Simple Linear Regression. How To Study Relation Between Two Quantitative Variables? Scatter Plot. Pearson s Sample Correlation.

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Coleen Webb
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1 Correlato & Regreo How To Study Relato Betwee Two Quattatve Varable? Smple Lear Regreo 6. A Smple Regreo Problem I there relato betwee umber of power boat the area ad umber of maatee klled? Year NPB( ) Nkll(y ) Maatee Klled 0 0 Scatter Plot Number of Power Boat 0 4 Correlato Pearo Sample Correlato The correlato, ρ, betwee two radom varable, X ad Y, defed a, ( X μ X ) ( Y μ ρ average Y σ X σy ) product of the tadard devate of X ad Y, quatfe the tregth of lear relatohp. 5 r y y y : Sample tadard devato of y : Sample tadard devato of y 6

2 Correlato & Regreo y y y y y y y Total.4. d. 9.9 y.9 r Mea 567. y y Number of People Klled (447, 3) 0 0 Scatter Plot Number of Hadgu Regtered 0 y : y: Oe of par (447, 3) Shortcut Formula y y r y y y y y yy, yy y y Pearo Sample Correlato a Dfferet Formula ( )( y y) r ( ) ( y y) y yy Correlato Coeffcet Σ 7945 (Σ) 635 Σy 4 (Σy) Σ Σy 56 Σy 475 S 99.5, S yy S y 37 r , r

3 Correlato & Regreo Iterpretato of r < r < It meaure the tregth ad drecto of the lear relato betwee two quattatve varable. r f all pot le eactly o a traght le. ρ he otato for populato correlato coeffcet. Correlato Coeffcet r cloe to r cloe to r cloe to zero r cloe to zero 3 4 Correlato Doe Not Imply Cauato How to Model Lear Relato? Eample: The umber of powerboat regtered may ot be the drect caue for the death of Maatee. 5 6 Maatee Klled 0 0 Graph wth a Ftted Le Maatee Klled 0 y? +? Leat Square Prcple Fd oluto of α ad β of a traght le that mmze the followg varablty meaure: [ ( ˆ α + ˆ β )] y ˆ α + ˆ β Number of Power Boat Number of Power Boat 7 8 3

4 Correlato & Regreo mmze q q α q β e ( )[ y ( ) [ y y α y α + β [ y ( α + β )] ( α + β )] 0 ( α + β )] 0 + β α? β? 9 The Equato of The Ftted Le y? +? The leae-quared etmate of α, β are deoted a αˆ ad βˆ ad they are ˆ y β, ˆ α y ˆ β 0 Other formula ˆ y β r, ˆ α y r y The Equato of a Ftted Le y ˆ α + ˆ β the ample tadard devato of y the ample tadard devato of y Ca be ued for etmato or predcto. The Equato of a Ftted Le y ˆ α + ˆ β Mea of y at 4 Maatee Eample y ˆ 37 β ˆ α Ca be ued for etmato or predcto. Gve the etmate of locato of mea repoe for varou. 3 The regreo (predcto) equato: ˆ α + ˆ β

Correlato & Regreo Data R Commader R for Smple Lear Regreo 5 6 R Output Call: lm(formula MANKILL ~ NPOWERBT, data Dataet)

589 0.0008 *** NPOWERBT 0.49 0.09 9.675 5.e-07 *** --- Sgf. code: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' y Redual tadard error: 4.

6 o ad DF, p-value: 5.9e-07 8 Equato of the regreo le: ˆ α + ˆ β ; 4.44 +.

5 Correlato & Regreo Data R Commader R for Smple Lear Regreo 5 6 R Output Call: lm(formula MANKILL ~ NPOWERBT, data Dataet) Redual: M Q Meda 3Q Ma Coeffcet: Etmate Std. Error t value Pr(> t ) (Itercept) *** NPOWERBT e-07 *** --- Sgf. code: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' y Redual tadard error: 4.76 o degree of freedom Multple R-Squared: , Adjuted R-quared: Coeffcet of determato R 7 F-tattc: 93.6 o ad DF, p-value: 5.9e-07 8 Equato of the regreo le: ˆ α + ˆ β ; A Etmato If at a certa year the umber of power boat regtered 0,000, etmate how may maatee o average would be klled The average repoe at

6 Correlato & Regreo Graph wth a Ftted Le Maatee Klled How log hould you wat tll et erupto? Number of Power Boat Durato ad Iter-erupto Tme 0 Itererupto Tme CDUR Iter-erupto Tme Durato of Erupto.5.0 Durato Durato ad Iter-erupto Tme Cauto 0 Avod uure etrapolato. Caualty? Iter-erupto Tme Durato

7 Correlato & Regreo Problem of etrapolato Problem of etrapolato Scope of data Scope of data 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 y at Scope of data Etmate y at 39 Regreo ad Caualty Eample: y female lfe epectacy GDP (Gro dometc product) Regreo telf provde o formato about caual patter ad mut upplemeted by addtoal aaly (wth deged ad cotrolled epermet) to obta ght about caual relatohp. Female lfe epectacy GDP per capta Before Traformato 4 7

8 Correlato & Regreo Eample: y female lfe epectacy GDP (Gro dometc product) Eample: y female lfe epectacy GDP (Gro dometc product) Female lfe epectacy 99 Female lfe epectacy Natural log of GDP Natural log of GDP ŷ ˆ α + ˆ β l() After l(gdp) Traformato Traformato Crcle of Power: p or y p y up Quadrat II Quadrat I Traformato For up or y up: try p > for p or y p Eample:, y, 3, y 3, or e, e y dow up For dow or y dow: try p < for p or y p Eample: -/, y -/, -, y -, or l(), l(y) Quadrat III y dow Quadrat IV Smple Lear Regreo t-tet for correlato Hypothe: H 0 : ρ ρ 0, v.. H a : ρ ρ 0 Tet Stattc: (If data are bvarate ormal.) 8.9 Tet Cocerg Regreo ad Correlato t r ρ0 ( r )/( ) ~ t-dtrbuto d.f. Deco rule: Reject H 0, f C.V. approach: t < t α/ or t > t α/ p-value approach: p-value < α

Correlato & Regreo I there a gfcat correlato? R for Correlato Eample: (Maatee) H 0 : ρ 0, v.. H a : ρ 0 t. 94 0 9. 65 (. 886)/( 4 ) d.f. 4 -, p-value <.0005, reject H 0, there gfcat lear relato.

9e-07 alteratve hypothe: true correlato ot equal to 0 95 percet cofdece terval: 0.84 0.986797 ample etmate: cor 0.

9 Correlato & Regreo I there a gfcat correlato? R for Correlato Eample: (Maatee) H 0 : ρ 0, v.. H a : ρ 0 t (. 886)/( 4 ) d.f. 4 -, p-value <.0005, reject H 0, there gfcat lear relato. r R Output wth t-tet for Zero Correlato Pearo' product-momet correlato data: Dataet$MANKILL ad Dataet$NPOWERBT t , df, p-value 5.9e-07 alteratve hypothe: true correlato ot equal to 0 95 percet cofdece terval: ample etmate: cor Eample: I a vetgato, coutre were cluded to tudy the relato betwee female lfe epectacy ad the brthrate. Frt Order Smple Lear Regreo Model Model aumpto: Female lfe epectacy r.87 y α + β + ε wth error, ε, depedet, detcally ad ormally dtrbuted a Ν (0, σ ), ad mea of y at μ y α + β. Brth per 00 populato,

10 Correlato & Regreo Model Aumpto Redual y Redual: e y y ( ˆ ˆ α + β ) Eample: Fd the redual at 4 ad the oberved y. Redual Sum of Square ŷ Predcted y The redual Redual Sum of ( or Square (SSRed) Error Sum of Square, SSE) ( y ) Meaure Square Error ad Stadard devato for regreo Etmato of σ : y MSE SSE / ( ) 8.87 (Degree of freedom ) Etmated Stadard Error of the regreo model: y Iferece for Regreo Coeffcet β (t-tet) Hypothe: H o : β β 0, v.. H a : β β 0 (It ofte tetg for Ho: β 0 v.. Ha: β 0.) Tet Stattc: ˆ β β 0 t e ˆ ( ˆ β ) ~ t-dtrbuto d.f., where e ˆ ( ˆ β ) y ( ).03 Deco rule: Reject H o, f C.V. approach: t < t α/ or t > t α/ p-value approach: p-value < α

11 Correlato & Regreo Iferece for Regreo Coeffcet α (t-tet) Hypothe: H o : α α 0, v.. H a : α α 0 (It ofte tetg for H o : α 0 v.. H a : α 0.) Tet Stattc: ˆ α α 0 t ~ t-dtrbuto d.f., e ˆ ( ˆ α) where e ˆ ( ˆ α) + y 7.4 ( ) 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) 0% cofdece terval for predctg the mea repoe at : t e ˆ ( ) / ± α where e ( ) ( ) d.f. ˆ ( ) + y Predcted Number of Maatee Klled o Average at 4 > 6.0 ± 3.9 > (.09, 9.9) 6 6 Predctg a Sgle New Repoe The (-α) 0% cofdece terval for predctg a dvdual outcome at : t e ˆ ( ~ y) / ± α Cofdece Iterval Bad where e ( ) ( ) d.f. ˆ ( ~ y) + + y Predcted Number of Maatee Klled at 4 > 6.0 ±. > (5.9, 6.) Number of maatee klled Number of Powerboat Evaluato of the Model Itererupto Tme CDUR Total Populato Coeffcet of Determato (R ): It the proporto of varato oberved y that ca be eplaed by the varable wth the lear regreo model Durato of Erupto 65 66

12 Correlato & Regreo R Output Call: lm(formula MANKILL ~ NPOWERBT, data Dataet) Redual: M Q Meda 3Q Ma Coeffcet: Etmate Std. Error t value Pr(> t ) (Itercept) *** NPOWERBT e-07 *** --- Sgf. code: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' Redual tadard error: 4.76 o degree of freedom Multple R-Squared: , Adjuted R-quared: Coeffcet of determato R F-tattc: 93.6 o ad DF, p-value: 5.9e Redual Plot A catter plot of the redual agat the predcted value of the repoe varable to verfy the aumpto behd the regreo model. Homogeety of varace Radom ormal error Appropratee of the lear model 68 Graph wth a Ftted Le Redual Plot.5 Scatterplot Depedet Varable: Number of maatee klle Maatee Klled 0 Regreo Stadardzed Redual Number of Power Boat Regreo Stadardzed Predcted Value 69 Redual Plot 0 0 Model ot a good lear ft Volato of the equal varace aumpto 7

r y Simple Linear Regression How To Study Relation Between Two Quantitative Variables? Scatter Plot Pearson s Sample Correlation Correlation

r y Simple Linear Regression How To Study Relation Between Two Quantitative Variables? Scatter Plot Pearson s Sample Correlation Correlation Maatee Klled Correlato & Regreo How To Study Relato Betwee Two Quattatve Varable? Smple Lear Regreo 6.11 A Smple Regreo Problem 1 I there relato betwee umber of power boat the area ad umber of maatee klled?