Standardized Coefficients

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Sabina O’Connor’
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1 Standadized Coefficient Ta. How do ou decide which of the X ae mot impotant fo detemining? In thi handout, we dicu one poile (and contoveial) anwe to thi quetion - the tandadized egeion coefficient. Fomula. Fit, we will give the fomula and then explain thei ationale: Geneal Cae: Two IV cae: = x 1 ( 1 ) = 1 1 ( 1 1) = 1 1 A thi fomula how, it i ve ea to go fom the metic to the tandadized coefficient. Thee i no need to actuall compute the tandadized vaiale and un a new egeion. Compae thi to the fomula fo the metic coefficient. Note that coelation tae the place of the coeponding vaiance and covaiance. 1 IV cae = x In the one IV cae, the tandadized coefficient impl equal the coelation etween and X Rationale. The paamete a, 1,, etc., ae often efeed to a the metic egeion coefficient. It i often difficult to a which of the X vaiale i mot impotant in detemining the value of the dependent vaiale, ince the value of the egeion coefficient depend on the choice of unit to meaue X. In the peent example, thi i not o polematic, ince oth education and jo expeience ae meaued in ea. But uppoe intead that ou independent vaiale wee education and IQ - how would we detemine which vaiale wa moe impotant? The value of the metic coefficient would tell u little, ince IQ and education ae meaued in ve diffeent wa. Fo example, uppoe the metic coefficient fo education wa.0, and the metic coefficient fo IQ wa 1.0. Thi would mean that each additional ea of education wa woth $000 on aveage, and each 1-point inceae in IQ wa woth $ ut we cetainl could not infe fom thi that education wa moe impotant than IQ in detemining eaning. Keep in mind, too, that IQ coe ae tpicall caled to have a mean of 100 and a tandad deviation of 16. Thi i an aita caling, howeve; the could jut a eail divide all IQ coe, giving a mean of 50 and an.d. of 8. Such an aita ecaling would change the value of the metic coefficient fo IQ; intead of equaling 1, the coefficient would equal. Standadized Coefficient - Page 1

Fit, we will give the fomula and then explain thei ationale: Geneal Cae: Two IV cae: = x 1 ( 1 ) = 1 1 ( 1 1) = 1 1 A thi fomula how, it i ve ea to go fom the metic to the tandadized coefficient.

2 One popoed olution (much le popula than it ued to e) ha een to etimate egeion model uing tandadized vaiale which ae metic-fee. Thi i done computing Z coe fo each of the dependent and independent vaiale. That i, = ( - µˆ )/, X 1 = (X 1 - µ )/ 1 1, X = (X - µ )/, etc. Conveel, = µˆ +, X1 = 1 µ + 1 X1, X = µ + X Each tandadized vaiale ha a mean of 0 and a vaiance of 1. Hence, fo example, if = 0, = µˆ = If =, that mean the individual ha a coe that i tandad deviation aove the mean fo ; that i, = µˆ + = = Fo the fit cae in the peent data et, = 5 ==> = ( )/9.79 = Fo the lat cae, = 48.3 ==> = ( )/9.79 =.44. Uing the tandadized vaiale, we etimate the model = 1 X1 + X + e whee 1 and ae the tandadized egeion coefficient. Note that we do not include the tem a. Thi i ecaue a = µˆ - 1 ˆ µ X - 1 ˆ µ X = = 0. Intepetation. We intepet the coefficient aing that an inceae of 1 in X1 (i.e. 1 tandad deviation) eult, on aveage, in an inceae of 1 in. Fo example, a we will ee momentail, 1 =.884. Hence, inceaing X (the tandad deviation of X1) inceae X1 1, which inceae (on aveage).884, o, equivalentl, inceae = (ou can confim thi noting 1 = 1.933, and = 8.65). Similal, an inceae of in X eult in an aveage inceae in of. Hence, tandadized coefficient tell ou how inceae in the independent vaiale affect elative poition within the goup. ou can detemine whethe a 1 tandad deviation change in one independent vaiale poduce moe of a change in elative poition than a 1 tandad deviation change in anothe independent vaiale. Computation. We could actuall compute the tandadized vaiale, and then epeat tep a and fom the Multiple Regeion handout. Given that we have made it thi fa, howeve, it i poal eaie to note that = x Standadized Coefficient - Page

Conveel, = µˆ +, X1 = 1 µ + 1 X1, X = µ + X Each tandadized vaiale ha a mean of 0 and a vaiance of 1. Hence, fo example, if = 0, = µˆ = 4.415.

3 Poof (Optional) Step - = a + 1 X 1 + X + e - Rationale Sutact fom oth ide = - 1 X 1 - X + 1 X 1 + X + e - Sutitute fo a = 1 (X 1 - X 1 ) + (X - X ) + e Reaange tem = 1 1 (X 1 - X 1 )/ 1 + (X - X )/ + e Multipl and divide.d. = 1 1 X 1 + X + e Sutitute tandadized X ==> ( - )/ = = 1 1 / X 1 + / X + e/ Divide oth ide = 1 X 1 + X + e Sutitute tandadized coefficient ==> = / Q.E.D. Hence, fo thi polem, 1 = 1 1 / = / 9.79 =.884 = / = / 9.79 =.36. Alo, it eail follow that, if H = the et of all the X (independent) vaiale, G = the et of all the X vaiale except X, then, = = x (1- R X G 1- RH ) (N - K - 1) Ego, 1 = 1 1 / = /9.79 =.096, = / = /9.79 =.096. O, equivalentl, 1 = %[(1 - R 1 5)/((1 - R 1 5) (N - K - 1))] = %[( )/(( ) 17)] =.096 Standadized Coefficient - Page 3

933 4.48 / 9.79 =.884 = / = 0.649 5.46 / 9.79 =.36.

4 (Note that, when thee ae onl independent vaiale, thei tandadized tandad eo will e the ame. Thi will geneall not e tue when thee ae moe than independent vaiale.) Altenative computation ( IV Cae onl!). Recall that, when thee ae two independent vaiale, 1 = ( ) / ( ) = ( ) / ( ) When vaiale ae in tandadized fom, the coelation matix i the ame a the covaiance matix. That i, the vaiance of the tandadized vaiale = 1, and the covaiance equal the coelation. Hence, when thee ae two independent vaiale, ou could alo compute 1 = ( 1-1 ) / (1-5 1 ) = ( ) / (1 - (-.107)5 =.874 /.989 =.884 = ( ) / ( ) = ( ) / (1 - (-.107)5 =.358 /.989 =.36 (Recall too that, in the ivaiate cae, = x / x 5. Hence, when thee i onl one independent vaiale, = x.) [Optional] Othe Anale with Standadized Vaiale. Futhe, if ou wee o inclined, ou could go though all the othe tep outlined in ou initial dicuion of multiple egeion. Among the thing ou would dicove ae SST = (n - 1), MST = 1, SSR = R5 (N - 1), MSR = R5 (N - 1)/K, and the value of the computed t and F ae unaffected the tandadization. In pactice, I don t thin thee would e much eaon fo wanting to do thi. Alo, if ou wee peented with the eult of an anali done with tandadized vaiale, and if ou new the.d. of the untandadized vaiale, it would e a fail taightfowad matte to compute the eult of the anali fo the untandadized vaiale. Jut eep in mind that SST = 5 SST and SSE = 5 SSE. Alo, SSR = R5 SST (egadle of whethe vaiale ae tandadized o not). Wh might ou want to do thi? Poil ecaue eult ae onl peented fo the tandadized vaiale, and ou want to figue out what the untandadized eult ae. (Thi i not an uncommon ituation.) Alo, computation ae much imple fo tandadized vaiale; depending on what ou ae inteeted in, it ma e eaie to wo thing out uing the tandadized vaiale and then convet ac to the metic coefficient at the end. Hence, eing ale to convet tandadized eult ac into metic eult can occaionall e ueful. Standadized Coefficient - Page 4

Recall that, when thee ae two independent vaiale, 1 = ( 5 1-1 ) / ( 1 5 5-1 5) = ( 1 5-1 1 ) / ( 5 1 5-1 5) When vaiale ae in tandadized fom, the coelation matix i the ame a the covaiance matix.

5 Going fom tandadized to metic. It i ve ea to convet tandadized coefficient ac into metic coefficient, povided ou now the tandad deviation. = x, = x Fo example, 1 = 1 / x1 = / 4.48 = 1.931, = / x = / 5.46 = 0.649, 1 = 1 / x1 = / 4.48 =.10, = / x = / 5.46 =.17 Computing R. Standadized coefficient povide an ea mean fo computing R. R = ; o, Ego, R = R = ( IV Cae) = =.844; o, R = = =.844 Caution aout tandadized coefficient: T The coefficient can often e le intuitivel meaningful T The ue of tandadized coefficient can mae it difficult to mae compaion aco goup - ecaue the tandadization i diffeent fo each goup. Fo excellent dicuion on tandadized vaiale and coefficient, ee Oti Dudle Duncan oo, Stuctual Equation Modeling. Alo ee Kim, J. & G. Feee Standadization in Caual Anali. Sociological Method and Reeach 10(): We will dicu thee iue much moe in Stat II. Standadized Coefficient - Page 5

R = ; o, Ego, R = R = + + 1 ( IV Cae) =.884.845+.36.68=.844; o, R = + + 1 =.884 +.36 +.884.36 -.107 =.

Semipartial (Part) and Partial Correlation

Semipartial (Part) and Partial Correlation Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated