Rasch Dichotomous Model vs. One-parameter Logistic Model (1PL 1-PL)

For most practical purposes these models are the same, despite their conceptual differences.

Aspect Rasch Dichotomous Model Item Response Theory:
One-Parameter Logistic Model
Abbreviation Rasch 1-PL IRT, also 1PL
For practical purposes When each individual in the person sample is parameterized for item estimation, it is Rasch. When the person sample is parameterized by a mean and standard deviation for item estimation, it is 1PL IRT.
Motivation Prescriptive: Distribution-free person ability estimates and distribution-free item difficulty estimates on an additive latent variable Descriptive: Computationally simpler approximation to the Normal Ogive Model of L.L. Thurstone, D.N. Lawley, F.M. Lord
Persons, objects, subjects, cases, etc. Person n of ability Bn, or
Person ν (Greek nu) of ability βn in logits
Normally-distributed person sample of ability distribution θ, conceptualized as N(0,1), in probits; persons are incidental parameters
Items, agents, prompts, probes, multiple-choice questions, etc.; items are structural parameters Item i of difficulty Di, or
Item ι (Greek iota) of difficulty δι in logits
Item i of difficulty bi (the "one parameter") in probits
Nature of binary data1 = "success" - presence of property
0 = "failure" - absence of property
1 = "success" - presence of property
0 = "failure" - absence of property
Probability of binary data Pni = probability that person n is observed to have the requisite property, "succeeds", when encountering item i Pi(θ) = overall probability of "success" by person distribution θ on item i
Formulation: exponential form
e = 2.71828
Formulation: logit-linear form
loge = natural logarithm
Local origin of scale: zero of parameter estimatesAverage item difficulty, or difficulty of specified item. (Criterion-referenced) Average person ability. (Norm-referenced)
Item discriminationItem characteristic curves (ICCs) modeled to be parallel with a slope of 1 (the natural logistic ogive)ICCs modeled to be parallel with a slope of 1.7 (approximating the slope of the cumulative normal ogive)
Missing data allowedYes, depending on estimation methodYes, depending on estimation method
Fixed (anchored) parameter values for persons and itemsYes, depending on softwareItems: depending on software. Persons: only for distributional form.
Fit evaluationFit of the data to the model
Local, one parameter at a time
Fit of the model to the data
Global, accept or reject the model
Data-model mismatchDefective data do not support parameter separability in an additive framework. Consider editing the data.Defective model does not adequately describe the data. Consider adding discrimination (2-PL), lower asymptote (guessability, 3-PL) parameters.
Differential item functioning (DIF) detectionYes, in secondary analysisYes, in secondary analysis
First conspicuous appearanceRasch, Georg. (1960) Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.Birnbaum, Allan. (1968). Some latent trait models. In F.M. Lord & M.R. Novick, (Eds.), Statistical theories of mental test scores. Reading, MA: Addison-Wesley.
First conspicuous advocateBenjamin D. Wright, University of ChicagoFrederic M. Lord, Educational Testing Service
Widely-authoritative currently-active proponentDavid Andrich, Univ. of Western Australia, Perth, AustraliaRonald Hambleton, University of Massachusetts
Introductory textbookApplying The Rasch Model.T.G. Bond and C.M. FoxFundamentals of Item Response Theory. R.K. Hambleton, H. Swaminathan, and H.J. Rogers.
Widely used softwareWinsteps, RUMM, ConQuestLogist, BILOG
Minimum reasonable sample size30200 (Downing 2003)

See also: Andrich, D. (2004) Controversy and the Rasch model: A characteristic of incompatible paradigms? Medical Care, 42, 7-16. Reprinted in E.V. Smith & R.M. Smith, Introduction to Rasch Measurement: Theory, Models and Applications. JAM Press, Minnesota. Ch. 7 pp 143-166.

Downing S.M. (2003) Item response theory: applications of modern test theory for assessments in medical education. Medical Education, 37:739-745.


Linacre J.M. (2005). Rasch dichotomous model vs. One-parameter Logistic Model. Rasch Measurement Transactions, 19:3, 1032



Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch
Applying the Rasch Model 2nd. Ed., Bond & Fox Best Test Design, Wright & Stone Rating Scale Analysis, Wright & Masters
Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr.
Statistical Analyses for Language Testers, Rita Green Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement

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