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

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Claud Sullivan
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1 Iter-erupto Tme Weght Correlato & Regreo 1 1 Lear Regreo 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) Heght: Eplaator varable (Predctor, Idepedet) 3 4 Durato ad Iter-erupto Tme Durato Iter-erupto tme: Repoe varable (Outcome, Depedet) Durato of erupto: Eplaator varable (Predctor, Idepedet) 5 How to predct umber of tudet have ee deae b of tudet pla vdeo game? # of tudet have ee deae => Repoe varable # of tudet pla vdeo game => Eplaator varable School VdeoG( ) EeD( ) CR - 1

2 Correlato & Regreo Equato of a Straght Le = Number of Hadgu Regtered 800 = + the lope the -tercept repoe or depedet varable eplaator, depedet, or predctor varable 7 8 Graph wth a Ftted Le 0 =? +? Leat Square Prcple Fd oluto of ad of a traght le that mmze the followg varablt meaure: [ ( ˆ ˆ )] ˆ ˆ ˆ Number of Hadgu Regtered 9 mmze q q q e [ ( )] ( )[ ( )] 0 ( ) [ ( )] 0?? 11 The Equato of The Ftted Le =? +? The leae-quared etmate of, are deoted a ˆ ad ˆ ad the are ˆ, ˆ ˆ 1 CR -

3 Correlato & Regreo 1 1 Sum of Square 1 1, ˆ r Other formula, ˆ r the ample tadard devato of the ample tadard devato of The Equato of a Ftted Le ˆ ˆ ˆ Mea of at = 4 Hadgu Eample 567., 9.43, 1.19, 91.91, r.941 ˆ ˆ Ca be ued for etmato or predcto. Gve the etmate of locato of mea repoe for varou. 15 The regreo (predcto) equato: ˆ ˆ ˆ ˆ A Etmato Graph wth a Ftted Le If at a certa ear the umber of hadgu regtered 700,000, etmate how ma people o average would be klled b gu. ˆ The average repoe at = Number of Hadgu Regtered CR - 3

4 Correlato & Regreo 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 19 0 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 m? + e Epectato of gve Model the relato betwee ad wth error, e, depedet, detcall ad ormall dtrbuted a N (0, ). 1 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 (0, ), ad mea of at m CR - 4

5 Correlato & Regreo Redual: Redual e ˆ ( ˆ ˆ ) Eample: Fd the redual at = 4 ad the oberved = 1. Predcted = = ŷ The redual = = Redual Sum of Square Mea Square Error ad Stadard devato for regreo Redual Sum of Square (SSRed) ( or Error Sum of Square, SSE) 1 ( ˆ ) Etmato of : = = MSE = SSE / ( ) = (Degree of freedom = ) Etmated Stadard Error of the regreo model: = = = Source of Varablt Repoe varable Eplaator varable Error Repoe Varable Varablt TotalSum of Square(SS To) 1 ( ) 9 CR - 5

6 Correlato & Regreo Error Varablt Redual Sum of Square (SSRed) ( or Error Sum of Square, SSE) 1 ( ˆ ) Regreo Varablt Regreo Sum of Square(SS R) 1 ( ˆ ) ŷ ŷ 31 3 Total Sum of SSTo - SSE Source of Varablt SSTo = SSR + SSE Square(SSTo) ( ) 1 Regreo Sum of Square(SSR) ( ˆ ) 1 ANOVA Table Source of Var. S.S. d.f. M.S. F Regreo SSR 1 SSR/1=MSR MSR/MSE Error SSE SSE/( )=MSE Total (corrected) SSTo 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. Square root of mea quare error Adjuted Coeffcet of Determato Redual tadard error: 4.76 o 1 degree of freedom Multple R-quared: , Adjuted R-quared: F-tattc: o 1 ad 1 DF, p-value: 5.9e-07 r 1 SSE SSTo SSR SSTo Coeffcet of Determato Model Utlt CR - 6

7 Correlato & Regreo Coeffcet: Etmate Std. Error t value Pr(> t ) (Itercept) *** HadGuRegtrato e-07 *** --- Sgf. code: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Regreo coeffcet t-tet for gfcace of regreo coeffcet Equato of the regreo le: ˆ ˆ ˆ ; ˆ Iferece for Regreo Coeffcet (t-tet) Hpothe: H o : = 0, v.. H a : 0 (It ofte tetg for Ho: = 0 v.. Ha: 0.) Tet Stattc: ˆ e ˆ ( ˆ ) t 0 ~ t-dtrbuto d.f. =, where e ˆ ( ˆ ) ( ) 1 =.013 Deco rule: Reject H o, f C.V. approach: t < t / or t > t / p-value approach: p-value < 38 Iferece for Regreo Coeffcet (t-tet) Hpothe: H o : = 0, v.. H a : 0 (It ofte tetg for H o : 0 v.. Ha: 0.) Tet Stattc: ˆ e ˆ ( ˆ ) t 0 ~ t-dtrbuto d.f. =, where 1 e ˆ ( ˆ ) ( ) = Deco rule: Reject H o, f C.V. approach: t < t / or t > t / p-value approach: p-value < 39 Predctg Mea Repoe The (1-) 0% cofdece terval for predctg the mea repoe at : ˆ t e ˆ ( ˆ) / where 1 ( ) e ˆ ( ˆ) d.f. = ( ) 1 Predcted Average Number of People Klled at = 4 => => (1.09, 19.9) Predctg a Sgle New Repoe The (1-) 0% cofdece terval for predctg a dvdual outcome at : ~ t ˆ ( ~ / e ) where 1 ( ) e ˆ ( ~ ) 1 ( ) d.f. = 1 Cofdece Iterval Bad 0 Predcted Number of People Klled at = 4 => => (5.91, 6.11) Number of Hadgu Regtered CR - 7

8 Correlato & Regreo 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 Redual Plot Redual Plot Scatterplot Depedet Varable: Number of People Klled No a good lear model Varace are ot homogeeou Regreo Stadardzed Predcted Value Eample: = female lfe epectac = GDP (Gro dometc product) Eample: = female lfe epectac = GDP (Gro dometc product) Before Traformato 47 After l(gdp) Traformato 48 CR - 8

9 Correlato & Regreo Eample: = female lfe epectac = GDP (Gro dometc product) Traformato Crcle of Power: p or p up Quadrat II Quadrat I dow up ŷ ˆ Quadrat III Quadrat IV dow e l() ˆ l() 49 Traformato For up or up: tr p > 1 for p or p Eample:,, 3, 3, or e, e For dow or dow: tr p < 1 for p or p Eample: -1/, -1/, -1, -1, or l(), l() 51 CR - 9

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