo a Redual Total a. Predctor: (Cotat), NPOWERBT b. Depedet Varable: MANKILL (Cotat) NPOWERBT a. Depedet Varable: MANKILL Utadardzed Coeffcet B Coeffcet a Std. Error Stadardzed Coeffcet Beta t Sg. Equato of the regreo le: ˆ αˆ + βˆ ; ˆ Iferece for Regreo Coeffcet Hpothe : Ho: β, v.. Ha: β Tet Stattc : βˆ t ~ t-dtrbuto d.f., where ( βˆ ) Deco rule : Reject Ho, f C.V. approach: t < t α/ or t > t α/ p-value approach: p-value < α ( β ˆ ).3 ( ) Hpothe : Ho: β v.. Ha: β Tet Stattc : βˆ t ~ t-dtrbuto d.f., where ( βˆ ) Deco rule : Reject Ho, f C.V. approach: t < t α/ or t > t α/ p-value approach: p-value < α ˆ ( β ) 7.4 ( ) + A. Chag 8
9 Correlato ad Regreo Iferece for Predcted Value The (-α) % cofdece terval for predctg the mea repoe at : ŷ ± tα / ( ŷ ) where ( ŷ ) + ( ) ( ) d.f. Predcted Number of Maatee Klled o Average at 46 > > (.9, 9.9) The (-α) % cofdece terval for predctg a dvdual outcome at : ŷ ± tα / ( ~ ) where ( ~ ) ( ) d.f. ( ) + + Predcted Number of Maatee Klled at 46 > 6.. > (5.9, 6.) 6 Cofdece Iterval Bad Number of maatee klled Number of Powerboat A. Chag 9
10 Correlato ad Regreo Evaluato of the Coeffcet of Determato It the proporto of varato oberved that ca be eplaed b the varable wth the lear regreo model. It the quare of the Pearo correlato coeffcet, r. r SSE/SSTo SSR/SSTo Redual Plot A catter plot of the redual agat the predcted value of the repoe varable to verf the aumpto behd the regreo model. (Homogeet of varace, radom ormal error, appropratee of the lear model.).5 Scatterplot Depedet Varable: Number of maatee klled Regreo Stadardzed Redual Regreo Stadardzed Predcted Value ot a good lear ft Volato of the equal varace aumpto A. Chag
11 Correlato ad Regreo Traformato The crcle of power rule. Whe the relato betwee ad ot lear, we beg lookg at traformato of the form p ad p. p, -3, -, -, -/, l, /,,, 3, Whe the catter plot how patter of data a Quadrat I, oe ca traform the varable wth a p fucto b creag power p to a umber larger tha oe, or traformato varable wth p fucto b creag power p to a umber large tha oe. Th what the up ad up mea. If the patter mlar to the catter plot Quadrat II, the ether creae the power of p traformato to a umber larger tha oe ( up) or decreae the power of p traformato to a umber le tha oe ( dow). up Quadrat II Quadrat I dow up Quadrat III Quadrat IV dow Eample : female lfe epectac, GDP (Gro dometc product) Plot of v.. Plot of v.. l() Female lfe epectac Female lfe epectac GDP per capta Natural log of GDP A. Chag
12 Correlato ad Regreo Multple Regreo Relatg a repoe (depedet, put) to a et of eplaator (depedet, output, predctor) varable,, 3,, k. A techque for modelg the relatohp betwee varable. Determtc compoet: Radom compoet: µ + e,, 3,..., k,, 3,..., k µ β + β + β + β β k k e Multple Lear Regreo : b + b + b + b b k k + e The Leat-Square Regreo Equato: ŷ β ˆ + βˆ + βˆ + βˆ βˆ 3 3 k k Eample : Ue Geder ad Age to predct bod fat percetage. : 3 b + b + b + e where Age, ad Geder ( or, dcator varable) FAT 3 6 GENDER M F 7 R Summar R Square Adjuted R Square Std. Error of the Etmate.845 a AGE a. Predctor: (Cotat), NGENDER, AGE ANOVA b Sum of Square df Mea Square F Sg. Regreo a Redual Total a. Predctor: (Cotat), NGENDER, AGE b. Depedet Varable: FAT Tet for gfcace of the model: Ho: gfcat (β I are all zero). Ha: gfcat (Some β I are all zero). p-value. <.5 (Cotat) AGE NGENDER Utadardzed Coeffcet B a. Depedet Varable: FAT Coeffcet a Std. Error Stadardz ed Coeffcet Beta Leat-Square Regreo Eequato: ˆ t Sg. Iferece for Regreo Coeffcet: Ho: β v.. Ha: β p-value.4 <.5 Ho: β v.. Ha: β p-value. <.5 Ho: β v.. Ha: β p-value. <.5 A. Chag
13 Correlato ad Regreo R Summar b R Square Adjuted R Square Std. Error of the Etmate.878 a a. Predctor: (Cotat), INT, AGE, NGENDER b. Depedet Varable: FAT : b + b + b + b 3 + e where Age, ad Geder ( or, dcator varable) ANOVA b Sum of Square df Mea Square F Sg. Regreo a Redual Total a. Predctor: (Cotat), INT, AGE, NGENDER b. Depedet Varable: FAT (Cotat) AGE NGENDER INT Utadardzed Coeffcet B a. Depedet Varable: FAT Coeffcet a Std. Error Stadardz ed Coeffcet Beta t Sg. I the model gfcat? Coeffcet of determato I there a teracto betwee Age ad Geder varable? Lt all the varable are gfcat the model? What the regreo equato: What the average percetage bod fat for a female aged? (Ue pot etmate.) Check the equal varace aumpto 3 Scatterplot Depedet Varable: FAT Vualze the teracto Regreo Stadardzed Redual FAT GENDER M F 7 Regreo Stadardzed Predcted Value AGE Utadardzed Redual Normalt tet for redual Tet of Normalt Kolmogorov-Smrov a *. Th a lower boud of the true gfcace. a. Lllefor Sgfcace Correcto Shapro-Wlk Stattc df Sg. Stattc df Sg..7 5.* A. Chag 3
14 Correlato ad Regreo Udertadg the female lfe epectac ad how t related wth eplaator varable: Brth Rate, Urbazato, Phoe, Doctor, ad GDP. After Log Traformato for Before Traformato Phoe, Doctor, ad GDP Female lfe epecta Female lfe epecta Brth per popu Brth per popu Percet urba, 99 Percet urba, 99 Phoe per peopl Natural log of phoe Doctor per, p Natural log of docto GDP per capta Natural log of GDP Summar b R R Square Adjuted R Square Std. Error of the Etmate Durb-Wato.934 a a. Predctor: (Cotat), Natural log of GDP, Percet urba, 99, Brth per populato, 99, Natural log of doctor per, Natural log of phoe per people b. Depedet Varable: Female lfe epectac 99 Idepedece of error Regreo Redual Total ANOVA b Sum of Square df Mea Square F Sg a a. Predctor: (Cotat), Natural log of GDP, Percet urba, 99, Brth per populato, 99, Natural log of doctor per, Natural log of phoe per people b. Depedet Varable: Female lfe epectac 99 Coeffcet a (Cotat) Brth per populato, 99 Percet urba, 99 Natural log of phoe per people Natural log of doctor per Natural log of GDP Utadardzed Coeffcet B Std. Error a. Depedet Varable: Female lfe epectac 99 Stadardz ed Coeffcet Beta Colleart Stattc t Sg. Tolerace VIF E Multcolleart Tolerace meaure the tregth of the lear relato betwee the depedet varable. It better to be hgher tha.. VIF the recprocal of Tolerace. A. Chag 4
15 Correlato ad Regreo ANOVA d 3 Regreo Redual Total Regreo Redual Total Regreo Redual Total Sum of Square df Mea Square F Sg a b c a. Predctor: (Cotat), Natural log of phoe per people b. Predctor: (Cotat), Natural log of phoe per people, Brth per populato 99 c. Predctor: (Cotat), Natural log of phoe per people, Brth per populato 99, Natural log of doctor per d. Depedet Varable: Female lfe epectac 99 Step-we electo 3 (Cotat) Natural log of phoe per people (Cotat) Natural log of phoe per people Brth per populato, 99 (Cotat) Natural log of phoe per people Brth per populato, 99 Natural log of doctor per Utadardzed Coeffcet B Std. Error a. Depedet Varable: Female lfe epectac 99 Coeffcet a Stadardz ed Coeffcet Beta Colleart Stattc t Sg. Tolerace VIF What are the gfcat factor that are related to the female lfe epectac? I tepwe regreo, a large umber of tet are performed ad lead to hgher probablt of Tpe I or Tpe II error. It hould be ued whe oe wat to determe mportat depedet varable from a large umber of potetall ueful varable the modelg proce. A. Chag 5
16 Correlato ad Regreo Ue of regreo aal. Decrpto (model, tem, relato) Relato betwee alar, rak, ear of ervce, Relato betwee lfe epectac, brth rate, GDP,. Cotrol Uderpad, overpad, ded too oug, 3. Predcto Salar for ew comer, future alar, lfe epectac 4. Varable creeg (mportat factor) What are the mportat factor that affectg alar or lfe epectac? Cotructo of regreo model:. Hpotheze the form of the model for,, 3,..., q a) Selectg predctor varable. b) Decdg fuctoal form of the regreo equato. c) Defg cope of the model (deg rage).. Collect the ample data (obervato, epermet). 3. Ue ample to etmate ukow parameter the model. 4. Specfg the probablt dtrbuto of the radom error. 5. Stattcall check the uefule of the model. 6. Appl the model deco makg. 7. Revew the model wth ew data. µ What lear model? Eample of a lear model: β + β + ε β + β + β + ε β + β + β + β 3 + ε β + β + β + β 3 + β 4 + β 5 + ε β + β l() + ε β + β e + ε lear term of t parameter. A. Chag 6
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