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

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Griselda Kennedy
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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 ) , ) Scatter Plot Ct NGR ) Nkll ) Correlato The correlato, r, betwee two radom varable, X ad Y, defed a, X X ) Y average Y ) r X Y 447, 3) Number of Hadgu Regtered 3 product of the tadard devate of X ad Y, quatfe the tregth of lear relatohp. 4 Pearo Sample Correlato r : Sample tadard devato of : Sample tadard devato of z : Sample tadard core of z z 567., , z z =.3).35) Ct NGR ) Nkll ) z : Sample tadard core of 5 6 SBD -

2 Summarzg Bvarate Data Total =.4. d. = r = Mea = , 3) Scatter Plot Number of Hadgu Regtered r Shortcut Formula r ) ) ) ), 9 Correlato Coeffcet = 7945 ) = 635 = 4 ) = = = 56 = 475 S = 99.5, S = S = 37 r = , r = Iterpretato of r 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. r he otato for populato correlato coeffcet. Correlato Stroger lear tred SBD -

3 Female lfe epectac 99 Death per people, 99 Summarzg Bvarate Data Correlato Coeffcet A 9 r =.968 r =.98 r 3 =.65 r 4 =.8 C r cloe to r cloe to Brth per populato, 99 GDP per capta B 9 D 9 r cloe to zero r cloe to zero - 3 Male lfe epectac 99 = 5 Doctor per, people 4 Correlato Doe Not Impl Cauato Eample: I a vetgato, coutre were cluded to tud the relato betwee female lfe epectac ad the brthrate. 9 The umber of hadgu regtered ma ot be the drect caue for the umber of people klled b gu. Brth per populato, Correlato Brth per populato, 99 Female lfe epectac 99 Pearo Correlato Sg. -taled) N Pearo Correlato Sg. -taled) N Brth per populato, 99 **. Correlato gf cat at the. level -taled). Female lfe epectac **.. -.8**.. Lear Regreo 7 8 SBD - 3

4 Weght Summarzg Bvarate Data Lear Relato If ou are a drug tore bug Tleol. A 4 cout Tleol cot $.. If the cot to get to the drug tore $3., the oe ca ue a determtc model = 3 + Ca heght formato be ued to fd out the weght of a dvdual? 3 9 Heght How to predct umber of people klled b gu b umber of hadgu regtered? umber of people klled b gu => Repoe varable umber of hadgu regtered => Eplaator varable Ct NGR ) Nkll ) Equato of a Straght Le = + Number of Hadgu Regtered = + the lope the -tercept repoe or depedet varable eplaator, depedet, or predctor varable 3 4 SBD - 4

5 Summarzg Bvarate Data SBD Graph wth a Ftted Le =? +? Number of Hadgu Regtered 6 Leat Square Prcple Fd oluto of ad of a traght le that mmze the followg varablt meaure: )] [ 7 e q )] [ mmze )] [ ) )] )[ q q?? 8 The Equato of The Ftted Le The leae-quared etmate of, are deoted a ad ad the are, =? +? 9, Sum of Square Other formula the ample tadard devato of the ample tadard devato of r r,

6 Summarzg Bvarate Data The Equato of a Ftted Le Mea of at = 4 Hadgu Eample 567., 9.43,.9, 9.9, r Ca be ued for etmato or predcto. Gve the etmate of locato of mea repoe for varou. 3 The regreo predcto) equato: A Etmato Graph wth a Ftted Le 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 34 Cauto Problem of etrapolato Caualt? Avod uure etrapolato. Error Etrapolated reult for a value out of the cope of A poble tred Scope of data Etmate at SBD - 6

7 Summarzg Bvarate 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. Regreo Model: Repoe, Outcome = Regreo Model? + e Epectato of gve Model the relato betwee ad wth error, e, depedet, detcall ad ormall dtrbuted a N, Frt Order Smple Lear Regreo Model Model aumpto: Model Aumpto Equal varace Normal error = + + e wth error, e, depedet, detcall ad ormall dtrbuted a N,, ad mea of at Redual: Redual e ) Eample: Fd the redual at = 4 ad the oberved =. Predcted = = 6. ŷ The redual = 6. = SBD - 7

8 Summarzg Bvarate Data Redual Sum of Square Mea Square Error ad Stadard devato for regreo Redual Sum of Square SSRed) or Error Sum of Square, SSE) ) Etmato of : = = MSE = SSE / ) = 8.87 Degree of freedom = ) Etmated Stadard Error of the regreo model: = = = Source of Varablt Repoe varable Eplaator varable Error Repoe Varable Varablt TotalSum of SquareSS To) ) Error Varablt Redual Sum of Square SSRed) or Error Sum of Square, SSE) ) Regreo Varablt Regreo Sum of SquareSS R) ) ŷ ŷ SBD - 8

9 Summarzg Bvarate Data Total Sum of SSTo - SSE Source of Varablt SSTo = SSR + SSE SquareSSTo) ) Regreo Sum of SquareSSR) ) ANOVA Table Source of Var. S.S. d.f. M.S. F Regreo SSR SSR/=MSR MSR/MSE Error SSE SSE/ )=MSE Total corrected) SSTo 49 Evaluato of the Model Coeffcet of Determato: It the proporto of varato oberved that ca be eplaed b the varable wth the lear regreo model. r SSE SSTo SSR SSTo Model Model Summar Adjuted Std. Error of R R Square R Square the Etmate.94 a a. Predctor: Cotat), Number of Hadgu Regtered Model Regreo Redual Total Correlato Coeffcet 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 Coeffcet of Determato Model Utlt Mea Square Error MSE) = 5 5 t-tet for gfcace of regreo coeffcet Model Cotat) Number of Hadgu Regtered Regreo coeffcet Coeffcet a Utadardzed Coeff cet a. Depedet Varable: Number of People Klled Stadard zed Coeff ce t B Std. Error Beta t Sg Model Cotat) Number of Hadgu Regtered Regreo coeffcet Coeffcet a Utadardzed Coeff cet a. Depedet Varable: Number of People Klled Stadard zed Coeff ce t B Std. Error Beta t Sg Equato of the regreo le: ; Equato of the regreo le: ; SBD - 9

10 Summarzg Bvarate Data 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 55 Number of Hadgu Regtered 56 Redual Plot Redual Plot Scatterplot Depedet Varable: Number of People Klled - - No a good lear model Varace are homogeeou Regreo Stadardzed Predcted Value Age ad umber of leep hour 9 Legth of leep Age Sere Lear Sere) Sere Pol. Sere) SBD -

11 Summarzg Bvarate Data Eample: = female lfe epectac = GDP Gro dometc product) Eample: = female lfe epectac = GDP Gro dometc product) GDP per capta Natural log of GDP Before Traformato 6 After lgdp) Traformato 6 Eample: = female lfe epectac = GDP Gro dometc product) 9 Traformato Crcle of Power: p or p up Quadrat II Quadrat I dow up Natural log of GDP ŷ Quadrat III Quadrat IV dow l) Traformato For up or up: tr p > for p or p Eample:,, 3, 3, or e, e For dow or dow: tr p < for p or p Eample: -/, -/, -, -, or l), l) 65 SBD -

Linear Regression. Can height information be used to predict weight of an individual? How long should you wait till next eruption?

Linear Regression. Can height information be used to predict weight of an individual? How long should you wait till next eruption? Iter-erupto Tme Weght Correlato & Regreo 1 1 Lear Regreo 0 80 70 80 Heght 1 Ca heght formato be ued to predct weght of a dvdual? How log hould ou wat tll et erupto? Weght: Repoe varable (Outcome, Depedet)