DOCUMENT RESUME

ED 326 572 TM 015 914

AUTHOR Gibbons, Robert D.; And OthersTITLE Full-Information Item Bi-Factor Analysis. ONR

Technica.L. Report. Biometric Lab Report No. 90-2.INSTITUTION Illinois State Psychiatric Inst., Chicago.SPONS AGENCY National Science Foundation, Washington, D.C.; Office

of Naval Research, Arlington, VA. Cognitive andNeural Sciences Div.

PUB DATE 31 Jul 90CONTRACT NSF-BNS-85-11774; ONR-N00014-89-J-1104NOTE 22p.

PUB TYPE Reports - Research/Technical (143)

EDRS PRICE MF01/PC01 Plus Postage.DESCRIPTORS *Equations (Mathematics); *Estimation (Mathematics);

Factor Analysis; *Item Analysis; Item ResponseTheory; Mathematical Models; Maximum LikelihoodStatistics; *Test Items

IDENTIFIERS *Bi Factor Analysis; Binary Respcnse; *FullInformation Factor Analysis

ABSTRACTA plausible "s"-factor solution for many types of

psychological and educational tests is one in which there is onegeneral factor and "s - 1" group- or method-related factors. Thebi-factor solution results from the constra4nt that each item has anon-zero loading on the primary dimension nalpha(sub jl)" and at mostone of the "s - 1" group factors. This structure has been termed the"bi-factor" solution by K. J.,Holzinger and F. Swineford (1937), butit also appears in the work of L. R. Tucker and K. G. Joreskog. Allattempts at estimating the parameters of this model have beenrestricted to continuously measured variabler; it has not beenpreviously considered in the context of item response theory (IRT).It is conceiva..le that the bi-factor structure might arise inIRT-related problems. The purpose of this paper is to derive abi-factor item response model for binary response data, and todevelop a corresponding method of parameter estimation. Thisrestriction leads to a major siWification of the likelihoodequations that: (1) permits the statistical evaluation of problems ofunlimited dimensionality; (2) permits conditional dependence amongdiscrete and previously identified subsets of items; and (3) in somecases, provides more parsimonious factor solutions than anunrestricted full-information item factor analysis might provide.(Author/RLC)

* Reproductions supplied by EDRS are the best that can be made *

* from the original document. *

S. OSP0'. NWT OF SOUCATIONOles m and Improwraera

EDUCATIONAL RESOURCES INFORMATIONMIER (BOP

Ehus document hes boon reproduced asreceived Pore the person or orgenastiononginatinp tt.

o t Aar diengse Save been made to improvereproduction teadda

POS. ot Weer or opinions staled this docu-ment do not ascsseer*, repeesest Oki&OEM poution or poky.

Full-Information Item Bi-Factor Analysis

ONR Technical Report

Robert D. GibbonsUnix ersity of Illinois at Chicago

Donald R. HedekerUniversity of Illinois at Chicago

R. Darrell Bock

University of Chicago

July 1990

Dr. R.D. GibbonsBi omet ri c La boratory

Illinois State Psychiatric Institute,1601 W. Taylor St.. Chkago, IL 60612. USA.

Supported in part by the Cognitive Science Program, Office cf Naval Research.

under Grant #N00014-89-.1-1104, and in part by NSF grant BNS 85-11174.Reproduction in whole or in part is p...mitted for any purpose of the United

States Government. Approved for public release; distribution unlimited.

2

Coin, CLASSIICATU5N Of THIS PAGE

REPORT DOCUMENTATION PAGEFonnApprovaiOW No 040 MI

1

70

Is RffatrIAMIeLASSIfICATIONlb RESTRICTIVE MARKINGS

2a SECURITY CLASSIFICATION AUTHORITY 3 DISTRIBUTION/AVAILABILITY OF REPORT

Approved for public release;distribution unlimited2b DECLASSIFICATION / DOWNGRADING SCHEDULE

4 PERFORMING ORGANIZAT:ON REPORT NuMBER(S)

Biometric Lab Report #90-2

5 MONITORING ORGANIZATION REPORT NL.MEIE4(5)Cognitive Science Research ProgramOffice of Naval Research (Code 1142C5

6a NAME OF PERFORMING ORGAMZATION

University of Illinois

6b OFF.CE SYMBOL(If applicable)

7a NAME OF MON 702ING OP.GANIZAT ON

6c. ADDRESS (City. State, and ZIPCode)

Illilois State Psychiatric Institute 529W1601 West Taylor StreetChicago, Illinois 60612

7b ADDRESS (City State. and ZIP Code)

800 N. Quincy StreetArlington, Virginia 22217-5000

9 PROC1.gEMENT iN5TRuMEt4T IDENTIFICATION NUMBER

N 00014-894-1104(la NAME OF FUNDING/SPONSORING

ORGANIZATIONfsb OFFICE SYMBOL

(If applicable)

13c. ADDRESS (City, State. and ZIP Code) 10 SCJRCE OF FLINDING %UMBERS

PROGRAMELEMENT NO61153N

PROIECTNORR042046 R04204-01

VMS aTNO

I 4421553

11 TITLE (Include Security Classification)

Full-Information Item Bi-Factor Analysis

12 PERSONAL AuTHOR(S) Robert D. Gibbons, R. Darrell Bock andionald Hedeker

13a TYPE Of REPORT 30 TIME ClvEREDFRom 7/31/89 To 7/31/90

14 DATE of REPORT (Year, MontitOay)July 31, 1990

ftc PAGE COL,NT

16 SUPPLEMENTARY NOTAT'ON

17 COSA". COOE :8 5L,B,CCT TEaMS Continue on reverse if necessary and identify by block number)

0UP 5..,8 GQ0uP

05 Og

-19 Ai35":. 'ontinue on reverse if netesSary and identify by blocs( ,ursbpI

A plausible s-factor solution for many types of psychological and educa- .

tional tests is one in which there is one general factor and .7, 1 group ormethod related factors. The bi-factor solution results from the constraint thateach item has a non-zero loading on the primary dimension cf.11 and at mostone of the s 1 group factors. This structure has been termed the -bi-factor-sohition by Holzinger k Swineford. but it also appears in the work of Tuckerand Joreskog. All attempts at estimating the parameters of this model havebeen restricted to continuously measured variables: it has not been previousl,-(:onsidered in the context of item-response theory (IRT). It is conceivable. how-ever, that the bi-factor structure might arise in IRT related probleros.

20 DISTROMON /AVAn.ABILiTY OF ABSTRACT

IMUNCLASSIFIED/UNLIMITED 0 SAME AS RPT 0 VIC USERS

21 ABSTRAZT SECURITY CLASSIFICATION

f U

22a NAME OF RESPONtrLE Ii... tRTNA Davis

22b TELEPHONE (I6Kire Area Ode)202/696-4 4 "calriE1s4Y9WL...

DO Form 1473, JUN 86 Previous editions are obsolete

S/N 0102-LF-014-6603

3

SECURITY CASSIFICATION OF THIS PAGE

4,11 - ' -' 9-

SECURI4Y CLASSIFICATION OF THIS PAGE

LAM/

''.7'1'7,"4-0411% t'

; '

The purpose of this paper is to derive a bi-fattor item-response Todd- forbinary response data. and to develop a corresponding tuethOd' of IYaratteterestimation. This restriction leads to a major sir4PlifiCatiOn orthe kketihoodequations that (1) permits the statistical evaluation of problems of ufiliMiteddimensionality. (2) permits conditionai dependenCe among discrete and previ-ously identified subsets of items, and (3) in some cases provides more parsimo-nious factor solutions than an unrestricted full-information item factor analysismight provide (e.g.. Bock and Aitkin. 1981);

DO Form 1473, JUN U (Reverse)

",

4

CURITY CLASSIFICATION Of THIS PAGE,

ABSTRACT

A plausible s-factor solution for many types of psychological and educa-tional tests is one in which there is one general factor and s 1 group ormethod related factors. Th bi-factor solution results from the constraint thateach item has a non-zero loading on the primary dimension ail and at mostone of the s 1 group factors. This structure has been termed the "bi-factor"s'Aution by Holzingei & Swineford, but it also appears in the work of Tuckerand Joreskog. All attempts at estimating the parameters of this model havebeen restricted to continuously measured variables; it has not been previouslyconsidered in the context of item-response theory (IRT). It is conceivable, how-ever, that the bi-factor structure might arise in IRT related problems.

The purpose of this paper is to derive a bi-factor item-response model forbinary response data, and to develop a corresponding method of parameterestimation. This restriction leads to a major simplification of the likelihoodequations that (1) permits the statistical evaluatioa of problems of unlimiteddimensionality, (2) permits conditional dependence among discrete and previ-ously identified subsets of items, and (3) in some cases provides more parsimo-nious factor solutions than an unrestricted full-information item factor analysismight provide (e.g., Bock and Aitkin, 1981).

1 IntroductionConsider the case in which, for n variables, an s-factor soiution exists in whichthere is one general factor and s 1 group or method related factors. The bi-factor solution constrains each item to have a non-zero loading on the primarydimension all and on not more than one of the s 1 group factors (i e.,(IA, h = 2.... , s). For four items, the factor-pattern matrix might be

an 0

022 0

0 033 1

0 043

This structure has been termed the "bi-factor" solution by Holzinger StSwineford (1937), inter-battery factor analysis by Tucker (1958), and is alsoone of the confirmatory factor analysis models considered by Joreskog (1969).In these applications, the model is restricted to test scores, assumed to be con-tinuously distributed. It is easy. however to conceive of situations where thebi-factor pattern might arise at the item level. It is plausible for paragraphcomprehension tests, for example. in which case the primary dimension de-scribes the targeted aptitude and the additional factors describe knowledge ofthe content area within the paragraphs. In this context, items would be condi-tionally independent between pa,-agraphs. but conditionally dependent withinspecific paragraphs.

The purpose of this paper is to derive an item-response model for binary re-sponse data that exhibit the bi factor structure and to develop a correspondingmethod of parameter estimation. Of course. other types of tests that consistof items tapping different content areas would also be suitable for this type ofanalysis. As we will show, this restricti n leads to a major simplification of thelikelihood eqaations that (1) permits the statistical evaluation oil' problems ofunlimited dimensionality. (2) permits conditional dependence among discreteand previously identified subsets of items, and (3) in some cases provides moreparsimonious factor solutions than an unrestricted full-information item factoranalysis might provide (e.g., Bock and Aitkin, 1981). In the following sections,we derive the likelihood and its first derivatives so that an EM solution to itembi-factor analysis may be obtained.

2 Likelihood EvaluationStuart (1958) showed that if n variables follow a standardized multivariate. nor-mai distribution where the correlation pu = E;,=, athajh and cr,h is nonzero for

only one h, then the probability that the respective variables are simultaneouslyless than -y3 is,

P = [.1nis11 - ahy)/(1 - c(h)1/2)1 f(y)dyh=1 =1

00

j

where

1f(t) = exp(--t2)/(2701/2

2

-y,

F(t) = f(t)dt

and nh is the number of items loading on limension h (h = 1, . . . ,$).Equation (1) follows from the fact that if each variate is related to only a

single dimension, then the s dimensions are independent, and the joint prob-ability is simply the product of the s unidimensionr.l probabilities. In thepresent context, this result only applies to the s - 1 "nuisance" dimensions(i.e., h = 2,....$); if a primary dimension exists, it will not be independent ofthe other s - 1 dimensions. To compute this probability therefore requires atwo-dimensional generalization of Stuart's (1958) original result.

To derive the two,dimensiolal result, we begin by noting that the proba-bility of the primary dimeirsion can be obtained using the formula of Dunnettand Sobel (1955),

P = [II F(611- (1,10/(l )h/2)1 f(y)dy,-00

2=1

(2)

which is valid as long as p,, = ataj. Of course, this directly implies a unidi-mensional probiem. Combining the two results yields.

Pr1 lc° rfl F(1'1- a'hY)] f(y)dy} f(z)dz,

°° h=2 3=1,2 ,2

which can be approximated to any practical degree of accuracy using Gauss-Rermite quadrature (Stroud and Sechrest, 1966). What is important aboutthis result is, if the assumptions are reasonable (as they clearly are for manyIRT applications), then the probability of any response pattern can be obtainedby a two-dimensional integration, regardless of the dimensionality s.

For example, if y3 = ELI a3hOh + Ej and we assume that

(3)

3

7

7

yi N(0,1),N(0,10, and

ejN(°' I 7 h=1(1jhb

then the unconditional probability of observing score pattern x = xt is,

Pe =-°° h=2 -00 j=t

TIEF(01, oh)Fej

which can be approximated by.

[1 Rohe/4re)AO/Odell f(01)(101.,

(4)

PeQ sn [

Q nhE H[F(.Vqt. Xqh )]1-0 [1 F(XqXqh)11) A(K9)]} A(4),

41 h=2 9h 3=1

(5)

where

F(X9 Xqh) = F °Y, Jag, «,h-Vqh

aJ2h

and Xq and A(;) are the nodes and corresponding weights of a Gauss-Hermitequadrature.

3 Marginal Maximum Likelihood Estimation- parameters of the item bi-factor analysis model can be estimated by the

method of marginal maximum likelihood using a variation of the approachdescribed by Bock St Aitkin (1981). The parameters of this model include n"thresholds" or "intercepts", n primary factor loadings or -slopes" and a totalof n factor loadings or slopes on the h = 2.... , s additional dimensions (i.e.,r1.2nh = n). The likelihood equations are derived as follows. Let

4

and

Nh(x) = E re[Eth(X91)]Leh(X91, Xqh)/ Pe. (13)1=1

It should be noted that these equations are similar to Caose in the unrestrictedcase, except that in the bi-factor case, the conditional probability of responsepattern xeh (i.e., responses to ;terns j = 1,...,nh in subsection h for respo...sepattern e) is weighted by the factor, Eth(4). Furthermore, since each itemonly appears in one subsection (h), the N now vary with h, in contrast tothe unrestricted case. As such, the Nh denote the effective sample size forsubset h at quadrature point (X11, :Q. When weighted by A(X) and summedover the quadrature nodes for each subsection, Nh yields the total number ofrespondents, whereas the corresponding weighting and summation for fj yieldsthe total number of respondents answering item j correctly.

From provisional parameter values, each E-Step yields fj and Arh, the expec-tations of the complete data statistics computed conditional on the incompletedata (see Bock, Gibbons, Sz Muraki, 198). The subsequent M-step solvesequation (10) using conventional maximum likelihood multiple probit analy-sis, substituting the provisional expectations of and Nh (see Bock St Jones,1968).

4 IllustrationTo illustrate the application of the bi-factor IRT model, we have evaluated 20items selected from an ACT natural science test, for a random mple of 1000examinees (we are indebted to Terry Ackerman and Mark Reckase for thesedata). Ts test involves a series of questions regarding each of four paragraphs.For the purpose of this illustration. we selected the first 5 items from each offour paragraphs.

Table 1 displays the unrestricted prornax-rotated 4-factor solution, whichadequately fit these data (improvement in fit of a four-factor model over athree-factor model was x?,7 = 31.59,p < .02; the improvement in fit of fivefactors over four factors was not significant (xL = 18.44,p < .30). Inspectionof Table 1 reveals that each factor is dominated by items from a particularparagraph. In contrast, the estimated factor loadings for the bi-factor model(see Table 2) with s = 5 (i.e., one primary dimension and four paragraph-specific dimensions) revealed a strong general ability dimension, as well asappreciable within pmagraph associations. The fit of the restricted model wasnot significantly different from the fit of either the four-factor (v245 = 23.83,p <.99) or the five-factor ( = 43.22,p < .95) unrestricted models. Inspection

6

9

'777.

pe

= LIfiai[Ficior"[1 Fi(e))1-x"f(911)dOhl f(00d01h.2

= Leh(0)f(Oh)dOh} f(01)dOlh=2 "is

Then the log likelihood is,

(6)

log L = E re log A (7)r=i

where S denotes the number of unique response patterns. The derivative ofthe log marginal likelihood with respect to a general item parameter vi is asfollows.Let

[nsh-2 fe Leh(0)f(Oh)dOh]Ethkul) rh nx fin Ijn

Joh Ldehkv 1.1 kvh luvh

Then

log L 5 re (OPt)_av,

EtPt Ov,

(8)

(9)

Eth(90{ xt, FJ(C)

)(8)t1 F.,(9))OF;(0)

fehLA(0) f(Oh)dehl f(et)de,.

Z-z1 Pt el

(10)

Following Bock and Aitkin (1981), the margina. likelihood equations canbe solved, usini the EM algotithm of Dempster, Laird & Rubin (1977), byreplacirg the integrals with Gauss-Hermite quadratures and rearranging termsinto the two-dimensional form:

Q Q fi(X) ICI,(X)F;(X) rvi(C))F, A(Xqh)A(4), (11)EE pc)[1 FAX)] ai91 qh

where

F.,(x)=Ere.re,[Eth(4)]Lth(Xqi.-Vgh)/131t.1

5

(12)

of the loadings within each paragraph reveals that the intra-paragraph itemassociations are quite variable.

As a computational note, we should point out that the numerical precisionof the bi-facter solution represents a major improvement over the unrestrictedsolution. Given that the bi-factor solution only requires approximation of atwo-dimensional integral, we were able to use 100 quadrature points (i.e., 10 ineach dimension) instead of the 243 quadrature points used in the unrestrictedfive factor solution, (i.e., 3 in each dimension). Five factors probably representsthe highest dimensional solution elat is compntationl tractable at this time.Parameters of the unrestricted models were estima,ed using the TESTFACTprogram (Wilson, Wood St Gibbons, 1984).

5 A Simple Structure ModelConsider an orthogonal simple structure factor model in which each item loadson one and only one of ti dimensions. This satisfies a complete simple struc-ture model as defined by Thurstone (1947), which for measurement data couldbe evaluated using methods for confirmatory factor analysis (Joreskog, 1969).This is, of course, a simplification of the bi-factor model in which there is noprimary dimension. In this case, the unconditional probability in (5) is reducedto the unidime..sional form,

, rQ nh

Pe _1=niE rEF(.vq4)).-0[1_fivqop-ro Aucqh,Lqh .)=1

where

F(Xqh) = F aJaqhv17-7ojh

(14)

that is, (5) reduces to the product of the s independent unidimensional prob-abilities. The likelihood equations in (11) can then be approximated by,

alogL ,if,(Xqh) gh(.3cqh)F,(xc) OFJ(Xqh)) Auc),Via F (.1c)(1 FJ(Xqh)) av,

where

FAXqh) = E rexejLeA(Xqh)lehl=1

7

(15)

(16)

and

sRh(Xq) . F. reLm(X, ), eh.

1=1

In this case, eh represents the constant

Q

eh = E Lth(Xu)A(Xqh),gh

and

Pe = H ehh=1

(17)

It is interesting to note that f, and :Crh now only contain information Lorathe specific subset of items (h) for which item j is a member. This is, ofcourse, due to the independence between the subsets that results from thesimple structure.

Application of the simple structure model to the ACT natural science testexample yields the it-rn-parameters displayed in Table 3. Inspection of theparameter estimates in Table 3 reveals that removal of the primary factor in-creases the mIgnitude of the loadings on the individual paragraph dimensions.In terms cf model fit both the bi-factor model ()do = 336,p < .0001) andthe unrestricted four-factor model (L = 361.p < .0001) provide significantimprovements in fit over the simple structure model, indicating that the test isin fact measuring a primary ability dimension and not merely four independentrealms of knowledge.

6 DiscussionThe bi-factor model presented here provides a natural alternative to the tradi-tional conditionally-independent unidirnensional IRT model. When potentialsources of conditional dependence are known in advance, as in the case ofparagraph comprehension tests or tests in which two or more methods of itempresentation are involved, the item bi-factor solution provides an excellent al-ternative. An attractive by-product of this model is that it requires only theevaluation of a two-dimensional integral. regardless of the number of potentialsubtests, paragraphs. or content areas. Thest, different concent areas are, ofcourse, assumed to be independent conditional on the primary ability dimen-sion that the test was designed to measure. As such, tit- limitations on thedimensionality of the full-information item factor analysis model embodied in

s

1ES

<

the TESTFACT proam (Wilson, Wood Sz Gibbons, 1984), do not apply. Ofcourse, the subsections (e.g., paragraphs) must be known in advance.

In certain situations, for example psychiatric measurement (Gibbons, 1985),the existence of a primary dimension (e.g., depression), is itself at question. Inthis case, comparison of the bi-factor and simple factor solutions presented hereis of particular interest. Item bi-factor analysis could therefore help answer thequestion of whether depression is a unitary disorder or a mixture of a series ofqualitatively distinct abnormalities; a question that has long plagued psychi-atric researchers. Comparison of the fit of the bi-factor and simple structuremodels provides a tool for investigating such problems in psychiatric researchand other areas as well.

Finally, those cases in which little is known about the structure of a partic-ular test, but little confidence can be placed in the assumption of conditionalindependence, the more general solution presented by Gibbons et. a/. (1989),using Clark's (1961) formulae for the moments of n jointly normal variables,could be used. This procedure uses a direct approximation to the multivariatenormal distribution that underlies the item-response function, without restric-tions on the form of the inter-item residual covariances. With it, the assump-tion of conditional independence is not required. Further work in this area isunderway.

9

1 3

,

References

Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimationof item parameters: application of an EM algorithm. Psychometrika, 46,443-459.

Bock. ft. D., Gibbons, R. D., Sz Muraki, E. (1988). Full-information itemfactor analysis. em Applied Psychological Measurement, 12, 261-280.

Bock, ft. D., & Jones, L. V. (1968). The measurement and prediction of judg-ment and choice. San Fransisco: Holcten-Day.

Clark. C. E. (1931). The greatest of a finite set of random variables. Opera-tions Research, 9, 145-162.

Dempster, A. P.. Laird, N. M.. & Rubin. D. B. (1977). Maximum likelihoodfrom incomplete data .ia the EM algorithm (with discussion). Journalof the Royal Statistical Society, Series B, 39, 1-38.

Dunnett, C. W. Sz Sobel. M. (1955). Approximations to the probability inte-gral and ce, tain percentage points of a multivariate analogue of Student'st-distrib-tion. Biometrika, 42, 258 260.

Gibbons, R. D.. Bock, R. D. & Hedeker, D. R. (1989). Conditional Depen-dence. ONR Final Report, No. 89-1.

Gibbons R.D., Clark D.C.. Cavanaugh S.V. (1985). Application of modernpsychometric theory in psychiatric research. Journal of of PsychiatricResearch. 19. 43-55.

Holzinger. K. J.. & Swineford, F. (1937). The bi-factor method. Psychome-trIka. 2. 41-54.

Joreskog. K. G. (1969). A general approach to confirmatory maximum likeli-hood factor analysis. Psychometrika. 34. 183-202.

Stroud. A. H.. Sz Sechrest, D. (1966). Gaussian Quadrature Formulas. Engle-wood Cliffs, NJ: Prentice Hall.

Stuart, A. (1958). Equally correlated variates and the multinormal integral.J. Roy. Statist. Soc. Ser. B. 20, :37:3-378.

Thurstone, L. L. Multiple Factor Analysis. Chicago: University of ChicagoPress. 1947.

10

1 4

Tucker, L. R. (1958). An inter-battery method of factor analysis. Psychome-trika, 23, 111-136.

Wilson, D., Wocd, R., St Gibbons, R. D. (1984). TESTFACT: Test scor-ing, item statistics, and item factor analysis. Mooresville, IN.: SCientificSoftware, Inc.

Li

I_ 5

.

Table 1

Full-Information Item Factor Analysis - Unrestricted Pre.,max SolutionACT Natural Sci mce Test 20 items and 1000 subjects

Item ..y; ail a22 ais t,./341 -.215 .401 -.005 -.036 .2162 -.385 .185 -.019 -.007 .1053 -.356 .667 -.070 -.081 -.0814 -.098 .619 .013 .044 -.0225 -.029 .562 -.092 -.059 .1196 -.582 .129 .068 .256 .0307 -.585 .184 -.211 .419 .1028 -.137 -.037 -.061 .025 .1729 -.246 .238 .063 .362 -.254

10 -.089 -.*)*)4 .128 .620 .06011 -.049 .382 .135 -.034 .31112 -.407 -.024 -.065 .124 .32013 -.265 .247 .082 .020 .17314 -.051 .137 .005 .007 .58513 .040 .224 .129 -.045 .29516 .345 .1.53 .289 -.122 -.10917 .167 -.007 .682 .089 -.04418 .119 -.096 .520 -.024 .12019 .543 .008 .500 .067 .09120 .672 -.07:3 -.010 .034 .163

12

16

,

Table 2

FLal-Information Item Bi-Factor AnalysisACT Natural Science Test - 20 items and 1000 subjects

Item 7.; ail aj2 j3 ai4 ais1 -.230 .524 .1292 -.392 .232 .1153 -.370 .411 .4274 -.118 .548 .2785 -.046 .489 .338

6 -.593 .311 .277

7 -.600 .376 .3148 138 .087 -.0199 -.259 .207 .390

10 -.103 .926 .47611 -.062 .484 .14112 -.41:3 .261 .13513 -.277 .493 .19914 -.066 .573 .18715 .625 .492 .26016 .340 .112 .26117 .150 .306 .66218 .160 .240 .57119 .528 .340 .49320 .671 .061 .031

13

7

Table 3

Full-Information Simple Structure Item Factor AnalysisAC I Natural Science Test - 20 items and 1000 sUbjects

Item Ili aj 1 OP aj3 aj41 -.224 .4822 -.391 .251:3 -.368 .5714 -.111 .6125 -.040 .5856 -.592 .4087 -.597 .4678 -.1:38 .0329 -.258 .429

10 -.102 .50911 -.056 .48919 -.412 .2971:3 -.27:3 .44914 -.058 .59115 .0:31 .56616 .341 .28217 .157 .7:3218 .16:3 .61619 .5:34 .59790 .671 .057

14

18

Destnbutson Last

Dr. Terry Adamen Dr. John B. Carroll Dr. Lou MaioEducatamal Psychology 400 Ekon Rd. Nat* CERL2M alumnae Oldg, Chapel Hig. NC 27314 Urmansty of MinosUrewmury of amass 103 Smith Mathews MenusChempsems, U. 41401 Dr. John M. Canna Wiens. U. 61101

IBM Wsteon Research CenterDr. James Mona User !Mesas Institute Dr. Dettprased DI,1403 Norman Hall P O. Bca 704 Center tee Naval AnalyseUniversity of Florida Yoettom Hostas. NY 10598 4401 Ford AmmoGammende. FL 32605 P O. Box 16214

Dr. Robert M. Carroll Aleandria, VA 223024264Dr. &Mg B. Andersen Chief of Naval OperationsDeferment of Stausuce 0P41112 Mr. Hal-IG DontStudiestraede 6 Washineton. DC 20350 Ball Communicaume Research1455 Cosuresein Room PYA1K207DENMARK Dr. R.aymond E. areal P.O. Bor 1320

UES LAMP Some Advisor Pscammy. NJ 01855-1320Dr. Ronald Armauong AFHRUMOELRutgers Unnerecy &roots AFB. TX 7E35 Dr. Frei DrawsGraduate School of Management Umensty of MoosNemo. NJ 07102 Mr. Hus Him Chung Department of PsyMolom

UnNersay of Minces 033 E. Denali St.Dr Eva L. Beta Department of Sumac. Chmspear. IL 61820UCLA Center for the Study 101 Mini Hall

of Evaluation 725 bouta Wreht St. Dr. Stephen Dunbar145 Moore Hall Champaign. IL 618M 22413 Lon** CenterUnwary of Califon'''. ler Meleurmentl.oe Metier. CA 90024 Dr Norman Oa Universal, of Iowa

Department of Psycholoily Iowa City, IA 52242Dr. Laura L Barns. Linn. of So CaliforniaCollege of Educerion Loa Angeles. CA 400841061 Dr. Jams A. UMUnnersty of Toledo

Air Fame Human Resources lab2801 W. Bancroft Street Director, Manpower Propane Brooks AFB. TX 7$235Toledo OH 43106 Center (or !Nava Analyses4401 Ford Avenue Dr. Susan EmanationDr William M. Bart P.0 Boa 1626$ Umendy of KammUnnenay of Minnesota Alemndna. VA 22302.0268

PII0b°1011, DePMmutDept. of Edw.. Pep:ming426 Frame

330 Burton Hall Director. Lavreus, KS MOO17$ Pillsbury Dr.. SI. Manpower Support andManespola. MN 55155 Read.ness Program Dr. George Engelhard Jr.

Center for Naval Maly= Demean ci Educetional StudioDr. Isaac Beyer 2000 North Beaureprd Street Emory UmenayMall Stop: 10.11 Alexandria. VA :1311 210 Fableurne Bldg.Educational Teem' g ServiceAtlanta. GA 30322Rosedale Road Dr Stanley Collver

Pnnceton. NJ 00511 Office o( Naval Technotom Dr. Balliaelft A. FastingCode 222 Opsrulonal Tectinotopm Corp.Dr Menucha &tribe= SOO N Quincy Street 502$ Cellegkart Suse r.sSchooi of Education Arington, VA 22217 5000 San Antonio. TX 78:128Tel Aviv Unntersiry

Ramat Avw One Dr. Hans F Cromoas Dr. P.A. FedencoISRAEL Faculty of Law Code 51University of Lmourg N PRDCDr Arthur S. Btarse: P 0 Bo* 616 San Dago. CA 92152-6800Code N712 Maastricht

Naval Training Systems Center The NETHERLANDS 6200 MD Dr. Leonard FM*Odando. FL 32813-7103

Lindquist CenterMs. Carolyn R. Crone (or Mals11116141.Dr Bruce &oases Johns Hopkins Umversiry Universiry of 10alaDefense Manpower Data Center Department of Psycholofy Iowa Coy. IA 52242

99 Pacific St. Charles A 34tb StreetSuite 15 SA Baltimore MD 212111 Dr Richard L Ferguson

Monterey. CA 939434231American College Testing

Dr. Timothy Davey P.O. Box MSCdt. Arnold Bohner American College Testing Prouam lows Ciry. IA 5:243Smut Pmcholopsch Ondervost P 0 Box 16$Retrucertemaen SMatementrum lova Ors. IA 52243 Dr. Gerhard Farb,:Kwaruer 1Coningen Avrid

Lielesprie 5/3Brunnsenst Dr C. M. Dayton A 1010 Vienna1133 &um" BELGIUM Department of Measurement AUSTRIAStatistics A Evaluation

Dr. Robert Breaux College of Educaucn Dr. Myron NWCode 2131 University of Maryland U.S. Army HeadquartersNaval Training Symms Center College Part, MD 11742 DAPEAIRROttando. FL 324264224

The PanusonDr Ralph J. DeAyals Washangton. DC 20310.0300Dr. Rohm Brennan Measurement. Statutes.

Amnon Carp Taming and Evsluation Prot Donald Fitageral.Programs Benjamin Bldg,. Ra. 4812 University a( New EnglandP. O. Bon 161 Universe,/ of Maryland Deferment of PmcbolomIowa City. IA $7243 College Part MD 28742 Anaidale. New South Wales 2351AUSTRALIADr Gregoty Candi*

CTIVI*Gratv-HillMr. Paul FOlay

2303 Garden RoadNavy Personnel RAD CenterMonterey, CA ONOSan Dow. CA 12152.6400

9

sil 311501- Ill 4 A 8 NI' II : I 114 II 1 11 1 ri I Isig 194pi 11 fail -a-68 t141); 7 2; 4.54 Vfi ,i; 8 d116 st 1_

211 !II gi; !Ili 710 ql !lift 110 III. a a ._ a z _la aal a2M7

g

§Si II. 4

Lib ilj! III! III tz ! e 231I'dCd vz

I zfljt II/axz 8 a al az iP0 a A& ad8gat k

ip

&Akb 310l

8I

el big

pl JONe;

144

2) Ai II4511 IPaz 5

r.;

i L. ir IP 1111- '1 1 1d41 di A! i 1-2 lip j3§I! 1116 !vvill if& 31 g .4 if 4

I !li !i1;1 nil Ili 114111 !Ilii ji!ii lib Mill !hi Ilg ill;

Cs?

1its

I-

-I

Oil;

Jill! ill A

)1!

!II

fjj '1g:

IPP

Wig

if,i

Tgli A

RoA3

113;44II Jo lisi

ii

Milli !hi

firi !lit!in D

m 61

we!

ta36

6D Ix 6 3t

t-

lig

gu -

II a§

IJ

0=J1

!II

11111

1I

ciov

4,m

."fn

!lig

Igi

g/2

81 d§

1(46 1 1."

$8

113§

Ilv

ilig

!III

141 41111

311.

Ibii !

!Tit

.g

x7.3

DaP

6. E0

A.te.

0,A6z3 li<22 0

42

4

I1,,

11

41

to.,1

3 1-Id=

ii,

ft

114 I sKi 111 In"

11Ri* A

1"

IT(

fie eg ei

3

-Iliisi ji

zovNJfee

vii55d

.16

I"

I

2

ri

gild/ ID

1*14t 1

4

WI

I

II

8V..5:1 8.

82..i 4115IA

88

.3

-ighl -aid :Li

,§bs

'24 ATI

*

tie17

.7*,

rre

07,

,

ghr,

Offr

hirt

.i fh

biri

f*E

bir

(59

rtt *

r'99

qF

9ri

f9.1

7.t

I O

il;;I

t ri 4

E1

dH

ign

iir

rvr

in9t

;a

1 g

gift

s11

ir11

if11

101

g11

1

.7; 4

4' .;

;;;;

-,P

.