Rasch Dichotomous Model vs Birnbaum 3-PL Three-Parameter Logistic Model

Headings:	Rasch Dichotomous Model	Birnbaum 3-PL Model
Item parameters	1 item parameter	3 item parameters
A-1 Binomial probability of person n succeeding on dichotomous item i:	log_e [P/(1-P)] = B_n - D_i	log_e [(P-c_i)/(1-P)] = 1.7 a_i (t - b_i)
A-2 Item characteristic curve (ICC):	Monotonic ogive with slope and lower asymptote to be estimated	Logistic ogive with specified slope and asymptotes
A-3 Person ability:	B_n measures person n B-distribution estimated from data	Subjects assumed to be sampled from N(0,1) or other arbitrary distribution
A-4 Item difficulty:	D_i calibrates item i	b_i estimates ICC inflection point
A-5 Item discrimination:	Specified at constant or unity. Misfit detects variation	a_i estimates ICC slope at b_i. Sample dependent
A-6 Guessing success on item by low ability persons:	Preset at constant or zero. Person fit detects lucky guessers	c_i estimates ICC lower asymptote. Sample dependent
B-1 Motivation:	Measurement construction	Data description
B-2 Ruled by:	Theory and intention	Data and chance
B-3 Substance of latent variable:	Definitive. Items uniquely ordered	Ambiguous. Item order varies with ability level because a_i and c_i variation causes ICCs to cross
B-4 Unidimensionality:	Specified by model	Assumed
B-5 Local independence:	Verified by fit analysis	Not evaluated
B-6 Sufficient statistics:	Unweighted raw scores	Weighted raw scores if and only if weights known a priori
B-7 Unit of calibration:	Log-odds unit (logit)	Normit-scaled logits (logit/1.7)
C-1 Estimation	Raw scores are sufficient. No arbitrary constraints needed	No sufficient statistics. Arbitrary constraints required to control parameter interactions
C-2 Standard errors:	Well defined	Skewed by arbitrary constraints
C-3 Fit statistics:	Based on asymptotic distributions of responses	Clouded by parameter interactions
C-4 Gross misfit between model and data:	Fit statistics identify invalid data and guide diagnosis and remediation	Hidden by over-parameterization and arbitrary constraints required for estimation
C-5 Person diagnosis and quality control:	Guided by individual person fits and specific item response residuals	Person estimates defined to be random events!?
C-6 Item diagnosis and quality control:	Guided by individual item fits and specific person response residuals	Hidden by over-parameterization and arbitrary constraints required for estimation
C-7 Random guessing: Response set: Scanning error:	Identified by misfit which cues remediation or elimination of error	Increases c_i and decreases a_i of whatever item encounters unexpected successes
C-8 Item miskeying: Many correct options: No correct options:	Identified by misfits which cue remediation or elimination of errors	Decreases unestimated upper asymptote. Decreases a_i
C-9 Duplicate test item:	Detected by model overfit	Increases a_i. Seems to improve test!?
C-10 Item bias: Different item function:	Size and significance estimated from person group residuals	Not detectable. Requires additional analysis
D-1 Missing data:	No problem	Biases estimates
D-2 Minimum useful data:	4 items by 10 persons	Said to be at least 1000 persons
D-3 Typical stable data:	20 items by 200 persons	Does not exist
D-4 Common-item equating: Item Banking: Computer-adaptive tests:	Straightforward	Impossible, unless b_i assumed to dominate (i.e. Rasch model approximated)
D-5 Common-person equating:	Straightforward	Only if person distributions match
D-6 Weighting to combine items of differing discriminations	Model holds when weights decided rather than estimated. Weight validity assessed by fit	When a_i pre-set, then approximates weighted Rasch analysis
D-7 Polytomous data:	Solved by Rating Scale model	Not addressed
D-8 Judge intermediation:	Solved by Facets model	Not addressed

Headings:

Rasch Dichotomous Model

Birnbaum 3-PL Model

Item parameters

1 item parameter

3 item parameters

A-1 Binomial probability of person n succeeding on dichotomous item i:

log_e [P/(1-P)] = B_n - D_i

log_e [(P-c_i)/(1-P)] = 1.7 a_i (t - b_i)

A-2 Item characteristic curve (ICC):

Monotonic ogive with slope and lower asymptote to be estimated

Logistic ogive with specified slope and asymptotes

A-3 Person ability:

B_n measures person n B-distribution estimated from data

Subjects assumed to be sampled from N(0,1) or other arbitrary distribution

A-4 Item difficulty:

D_i calibrates item i

b_i estimates ICC inflection point

A-5 Item discrimination:

Specified at constant or unity. Misfit detects variation

a_i estimates ICC slope at b_i. Sample dependent

A-6 Guessing success on item by low ability persons:

Preset at constant or zero. Person fit detects lucky guessers

c_i estimates ICC lower asymptote. Sample dependent

B-1 Motivation:

Measurement construction

Data description

B-2 Ruled by:

Theory and intention

Data and chance

B-3 Substance of latent variable:

Definitive. Items uniquely ordered

Ambiguous. Item order varies with ability level because a_i and c_i variation causes ICCs to cross

B-4 Unidimensionality:

Specified by model

Assumed

B-5 Local independence:

Verified by fit analysis

Not evaluated

B-6 Sufficient statistics:

Unweighted raw scores

Weighted raw scores if and only if weights known a priori

B-7 Unit of calibration:

Log-odds unit (logit)

Normit-scaled logits (logit/1.7)

C-1 Estimation

Raw scores are sufficient. No arbitrary constraints needed

No sufficient statistics. Arbitrary constraints required to control parameter interactions

C-2 Standard errors:

Well defined

Skewed by arbitrary constraints

C-3 Fit statistics:

Based on asymptotic distributions of responses

Clouded by parameter interactions

C-4 Gross misfit between model and data:

Fit statistics identify invalid data and guide diagnosis and remediation

Hidden by over-parameterization and arbitrary constraints required for estimation

C-5 Person diagnosis and quality control:

Guided by individual person fits and specific item response residuals

Person estimates defined to be random events!?

C-6 Item diagnosis and quality control:

Guided by individual item fits and specific person response residuals

Hidden by over-parameterization and arbitrary constraints required for estimation

C-7 Random guessing: Response set: Scanning error:

Identified by misfit which cues remediation or elimination of error

Increases c_i and decreases a_i of whatever item encounters unexpected successes

C-8 Item miskeying: Many correct options: No correct options:

Identified by misfits which cue remediation or elimination of errors

Decreases unestimated upper asymptote. Decreases a_i

C-9 Duplicate test item:

Detected by model overfit

Increases a_i. Seems to improve test!?

C-10 Item bias: Different item function:

Size and significance estimated from person group residuals

Not detectable. Requires additional analysis

D-1 Missing data:

No problem

Biases estimates

D-2 Minimum useful data:

4 items by 10 persons

Said to be at least 1000 persons

D-3 Typical stable data:

20 items by 200 persons

Does not exist

D-4 Common-item equating: Item Banking: Computer-adaptive tests:

Straightforward

Impossible, unless b_i assumed to dominate (i.e. Rasch model approximated)

D-5 Common-person equating:

Straightforward

Only if person distributions match

D-6 Weighting to combine items of differing discriminations

Model holds when weights decided rather than estimated. Weight validity assessed by fit

When a_i pre-set, then approximates weighted Rasch analysis

D-7 Polytomous data:

Solved by Rating Scale model

Not addressed

D-8 Judge intermediation:

Solved by Facets model

Not addressed

Rasch Dichotomous Model vs Birnbaum 3-PL Three-Parameter Logistic Model. Wright BD. … Rasch Measurement Transactions, 1992, 5:4 p.178

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 3rd. 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	Rasch models for measurement, David Andrich	Constructing Measures, Mark Wilson	Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish:	Análisis de Rasch para todos, Agustín Tristán	Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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 3rd. 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

Rasch models for measurement, David Andrich

Constructing Measures, Mark Wilson

Rasch Analysis in the Human Sciences, Boone, Stave, Yale

in Spanish:

Análisis de Rasch para todos, Agustín Tristán

Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

Forum

Rasch Measurement Forum to discuss any Rasch-related topic

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.

Coming Rasch-related Events
May 17 - June 21, 2024, Fri.-Fri.	On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 12 - 14, 2024, Wed.-Fri.	1st Scandinavian Applied Measurement Conference, Kristianstad University, Kristianstad, Sweden http://www.hkr.se/samc2024
June 21 - July 19, 2024, Fri.-Fri.	On-line workshop: Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 5 - Aug. 6, 2024, Fri.-Fri.	2024 Inaugural Conference of the Society for the Study of Measurement (Berkeley, CA), Call for Proposals
Aug. 9 - Sept. 6, 2024, Fri.-Fri.	On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 4 - Nov. 8, 2024, Fri.-Fri.	On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 17 - Feb. 21, 2025, Fri.-Fri.	On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 16 - June 20, 2025, Fri.-Fri.	On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 20 - July 18, 2025, Fri.-Fri.	On-line workshop: Rasch Measurement - Further Topics (E. Smith, Facets), www.statistics.com
Oct. 3 - Nov. 7, 2025, Fri.-Fri.	On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com