transformation of factors by artificial personal probability functions

37 Pages·2006·1.37 MB·English

TRANSFORMATION OF FACTORS BY ARTIFICIAL
PERSONAL PROBAGILITY FUNCTIONS

se THM

seoyard a Tuseer

| E |
E
Pr
iT

Eoncat ove
Teansfarration of Factors ty Artificial

sonal Frubabitity Suastions

word Tucker

Universily a= [Tunois at
end
Edueaticna] Tesing Service
Carl f. Finkbetner

Tne Procter and Panols Somsany

Eauzalines? Testing Sew
Primevsa, New censeg
1yel
Copyrtaht @) soul, Wlacwetonal Teacing Yersice. ATT

gate zacervad,
Aesuract

Use of Judgnentel end heuristic principles has Ted to devetopnent
of an autonaticed canputer method for transforuetian (ratetion} of factors,

Jn this nethod the notion of an attribute vector being "in an hyperpTone’

is fomsaliged “nan art*fieial personal probability funetfon and the hyver-
slune Togetton “6 defined by a wefghted least scuares orinciple using the
aersona! orobaliiLivs as weights. The method has yieided supertor results
in nueraus Urta1s with real and Honte Carta data for positive wanifold
situations, Thy nethed has not been studied as thoroughly in the sore generat

situation tnvolving boch pos

ve and negative transforsed factor Ioudings but

appears 29 yfald good resul*
ransformeUian af Factors by SrLifiviat Parsonal Protabisty Functions
In the perind of tine since ThuesLone 11939] first proposed the
principle of siwole struccare to resolve the indecerwinancy in faczor zruns~
formations 4 veriely af procedures have heen devised and sciltzed to attenp!
to sacisty cais principle, Thurstune used geaphical rotettans of axes in
suber of his stuttes {S02 1233, 134] for examples} and descr tbed tre proced-

ares ta uTkipte Fi

rar Aoatysis (19/1. Dh}

fons to the graahical raced.

ures have focussed mainly an the extent vf handvark and on the axtent of 3

donce on sunjective judywents frvolved. Tucxer

f1990%, JBreskea (Nete 1) und
Maradith (19/0) indesendently sraposed a wethod Lil izing tne Teast, orinctoal
avis ef solected lest vectors wblek were to have war-zero transTarmed Yeodings

on each factor $0 obviate

subjective Suarients af exact locations of syper
planes. ilorst (1942) aranosed a nonegrashical method for selecting non-zera

fed luadlings faTlened ly use of a yeriant cf the preced-

Sng mathec, Tuckee (1849) reportec a scmi-urviytical wetted oF

tor rotaLior

which wes clisely rateted to hth his aarlier suscestior ané ta Horst's propo
sé]. Conpletely automatic eaceduves for transformations in uncorrelated
Factors ware orovosed 3y Carrol] (1953), Newhans and liriuley (1354), Keiser

[°Y59), Ssunders (dota 4), and Sehaersna (°ys6}. Autowt ix rethods fry Leans-

Fon aniens ce correlated farcurs fallewed by Pinsca & Saunders ihote 31, farval!
LI9B7hs Haves and Rafsor (1964), Hendrickson & Unite (1964), Coer (2866}, Jenn
rich and Sarpson (3966), Cranfnrd and Fergusan (3970), Jenurich (1973), <atz

4k fuhIe (1978), and taisor & ican {R

2), Asecent propacal by Kaiser

and Geeny (19/8) rate

Tuckae's (1889) arvaagal asing Lhe Teast principal

axis fur tests

ted automatically by means af orthanex solution, Most of

these upproaches vbiactify rotation by proposing weLazrstise) ful

pear areTegeus se sie Ccaccres pf Thuvscere’s Af ple struc.see orincisle,
The use of wany oF these methods was iTTustrated by application te

bodies of data and,

‘oquently, couperison was made to results obtained by
prephrica’t procedures, Wokstien (19/1) reported # major comparative evelua

t

oF sevara’ prottneat wethods

oblique (correlated) factar transforma

roms fa wh

he dsed tne graahfcal rotation results as one crfterton ta

qualliy ef casalts obtained. Thus, the role of cunjestive Judunent

transforuations has aot»

n elinizated fn the automatte methadss
VatNe', EMS role fas hecome one oF criterion upon usicn to Sudge the qual Fey
of vesulls ootsined ay various pencedures.

Rather than offer anetaer attenme te touasTste the simple structure arto
cipal into a mathematica! function, this report propuses that detai ied judgnents
rawolyed in tae graphical praceduve be autonet znd

Povconal and ArLificia?, Personal Probab ey

Persone] ProkaliT ey,

Th

2 [rst qudgewata? tapte concurs “parsunad arubabiTsty" and “artificial
personal prakabitity" of the vector fas on atteaute heing in an Ayperplane.
The concept uF 9 vector heing in an hyperplane is oF Tany standing Free geaps-
{cal rotation af axes. Far instance, fucker (1990, 1955) combined tais ulta
least squares solutions ta define the Incativa of hyyerpTanes. ‘this concest

is unint

wed here for “acter matrices af such thenretical Toedinys as éntro-

skwced by Tucker, Kovpuaa and Lon (1969) fur major doreias. However, analyses

real uorld data do aot dee} directly with theoretic Toadings ond 0 the

concep? cannot he maiateined strictly in practice. Divergence of vaservei

dca factor loastings from theoretic factor loadings vecurs as a result oF
several inflvences: there 4s not oniy the influence of analyses being based
on observations of a sample of entities (Ie, sampling esrpr) but also. Lhe

aiferences

wan theoretic constructs and actual phenowena
Persone? probability concerns the inverse probability cf a thecretic
attribute vector eirg ir ar hyperplane giver the attritute vector, Ir gragk-

ical votatior of factor axes, duccwert of such a perseral probability 1s mide

hiary tines in teres of the concert of kyperslane width defined as an ‘ntervil

of observec factor Tocdings round the zero value. Attributes ere fudged
Likely er undikely to be in the hyperplane (i.e. have Tosdings in the interval

‘around zero) for better dota than tne deta being analyzed. Mare w

given in determining the ec

n of the hyperplane lo vectors more 1izeiy ta

Le in tha hyperglane than ‘s given to vectors lees '*kely to a in the

hyserslave

Tre corcest of probability of an attribute vector being tn an typerplane

1

ay 2¢ given same reaming in Uke folowing fashion. Let: be # varisb‘e

representing theuretiy facLar Toauings ia some theorevicsl dona‘n. Attr“bute
vactars heving values uf S within an interval about zero for sone factor are
considered as being in the Syperplane for that factor, “hese attribute vectors

WiTl be designated nere az aeing in class 0 for the factor vinila 21% other attri-

bute vacters wi11 be designated as being “ class 1. Let the Tistizs o:

class 0 interva: be ¢ 4, Tn 2 common usane the hyperplane widah 13 26,

Let b ae a variatTe vepraserting ob:

ved factor Toacings, The prebi~
bility that an attribuie vecuor ts in class 9 given an observed factor Inoding

aay be synbatized by pice Ob). Yersorat sropab:

ly rafers to 2 judgment of

this probebilicy (a Jaxlyrenl wnich 15 fdementoT ta gesphical sotative!
justyation of

Mlestrative materia! conesening ple=nity} sway be developed in the

Following Yasiton, Let Tb 1} be a endittenal proban icy censity fmet-on

(b-d..1 of b given 6. The Afvuriate 2.d.f. of band gis:

Hibs) = FiRl-rth a) oy

where 118 Ihe p.d.f, of 2a sare etace of s

tudies. For 9 giver

value of by cer

snp somaya @i
Fasids "ety 2)
3.
then:
Pie=d bt = (a)

Aeco {aot pic~d b) 4s a Function Gf the otseruad facter Inading » and
opens on the density function of thearetic loadings 11+), the value of

by Sil the condizional density function f(b
Figures | and 2 present i: ‘ustrattoas of the functional relation of

tok Ta devalapment of these 4!lustretéons, the conditional

plod 3
furction ((b 4) 18 taker ta ba aoewl wth wean = and stanerd deviation
Se. Class 4 intervel Twit wes taken Lo beey = 01. Tor the teo-sided ease
‘in Figure 7 the dersity function f(s] wos taken te be uniform *n the inierveis

1.2 t0-.1 ond 1 to 1.0 wit cousT ordinate ip these tuo intervals, ihese

being the mtervals far class i. Furtner, # (2) was token te be uniform with

separate ordinate ta the class £ irterval trom - «1

Ouiside the

Fang -1.0 vo 1.0 f(@} was takeu lo be aera. Ir the top functions of
Figure 1 the ardiwtes for classes G and 1 ware squat so thet the probe:

bility oF © raidou s being 1a class O mas 1, Unis prebabiTtty being cesiy

In the bottan functions the ordinstes fer class &
va] and class | Intervals ware 2.5 and 277778, ease:

ve se

tret picsd) equals 6, Tne functions an che Teft are far

085 and those an che right are for sj - 120. Mote tml the functions

b
on Lhe ieft for a, = 025 have steeper sTapes than do the furetians an the

vrighl for, - 100, As sjy bocanes intirites inal, the function world amproach

a stem function

fa. Anather point tp note 1s that the buttow functions for

piced) = .5 are ore spread out then are the top functions for pis
This spread af the functicns depends not only on the width of the class &
interval , but alsa an the retalive probability of p's being ir class 0

ert Piguré | about here

Figure © presents resutls obtzined when

3 neyative 6's were excluded.
hw class interval was rw- -0 ta «1 and tne class 1 interval was trom 7
ta Tul Teatures of the oleed|b! Timotions are sivilar for positve b's to the

cwo-s! ed

se i Pgure 1. However, fos negative Ls the functions approach

uit,

Tasert -iqure 2 abut here

ArLIieis1 Parsons] Probebiiit
The precise value of s{c-O|b} fer any given ate"bute vector with

Tossing han 2 factor depends on the density “unction fit}, the hyzerslare

width parameter gy and the conditional dens‘ty Funct on <ia;3!. These

density Funetions are xnayn only vaguclys however, a judgment way be aade ty

gn aneTysé as (a the value of piced]b), This may be cons‘uered to be te

anolyst"s personal probability of an attribute vector beiny in a factor
hyperplane. 4 funeticnat representation of such personal probabilities as
Hependent or. the chserved factor “oadings is designated hare as an arvificial

frersonal pr

sab Iéty funct‘on, “his Function cay by quize arbisrary but should

have the lynes of forms grasented in Figures 1 and 2. A convenient form for

computer apoiicat‘ons is given in eyvacion 5,
age = 180 + abi), s}
here APP 4s the value of the artifivial parsenai prosattitty, @ ard ¢ ava

paranste

of Une Function, For 2 one sided function the value af APE is
detined as nity fer ai? negative values of b. Figure 3 presents 1lustrations
bf Sais ABP function tole that the {Tustraled fuactiens are very strlen te

the plerG]b} funerion

et the lower left pt Figures T and 2

Yalune oF the perameters a ans coef the APP functioe oF eavation (E}

way be rolated te the values af e tor APP = 5 and

Let b be the posh

tive value of & for APP = .5 and > . be ihe positive value of b for APF >

The ratia af b | to b | is tudicetive of the slaps of the APP funetian. At

a vatia of uaviy the function wuld ae @

ep function; se the ratio ine-eases

‘the siupe of the Function would decrease. The following equations give the

relations of the paroueters to these

bes oF Bs

aCoi/Ia(y 4/9 4} = TWEOVFLIa(D y} = Talb gs (e

fhotee oF

© soweliet aveitrary ond, within a liwited range does aot have

substantéal e*feet an results, fn experinental trials volses as Tow as c= 9

and as high as = 10 have deen found to yield good results, Gur exver Sens

38 oe

© = 10 gives soncuhas sharper resuTts. This corresponds Lp ¢ ratio
of h fb, equal en appronimaceiy 7.25. I Figure 3. bg was taken at.
se that a = “0,002,000, 000.

Hetahted Least Squares Factor Treas‘

lations

Retore continuing with d?scuss‘on of the use ef personal probabiliries

and

evferal persona! probabilities in tector cransformations, procedures

for eigited Teast squatas factmr teansfarmations ove discussed, “he persona?

ba ELIAS and artificial personal probsbriities will be used, subsequent ly.

weights. luo procedures wil? be discussed: First, a wefaited least squares

oF project

cn namals to hyperplanes; second, @ weightes ‘least squares of
Factor jaadings (in the class of precceures dennrich anc Senpson (1966) termed

divact factor retatian}. Keiohts are vestyt:

ted te beina aoneneuative