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It called the unexplained error that remains after the regression is, therefore,sometimes The differencebetweenthe two quantities,S, - $ , quantifies imthe sum of the squares provement or error reduction due to describing the data in terms of a straight line rather than as an averagevalue Becausethe magnitudeof this quantity is scale-dependent, the differenceis normalizedto S, to vield
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where 12 is called the coefficient of determination and r is the correlation coefficient /- fit, of \: J r)) For a peri-ect t : 0 and 12 : l, signifying that the line explains100% the An variability of the data For 12 : 0, S, : Sr and the fit represents improvement, no is alternativeformulation for r that is more convenientfor computerimolementation r:
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EXAMPLE 133 Estimotion Errors theLineor of for Leqsl-Souores Fit Problem Stqtement Compute totalstandard the deviation standard of the the error estimate, thecorrelation for 132 and coefficient thefit in Example lrlyl' : Thedatacanbe setup in tabular form andthenecessary computedin sums as
I 32 LINEAR LEAST-SQUARES REGRESSION
TABTE135
Dotoond summotions needed compute goodness-of-fit to the stotistics f o r t h ed o t of r o mT o b l e 3 , ] 1
li eo i atxi (yi - t)2 (!i - ao - a$)2
2 3 4 5 6 7 B
t0 20 30 4A 50 60 7A BO
25 7A 380 550 610 t 224 830 I ,454
-39 58 t55t2 349 82 54452 73523 9 3 39 3 I,t2863 1 , 3 23 3
380,535 327,441 68,579 8,441 1,0r6 3 3 42 2 9 3 53 9 r 6 53 , 0 6 6 t ,808,297
4,171 7,245 9il 30 I6,699 B ], 8 3 7 8 9 ,I B O 16,444 216,1tB
The standard deviation [Eq (133)] is
: 50826
and the standard errorof the estimate [Eq (1319)] is
: 18979
Thus, because sr,/r( s,,,the linear regression model has merit The extentof the improvement is quantified [Eq ( 1320)l by
1 8 0 8 , 2- 7 1 6 1 1 8 92 : 0880-5 l 808 297
of or r : \/088[5 : 09383 Theseresultsindicatethat 8805clc the orisinal uncertaintv hasbeenexplained the linearmodel by
Before proceeding, word of cautionis in orderAlthough the coefficientof determia provides a handy measureof goodness-of'-fit, should be careful not to ascribe you nation more meaningto it than is warranted Justbecause is "close" to I doesnot meanthat the r'2 "good" For example, is possible obtaina relativelyhigh valueof r'l fit is necessarily it to when the underlying relationshipbetween-y and ;r is not even linear Draper and Smith ( 198l) provide guidanceand additionalmaterialregardingassessment resultsfor linear of regression addition,at the rninimum,you shouldalwaysinspecta plot of the dataalong ln with your regression curve A n i c e e x a r n p l e w a s d e v e l o p e d b y A n s c o m b e ( 1 9 7 3 ) A1 3n lF i,g e c a m e u p w i t h si 0 h four datasetsconsisting I I datapointseachAlthough their graphsare very different,all of have the samebest-fit equation,)': 3 + 05r, and the samecoefficientof determination,
LINEAR EGRESSION R
F T G U R E l O t3 Anscombe's doto olong ihebestfit ,y: 3 + O5x four sets with Jine,
I33
TINEARIZATION NONTINEAR RETATIONSHIPS OF
Linear regression providesa powerful techniquefor fitting a bestIine to dataHowever, it is predicated the fact that the relationship on betweenthe dependent and independent variables is linear This is not always the case,and the first step in any regression analysis shouldbe to plot and visually inspectthe datato ascertain whethera linear model applies In some cases, techniques such as polynomial regression, which is described Chap14, in are appropriate For others,transformations be usedto expressthe datain a form that can w iscompatible ith linear egression r One exampleis the exltonential model:
! : olreP"
FIGU {o)Th (fJore
(l322)
where a1 and B1 are constants This model is usedin many lields of engineering sciand (positiveB1 or decrease enceto characterize quantities (negativeB1 at a rate that increase ) ) that is directly proportionalto their own magnitudeFor example,populationgrowthor radioactivedecaycan exhibit suchbehaviorAs depictedin Fig 1311a,the equation representsa nonlinearrelationship(for fu l0) between,vand x Another exampleof a nonlinearmodel is the simplepower equation:
)- : a2xP)
( 1323)
wherea2 andB2 areconstant coefficients This model haswide applicabilityin all fields of engineeringand scienceIt is very frequently used to fit experimentaldata when the underlyingmodel is not known As depicted Fig 1311b, equation (for B2 l0)is in the nonlinear
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