Fred Lord's three-parameter-logistic Item Response Theory (3-PL
IRT) model (Birnbaum, 1968) incorporates an item discrimination
parameter, modeling the slope of the item characteristic curve, and
a lower asymptote parameter modeling "guessing" or, better, "item
guessability". Here is a 3-PL model, written in log-odds format,
with c_{i} as the lower asymptote, a_{i} as the item
discrimination, θ_{n} as the person ability and
b_{i} as the item difficulty:
(1) |
Lord's 4-PL model (Barton & Lord, 1981) incorporates an
upper asymptote parameter for item-specific "carelessness". Here is
a "carelessness" model, written in log-odds format, with
d_{i} as the upper asymptote:
(2) |
4-PL IRT Item Characteristic Curve |
Upper and lower asymptotes are notoriously difficult to estimate, so it appears that Lord abandoned his 4-PL model, and the value of ci in the 3-PL model is, on occasion, imputed from the number of options in a multiple-choice item, instead of being estimated directly from the data. Even the estimation of item discrimination usually requires constraints, such as "a_{i} cannot be negative or too big."
The dichotomous Rasch model, however, provides an opportunity to estimate a first approximation to these parameters. These estimates can be useful in diagnosing whether the behavior they reflect could be distorting the Rasch measures. In the dichotomous Rasch model, c_{i}=0, d_{i}=1 and a_{i}=1. We can, however, treat the Rasch values as starting values in a Newton-Raphson iterative processed apparently intended to find the maximum-likelihood values of each of these parameters, in a context in which all other parameter values are known.
Following Wright & Masters (1982, 72-77), and using the standard approach of first and second derivatives of the log-likelihood of the data with respect to the parameter of interest, we obtain the following Newton-Raphson estimation equations for the first approximations:
Item discrimination (ICC slope):
(3) |
with S.E. = 1/ sqrt ( | ) | (4) |
The Rasch expectation of a_{i} is 1.
A corollary is that, when data fit the dichotomous Rasch model, there is zero correlation between the observation residuals and their generating measure differences.
There is a similar result for polytomous items. The Generalized
Partial Credit can be written:
(5) |
The "generalized" item discrimination (ICC slope), equivalent to a Rasch item discrimination index, is:
(6) |
where | and M_{ni0} = 0 | (7) |
And for the discrimination of polytomous inter-category
"generalized" thresholds:
(8) |
the "generalized" threshold discrimination is:
(9) |
Returning to the dichotomous model, the lower asymptote (guessability) is:
(10) |
where 0 <= c_{i} <= 1
with S.E. = 1/ sqrt | (11) |
The Rasch expectation of c_{i} is 0.
The upper asymptote (carelessness) is:
(12) |
where 0 <= d_{i} <= 1
with S.E. = 1/ sqrt | (13) |
The Rasch expectation of d_{i} is 1.
In practice, it is convenient to use only observations in the lower tail for estimating the lower asymptote, in the center for estimating discrimination, and in the upper tail for estimating the upper asymptote.
John Michael Linacre
Birnbaum A. (1968) Some latent trait models and their uses in inferring an examinee's ability. In F.M. Lord & M.R. Novick, Statistical theories of mental test scores (pp. 395-479). Reading, MA: Addison-Wesley.
Barton M.A. & Lord F.M. (1981) An upper asymptote for the three-parameter logistic item-response model. Princeton, N.J.: Educational Testing Service.
Discrimination, Guessing and Carelessness Asymptotes: Estimating IRT Parameters with Rasch. Linacre J.M. … Rasch Measurement Transactions, 2004, 18:1 p.959-960
Discrimination, Guessing and Carelessness Asymptotes: Estimating IRT Parameters with Rasch, Linacre J.M. … Rasch Measurement Transactions, 2004, 18:1 p.959-960
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