Example. Row Hydrogen Carbon
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1 SMAM 39 Least Squares Example. Heatg ad combusto aalyses were performed order to study the composto of moo rocks collected by Apollo 4 ad 5 crews. Recorded c ad c of the Mtab output are the determatos of hydroge (H) ad carbo (C) parts per mllo (PPM) for specmes. Row Hydroge Carbo x = x y y xy = = 6 = = Some of the above tems wll be computed o a had held calculator. The computg formula for xy x y = = = b = x ( x) = = ( ) ( )( 6) = =. 79 ( ) ( )
2 a = ( y bx) a = ( 6 (. 79)( )) = The least square equato s y = x The predcted values are obtaed by substtutg the x value to the least square equato to obta the predcted y. The dfferece betwee the observed ad the predcted values are called the resduals. For example whe x = 8 the observed value of y =. The predcted value s y = () = The resdual s = Oe way to fd SSE s to fd the sum of the squares of the resduals. The SSR may be foud after fdg SST by subtracto. Ths method s proe to roudoff errors. It s better to fd SSR frst. Use the followg computg formulae S S S xx yy xy ( x ) = x ( y ) = y = xy x y SSR =bs xy = (.79)(365.4)= SST = S yy = SSE = R =00( /3566.3)=79.4% of the varato s accouted for.
3 Worksheet sze: cells MTB > ame c='hydroge'\ MTB > ame c='carbo' MTB > set c DATA> DATA> ed MTB > set c DATA> DATA> ed MTB > prt c c Data Dsplay Row Hydroge Carbo Cosder the scatterplot. MTB > plot c*c
4 x 05+ x x Carbo x 70+ x x x x 35+ x Hydroge The graph of a straght le mght be a reasoable ft. The correlato coeffcet s gve the computato. MTB > corr c c Correlatos (Pearso) Correlato of Carbo ad Hydroge = The regresso le s gve below MTB > regress c o c Regresso Aalyss The regresso equato s Carbo = Hydroge Predctor Coef Stdev trato p Costat Hydroge s = 7.59 Rsq = 79.5% Rsq(adj) = 77.% Aalyss of Varace SOURCE DF SS MS F p Regresso Error Total
5 Uusual Observatos Obs. Hydroge Carbo Ft Stdev.Ft Resdual St.Resd R R deotes a obs. wth a large st. resd. The regresso le accouts for 79.5% of the varato. We ca fd the predcted values ad plot them. MTB > let c5= *c MTB > ame c5='predct' MTB > prt c c5 Row Hydroge predct I the plot below the letter b are the predcted values ad the letter a are the observed values. TB > gstd * NOTE * Stadard Graphcs are eabled. Professoal Graphcs are dsabled. Use the GPRO commad to eable Professoal Graphcs. MTB > mplot c*c c5*c
6 Character Multple Plot A B 05+ A A B BB A 70+ B A A A A = Carbo vs. Hydroge B = C5(predct) vs. Hydroge Ths gves a dea of how good the ft s. Recorded here are the scores of 6 studets o a mdterm ad fal exam s statstcs. Data Dsplay Row mdterm fal MTB > Aga lets make a scatter plot MTB > ame c3='mdterm' MTB > ame c4='fal' MTB > set c3 DATA> DATA> ed MTB > set c4
7 DATA> DATA> ed MTB > plot c4*c3 haracter Plot 00+ x Fal x x x x x x 75+ x x x x x x 50+ x x x Mdterm MTB > GPro. MTB > Observe that oe of the observatos (8,56) s way out. MTB > corr c4 c3 Correlatos (Pearso) Correlato of fal ad mdterm = MTB > regress c4 o,c3 Regresso Aalyss The regresso equato s fal = mdterm Predctor Coef Stdev trato p Costat mdterm s = 6. Rsq = 34.0% Rsq(adj) = 9.3% Aalyss of Varace
8 SOURCE DF SS MS F p Regresso Error Total Uusual Observatos Obs. mdterm fal Ft Stdev.Ft Resdual St.Resd RX Oly 34% of the varato s accouted for. Redog the regresso wthout the uusual observato mproves the ft cosderably but ot eough to make t worthwhle. MTB > let c6=c3 MTB > let c7=c4 The uusual observato was deleted from the colums o the worksheet. MTB > regress c7 o,c6 Regresso Aalyss The regresso equato s C7 = C6 5 cases used cases cota mssg values Predctor Coef Stdev trato p Costat C s = 3.00 Rsq = 59.4% Rsq(adj) = 56.3% Aalyss of Varace SOURCE DF SS MS F p Regresso Error Total Theoretcal Devlopmet Gve a set of data pots (X, Y ) the objectve s to fd the straght le such that the sum of the squares of the dfferece betwee the observed values ad
9 those that would be predcted by the regresso equato s a mmum. Ths amouts to fd the values of the slope ad the y tercept such that Fab (, ) = ( Y a bx) = s mmzed. A o calculus dervato of the LS Equato s gve o the ext page.
10 Dervato of Least Square Formula wthout Calculus Notato S xx = Ú =ƒ Hx x L S xy = Ú =ƒ Hx x L Hy y ) S yy = Ú =ƒ Hy y L. The goal s to fd a ad b so that F(a,b)=Ú =ƒ Hy a bx L s mmmzed. Ths represets the dfferece betwee the observed values ad those predcted by the best fttg equato. Now addg ad subtractg y ad bx Ú =ƒ Hy a bx L = Ú yl + Hy a b xl bhx xme = Ú =ƒ Hy y L +Hy a b xl +b Ú =ƒ = S yy + Hy a b xl +b S xx bs xy = S yy + Hy a b xl +S xx Jb bs xy = S yy IS xym Sxx Sxx + Iy a bxm +S xx Jb S xy Sxx N The above expresso s mmzed whe Hx x M bú =ƒ Hx xl Hy y ) + J S xy Sxx N IS xym ) Sxx b= S xy Sxx ad a = y bx.
11 Oce the regresso equato s derved the corrected sum of squares ca be broke up to two parts a sum of squares due to regresso ad a sum of squares due to error. SST =Ú = Hy yl =Ú = Hy a bx + a + bx yl =Ú = Hy a bx L +Ú = Ha + bx yl +Ú = Hy a bx L Ha + bx yl Sce a = y bx. y a bx =y y +bx bx Ú = Hy a bx L Ha + bx yl=ú = Hy y + b x bx L Hbx b xl = bs xy b S xx = S xy Sxx S xy Sxx Sxx=0 The cross term s therefore zero ad SST =SSR +SSE SSR = Ú = Ha + bx yl = b Ú = Hx xl = bs xy The quatty R = SSR represets the proporto of the SST varato accouted for by the regresso le.
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