Gregory Carey, 1998 Linear Transformations & Composites  1. Linear Transformations and Linear Composites


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1 Gregory Carey, 1998 Liear Trasformatios & Composites  1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio takes the form of creatig a ew variable from the old variable usig the equatio for a straight lie: ew variable a + b* (old variable) where a ad b are mathmatical costats What is the mea ad the variace of the ew variable? To lve this let X deote the old variable ad assume that it has a mea of X ad a variace of S X Let X* deote the ew variable The X* a + bx The mea of X* is X * X * ( a + bx) a +bx a + bx a + b X a + b X a + bx X * a + bx The variace of the ew variable, X*, ca be foud i a similar way The variace i X* is simply the variace of the quatity (a + bx) So, we merely substitute ad use me algebra: S X* ( ) X * X * ( ) a + bx a bx ( ) bx bx 1
2 Gregory Carey, 1998 Liear Trasformatios & Composites  [ ( )] b X X ( ) b X X ( ) b X X b S X S X* b S X Oe of the most importat applicatios of liear trasforms comes i stadardizatio To remove the effects of, say, school grade i a data set o readig ability, oe could first rt the data by grade ad the stadardize the variables that each grade has the same mea ad the same stadard deviatio Oe of the most frequetly used methods of stadardizatio is to Z trasform a variable that the mea ad stadard deviatio of the ew variable are, resepctively, 0 ad 1 The familiar formula for this is Z ( X X ) With a little bit of algebra, we ca rework this formula to This is the same as the liear equatio Z X + 1 X ew variable a + b * old variable The ew variable is Z ad the old variable i The value of a is ad the value of b is Hece the mea of Z must be X 1 Z a + bx X + 1 X 0
3 Gregory Carey, 1998 Liear Trasformatios & Composites  3 ad the variace of Z will be s Z b 1 s X 10 Trasformig data to have a desired mea ad/or stadard deviatio The formulas give above may be used to demostrate how to trasform variables to have a desired mea ad stadard deviatio For example, suppose that we had raw scores o a ewly developed MMI scale ad would like to express these scores i the customary metric of the MMI, T scores with a mea of 50 ad a stadard deviatio of 10 Let X deote the variable i raw uits with observed mea X ad observed stadard deviatio Let X deote the desired mea ad s d d deote the desired stadard deviatio Takig the square root of equatio give above for the variace of a trasformed variable gives s d b b s d Thus the slope is simply the desired stadard deviatio divided by the observed stadard deviatio The itercept may be foud by substitutig this expressio ito the equatio for the mea of a trasformed variable: X d a + bx a + a X d s d s d X X uttig these expressios for a ad b together (plus doig a little algebra) gives the formula for the desired trasformatio: X X X d X d + s d I plai Eglish, to trasform a variable to have a desired mea ad a desired stadard deviatio, simply take the Ztrasform of the origial variable, multiply it by the desired stadard deviatio, ad the add the desired mea I the case of the MMI, where we wated scores with a mea of 50 ad a stadard deviatio of 10, we would simply fid the Z trasform of the origial score, multiply that by 10, ad the add 50 3
4 Gregory Carey, 1998 Liear Trasformatios & Composites  4 Covariace ad Correlatio of Two Liearly Trasformed Variables What is the covariace betwee two variables that have bee liearly trasformed? Here, let the old variables be X ad Y ad the ew variables be deoted as, respectively, X* ad Y* The the trasformatio will take the form ad The covariace is defied as cov X*,Y * X* a + bx Y* c + dy ( ) X * X * ( )( Y * Y *) Oe agai, substitute ad do me algebra: ( a + bx a bx )( c + dy c dy ) bx bx b ( X X ) bd X X ( )( dy dy ) [ ][ d ( Y Y )] ( )( Y Y ) bd X X ( )( Y Y ) bd cov(x,y) cov( X*,Y *) bd cov(x,y) Thus, a liear trasformatio will chage the covariace oly whe both of the old variaces are multiplied by methig other tha 1 If we simply add methig to both old variables (ie, let a ad c be methig other tha 0, but make b d 1), the the covariace will ot chage Although a liear trasformatio may chage the meas ad variaces of variables ad the covariaces betwee variables, it will ever chage the correlatio betwee variables Cosider X* ad Y* as give above We have already show that the variaces of these two variables are ad S X* b S X S Y* d SY We have al demostrated that the covariace betwee the two trasformed variables is 4
5 Gregory Carey, 1998 Liear Trasformatios & Composites  5 cov( X *,Y *) bd cov(x,y) The correlatio betwee the trasformed variables will be corr( X *,Y *) Agai, we substitute ad perform me algebra: which is the correlatio betwee X ad Y II Liear Composites corr(x *,Y *) b S X bd cov X,Y ( ) b d S X S Y bd cov( X,Y) bd S X S Y cov X,Y ( ) S X S Y cov( X *,Y *) S X *S Y * bd cov( X,Y ) d S Y Mea of a Liear Composite Here, we wish to examie what happes whe a etirely ew variable is costructed as a liear fuctio of several old variables Let X i deote the ith old variable ad Y the ew variable We ca make the case mewhat more geeral by assumig that we add a residual, U, that is actually a radom umber take from a stadard ormal distributio with mea of 0 ad stadard deviatio of 1 The equatio for the ew variable is Y a + b 1 + b X +b p + uu (If this busiess of the radom variable, U, is botherme, the simply let the quatity u equal 0 i the equatio ad i all that follows othig of substace will chage) We ca ow go through the same algebra that we used above i the trasformatio of a old variable ito a ew variable to calculate the variace of variable Y The oly trick here is to recall that variable U will have a mea of 0, ad because it is radom, will be ucorrelated with all the Xs Cosider the mea The mea of Y is the mea of a+ b 1 + b X + b p + uu 5
6 Gregory Carey, 1998 Liear Trasformatios & Composites  6 i1 (a + b 1 i + b X i + K b p i + uu ) a + b 1 X + b 1i X + K b i p X + u U pi i a + b 1 i1 i1 i i1 X i i + b i1 i1 + K b p + u a + b 1 + b X + K b p + uu i1 a + b 1 + b X + K b p + 0 Y a + b 1 + b X + b p i1 U i i1 Variace of a Liear Composite Similar logic will write the variace of Y as a fuctio of the variables o the right side of the equatio We will ot go through the elaborate algebra, but istead give the result: S Y i 1 j 1 b i b j cov( X i, X j ) + u ote that the term u is OT icluded i this summatio For example, suppose that the ew variable is a liear composite of three variables, or The Y a + b 1 + b X + b 3 X 3 + uu S Y b1 SX1 + b SX + b 3SX3 + b 1 b cov(,x ) + b 1 b 3 cov(, X 3 ) + b b 3 cov( X,X 3 ) + u (Recall, here, that cov( X i, X i ) S Xi ) Covariace of Two Liear Composites With similar algebra, it ca be show that the covariaces betwee ay two liear composites ca be writte i terms of the bs, ad the covariaces amog the Xs ad the Us Let ad Y 1 a 1 + b 11 + b 1 X +b 1p + u 1 U 1 Y a + b 1 + b X + b p + u U The 6
7 Gregory Carey, 1998 Liear Trasformatios & Composites  7 ( ) cov Y 1,Y b1i b j cov X i,x j i1j1 ( ) + u 1 u cov U 1,U ( ) If all the variables are stadardized, the bs become β coefficiets, all of the covariaces become correlatios, ad all variaces become 10 The S Y 1 i 1 j 1 β i β j corr( X i, X j ) +u ad covy ( 1,Y ) corr( Y 1,Y ) i 1 j 1 β 1i β j corr( X i, X j ) +u 1 u corr(u 1,U ) III Trasformatios ad Liear Composites i Matrix Algebra Trasformatios of variables ca be ecoomically writte usig matrix algebra Let X deote the old variable ad Y deote the ew variable We have see that the trasformatio for the ith idividual takes the form Y i a + bx i ow let x deote a colum vector of old variable values ad y a colum vector of ew variable values Equatio ( ) above may ow be writte as or Y 1 a Y a a Y + X X b y a + xb If we wish to make a ew variable as a liear composite of several old variables, the let X deote a matrix of the old variable values The rows of X correspod to the observatios ad the colums to the variables Let b deote a colum vector of weights The equatio becomes or Y 1 a Y a a Y 1 p X X p + X 1 X y a + Xb b 1 b b p 7
8 Gregory Carey, 1998 Liear Trasformatios & Composites  8 A more geeral formulatio permits a liear trasformatio of oe set of variables ito aother, ew set of variables That is, istead of a colum vector of Ys, there is ow a matrix of Ys Let X ij deote the score of the ith per o the jth old variable ad let Y ij deote the score of the ith per o the jth ew variable Let a j deote the costat for the jth variable, ad let b ij deote the weight used to multiply the ith X variable for the jth Y variable The trasformatio is or Y 11 Y 1 Y 1q a 1 a a q 1 p b 11 b 1 b 1q Y 1 Y Y q a 1 a a q X X p b 1 b b q + Y 1 Y Y q a 1 a a q X 1 X b p1 b p b pq Y A + XB The geeral case for the mea ad the variacecovariace matrix of the trasformed variables ca ow be writte Let or Y 1 Y Y q a 1 a a q b 11 b 1 b p1 b 1 b b p + b 1 q b q b pq y a + B t x Likewise, the covariace matrix may be writte i a geeral form Let C ij deote a covariace matrix betwee the i variables (the rows of the matrix) ad the j variables (the colums) The variacecovariace matrix for the ew variables is C yy B t C xx B ad the covariace matrix betwee the trasformed variaces ad the origial variables is C yx B t C xx Oce agai, the trasformatio of several X variables ito a sigle Y variable is a special istace of this equatio where B becomes a colum vector If oe trasforms a sigle X ito a sigle Y, the B is a colum vector ad matrix C xx becomes a scalar equal to the variace of X X 8
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