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**What is applied psychometrics?**

Tim Croudace Department of Psychiatry John Rust The Psychometrics Centre University of Cambridge

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**What is applied psychometrics?**

Professor John Rust 2

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**Overview About the Centre What is psychometrics? Psychometrics today**

What we are doing now What we are going to do

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**The Psychometric Centre**

Educational and diagnostic eg Wechsler Organisational eg Watson-Glaser, Orpheus Statistical, IRT and AI techniques Computer languages eg Mplus, Stata, R Web based assessment BPS Level A and B courses Seminars, workshops and summer schools PhDs in psychometrics or related areas Tutorial materials on website 4 4

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**Current activities Who we are (people)**

Announcement about summer schools Announcement about forthcoming workshops

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**What is psychometrics? “The science of psychological assessment”**

Much assessment is “high stakes” Questionnaires and social surveys Recruitment and staff development Licensing and chartering (eg Accountants, Surgeons) School and University examinations Psychiatric and ‘special needs’ diagnosis Credit ratings Career guidance Social awareness

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**Types of assessment First impressions Application forms and references**

Objective tests (on or off line) Projective tests Interviews Essays and examinations Research questionnaires and semi-structured interviews 7 7

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**The 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: London 8 8

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**Can 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

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**The 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

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**The 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

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**What is applied psychometrics?**

Tim Croudace Department of Psychiatry University of Cambridge

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**psychometry psycho·met·rics (sī′kō me′triks)**

Etymologically (from the Greek) psychometry means measuring the mind P. Kline (1979) “The meaning of psychometrics” p1

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**-definitions-definitions-definitions-**

Collins English Dictionary Psychometrics definition : psychometrics n the branch of psychology concerned with the design and use of psychological tests application of statistical & mathematical techniques to psychological testing dictionary.reverso.net/english-definition/psychometrics

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What 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

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**[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 perception Psychometrics 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.

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**What 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

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**What 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 scale Fayers and Hays p55 Assessing Quality of Life in Clinical Trials. Oxford Univ Press: Chapter on Applying IRT for evaluating questionnaire item and scale properties.

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**Psychometric (Measurement) Theory : 2 main schools, old & new**

Classical Test Theory Associated with use of traditional (old) psychometric methods linear factor analysis Cronbach’s alpha (internal consistency), summing items and simple sum scores Item response theory Modern test theory A 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

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**Classical Test Theory Reliability estimation**

Reliability coefficient Major error source Data-gathering procedure Statistical data analysis 1. Stability coefficient Changes over time Test-retest Produce-moment correlation 2. Equivalence coefficient Item sampling: from test form to test form Given form j, form k 3. Internal consistency coefficient Item sampling: test heterogeneity A single administration Split-half correlation/ Spearman Brown correction, coefficient alpha Factor loadings Other Table 4.1 p26 Dato M.N. De Gruiter and Leo J. Th. Van der Kamp (2008)

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**Reliability coefficients STATA alpha and cialpha commands**

Continuous outcomes: Guttman-Cronbach alpha Test scale = mean(unstandardized items) Average interitem covariance: Number of items in the scale: Scale reliability coefficient: Cronbach's alpha one-sided confidence interval Items | alpha [95% Conf.Interval] Test | >=

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**Exploratory Factor Analysis (ML): STATA factor command**

factor v1-v8, factors(2) ml Factor analysis/correlation Number of obs = 87 Method: maximum likelihood Retained factors = 2 Rotation: (unrotated) Number of params = 15 Schwarz's BIC = Log likelihood = (Akaike's) AIC = Factor | Eigenvalue Difference Proportion Cumulative Factor1 | Factor2 | LR test: independent vs. saturated: chi2(28) = Prob>chi2 = LR test: 2 factors vs. saturated: chi2(13) = Prob>chi2 = Factor loadings (pattern matrix) and unique variances Variable | Factor1 Factor2 | Uniqueness v1 | | v2 | | v3 | | v4 | | v5 | | v6 | | v7 | | v8 | | rotate, bentler bl(.35) Rotation: orthogonal bentler (Kaiser off) Number of params = 15 Factor | Variance Difference Proportion Cumulative Factor1 | Factor2 | Rotated factor loadings (pattern matrix) and unique variances v1 | | v2 | | v3 | | v4 | | v5 | | v6 | | v7 | | v8 | | (blanks represent abs(loading)<.35) Factor rotation matrix | Factor1 Factor Factor1 | Factor2 |

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**(2) Exploratory Factor Analysis (ML): STATA rotate command**

. rotate, bentler bl(.35) Rotated factor loadings (pattern matrix) and unique variances Variable | Factor1 Factor2 | Uniqueness v1 | | v2 | | v3 | | v4 | | v5 | | v6 | | v7 | | v8 | | (blanks represent abs(loading)<.35) Factor rotation matrix | Factor1 Factor Factor1 | Factor2 |

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**Confirmatory 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] =

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**Single 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] =

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**Confirmatory Factor Analysis (ML): STATA estat fitindices commands**

Fit indices RMSEA = % CI= (0.1868, ) RMSR = TLI = CFI = AIC = BIC =

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**Multidimensional 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] =

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**Single factor model (ML): STATA confa commands**

. estat fitindices Fit indices RMSEA = , 90% CI= (0.0637, ) RMSR = TLI = CFI = AIC = BIC =

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**Reliability 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 =

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**Reliability 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).

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**What is applied psychometrics?**

Tim Croudace Department of Psychiatry John Rust The Psychometrics Centre University of Cambridge

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Message TRI IRT

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**Latent Trait Modelling Note: IRT = IRM = LTM = CDFA***

Latent trait modelling = factor analysis of categorical (binary/ordinal/nominal) data Unidimensional LTM is widely used to measure variables/constructs such as Personality Dimensions and Intelligence Ability: Mathematical / Verbal / Spatial Social and political attitudes Consumer preferences Health, 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

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**Here 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

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**8 IRT models you might see …**

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Rasch 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 Discriminations GHQ GHQ GHQ GHQ GHQ GHQ GHQ GHQ GHQ GHQ Item Difficulties GHQ1$ GHQ5$ GHQ12$ GHQ11$ GHQ26$ GHQ4$1 GHQ20$ GHQ9$ GHQ10$ GHQ6$

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**IRT 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

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**Running 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, PhD Richard N Jones, ScD

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**Running Commercial IRT software from Stata runparscale**

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**Running Commercial IRT software from Stata runparscale**

PARSCALE ITEM PARAMETERS item slope (se) location (se) GHQ (0.091) (0.063) 2 GHQ (0.060) (0.124) 3 GHQ (0.056) (0.287) 4 GHQ (0.091) (0.064) 5 GHQ (0.087) (0.065) 6 GHQ (0.100) (0.081) 7 GHQ (0.139) (0.044) 8 GHQ (0.122) (0.048) 9 GHQ (0.075) (0.156) 10 GHQ (0.101) (0.077) 11 GHQ (0.088) (0.068) 12 GHQ (0.124) (0.048)

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parscale ITEM FIT STATISTICS [not to be trusted for short tests, illustrative only] | BLOCK | ITEM | CHI-SQUARE | D.F. | PROB. | | GHQ1 | 0001 | | 7. | | | GHQ2 | 0002 | | 9. | | | GHQ3 | 0003 | | 10. | | | GHQ4 | 0004 | | 8. | | | GHQ5 | 0005 | | 8. | | | GHQ6 | 0006 | | 9. | | | GHQ7 | 0007 | | 7. | | | GHQ8 | 0008 | | 7. | | | GHQ9 | 0009 | | 10. | | | GHQ10 | 0010 | | 9. | | | GHQ11 | 0011 | | 8. | | | GHQ12 | 0012 | | 7. | | | TOTAL | | | 99. | |

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X-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

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**Non-parametric IRT Mokken Analysis STATA loevH command**

. loevH GHQ1-GHQ12 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 GHQ GHQ GHQ Scale loevH 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.

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**(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).

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**(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).

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(1) Rasch model in STATA Estimation 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

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**(2) Rasch model in STATA raschtest**

Ability Expected Group Score parameters std Err. Freq. Score ll

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**Running 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 indicators runmplus y1-y12, type(efa 1 4) Exploratory factor analysis with categorical indicators runmplus y1-y12, type(efa 1 4) categorical(all) Exploratory factor analysis with a mixture of categorical and continuous indicators runmplus y1-y12,type(efa 1 4) categorical(y1 y3 y5 y7 y9 y11) Confirmatory factor analysis with continuous indicators runmplus y1-y6, model(f1 by y1-y3; f2 by y4-y6;)

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**And finally … think useR**

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**IR : irtoys package example plots (from manual)**

Author: Ivailo Partchev

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**Extract 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.

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**(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.

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**(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.

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Psychometrics in R Special volume of the Journal of Statistical Software Volume 20 Multilevel Rasch Correspondence Analysis Rasch Multilevel IRT Multidimensional Rasch Extended Rasch Marginal Maximum Likelihood IRT Mokken scale analysis …

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**Free 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 Imputation Rasch model for which Goodness of Fit is assessed using a parametric Bootstrap version of the Pearson chi-squared.

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**Free software Factor/M-IRT MIRT Factor NOHARM**

Urbano Lorenzo-Seva & Pere J. Ferrando MIRT NOHARM

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**FACTOR //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.

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**Interesting Journals …**

Psychological Assessment Psychological Methods Multivariate Behavioural Research Applied Psychological Measurement Journal of Educational and Behavioural Statistics Structural Equation Modeling Psychometrika Educational and Psychological Measurement

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**Running Mplus www.statmodel.com from Stata runmplus**

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**Running Mplus www.statmodel.com from Stata runmplus**

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**Running Mplus www.statmodel.com from Stata runmplus**

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**Running Mplus www.statmodel.com from Stata runmplus**

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**Running Mplus www.statmodel.com from Stata runmplus**

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**Running Mplus www.statmodel.com from Stata runmplus**

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**Running Mplus www.statmodel.com from Stata runmplus**

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**Running Mplus www.statmodel.com from Stata runmplus**

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**Excellent book chapter (non-technical)**

Application oriented book Assessing Quality of Life in Clinical Trials; Methods and Practice Edition: 2nd Author(s): Peter Fayers; Ron Hays ISBN: see Chapter by Reeve and Fayers Applying item response theory modelling for evaluating questionnaire item and scale properties download for free from

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**££££££££££££££££££££££**

And out there in commerce, money talks…

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**As Test-Taking Grows, Test-Makers Grow Rarer, May 5, 2006, NY Times.**

Psychometrics, one of the most obscure, esoteric and cerebral professions in America …. is now also one of the hottest

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Chapter 7 Sampling and Sampling Distributions

Chapter 7 Sampling and Sampling Distributions

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