Comparing the Normal Ogive and Logistic Item Characteristic Curves
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In item response theory, the relationship between a latent ability () and the probability of a correct response (
) on a test item is modelled by an item characteristic curve. This Demonstration plots the item characteristic curve of a single dichotomous item under two different models: the normal ogive model and the logistic model. The parameters
,
, and
represent item properties related to discrimination, difficulty, and guessing. The constant
is used to scale the logistic curve. Notice that the two curves are nearly identical when
.
Contributed by: Vincent Kieftenbeld (March 2011)
Open content licensed under CC BY-NC-SA
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The probability that a person with ability level gives a correct response (
) to an item with discrimination parameter
, difficulty parameter
, and pseudo-guessing parameter
is modelled in the normal ogive model as
.
Alternatively, in the three-parameter logistic model,
.
The constant D is used to scale the logistic curve and represents the relationship between logits and probits. When , the models agree closely; that is, 1 logit is approximately equal to 1.7 probit. In fact,
minimizes the maximum difference between the normal ogive and logistic curves.
Reference:
G. Camilli, "Origin of the Scaling Constant d=1.7 in Item Response Theory," Journal of Educational and Behavioral Statistics, 19(3), 1994 pp. 293–295.
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