Correlation. Pearson s Sample Correlation. Correlation and Linear Regression. Scatter Plot

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Correlato ad Lear Regreo Dr. Thoma Smotzer Eame Relato Betwee Two Quattatve Varable I there a relato betwee the umber of hadgu regtered ad the umber of people klled b gu?

1 Correlato ad Lear Regreo Dr. Thoma Smotzer Eame Relato Betwee Two Quattatve Varable I there a relato betwee the umber of hadgu regtered ad the umber of people klled b gu? ear #reg #kll Scatter Plot Correlato Number of People Klled The correlato betwee two radom varable X ad Y defed a the epectato of the ( X µ X ) ( Y µ Y ) ρ avg σ σ X Y product of the tadard devate of X ad Y. Quatfe the tregth of lear relatohp. Number of Hadgu Regtered (447, ) 4 Pearo Sample Correlato r ( )( ( ) ) ( ), 5 6

2 Correlato Coeffcet Σ 7945 (Σ) 65 Σ 4 (Σ) Σ Σ 56 Σ 475 S 99.5, S S 7 r , r Iterpretato of r r It meaure the tregth ad drecto of the lear relato betwee two quattatve varable. r or - f all pot le eactl o a traght le. ρ the otato for populato correlato coeffcet. 7 8 Correlato Coeffcet r cloe to r cloe to Correlato Doe Not Impl Cauato r cloe to zero r cloe to zero The umber of hadgu regtered ma ot be the drect caue for the umber of people klled b gu. 9 t-tet for correlato Hpothe: Ho: ρ ρ v.. Ha: ρ ρ Tet Stattc: r ρ t ~ t-dtrbuto d.f. ( r )/( ) Deco rule: Reject Ho f t < t α/ or t > t α/ p-value < α I there a gfcat correlato? Eample: (Hadgu) Ho: ρ, v.. Ha: ρ t (. 886)/( 4 ) d.f. 4 p-value <.5 reject Ho There gfcat lear relato.

3 Spearma Rak Correlato Coeffcet r ( r r )( r r ) 6 d ( ) ( ) ( ) r r Spearma correlato coeffcet calculated baed o the rak of the obervato. r rak of amog all r rak of amog all, d r r. For te ue average rak. r r Year #reg( ) Rk #kll( ) Rk Σ d ( ) + ( 5) + ( 6.5) + 57 r t ( 4 ) p-value <.5 <.5 There gfcat correlato. 4 Tet for correlato If ample ze le tha ue pecal table. If large ample ze > ue t-dtrbuto table. The t-tet for rak data the ame a the t-tet for o-rak data. The rak t-tet le etve to outler wth the dadvatage of lo of formato. 5 Two Quattatve varable Data: Temperature Mortalt Ide Eample: The relato betwee the average aual temperature ad the mortalt de for a tpe of breat cacer wome certa rego of Europe. SPSS output Summar 9 R R Square Adjuted R Square.94 a a. Predctor: (Cotat), Average Temperature Std. Error of the Etmate Mortalt Ide Average Temperature 7 8

4 Eample: coutre were vetgated to tud the relato betwee female lfe epectac ad brthrate. SPSS output 9 Summar Female lfe epectac 99 R R Square Adjuted R Square Std. Error of the Etmate.8 a a. Predctor: (Cotat), Brth per populato, 99 Brth per populato, 99 9 Number of People Klled Lear Regreo Graph wth a Ftted Le Lear Relato If ou are a drug tore bug Tleol. A 4 cout Tleol cot $.., f (Bu oe cot $.) Number of Hadgu Regtered Lear Relato Lear Relato If ou are a drug tore bug Tleol. A 4 cout Tleol cot $.. 4, f (Bu two cot $4.) If ou are a drug tore bug Tleol. A 4 cout Tleol cot $.. 6, f (Bu oe cot $6.)

5 Lear Relato If the cot to get to the drug tore $., the + Equato of a Straght Le α + β 5 α + β β the lope α the -tercept repoe or depedet varable eplaator, depedet, or predctor varable 6 Number of People Klled Graph wth a Ftted Le α + β Leat Square Prcple Fd value of α ad β of a traght le that mmze the followg varablt meaure: [ ( α + β )] α + β Number of Hadgu Regtered 7 8 mmze q q α q β e ( )[ ( ) [ α α + β [ ( α + β )] ( α + β )] ( α + β )] + β 9 The Equato of The Ftted Le α + β The leat-quare etmate of α ad β are deoted b αˆ ad βˆ ˆ β, ˆ α ˆ β 5

6 Importat Sum of Square, Other formula ˆ β r, ˆ α r the ample tadard devato of the ample tadard devato of The equato of a ftted le ˆ α + ˆ β 4 Mea of at 4 Ca be ued for etmato or predcto. Gve a etmate for the locato of the mea repoe for varou value. Hadgu eample ˆ 7 β ˆ α The regreo (predcto) equato: ˆ α + ˆ β Graph wth a Ftted Le A Etmato Number of People Klled If at a certa ear the umber of hadgu regtered, etmate how ma people o average would be klled b gu The average repoe at Number of Hadgu Regtered 5 6 6

7 Cauto Problem of etrapolato Avod uure etrapolato. Caualt? Scope of data 7 8 Problem of etrapolato Problem of etrapolato Etrapolated reult for a value out of the cope of Etrapolated reult for a value out of the cope of A poble tred Scope of data Etmate at Scope of data Etmate at 9 Regreo ad Caualt Regreo telf provde o formato about caual patter ad mut be upplemeted b addtoal aal (wth deged ad cotrolled epermet) to obta ght about the caual relatohp. Frt Order Smple Lear Regreo aumpto: α + β wth error ε depedet, detcall ad ormall dtrbuted a Ν (, σ ), ad the mea of at µ α + β

8 Aumpto e Redual ( ˆ α + ˆ β ) Fd the redual at 4 ad the oberved. Predcted ŷ Redual Sum of Square Redual Sum of Square (SSE) or Error Sum of Square ( ) The redual Mea Square Error ad Stadard devato for regreo Etmato of σ : MSE SSE / ( ) 8.87 (degree of freedom ) Source of Varablt Repoe varable Eplaator varable Error Etmated Stadard Error of the regreo model:

9 Total Sum of ( ) Source of Varablt SSTSSR+SSE Square (SST) SST - SSE Error Sum of Square (SSE) Regreo Sum of Square (SSR) ( ) ( ) 49 ANOVA Table Source of Var. S.S. d.f. M.S. F Regreo SSR SSR/MSR MSR/MSE Error SSE SSE/( )MSE Total (corrected) SST Summar Adjuted Std. Error of R R Square R Square the Etmate.94 a a. Predctor: (Cotat), Number of Hadgu Regtered Regreo Redual Total Coeffcet of Determato ANOVA b Sum of Square df Mea Square F Sg a a. Predctor: (Cotat), Number of Hadgu Regtered b. Depedet Varable: Number of People Klled SPSS output Mea Square Error (MSE) 5 (Cotat) Number of Hadgu Regtered Regreo coeffcet Coeffcet a Utadardzed Coeffcet a. Depedet Varable: Number of People Klled Equato of the regreo le: Stadard zed Coeffce t B Std. Error Beta t Sg ˆ α + ˆ β ; (Cotat) Number of Hadgu Regtered Regreo coeffcet t-tet for gfcace of regreo coeffcet Coeffcet a Utadardzed Coeffcet a. Depedet Varable: Number of People Klled Equato of the regreo le: Stadard zed Coeffce t B Std. Error Beta t Sg ˆ α + ˆ β ; Iferece for Regreo Coeffcet β (t-tet) Hpothe: H o : β β, v.. H a : β β (It ofte tetg for Ho: β v.. Ha: β.) Tet Stattc: ˆ β β t e ˆ ( ˆ β ) ~ t-dtrbuto d.f., where e ˆ ( ˆ β ) ( ). Deco rule: Reject H o, f C.V. approach: t < t α/ or t > t α/ p-value approach: p-value < α 54 9

10 Iferece for Regreo Coeffcet α (t-tet) Hpothe: H o : α α, v.. H a : α α (It ofte tetg for H o : α v.. Ha: α.) Tet Stattc: ˆ α α t ~ t-dtrbuto d.f., e ˆ ( ˆ α ) where e ˆ ( ˆ α) ( ) Deco rule: Reject H o, f C.V. approach: t < t α/ or t > t α/ p-value approach: p-value < α Predctg Mea Repoe The (-α) % cofdece terval for predctg the mea repoe at : t e ˆ ( ) / ± α where e ( ) ( ) d.f. ˆ ( ) + Predcted Number of People Klled o Average at 4 > 6. ±.9 > (.9, 9.9) Predctg a Sgle New Repoe The (-α) % cofdece terval for predctg a dvdual outcome at : Cofdece Iterval Bad t e ˆ ( ~ ) / ± α where e ( ) ( ) d.f. ˆ ( ~ ) + + Number of People Klled Predcted Number of People Klled at 4 > 6. ±. > (5.9, 6.) 57 Number of Hadgu Regtered 58 Evaluato of the Coeffcet of Determato: It the proporto of varato oberved that ca be eplaed b the varable wth the lear regreo model. SSE r SST SSR SST Summar Adjuted Std. Error of R R Square R Square the Etmate.94 a a. Predctor: (Cotat), Number of Hadgu Regtered Regreo Redual Total Coeffcet of Determato ANOVA b Sum of Square df Mea Square F Sg a a. Predctor: (Cotat), Number of Hadgu Regtered b. Depedet Varable: Number of People Klled Mea Square Error (MSE) 59

11 Redual Plot Graph wth a Ftted Le 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 Number of People Klled 6 Number of Hadgu Regtered 6 Redual Plot Redual Plot Scatterplot Depedet Varable: Number of People Klled Regreo Studetzed Redual ot a good lear ft Volato of the equal varace aumpto Regreo Stadardzed Predcted Value 6 64 Eample: female lfe epectac GDP (Gro dometc product) 9 Eample: female lfe epectac GDP (Gro dometc product) 9 Female lfe epectac 99 - Female lfe epectac GDP per capta Before Traformato 65 Natural log of GDP After l(gdp) Traformato 66

Eample: female lfe epectac GDP (Gro dometc product) 9 Traformato Crcle of Power: p or p up Quadrat II Quadrat I Female lfe epectac 99 4 5 6 7 8 9 Natural log of GDP dow ˆ α + ˆ l() 67 68 ŷ β dow up

12 Eample: female lfe epectac GDP (Gro dometc product) 9 Traformato Crcle of Power: p or p up Quadrat II Quadrat I Female lfe epectac Natural log of GDP dow ˆ α + ˆ l() ŷ β dow up Quadrat III Quadrat IV Traformato For up or up: tr p > for p or p Eample:,,,, or e, e For dow or dow: tr p < for p or p Eample: -/, -/, -, -, or l(), l() 69 Multple Regreo Relatg a repoe (depedet, put) to a et of eplaator (depedet, output, predctor) varable,...,, A techque for modelg the relatohp betwee varable. µ,,,..., q α + β + β q β q q α + β + β + β βq q mmze q β etmated b ˆ β ad α b ˆ α the leat quare etmator. ˆ α + ˆ β + ˆ β + ˆ β ˆ β ˆ q q the leat quare regreo le. e [ ( α + β β )] q q Eample: Stud weght () ug age ( ) ad heght ( ). Data: Age (moth), heght (che), ad weght (poud) were recorded for a group of chool chldre. 7 7

13 SPSS output Weght Age Weght Heght Scatter plot how that both age ad heght are learl related to weght : α + β + β wth weght, age, ad heght 7 Summar Adjuted Std. Error of R R Square R Square the Etmate.794 a a. Predctor: (Cotat), Heght, Age Coeffcet of determato: the percetage of varablt the repoe varable (Weght) that ca be decrbed b predctor varable (Age, Heght) through the model. 74 Regreo Redual Total a. Predctor: (Cotat), Heght, Age b. Depedet Varable: Weght ANOVA b Sum of Square df Mea Square F Sg a etmato: SPSS output (Cotat) Age Heght a. Depedet Varable: Weght Utadardzed Coeffcet Coeffcet a Stadard zed Coeffce t Colleart Stattc t Sg. Tolerace VIF B Std. Error Beta Tet for gfcace of the model: p-value. <.5 H : gfcat (β are all zero). H a : gfcat (Some β are all zero). 75 Iferece Regreo Coeffcet Ho: α v. Ha: α p-value. <.5 Ho: β v. Ha: β p-value. <.5 Ho: β v. Ha: β p-value. <.5 Colleart * tattc: If the VIF (Varace Iflato Factor) greater tha the we have erou colleart. 76 Coeffcet a (Cotat) Age Heght a. Depedet Varable: Weght Utadardzed Coeffcet Stadard zed Coeffce t Colleart Stattc t Sg. Tolerace VIF B Std. Error Beta Leat quare regreo equato: The average weght of chldre 44 moth old ad whoe heght 55 che would be: (44) +.9(55) lb (etmated b the model) 77 varable are. How to terpret α, β ad β. : wth weght, age, ad heght. α the average repoe whe both predctor β the rate of chage of epected weght per ut chage of age adjuted for the heght varable. β the rate of chage of epected weght per ut α + β + β chage of heght adjuted for the age varable. 78

Coeffcet Etmato wth Iteracto Betwee Age ad Heght : α + β + β + β wth weght, age, ad heght Utadardzed Coeffcet Coeffcet a Stadard zed Coeffce t B Std. Error Beta (Cotat) 66.996 6.89.6.59 Age -.97.6 -.

14 Coeffcet Etmato wth Iteracto Betwee Age ad Heght : α + β + β + β wth weght, age, ad heght Utadardzed Coeffcet Coeffcet a Stadard zed Coeffce t B Std. Error Beta (Cotat) Age Heght -.E INTAG_HT.96E a. Depedet Varable: Weght Colleart Stattc t Sg. Tolerace VIF Hgh VIF mple ver erou colleart Iteracto hould ot be ued the model. 79 Weght Idcator Varable: Bar varable that take ol two poble value, ad, ad ca be ue for cludg categorcal varable the model. Geder Male Female Male: Female: Group Stattc Std. Error N Mea Std. Devato Mea Ue of Idcator Varable the Regreo wth Age, Heght ad Geder varable a Predctor Varable : wth α + β weght age heght geder ( + β + β female, male) W e g h t Age Heght Geder Male 8 Female 8 (Cotat) Age Heght Geder a. Depedet Varable: Weght Utadardzed Coeffcet Coeffcet a Stadard zed Coeffce t Colleart Stattc t Sg. Tolerace VIF B Std. Error Beta Wth Age ad Heght varable the model, the Geder varable become gfcat. Geder ad Age a Predctor Varable : α + β + β wth weght, ( female, age, ad male) geder Whe comparg the dfferece average weght betwee geder, adjuted for age ad heght varable, the dfferece tattcall gfcat. Weght Geder Male Female 8 84 Age 4

(Cotat) Age Geder a. Depedet Varable: Weght Utadardzed Coeffcet Coeffcet a Stadard zed Coeffce t Colleart Stattc t Sg. Tolerace VIF B Std. Error Beta -.8

15 (Cotat) Age Geder a. Depedet Varable: Weght Utadardzed Coeffcet Coeffcet a Stadard zed Coeffce t Colleart Stattc t Sg. Tolerace VIF B Std. Error Beta Age ad Geder are both gfcat varable for predctg weght. There gfcat dfferece average weght betwee geder f adjuted for age varable. Commo mtake Ue of the terall coded value of a categorcal eplaator varable drectl lear regreo modelg calculato. The proper wa to clude a categorcal varable to ue dcator varable. For havg a categorcal varable wth k categore, oe hould et up k dcator varable Eample: A urve queto aked for Race wth poble repoe, Whte, Black, Hpac. Oe ca et up a dcator varable o repreet Whte, otherwe, ad aother dcator uch that repreet Black otherwe, ad ad repreet Hpac. Th urve alo aked for Your Bod Fat Percetage ad for our Number of hour of eerce per week. : Bod Fat Percetage Number of hour of eerce per week α + β + β + β Race : α + β + β + β Bod Fat Percetage Race: Whte Race: Black Number of hour of eerce per week Race Iterpretato of the model: ad, α + β + β ad, α + β + β Race: Hpac ad, α + β Suppoe that the leat quare regreo equato for the model above Etmate the avg. bod fat for a whte pero eerce hour per week: Stud female lfe epectac ug percetage of urbazato ad brth rate. 9 9 Etmate the avg. bod fat for a black pero eerce hour per week: Etmate the avg. bod fat for a hpac pero eerce hour per week: Female lfe epectac 99 Brth per populato, 99 Female lfe epectac 99 Percet urba,

16 lfe epectac, : α + β + β brth rate, Summar Adjuted Std. Error of R R Square R Square the Etmate.94 a a. Predctor: (Cotat), Brth per populato, 99, Percet urba, 99 percet urbazed Regreo Redual Total ANOVA b Sum of Square df Mea Square F Sg a a. Predctor: (Cotat), Brth per populato, 99, Percet urba, 99 b. Depedet Varable: Female lfe epectac 99 Tet for gfcace of the model: p-value. <.5 Coeffcet of determato: the percetage of varablt the repoe varable (female lfe epectac) that ca be decrbed b predctor varable (brth rate, percetage of urbazato) through the model. 9 H : gfcat (β are all zero). H a : gfcat (Some β are all zero). 9 etmato: (SPSS output) (Cotat) Brth per populato, 99 Percet urba, 99 Utadardzed Coeffcet a. Depedet Varable: Female lfe epectac 99 Iferece for Regreo Coeffcet: Ho: α v.. Ha: α p-value. <.5 Ho: β v.. Ha: β p-value. <.5 Ho: β v.. Ha: β p-value. <.5 Coeffcet a Stadard zed Coeffce t Colleart Stattc B Std. Error Beta t Sg. Tolerace VIF Colleart * tattc: If the VIF (Varace Iflato Factor) greater tha the we have erou colleart. 9 Leat quare regreo equato for etmatg epected repoe value The average female lfe epectac for the coutre whoe brth rate per ad whoe percetage of urbazato would be () +.54() Ue of regreo aal Decrpto (model, tem, relato): Relato betwee lfe epectac, brth rate, GDP, Relato betwee alar, rak, ear of ervce, Cotrol:Ded too oug, uderpad, overpad, Predcto: Lfe epectac alar for ew comer, future alar, Varable creeg (mportat factor): What are the mportat factor for lfe epectac? Cotructo of regreo model µ. Hpotheze the form of the model for Selectg predctor varable.,,,..., Decdg fuctoal form of the regreo equato. Defg cope of the model (deg rage).. Collect the ample data (obervato, epermet).. Ue ample to etmate ukow parameter the model. 4. Udertad the 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. q

Summarizing Bivariate Data. Correlation. Scatter Plot. Pearson s Sample Correlation. Summarizing Bivariate Data SBD - 1

Summarizing Bivariate Data. Correlation. Scatter Plot. Pearson s Sample Correlation. Summarizing Bivariate Data SBD - 1 Summarzg Bvarate Data Summarzg Bvarate Data - Eamg relato betwee two quattatve varable I there relato betwee umber of hadgu regtered the area ad umber of people klled? Ct NGR ) Nkll ) 447 3 4 3 48 4 4