Presentation on theme: "What is applied psychometrics?"— Presentation transcript:
1What is applied psychometrics? Tim CroudaceDepartment of PsychiatryJohn RustThe Psychometrics CentreUniversity of Cambridge
2What is applied psychometrics? Professor John Rust2
3Overview About the Centre What is psychometrics? Psychometrics today What we are doing nowWhat we are going to do
4The Psychometric Centre Educational and diagnostic eg WechslerOrganisational eg Watson-Glaser, OrpheusStatistical, IRT and AI techniquesComputer languages eg Mplus, Stata, RWeb based assessmentBPS Level A and B coursesSeminars, workshops and summer schoolsPhDs in psychometrics or related areasTutorial materials on website44
5Current activities Who we are (people) Announcement about summer schoolsAnnouncement about forthcoming workshops
6What is psychometrics? “The science of psychological assessment” Much assessment is “high stakes”Questionnaires and social surveysRecruitment and staff developmentLicensing and chartering (eg Accountants, Surgeons)School and University examinationsPsychiatric and ‘special needs’ diagnosisCredit ratingsCareer guidanceSocial awareness
7Types of assessment First impressions Application forms and references Objective tests (on or off line)Projective testsInterviewsEssays and examinationsResearch questionnaires and semi-structured interviews77
8The Psychometric Principles Maximizing the quality of assessment Reliability (freedom from error)Validity ( ‘... what is says on the tin’)Standardisation (compared with what?)Equivalence (is it biased?)Rust, J. & Golombok, S. (2009) Modern Psychometrics(3rd Edition): Taylor and Francis: London88
9Can everything be measured? “If anything exists it must exist in some quantity and can therefore be measured”. (Lord Kelvin 1824, 1907)In 1900, Lord Kelvin claimed "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement."[9
10The theory of true scores Whatever precautions have been taken to secure unity of standard, there will occur a certain divergence between the verdicts of competent examiners.If we tabulate the marks given by the different examiners they will tend to be disposed after the fashion of a gendarme’s hat.I think it is intelligible to speak of the mean judgment of competent critics as the true judgment; and deviations from that mean as errors.This central figure which is, or may be supposed to be, assigned by the greatest number of equally competent judges, is to be regarded as the true value ..., just as the true weight of a body is determined by taking the mean of several discrepant measurements.Edgeworth, F.Y. (1888). The statistics of examinations. Journal of the Royal Statistical Society, LI,10
11The evolution of the Latent Trait Edgeworth, F.Y. (1888). The statistics of examinations. Journal of the Royal Statistical Society, LI, With two measures of the same characteristic we can estimate true values.Melvin Novik and Frederick Lord (1968) “Statistical theories of mental test scores” use Classical Test Theory to derive Latent Trait Theory. Allan Birnbaum, in his supplement, established Item Response Theory of which Rasch Scaling is a special case.Today Latent Variable Analysis (LVA) is an integral part of statistical modelling in Psychometrics, Econometrics and Statistics.11
12What is applied psychometrics? Tim CroudaceDepartment of PsychiatryUniversity of Cambridge
13psychometry psycho·met·rics (sī′kō me′triks) Etymologically (from the Greek)psychometry meansmeasuring the mindP. Kline (1979)“The meaning of psychometrics” p1
14-definitions-definitions-definitions- Collins English DictionaryPsychometrics definition : psychometrics nthe branch of psychology concerned with the design and use of psychological testsapplication of statistical & mathematical techniques to psychological testingdictionary.reverso.net/english-definition/psychometrics
15What is psychometrics?The Science of Psychological Assessment “the branch of psychology dealing with measurable factors” Modern Psychometrics. by J. Rust & S. Golombok. Routledge. P 4
16[From Wikipedia, the free encyclopedia] Psychometrics – Even Wikipedia has something to say … it doesn’t begin too promisingly!!![From Wikipedia, the free encyclopedia]Psychometrics –Not to be confused with psychrometrics, the measurement of the heat and water vapor properties of air. For other uses of this term and similar terms, see (disambiguation).Psychometry [Redirected from Psychometry (disambiguation)] may refer to:Psychometry (paranormal) a form of extrasensory perceptionPsychometrics a discipline of psychology and education (getting warmer!!)And finally it begins to make sense …Psychometrics is the field of study concerned with the theory and technique of educational and psychological measurement, which includes the measurement of knowledge, abilities, attitudes, and personality traits. The field is primarily concerned with the construction and validation of measurement instruments, such as questionnaires, tests, and personality assessments.
17What is ? [Psychometric] Test Theory Psychometric Test Theory …is essentially a collection of mathematical concepts that formalize and clarify certain questions about constructing and using tests [and scales] and then provide methods for answering them R.P. McDonald (1999) Test Theory: a unified treatment. LEA. P 9
18What is psychometrics. Item Response Theory (IRT) What is psychometrics? Item Response Theory (IRT) Item Response Modelling (IRM)IRT refers to a set of mathematical models that describe, in probabilistic terms, the relationship between a person’s response to a survey question/test item and his or her level of the ‘latent variable’ being measured by the scaleFayers and Hays p55Assessing Quality of Life in Clinical Trials. Oxford Univ Press:Chapter on Applying IRT for evaluating questionnaire item and scale properties.
19Psychometric (Measurement) Theory : 2 main schools, old & new Classical Test TheoryAssociated with use of traditional (old) psychometric methodslinear factor analysisCronbach’s alpha (internal consistency),summing items and simple sum scoresItem response theoryModern test theoryA set or family of mathematical / probability models that describe the relationship between a person’s [response / answer] to a [questionnaire survey / test item] and his or her level of the latent variable being measured
20Classical Test Theory Reliability estimation Reliability coefficientMajor error sourceData-gathering procedureStatistical data analysis1. Stability coefficientChanges over timeTest-retestProduce-moment correlation2. Equivalence coefficientItem sampling: from test form to test formGiven form j, form k3. Internal consistency coefficientItem sampling: test heterogeneityA single administrationSplit-half correlation/ Spearman Brown correction,coefficient alphaFactor loadingsOtherTable 4.1 p26 Dato M.N. De Gruiter and Leo J. Th. Van der Kamp (2008)
21Reliability coefficients STATA alpha and cialpha commands Continuous outcomes: Guttman-Cronbach alphaTest scale = mean(unstandardized items)Average interitem covariance:Number of items in the scale:Scale reliability coefficient:Cronbach's alpha one-sided confidence intervalItems | alpha [95% Conf.Interval]Test | >=
26Confirmatory Factor Analysis (ML): STATA cfa1 command Log likelihood = Number of obs = 87 | Coef. Std. Err. z P>|z| [95% Conf. Interval] Lambda | v1 | v2 | v3 | v4 | v5 | v6 | v7 | v8 | Var[error] | v1 | v2 | v3 | v4 | v5 | v6 | v7 | v8 | Var[latent] | phi1 | Goodness of fit test: LR = ; Prob[chi2(20) > LR] = Test vs independence: LR = ; Prob[chi2( 8) > LR] =
27Single factor model (ML): STATA confa commands . confa (f: v1-v8), from(2SLS) log likelihood = Number of obs = 87 | Coef. Std. Err. z P>|z| [95% Conf. Interval] Loadings | f | v1 | v2 | v3 | v4 | v5 | v6 | v7 | v8 | Var[error] | v1 | v2 | v3 | v4 | v5 | v6 | v7 | v8 | Goodness of fit test: LR = ; Prob[chi2(20) > LR] = Test vs independence: LR = ; Prob[chi2( 8) > LR] =
28Confirmatory Factor Analysis (ML): STATA estat fitindices commands Fit indices RMSEA = % CI= (0.1868, ) RMSR = TLI = CFI = AIC = BIC =
29Multidimensional factor model (ML): STATA confa command (2 factors) confa (f1: v1-v4) (f2: v5-v8), from(2SLS) log likelihood = Number of obs = 87 | Coef. Std. Err. z P>|z| [95% Conf. Interval] Means | v1 | v2 | v3 | v4 | v5 | v6 | v7 | v8 | Loadings | v1 | v2 | v3 | v4 | v5 | v6 | v7 | v8 | Factor cov. | f1-f1 | f2-f2 | f1-f2 | Goodness of fit test: LR = ; Prob[chi2(19) > LR] = Test vs independence: LR = ; Prob[chi2( 9) > LR] =
30Single factor model (ML): STATA confa commands . estat fitindices Fit indices RMSEA = , 90% CI= (0.0637, ) RMSR = TLI = CFI = AIC = BIC =
31Reliability coefficients STATA kr20 command Kuder-Richardson KR20 Kuder-Richarson coefficient of reliability (KR-20) Number of items in the scale = 12 Number of complete observations = 6299 Item Item Item-rest Item | Obs difficulty variance correlation GHQ1 | GHQ2 | GHQ3 | GHQ4 | GHQ5 | GHQ6 | GHQ7 | GHQ8 | GHQ9 | GHQ10 | GHQ11 | GHQ12 | Test | KR20 =
32Reliability coefficients STATA kr20 command Computes the reliability coefficient of a set of dichotomous items, [Cronbach's alpha is used for multipoint scales] In addition, kr20 computes: - the item difficulty (proportion of 'right' answers), - the average value of item difficulty, - the item variance, - the corrected item-test point-biserial correlation coefficients, - the average value of corrected item-test correlation coefficients. The items must be coded as: - '0' for a wrong answer (unexpected answer), - '1' for a right answer (expected answer).
33What is applied psychometrics? Tim CroudaceDepartment of PsychiatryJohn RustThe Psychometrics CentreUniversity of Cambridge
35Latent Trait Modelling Note: IRT = IRM = LTM = CDFA* Latent trait modelling = factor analysis of categorical (binary/ordinal/nominal) dataUnidimensional LTM is widely used to measure variables/constructs such asPersonality Dimensions and IntelligenceAbility: Mathematical / Verbal / SpatialSocial and political attitudesConsumer preferencesHealth, Quality of life, Severity of disorder or symptoms e.g. in depression, back pain, fatigue etc…Multidimensional IRT is statistically developed but is less widely used presently
36Here the criterion 1 – 4 are binary but the latent variable (x-axis) is continuous (gaussian normal) From Muthen, B.O (1991). Latent variable epidemiology.Alcohol Research World
38Rasch model (logistic mixed model) (1 random effect (individual differences – x – axis)) 12 fixed effects – item thresholds (location of s-shapes along x) [Stata raschtest mixed effects logistic regression [inc gllamm]Item DiscriminationsGHQGHQGHQGHQGHQGHQGHQGHQGHQGHQItem DifficultiesGHQ1$GHQ5$GHQ12$GHQ11$GHQ26$GHQ4$1GHQ20$GHQ9$GHQ10$GHQ6$
39IRT in the Stata Journal J-7-3 st Est. dichotomous & ordinal item response models with gllamm By X. Zheng and S. Rabe-Hesketh Q3/07 SJ 7(3):313—333 describes the one- and two-parameter logit models for dichotomous items the partial-credit and rating scale models for ordinal items, and an extension of these models where the latent variable is regressed on explanatory variables SJ-7-1 st0119 Rasch analysis: Estimation and tests with raschtest By J. Hardouin Q1/07 SJ 7(1): command for estimating the Rasch model, the best known item response theory model for binary responses
40Running Commercial IRT software from Stata runparscale runparscale: runparscale brings the IRT analysis framework of PARSCALE into the Stata enviroment. While runparscale does little more than data reformat and ascii file creation, it removes a lot of the hassle of estimating IRT models.Authors: runparscale was written by Laura Gibbons, PhD and Richard Jones, ScD, under the direction of Paul Crane, MD MPH. We appreciate the assistance of Tom Koepsell, MD MPH.Please see runparscale.ado for UW License information.Laura Gibbons, PhDRichard N Jones, ScD
41Running Commercial IRT software from Stata runparscale
44X-axis Latent Trait value (IRT thresholds zero centred) Y-axis conditional standard error of measurement (s.e.m. varies with score value under Item Response Theory). Lower s.e.m = greater precision of measurement
45Non-parametric IRT Mokken Analysis STATA loevH command . loevH GHQ1-GHQ12Observed Expected NumberEasyness Guttman Guttman Loevinger H0: Hj<=0 of NSItem Obs P(Xj=1) errors errors H coeff z-stat. p-value HjkGHQGHQGHQGHQGHQGHQGHQGHQGHQGHQGHQGHQScaleloevH by [Websites AnaQol and FreeIRT] allows verifying the fit of data to the Monotonely Homogeneous Mokken Model or to the Doubly Monotone Mokken Model. It computes the Loevinger H scalability coefficients, and several indexes in the field of the Non parametric Item Response Theory.
46(1) Non-parametric IRT Mokken Analysis STATA msp command . msp GHQ1-GHQ12, c(.4) The two first items selected in the scale 1 are GHQ7 and GHQ8 (Hjk=0.7357) The item GHQ6 is selected in the scale 1 Hj= H= The following items are excluded at this step: GHQ3 The item GHQ12 is selected in the scale 1 Hj= H= The item GHQ10 is selected in the scale 1 Hj= H= The item GHQ11 is selected in the scale 1 Hj= H= The item GHQ1 is selected in the scale 1 Hj= H= The item GHQ4 is selected in the scale 1 Hj= H= The item GHQ5 is selected in the scale 1 Hj= H= None new item can be selected in the scale 1 because all the Hj are lesser than .4 or none new item has all the related Hjk coefficients significantly greater than 0 Observed Expected Number Easyness Guttman Guttman Loevinger H0: Hj<=0 of NS Item Obs P(Xj=1) errors errors H coeff z-stat. p-value Hjk GHQ GHQ GHQ GHQ GHQ GHQ GHQ GHQ GHQ Scale Scale: Significance level: The two first items selected in the scale 2 are GHQ2 and GHQ3 (Hjk=0.4111) Significance level: None new item can be selected in the scale 2 because all the Hj are lesser than .4 or none new item has all the related Hjk coefficients significantly greater than 0 . GHQ GHQ Scale There is only one item remaining (GHQ9).
47(2) Non-parametric IRT Mokken Analysis STATA msp command Scale: Significance level: The two first items selected in the scale 2 are GHQ2 and GHQ3 (Hjk=0.4111) Significance level: None new item can be selected in the scale 2 because all the Hj are lesser than .4 or none new item has all the related Hjk coefficients significantly greater than 0 . Observed Expected Number Easyness Guttman Guttman Loevinger H0: Hj<=0 of NS Item Obs P(Xj=1) errors errors H coeff z-stat. p-value Hjk GHQ GHQ Scale There is only one item remaining (GHQ9).
48(1) Rasch model in STATAEstimation method: Conditional maximum likelihood (CML) Number of items: 9 Number of groups: 10 (8 of them are used to compute the statistics of test) Number of individuals: 548 Number of individuals with missing values: 0 (removed) Number of individuals with nul or perfect score: 111 Conditional log-likelihood: Log-likelihood: Difficulty Standardized Items parameters std Err. R1c df p-value Outfit Infit U GHQ GHQ GHQ GHQ GHQ GHQ GHQ GHQ GHQ12* R1c test R1c= Andersen LR test Z= *: The difficulty parameter of this item had been fixed to 0
49(2) Rasch model in STATA raschtest Ability Expected Group Score parameters std Err. Freq. Score ll
50Running Mplus www.statmodel.com from Stata runmplus Runmplus [Author: Richard N Jones, ScD ]Builds an Mplus data file, command file, executes the command file and display Mplus log file (output) in the Stata results window.Factor analysis syntax examples:Exploratory factor analysis with continuous indicatorsrunmplus y1-y12, type(efa 1 4)Exploratory factor analysis with categorical indicatorsrunmplus y1-y12, type(efa 1 4) categorical(all)Exploratory factor analysis with a mixture of categorical and continuous indicatorsrunmplus y1-y12,type(efa 1 4) categorical(y1 y3 y5 y7 y9 y11)Confirmatory factor analysis with continuous indicatorsrunmplus y1-y6, model(f1 by y1-y3; f2 by y4-y6;)
53Extract from //cran.r-project.org/web/views/Psychometrics.html Classical Test Theory (CTT)The CTT package can be used to perform a variety of tasks and analyses associated with classical test theory: score multiple-choice responses, perform reliability analyses, conduct item analyses, and transform scores onto different scales.The CMC package calculates and plots the step-by-step Cronbach-Mesbach curve, that is a method, based on the Cronbach alpha coefficient of reliability, for checking the unidimensionality of a measurement scale.The package psychometric contains functions useful for correlation theory, meta-analysis (validity-generalization), reliability, item analysis, inter-rater reliability, and classical utility. Cronbach alpha, kappa coefficients, and intra-class correlation coefficients (ICC) can be found in the psy package.A number of routines for scale construction and reliability analysis useful for personality and experimental psychology are contained in the packages psych and MiscPsycho.Additional measures for reliability and concordance can be computed with the concord package.
54(2) Extract from //cran.r-project.org/web/views/Psychometrics.html Item Response Theory (IRT):The eRm package fits extended Rasch models, i.e. the ordinary Rasch model for dichotomous data (RM), the linear logistic test model (LLTM), the rating scale model (RSM) and its linear extension (LRSM), the partial credit model (PCM) and its linear extension (LPCM) using conditional ML estimation. Missing values are allowed.The package ltm also fits the simple RM. Additionally, functions for estimating Birnbaum's 2- and 3-parameter models based on a marginal ML approach are implemented as well as the graded response model for polytomous data, and the linear multidimensional logistic model.Item and ability parameters can be calibrated using the package plink. It provides unidimensional and multidimensional methods such as Mean/Mean, Mean/Sigma, Haebara, and Stocking-Lord methods for dichotomous (1PL, 2PL and 3PL) and/or polytomous (graded response, partial credit/generalized partial credit, nominal, and multiple-choice model) items. The multidimensional methods include the Reckase-Martineau method and extensions of the Haebara and Stocking-Lord method.The difR package contains several traditional methods to detect DIF in dichotomously scored items. Both uniform and non-uniform DIF effects can be detected, with methods relying upon item response models or not. Some methods deal with more than one focal group.The package lordif provides a logistic regression framework for detecting various types of differential item functioning (DIF).The package plRasch computes maximum likelihood estimates and pseudo-likelihood estimates of parameters of Rasch models for polytomous (or dichotomous) items and multiple (or single) latent traits. Robust standard errors for the pseudo-likelihood estimates are also computed.A multilevel Rasch model can be estimated using the package lme4 with functions for mixed-effects models with crossed or partially crossed random effects.Other packages of interest are: mokken to compute non-parametric item analysis, the RaschSampler allowing for the construction of exact Rasch model tests by generating random zero-one matrices with given marginals, mprobit fitting the multivariate binary probit model, and irtoys providing a simple interface to the estimation and plotting of IRT models. Simple Rasch computations such a simulating data and joint maximum likelihood are included in the MiscPsycho package.The irtProb is designed to estimate multidimensional subject parameters (MLE and MAP) such as personnal pseudo-guessing, personal fluctuation, personal inattention. These supplemental parameters can be used to assess person fit, to identify misfit type, to generate misfitting response patterns, or to make correction while estimating the proficiency level considering potential misfit at the same time.Gaussian ordination, related to logistic IRT and also approximated as maximum likelihood estimation through canonical correspondence analysis is implemented in various forms in the package VGAM.Two additional IRT packages (for Microsoft Windows only) are available and documented on the JSS site. The package mlirt computes multilevel IRT models, and cirt uses a joint hierarchically built up likelihood for estimating a two-parameter normal ogive model for responses and a log-normal model for response times.Bayesian approaches for estimating item and person parameters by means of Gibbs-Sampling are included in MCMCpack. In addition, the pscl package allows for Bayesian IRT and roll call analysis.The latdiag package produces commands to drive the dot program from graphviz to produce a graph useful in deciding whether a set of binary items might have a latent scale with non-crossing ICCs.
55(3) Extract from //cran.r-project.org/web/views/Psychometrics.html Structural Equation Models, Factor Analysis, PCA:Ordinary factor analysis (FA) and principal component analysis (PCA) are in the package stats as functions factanal() and princomp(). Additional rotation methods for FA based on gradient projection algorithms can be found in the package GPArotation. The package nFactors produces a non-graphical solution to the Cattell scree test. Some graphical PCA representations can be found in the psy package.The sem package fits general (i.e., latent-variable) SEMs by FIML, and structural equations in observed-variable models by 2SLS. Categorical variables in SEMs can be accommodated via the polycor package. The systemfit package implements a wider variety of estimators for observed-variables models, including nonlinear simultaneous-equations models. See also the pls package, for partial least-squares estimation, the gR task view for graphical models and the SocialSciences task view for other related packages.The package lavaan can be used to estimate a large variety of multivariate statistical models, including path analysis, confirmatory factor analysis, structural equation modeling and growth curve models. It includes the lavaan model syntax which allows users to express their models in a compact way and allows for ML, GLS, WLS, robust ML using Satorra-Bentler corrections, and FIML for data with missing values. It fully supports for meanstructures and multiple groups and reports standardized solutions, fit measures, modification indices and more as output.SEMModComp conducts tests of difference in fit for mean and covariance structure models as in structural equation modeling (SEM)The package FAiR performs factor analysis based on a genetic algorithm for optimization. This makes it possible to impose a wide range of restrictions on the factor analysis model, whether using exploratory factor analysis, confirmatory factor analysis, or a new estimator called semi-exploratory factor analysis (SEFA).FA and PCA with supplementary individuals and supplementary quantitative/qualitative variables can be performed using the FactoMineR package whereas MCMCpack has some options for sampling from the posterior for ordinal and mixed factor models.The homals package provides nonlinear PCA and, by defining sets, nonlinear canonical correlation analysis (models of the Gifi-family).Independent component analysis (ICA) can be computed using fastICA. Independent factor analysis (IFA) with independent non-Gaussian factors can be performed with the ifa package.A desired number of robust principal components can be computed with the pcaPP package.The package psych includes functions such as fa.parallel() and VSS() for estimating the appropriate number of factors/components as well as ICLUST() for item clustering.
56Psychometrics in RSpecial volume of the Journal of Statistical SoftwareVolume 20Multilevel RaschCorrespondence AnalysisRaschMultilevel IRTMultidimensional RaschExtended RaschMarginal Maximum Likelihood IRTMokken scale analysis …
57Free R software The program LTM is available for R from It is available as an R version and S-Plus version.ltm fits the logit-probit (normal latent trait; logistic link function) models with one- [and two] factors.In a very recent (but complex) development it also allows for inclusion of nonlinear terms (e.g., interaction and quadratic terms). Extra features:computation of factor scores using Multiple ImputationRasch modelfor which Goodness of Fit is assessed using a parametric Bootstrap version of the Pearson chi-squared.
58Free software Factor/M-IRT MIRT Factor NOHARM Urbano Lorenzo-Seva & Pere J. FerrandoMIRTNOHARM
59FACTOR //psico.fcep.urv.es/utilitats/factor/ Factor is a program developed to fit the Exploratory Factor Analysis model. Below we describe the methods used. Univariate and multivariate descriptives of variables: Univariate mean, variance, skewness, and kurtosis Multivariate skewness and kurtosis (Mardia, 1970) Var charts for ordinal variables Dispersion matrices: User defined tipo matrix Covariance matrix Pearson correlation matrix Polychoric correlation matrix with optional Ridge estimates Procedures for determining the number of factors/components to be retained: MAP: Minimum Average Partial Test (Velicer, 1976) PA: Parallel Analysis (Horn, 1965) PA - MBS. It is an extension of Parallel Analysis that generates random correlation matrices using marginally bootstrapped samples (Lattin, Carroll, & Green, 2003) Factor and component analysis: PCA: Principal Component Analysis ULS: Unweighted Least Squares factor analysis (also MINRES and PAF) EML: Exploratory Maximum Likelihood factor analysis MRFA: Minimum Rank Factor Analysis (ten Berge, & Kiers, 1991) Schmid-Leiman second-order solution (1957) Factor scores (ten Berge, Krijnen, Wansbeek, & Shapiro, 1999)In ULS factor analysis, the Heywood case correction described in Mulaik (1972, page 153) is included: when an update has sum of squares larger than the observed variance of the variable, that row is updated by constrained regression using the procedure proposed by ten Berge and Nevels (1977).Some of the rotation methods to obtain simplicity are:Quartimax (Neuhaus & Wrigley, 1954)Varimax (Kaiser, 1958)Weighted Varimax (Cureton & Mulaik, 1975)Orthomin (Bentler, 1977)Direct Oblimin (Clarkson & Jennrich, 1988)Weighted Oblimin (Lorenzo-Seva, 2000)Promax (Hendrickson & White, 1964)Promaj (Trendafilov, 1994)Promin (Lorenzo-Seva, 1999)Simplimax (Kiers, 1994)Some of the indices used in the analysis are:Test on the dispersion matrix: Determinant, Bartlett's test and Kaiser-Meyer-Olkin (KMO)Goodness of fit statistics: Chi-Square Non-Normed Fit Index (NNFI; Tucker & Lewis); Comparative Fit Index (CFI); Goodness of Fit Index (GFI); Adjusted Goodness of Fit Index (AGFI); Root Mean Square Error of Approximation (RMSEA); and Estimated Non-Centrality Parameter (NCP)Reliabilities of rotated components (ten Berge & Hofstee, 1999)Simplicity indices: Bentler’s Simplicity index (1977) and Loading Simplicity index (Lorenzo-Seva, 2003)Mean, variance and histogram of fitted and standardized residuals. Automatic detection of large standardized residuals.
63Running Mplus www.statmodel.com from Stata runmplus
64Running Mplus www.statmodel.com from Stata runmplus
65Running Mplus www.statmodel.com from Stata runmplus
66Running Mplus www.statmodel.com from Stata runmplus
67Running Mplus www.statmodel.com from Stata runmplus
68Running Mplus www.statmodel.com from Stata runmplus
69Running Mplus www.statmodel.com from Stata runmplus
70Running Mplus www.statmodel.com from Stata runmplus
71Excellent book chapter (non-technical) Application oriented bookAssessing Quality of Lifein Clinical Trials; Methods and Practice Edition: 2nd Author(s): Peter Fayers; Ron Hays ISBN: see Chapter byReeve and FayersApplying item response theory modelling for evaluating questionnaire item and scale propertiesdownload for free from