Dichotomous Rasch Model derived from Counting Right Answers:
Raw Scores as Sufficient Statistics

In statistical terminology, "sufficient" means "all the information that this dataset has about ...". So, from a Rasch perspective, the "raw score" has all the information there is in the dataset about the "ability" of the respondent. This is also the Classical Test Theory perspective. But this is not the 2-PL or 3-PL perspective. They say that the "pattern of responses" has all the information there is in the dataset about the "ability" of the respondent.

1. Person n of ability Bn responds to test items (i) with difficulties (Di) with scored (0,1) responses (Xni). For the raw score Rn = ΣXni to be sufficient for estimating ability, indeed, for counting right answers to be useful at all, Rn must extract ability Bn out of (Xni).

2. Bayes says that the probability of person n's response vector, given person ability Bn and items of difficulty (Di), is the probability of the response vector given the raw score Rn multiplied by the probability of the raw score given the ability:

P{(Xni)|Bn,(Di)} = P{(Xni)|Rn,(Di)} * P{Rn|Bn,(Di)}

3. If Pni = f(n,i) is the probability that n succeeds on i, then we can obtain the probability of person n's method of obtaining a score of Rn, and also all other methods of obtaining the same score.


4. Rn points unequivocally to ability Bn only if P{(Xni)|Rn, (Di)}, the probability of the particular responses given the raw score, is entirely free from variation due to Bn. Thus the remainder must be locally independent of Bn so that it does not matter which particular (Xni) are 1's and which are 0's provided that ΣXni = Rn. Thus the effect of Bn on the (Xni) can be factored out of (3) and cancelled.

[This is saying that the set of response vectors are locally independent given the person abilities and the item difficulties. Georg Rasch called this property "specific objectivity".]

5. A sufficient factoring is obtained by asserting that

Cni = Bn + Di,

Pni = Bn / Cni, and 1- Pni = Di / Cni, then



so that


is seen to be free of Bn.

6. Thus a sufficient parameterization is


Reparameterizing,


produces the Rasch model, showing that this form of f(n,i) supports the use of Rn for estimating the ability of person n.

7. The necessity of this formulation can be demonstrated by dividing the numerator into the denominator in (3) to produce the following term in the denominator, which must not vary with Bn:


8. In T there are the same number of Pni and Pnj terms in each product, because the raw score is invariant across the summation. The general condition for T not to vary with Bn, (apart from irrelevant specifics, such as when Pni and Pnj are constants), is that


which takes us to step (2) of Model Necessary for a Thurstone Scale, RMT 2:1 p. 9-10, and thence to the Rasch model as the necessary f(n,i).

Benjamin D. Wright

Note: Since most of Classical Test Theory (CTT) regards the raw scores as sufficient for decision-making, CTT implicitly assumes that the Rasch model holds for their data.


Dichotomous Rasch Model derived from Counting Right Answers: Raw Scores as Sufficient Statistics. Wright BD. … Rasch Measurement Transactions, 1989, 3:2 p.62


  1. The Rasch Model derived from E. L. Thorndike's 1904 Criteria, Thorndike, E.L.; Linacre, J.M. … 2000, 14:3 p.763
  2. Rasch model derived from consistent stochastic Guttman ordering, Roskam EE, Jansen PGW. … 6:3 p.232
  3. Rasch model derived from Counts of Right and Wrong Answers, Wright BD. … 6:2 p.219
  4. Rasch model derived from counting right answers: raw Scores as sufficient statistics, Wright BD. … 1989, 3:2 p.62
  5. Rasch model derived from Thurstone's scaling requirements, Wright B.D. … 1988, 2:1 p. 13-4.
  6. Rasch model derived from Campbell concatenation: additivity, interval scaling, Wright B.D. … 1988, 2:1 p. 16.
  7. Dichotomous Rasch model derived from specific objectivity, Wright BD, Linacre JM. … 1987, 1:1 p.5-6

Please help with Standard Dataset 4: Andrich Rating Scale Model



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

To be emailed about new material on www.rasch.org
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from Rasch.org

www.rasch.org welcomes your comments:

Your email address (if you want us to reply):

 

ForumRasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

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
Sept. 15-16, 2017, Fri.-Sat. IOMC 2017: International Outcome Measurement Conference, Chicago, jampress.org/iomc2017.htm
Oct. 13 - Nov. 10, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Oct. 25-27, 2017, Wed.-Fri. In-person workshop: Applying the Rasch Model hands-on introductory workshop, Melbourne, Australia (T. Bond, B&FSteps), Announcement
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
April 13-17, 2018, Fri.-Tues. AERA, New York, NY, www.aera.net
May 25 - June 22, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 29 - July 27, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 10 - Sept. 7, 2018, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 12 - Nov. 9, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
The HTML to add "Coming Rasch-related Events" to your webpage is:
<script type="text/javascript" src="https://www.rasch.org/events.txt"></script>

 

The URL of this page is www.rasch.org/rmt/rmt32e.htm

Website: www.rasch.org/rmt/contents.htm