LLTM - Linear Logistic Test Model

The Linear Logistic Test Model, LLTM, with integer "basic parameter" weights can be modeled with Facets easily.


For non-integer weights, multiply all the weights for all the items by the same value until all the weights approximate integers. Then use the integer weights.


A standard LLTM model for two "basic parameters" (components of item difficulty) looks like

 Bn - W1i * P1 - W2i *P2 -> Xni

where Bn is the ability of person n

P1 and P2 are the difficulties of basic parameters 1 and 2.

W1i and W2i are the pre-assigned weights of P1 and P2 in item i.


The standard LLTM can be estimated with Facets if W1i and W2i are integers. The Facets model is

Bn - P1 - P1 - ...  - P2 - P2 - ....  -> Xni

with W1i "P1" facets and W2i "P2" facets


Here is the procedure with Facets:


1. The "basic parameters", BP, the components of the items, are elements of a "basic parameter" facet.


2. The items are a dummy facet. The item labels include the element numbers of the BP elements.


3. There are as many BP facets in the data as the largest sum of all the integer BP weights for any item = BPL. For example, BPL= 7


4. The Facets specifications:

Facets = 9  ; Person+item+BPL facets in the data

Entered-in-data = 1, 2, 3, 3, 3, 3, 3, 3, 3  ; BPL=7 facets in the data for facet 3

Models = ?, ?, ?,?,?,?,?,?,?, D ; 2+BPL=9 facets


1, Persons

1= ...



2, Items, D ; a dummy facet

1=1124000 ; the BP elements for item 1 are 2*BP1 + BP2 + BP4. 0 = no BP element.

2=1222344 ; the BP elements for item 2 are BP1+3*BP2+BP3+2*BP4




3, Basic Parameters (BP)

1 = basic parameter one


4 = basic parameter four



3,2,1,1 ; the element number for facet 3 in the data (BP) is the first character in the label for the element in facet 2 (item)








; only facets 1 and 2 are needed in the data, Facets 3-9 are imputed by Dvalues=


1,1-10, 0,0,1,1,1,0,1,1,0,1 ; person 1, items 1-10, dichotomous scored responses



5. In the output,

The BP estimates are the estimates for the "basic parameters".

The item displacement is the difference between the LLTM estimate and the item estimate from the data.

The sum of the weighted BP for an item is -"sum of the measures" in the Residualfile= for a Person with measure 0.00, or add in the Person measure to the -"sum of the measures".


Fischer, G. H., 'The linear logistic test model as an instrument in educational research', Acta Psychologica 37 (1973), 359-74.



Simple LLTM example:


LLTM is similar to a Facets-style model. It models the item difficulty as a combination of components. Each component has a difficulty and components are weighted to reflect their contribution to the item. In the simplest case, the weighting is 0 = does not contribute, 1 = does contribute.


For a Facets analysis, let's say there are 4 components. In the Facets "Labels=" specification, we set up a "component" facet with 4 elements. In the data, we have 4 component facets, all referencing the one Labels= component facet.


The Facets specification file:

Facets = 6  ; person, item, 4 components facets in data

Entered-in-data = 1, 2, 3, 3, 3, 3 ; all 4 components facets in the data reference the same "component" facet

Models= ?,?,?,?,?,?, D ; dichotomous right/wrong items


1, Persons

1 = Jose


23 = Maria



2, items, D ; this is a dummy facet for specifying the components in each item

1 = 1300 item1 with active components 1, 3. Inactive components specified with 0.


20 = 2340 item 20 with active components 2,3,4


3, components of item difficulty

1= recognition

2= comprehension

3= computation

4= realization



2, 1-20 ; facet 2 in the data: all persons respond to a 20 item test

3, 2, 1 ; facet 3 in the data: decode the components from the item labels

4, 2, 2

5, 2, 3

6, 2, 4



23,  1, 0, 1, ... ; success by person 23, Maria, on item 1, failure on item 2, etc. of the 20 item test


Help for Facets Rasch Measurement Software: www.winsteps.com Author: John Michael Linacre.

For more information, contact info@winsteps.com or use the Contact Form

Facets Rasch measurement software. Buy for $149. & site licenses. Freeware student/evaluation download
Winsteps Rasch measurement software. Buy for $149. & site licenses. Freeware student/evaluation download

State-of-the-art : single-user and site licenses : free student/evaluation versions : download immediately : instructional PDFs : user forum : assistance by email : bugs fixed fast : free update eligibility : backwards compatible : money back if not satisfied
Rasch, Winsteps, Facets online Tutorials


Forum Rasch Measurement Forum to discuss any Rasch-related topic

Click here to add your email address to the Winsteps and Facets email list for notifications.

Click here to ask a question or make a suggestion about Winsteps and Facets software.

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 with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments, George Engelhard, Jr. & Stefanie Wind 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
Winsteps Tutorials Facets Tutorials Rasch Discussion Groups


Coming Rasch-related Events
Jan. 5 - Feb. 2, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 10-16, 2018, Wed.-Tues. In-person workshop: Advanced Course in Rasch Measurement Theory and the application of RUMM2030, Perth, Australia (D. Andrich), Announcement
Jan. 17-19, 2018, Wed.-Fri. Rasch Conference: Seventh International Conference on Probabilistic Models for Measurement, Matilda Bay Club, Perth, Australia, Website
Jan. 22-24, 2018, Mon-Wed. In-person workshop: Rasch Measurement for Everybody en español (A. Tristan, Winsteps), San Luis Potosi, Mexico. www.ieia.com.mx
April 10-12, 2018, Tues.-Thurs. Rasch Conference: IOMW, New York, NY, www.iomw.org
April 13-17, 2018, Fri.-Tues. AERA, New York, NY, www.aera.net
May 22 - 24, 2018, Tues.-Thur. EALTA 2018 pre-conference workshop (Introduction to Rasch measurement using WINSTEPS and FACETS, Thomas Eckes & Frank Weiss-Motz), https://ealta2018.testdaf.de
May 25 - June 22, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 27 - 29, 2018, Wed.-Fri. Measurement at the Crossroads: History, philosophy and sociology of measurement, Paris, France., https://measurement2018.sciencesconf.org
June 29 - July 27, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
July 25 - July 27, 2018, Wed.-Fri. Pacific-Rim Objective Measurement Symposium (PROMS), (Preconference workshops July 23-24, 2018) Fudan University, Shanghai, China "Applying Rasch Measurement in Language Assessment and across the Human Sciences" www.promsociety.org
Aug. 10 - Sept. 7, 2018, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Sept. 3 - 6, 2018, Mon.-Thurs. IMEKO World Congress, Belfast, Northern Ireland www.imeko2018.org
Oct. 12 - Nov. 9, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com


Our current URL is www.winsteps.com

Winsteps® is a registered trademark

Concerned about aches, pains, youthfulness? Mike and Jenny suggest Liquid Biocell