Factor Analysis and Latent Structure-IRT and Rasch Models

Jo reskog K G1967Some contributions to maximum likelihood factor analysis.Psychometrika32:443–82Jo reskog K G,So rbom D1996LISREL8.Scientific Software International,ChicagoLoehlin J C1998Latent Variable Models:An Introduction to Factor,Path,and Structural Analysis,3rd wrence Erlbaum Associates,Mahwah,NJMarsh H W,Balla J R,McDonald R P1988Goodness-of-fit indexes in confirmatory factor analysis:The effect of sample size.Psychological Bulletin103:391–410Miller M B1995Coefficient alpha:A basic introduction from the perspectives of classical test theory and structural equation modeling.Structural Equation Modeling:A Multidisciplinary Journal2:255–73Mueller R O1996Basic Principles of Structural Equation Modeling:An Introduction to LISREL and EQS.Springer-Verlag,New YorkMulaik S A1972The Foundations of Factor Analysis.McGraw-Hill,New YorkMulaik S A,James L R,Van Alstine J,Bennett N,Lind S, Stilwell C D1989Evaluation of goodness-of-fit indexes for structural equation models.Psychological Bulletin105:430–45 Mulvey P W,Miceli M P,Near J P1992The Pay Satisfaction Questionnaire:A confirmatory factor analysis.Journal of Social Psychology132:139–42O’Grady K E1989Factor structure of the WISC-R.Multi ariate Beha ioral Research24:177–93Raykov T1997Estimation of composite reliability for con-generic measures.Applied Psychological Measurement21: 173–84Steiger J H,Lind J M1980Statistically Based Tests for The Number of Common Factors.Paper presented at the annual meeting of the Psychometric Society.Iowa City,IA Stevens J J1995Confirmatory factor analysis of the Iowa Test of Basic Skills.Structural Equation Modeling:A Multidisciplinary Journal2:214–31Tanaka J S1993Multifaceted conceptions offit in structural equation models.In:Bollen K A,Long J S(eds.)Testing Structural Equation Models.Sage,Newbury Park,CA,pp. 10–39Windle M,Dumenci L1999The factorial structure and construct validity of the Psychopathy Checklist-Revised(PCL-R) among alcoholic inpatients.Structural Equation Modeling:A Multidisciplinary Journal6:372–93R.O.Mueller and G.R.Hancock Factor Analysis and Latent Structure:IRT and Rasch ModelsIn education and the social sciences,we often ask subjects to respond to a set of items(questions, statements,or tasks)on survey forms,self-report inventories,and mental tests,that are coded as discrete—often dichotomous—variables.In many settings it is natural to think,in analogy with factor analysis(see Factor Analysis and Latent Structure: O er iew;Factor Analysis and Latent Structure,Con-firmatory),that there is one or more continuous latent variables for each subject—such as political agency, extrovertedness,or ability in some area of mathe-matics—that can be measured or estimated using the positive and negative responses to these items.Item Response Theory(IRT),and specializations such as the Rasch Model,refer to a set of statistical modeling and estimation tools for making inferences about one or more continuous latent variables from this kind of multivariate discrete data.1.Introduction:the Rasch ModelAs an example,consider a test of transitive reason-ing in school children(e.g.,Sijtsma and Verweij 1999).An item may consist of three rods,A,B, and C,of differing lengths.The child is given evi-dence that Length(A) Length(B),and Length(B) Length(C),and is asked to deduce a relationship between Length(A)and Length(C);his or her answer is scored1if it is correct and0if it is incorrect.Let us say that J such items,involving different types of objects and attributes(balls of varying weight,disks of varying area,etc.)and different numbers of objects, make up a test given to N school children.If the scored responses were continuous we might build a two-way additive ANOVA model in which child i’s response to item j,Xij,is the sum(or difference)of the child’s achieved transitive reasoning level,and the item’s innate difficulty,as determined by the attribute on which objects are compared and the number of objects being compared.However,since the responses are dichotomous Xijl0or1,we need a model specifying the probability that Xijl1,in terms of the child’s transitive reasoning ability and the item’s difficulty.One of the earliest such models,developed by the Danish mathematician Georg Rasch(Rasch1960),specifiesPj(θi) P[Xijl1Qθi,βj]lexp(θikβj)1j exp(θikβj),i l1,…,N,j l1,…,J(1)whereθirepresents child i’s transitive reasoning level, andβjrepresents the difficulty of the item:as the latent variableθiincreases,the probability of getting the item right increases,and as the difficulty parameterβj increases,the probability of getting the item rightdecreases.The Rasch model,together with similar models using different forms for Pj(θi)developed by Jane Loevinger(Loevinger1948)and Frederic M Lord(Lord1952),became the core of what is now item response theory.On the logit scale,the Rasch model has exactly the two-way additive ANOVA structure we sought:logit Pj(θi)l lnPj(θ)1k Pj(θi)lθikβj.We further assume local independence among item responses,i.e.,all Xijare conditionally independent,5244Factor Analysis and Latent Structure,Confirmatorygiven θs and βs.Letting x ij denote the observed values of X ij ,we may write the likelihood for the Rasch model asP [X ij l x ij ,i l 1,…,N ,j l 1,…,J Q θ",…,θN ;β",…,βJ ]l Ni =" Jj ="P j (θi )x ij [1k P j (θi )]"−x ij .(2)An up-to date account of the Rasch model and its various extensions is given by Fischer and Molenaar (1995),to whom we also refer for historical notes and references.The Rasch model in Eqn.1has the interesting invariance property that the odds ratio comparing two items j and k isP j (θi )1k P j (θi ):1k P k (θi )P k (θi )l exp βk k βjindependent of θi .Hence in our example,the ‘dif-ficulty’of two items involving different transitive reasoning settings can be compared,in principle,using any convenient sample of subjects regardless of their transitive reasoning level (in practice,subjects still contribute at least ‘small sample bias’to the estimated βs).Similarly,the odds ratio comparing two subjects is independent of the item difficulty parameters.These two invariance properties are instances of ‘specific objectivity,’a property Rasch considered important in defining a measurement model and from which Eqn.1can be derived (see Fischer and Molenaar 1995).The Rasch model is also closely related to the Brad-ley–Terry model of paired comparisons.Consider two items j and k for which βj βk .For any subject i who got only one item right,it follows from Eqn.1that the probability that the easier item was correct is P [X ij l 1Q X ij j X ik l 1,θi ,βj ,βk ]lexp(βk k βj )1j exp(βk k βj )(3)independent of θi ;this is exactly the Bradley–Terry model with parameters φj lk βj .The lack of de-pendence on θi on the right can also be used to construct empirical tests of the Rasch model,since the probability in Eqn.3can be estimated without assuming the Rasch model and should be the same in any subpopulation of subjects in which the Rasch model applies.2.Estimation in the Rasch ModelTwo estimation methods are traditionally associated with the Rasch model,joint maximum likelihood (JML)and conditional maximum likelihood (CML).One can maximize Eqn.2jointly in the θs and βs,by numerically solving the ‘normal equations’obtained by setting all partial derivatives of the logarithm of Eqn.2equal to zero;the maximizing values θi and βj are called joint maximum likelihood estimators (JMLEs).However,the JMLEs turn out to be in -consistent ,or asymptotically biased:suppose we are primarily interested in estimating βj s;it need not be the case that each βj converges to βj as we increase the subject sample size N ,keeping the number of items J fixed.Indeed,as N increases,more θi s are also added to the model:intuitively,some information must be expended on estimating them,and there is not enough information left over to improve the estimates of the βi s.One solution to this problem is to exploit the fact that in the Rasch model,a sufficient statistic for each θi is subject i ’s total number correct score,X i +l ΣJ j ="X ij.If the likelihood in Eqn.2is divided by the joint likelihood for the observed x i +s,the resulting conditional likelihood contains only the βj parameters,and the conditional maximum likelihood estimators (CMLEs)βj obtained by maximizing this new like-lihood are consistent (asymptotically unbiased)as N grows and J stays fixed.For this reason,CML estimates are usually preferred over JML estimates.Additional details are provided by Andersen (1980),whose earlier work demonstrated the inconsistency problem with JML,and established consistency of CML for the Rasch model.Another approach to solving the inconsistency problem with JML is to assume the θi are independent random effects following a common (discrete or continuous)distribution f (θQ λ).Integrating over θi for each subject yields the marginal likelihoodNi ="&Jj ="P j (θi )x ij [1k P j (θi )]"−x ij f (θi Q λ)d θi (4)and maximizing this with respect to βi s and λ(using an E-M algorithm,for example)yields what are called maximum marginal likelihood (MML)estimates βj and λ.These MML estimates are also consistent (asymptotically unbiased).The method based on Eqn.4can be interpreted as an empirical Bayes method;it also links the Rasch model directly with other latent variable approaches such as factor analysis,where the latent variable θi is treated as an unobserved random variable,or equivalently as completely missing data (see Factor Analysis and Latent Structure:O er iew ).Finally,it can be shown that under Eqn.4,the marginal distribution of the data follows a log-linear model for the 2J table of the formln p (x i ",…,x iJ )l αk Jj ="βj x ij j Jk =!γk χ(x i +l k )where χ(S )l 1if statement S is true,and 0otherwise.This is the log-linear model of quasisymmetry;the connection between the Rasch model and the quasi-5245Factor Analysis and Latent Structure:IRT and Rasch Modelssymmetry model has been independently discovered innumerable times in the literature.Lindsay et al.(1991)give a reasonably complete account of the connection with quasisymmetry and consequences for the identifiability of the distribution f (θQ λ)in a semiparametric formulation of Eqn.4.3.Parametric IRTOther parametric IRT models have been developed for situations in which the Rasch model does not fit,while retaining the ideas of dichotomous (X ij ?o 0,1q )or at least polytomous (X ij ?o 0,1,…,K k 1q )item scores,a unidimensional (or low-dimensional)latent variable θi characterizing subjects,a low-dimensional parameter βj characterizing items,and local (conditional)in-dependence of all item responses X ij given all par-ameters.An accessible review of developments in parametric IRT in the 1960s,1970s and 1980s is given by Hambleton et al.(1991).A simple generalization of the Rasch model that is often useful for dichotomously-scored items,P j (θi )lexp[αj (θi k βj )]1j exp[αj (θi j )](5)is called the two-parameter logistic (2PL)model.In the 2PL model,the slope parameters αjk are called ‘discrimination parameters,’and they control the Fisher information for estimating the θs.Again,the location parameters βj play the role of ‘difficulty parameters’;they also determine the value of θmaximizing the Fisher information.The three-par-ameter logistic (3PL)model adds a nonzero lower asymptote to Eqn.(3),to model random or exchange-able response behavior by low-θsubjects.Further generalizations,discussed in the next several para-graphs,are treated at length by Fischer and Molenaar (1995)and van der Linden and Hambleton (1997).Additional covariates and other structures can be incorporated into these models as regression terms.For example,in the linear logistic test model (LLTM),the difficulty parameters βj in the Rasch model are rewritten as linear combinations of K basic parameters ηk with weights q jk ,andlogit P j (θi )l θi k Kk ="q jk ηk .The matrix Q l [q jk ]is usually obtained as a regression model matrix from a priori analysis of the items;for example,in the transitive reasoning example described in Sect.1,the q jk might indicate what kind and how many objects were used for each item.By contrast,multidimensional compensatory IRT models decom-pose the unidimensional θi parameter into an item-dependent linear combination of underlying traits,e.g.,logit P j (θi )l Dd ="αjd θid k βj .For example,mathematics word problems may in-volve both a math component,θi ",and a verbalcomponent θi #in different proportions in differentproblems,as determined by the discrimination par-ameters αjd .In some settings,the αjd may be fixed apriori ,like the q jk in the LLTM model,and in other settings they may be estimated analogously to factor loadings in Factor Analysis.Multiplicative or con-junctive IRT models combine unidimensional models for components of response in a product P j (θi )l ΠD dl 1P jd (θid )where P jd (θid )are parametric unidimen-sional dichotomous response functions.The usual interpretation is that the P jd (θid )are probabilities of correctly performing component skills or subtasks,which must be done in conjunction in order to generate a correct response to the item itself;this model is sometimes referred to as the multicomponent latent trait model (MLTM).Probit versions of all the above models,in which the logit function is replaced with Φ−"(p ),the inverse standard normal distribution function,are also in use.Aside from a trivial rescaling of the parameters,there is little difference in practice between the probit and logit models;the choice depends mostly on computa-tional expedience.IRT model can also be extended to handle polyto-mous or multicategory items,X ij ?o 0,1,2,…,K k 1q ,in much the same way that logistic regression models are extended polytomous data:in the ‘graded response model’(GRM)the cumulative logit,logit P [X ij c Q θi ,βj ],is linear in θi ;in the ‘sequential model’(SM)the continuation-ratio logit,logit P [X ij c Q X ij c k 1,θi ,βj ],is linear in θi ;and in the ‘partial credit model’(PCM)the adjacent-category logit,logit P [X ij l c j 1Q X ij ?o c ,c j 1q ,θi ,βj ],is linear in θi .Again,probit reformulations of these models are straightforward.When the discrimination parameters are all non-negative,the cumulative response curves in any of these models is nondecreasing (Hemker et al.1997),a condition referred to as ‘monotonicity.’By contrast,IRT formulations of direct-response prob-abilistic unfolding models,also known as proximity models or parallelogram models,have unimodal response curves that peak near the location βj of the item.As was the case for the Rasch model discussed in Sect.2,JML estimates for IRT models are relatively easy to set up but generally inconsistent (asymp-totically biased);CML methods can be used when simple sufficient statistics are available.MML meth-ods based on applying an E-M algorithm to Eqn.4,with P j (θi )redefined according to one of the above5246Factor Analysis and Latent Structure:IRT and Rasch ModelsFigure 1Two-way hierarchical structure for N individuals and J dichotomous response variables.Factors in the first level are independent and are multiplied together to form a likelihood for the response data matrix [X ij ].Factors in the second and third levels are alsoindependent,and covariates and special structure for the latent variables and item parameters may be introduced in these levelsparametric forms,are most common,however.The recent development of Markov Chain Monte Carlo (MCMC)integration methods for Bayesian statistics (see Marko Chain Monte Carlo Methods )provides an alternative to these methods.Patz and Junker (1999)sketch a general MCMC methodology for parametric IRT models,and Johnson and Albert (1999)survey a range of related applications from a working Bayesian statistician’s perspective.The IRT toolbox has been pushed forward in recent years by the needs of both large-scale educational assessment surveys (e.g.,Johnson et al.1994)and cognitively based assessment models (e.g.,Nichols et al.1995).To accommodate finer structure in the latent space,as well as to incorporate subject and item covariates in the model,it is useful to reformulate the IRT framework as a two-way hierarchical Bayes structure for N individuals and J response variables,as in Fig.1.In addition to the item response functions P j (θi )l P (θi ;βj )in Fig.1,f i and g j are prior distribu-tions on the latent variables θand item parameters βrespectively (both of which may be multidimensional),and λf and λg represent sets of hyperparameters needed to specify these prior distributions.The model in Fig.1is expressed for dichotomous items but can be generalized easily to polytomous items or combina-tions of item types,incomplete data,and so forth (e.g.,Patz and Junker 1999).The MML formulation dis-cussed in the previous paragraph is obtained by taking f i (θ)to be a single population distribution not de-pendent on j for the latent trait,and taking g j to be flat priors not depending on j for the item parameters.4.Nonparametric IRTNonparametric IRT refers to several related method-ologies for working with item response data without fully committing to any particular well-known family of parametric item response theory models.Some topics in nonparametric IRT are considered in van der Linden and Hambleton (1997)and Boomsma et al.(2001),and several current issues are surveyed in a special issue of Applied Psychological Measurement (Junker and Sijtsma 2001).Modern nonparametric scaling methods include the ‘essential independence’(EI)approach of Stout and his students and colleagues (e.g.,Stout et al.1996),and the Mokken scaling approach (e.g.,Hemker et al.1995);both methods are related to Cronbach’s alpha bound on reliability.Mokken scaling is very good at finding groups of items that are highly discriminating among subjects;Stout’s methods tend to identify groups of items that narrowly satisfy LI \EI,M,and U better.Ramsay (e.g.,Ramsay 1996)has developed computational tools for estimating item response functions nonparametrically,and for visualizing the dimensionality of θin terms of the surface in the J -dimensional unit cube generated by the joint like-lihood for X i ",…,X iJ as θi varies.5.Some ApplicationsParametric and nonparametric IRT methods are routinely used to assess the quality of individual items and sets of items in educational measurement work (Linn 1989)and to elucidate experimental design and statistical estimation issues when performances of examinees who sit for different versions of the same exam must be compared;such work goes under the name test equating.A similar problem arises in the scoring of sequentially-designed,computerized adap-tive tests (e.g.,Sands et al.1997).IRT modeling has also elucidated research into the assessment of so-ciological bias (differential item functioning)on stan-dardized test items (Holland and Wainer 1993)and the detection and diagnosis of subject outliers (Meijer 1996).Related work includes the study of rater effects (e.g.,Patz et al.2000)and matrix-sampling and the incorporation of group effects and other item and subject covariates in educational assessment survey work (Johnson et al.1994).In the broader social sciences and related areas,IRT has also played a substantial role.The Rasch model has long been applied in social survey work;see,for example,Duncan (1985).The Rasch model and other5247Factor Analysis and Latent Structure:IRT and Rasch ModelsIRT models can also be applied to model subject heterogeneity in estimating a closed population from multiple-recapture census data(see Censuses:History and Methods);Fienberg et al.(1999)recently surveyed this methodology and compared Bayesian IRT models with standard log-linear models in this context.Appli-cations to multiple outcomes of designed experiments and to panel data are indicated in the volumes by Fischer and Molenaar(1995)and van der Linden and Hambleton(1997).Sijtsma and Verweij(1999)apply nonparametric IRT to scale construction in develop-mental psychology;and Nichols et al.(1995)collect many applications of IRT and related Bayesian infer-ence network models(see Bayesian Graphical Models and Networks;Mislevy1996)in cognitive diagnosis. Applications of parametric and nonparametric IRT to psychiatric scales include Gibbons et al.(1985),Santor et al.(1995)and Kim and Pilkonis(1999).In bio-statistics Legler and Ryan(1997)use IRT to model multiple physical outcomes in the study of birth defects.6.SummaryItem response theory(IRT)has grown from its roots in postwar mental-testing problems,through intensive use in educational measurements in the1970s,1980s, and1990s,to become a mature statistical toolkit for modeling of multivariate discrete response data using subject-level latent variables.Applications of IRT can be found throughout the social sciences and related areas,from education,psychology,economics,and demography to medical research.Most parametric IRT models would be recognizable by modern statisticians as mixed-effects multivariate generalized linear models,but IRT has benefited from interaction with all parts of the statistical community. Almost any assessment phenomenon—from between-subject dependence due to institutional or sociological factors,to behavioral aspects of raters,to the analysis of item responses in terms of requisite subject or item features—can be expressed in the hierarchical mix-ture\Bayes modeling framework of Fig.1,because of its conceptual simplicity.Recent advances in com-putation,and MCMC methods in particular,have made it possible to estimate a vastly wider variety of these models than would have been imaginable even in the early1990s however the more complicated models typically also make higher sample-size de-mands.Nonparametric IRT approaches also rely on computationally intensive methods,including spline and kernel smoothing and bootstrap techniques,to estimate and test probability inequalities,stochastic ordering properties,and similar features of the models. Questions motivating both parametric and non-parametric IRT modeling inevitably involve identi-fying the phenomena that are worth detailed modeling, and seeing if the computational and data collection machinery can be pushed to be informative about reasonable models of these phenomena. Hambleton et al.(1991)provide a fairly straight-forward introduction to IRT in educational testing; Andersen(1980)presents the Rasch model in the context of other statistical models for the social sciences;and Fischer and Molenaar(1995)and van der Linden and Hambleton(1997)provide modern technical accounts.A broad range of current research issues in IRT is collected in Boomsma et al.(2001).BibliographyAndersen E B1980Discrete Statistical Models with Social Science Applications.North-Holland,New York Boomsma A,Snijders T A B,van Duijn M A J(eds.)2001 Essays on Item Response Theory.Springer-Verlag,New YorkDuncan O D1985Some models of response uncertainty for panel analysis.Social Science Research14:126–41 Fienberg S E,Johnson M S,Junker B W1999Classical multi-level and Bayesian approaches to population size estimation using multiple lists.Journal of Royal Statistical Society:Series A162:383–405Fischer G H,Molenaar I W(eds.)1995Rasch Models:Founda-tions,Recent De elopments,and Applications.Springer-Verlag, New YorkGibbons R D,Clark D C,von Ammon Cavanaugh S,Davis J M 1985Application of modern psychometric theory in psy-chiatric research.Journal of Psychiatric Research19:43–55 Hambleton R K,Swaminathan H,Rogers H J(eds.)1991 Fundamentals of Item Response Theory.Sage,Newbury Park, CAHemker B T,Sijtsma K,Molenaar I W1995Selection of unidimensional scales from a multidimensional item bank in the polytomous Mokken IRT model.Applied Psychological Measurement19:337–52Hemker B T,Sijtsma K,Molenaar I W,Junker B W1997 Stochastic ordering using the latent trait and the sum score in polytomous IRT models.Psymka62:331–47Holland P W,Wainer H1993Differential Item Functioning. Erlbaum,Hillsdale,NJJohnson V E,Albert J1999Ordinal Data Modeling.Springer-Verlag,New YorkJohnson E G,Mislevy R J,Thomas N1994Theoretical back-ground and philosophy of NAEP scaling procedures.In: Johnson E G et al.(eds.)Technical Report of the NAEP1992 Trial State Assessment Program in Reading.OERI,US Dept. of Ed.,Washington,DC,Chap.8,pp.133–46Junker B W,Sijtsma K2001Nonparametric IRT in action:An overview of the special issue.Applied Psychological Measure-ment25:Kim Y,Pilkonis P A1999Selecting the most informative items in the IIP scales for personality disorders:An application of item response theory.Journal of Personality Disorders13: 157–74Legler J M,Ryan L M1997Latent variable models for teratogenesis using multiple binary outcomes.Journal of the American Statistical Association92:13–20Linn R L(ed.)1989Educational Measurement,3rd edn. Macmillan,New YorkLindsay B,Clogg C C,Grego J1991Semiparametric estimation in the Rasch model and related exponential response models,5248Factor Analysis and Latent Structure:IRT and Rasch Modelsincluding a simple latent class model for item analysis.Journal of the American Statistical Association86:96–107Lord F M1952A theory of test scores.Psychometric Society, New YorkLoevinger J1948The technique of homogeneous tests compared with some aspects of‘scale analysis’and factor analysis. Psychological Bulletin45:507–30Meijer R R1996Person-fit research:An introduction.Applied Measurement in Education9:3–8Mislevy R J1996Test theory reconceived.Journal of Educational Measurement33:379–416Nichols P D,Chipman S F,Brennan R L(eds.)1995Cogniti ely Diagnostic Assessment.Erlbaum,Hillsdale,NJPatz R J,Junker B W1999Applications and extensions of MCMC in IRT:Multiple item types,missing data,and rated responses.Journal of Educational and Beha ioral Statistics24: 342–66Patz R J,Junker B W,Johnson M S submitted The hierarchical rater model for rated test items and its application to large-scale educational assessment data.Journal of Educational and Beha ioral StatisticsRamsay J O1996A geometrical approach to item response theory.Behmka23:3–17Rasch G1960Probabilistic models for some intelligence and attainment tests.University of Chicago Press,Chicago Sands W A,Waters B K,McBride J R1997Computerized Adapti e Testing:From Inquiry to Operation.American Psychological Association,Washington,DCSantor D A,ZuroffD C,Ramsay J O,Cervantes P,Palacios J 1995Examining scale discriminability in the BDI and CES-D as a function of depressive severity.Psychiatry Assessment7: 131–9Sijtsma K,Verweij A C1999Knowledge of solution strategies and IRT modeling of items for transitive reasoning.Applied Psychological Measurement23:55–68Stout W,Habing B,Douglas J,Kim H R,Roussos L,Zhang J1996Conditional covariance-based nonparametric multidi-mensionality assessment.Applied Psychological Measurement 20:331–54van der Linden W J,Hambleton R K(eds.)1997Handbook of Modern Item Response Theory.Springer-Verlag,New YorkB.W.Junker Factor Analysis and Latent Structure: OverviewThe terms factor analysis and latent structure refer to two aspects of essentially the same problem.Both are concerned with statistical problems in which some of the variables are latent,meaning that they are rmation about the latent variables has,therefore,to be obtained indirectly through indicators,also known as manifest or observable variables.The terminology of the subject reflects the diverse origins of the subject accumulated over almost a century.‘Factor’in this context is a synonym for latent variable.From a theoretical perspective,the only distinctive thing about a latent variable model is the presence of unobservable variables.In all other respects the normal methods of statistics apply. Indeed,a latent variable problem can be regarded as a standard statistical problem in which the data on the latent variables are missing.There is no essential difference,for purposes of analysis,between a problem in which some of the data are never obtained and one in which some are lost.1.IntroductionLatent variables arise mainly,but not exclusively,in the social sciences.This is because social science often deals in concepts which are constructs rather than the directly measurable variables which are typical of physical sciences.The earliest example,and still one of the most important,is general intelligence or‘g.’This goes back to Spearman(1904)and was constructed to describe the variation among individuals in what appeared to be common to a wide range of mental tests.Psychology and sociology abound in such latent variables.Attitudes as well as abilities are all spoken of in the discourse of these subjects as things which occur in varying amounts and which,therefore,appear in the theory as quantitative variables.In economics, variables like business confidence play a similar role.It could be justly claimed that the aspiration of these subjects to be regarded as sciences depends on the success with which latent variables can be quantified. Latent variables can be classified into several types. Many,like intelligence,are conceived of as continuous in which case we are looking for a scale on which individuals can be located.In other contexts it is more appropriate to think of the latent variable as cat-egorical.In that case individuals are supposed to belong to one of several classes which may or may not be ordered.What is true of the latent variables is,of course,true for the manifest variables and the only essential difference between the various methods is in the types of variables for which they are appropriate.A convenient way of displaying the relationship between the methods is to introduce the fourfold classification of Table1.We classify the manifest and the latent variables as metrical or categorical.In the former case we mean that they take values on someTable1Classification of latent variable methodsManifest variablesMetrical Categorical Latent variablesMetrical Factor analysis Latent traitanalysis Categorical Latent profileanalysisLatent classanalysis5249 Factor Analysis and Latent Structure:O er iewCopyright#2001Elsevier Science Ltd.All rights reserved.International Encyclopedia of the Social&Behavioral Sciences ISBN:0-08-043076-7。