Item Information Function

In item response theory, the item information function is a measure of how much statistical information a test item provides. Item information is a function of θ, the ability, proficiency, skill, or trait measured by the examinee’s responses to the test items. Figure 1 shows the information functions [Page 899]for four items. Items 1 and 2 are more informative at low values of θ than high values, and Items 3 and 4 are most informative at higher values of θ.

Figure 1 Information for four items. Items 1 and 2 have the same difficulty but Item 1 is more discriminating. Items 3 and 4 are more difficult than Items 1 and 2. Item 4 has the same parameters as Item 3, except that the lower asymptote is higher for Item 4.

The amount of information provided by an item, relative to θ, depends on the item’s parameters. The items in Figure 1 are dichotomous items (scored 0 or 1), so they have up to three item parameters (plus an examinee parameter) describing the probability of correct response:

$$P_i({\theta }_j)=c_i+(1-c_i)\frac{e^{D{a}_i({\theta }_j-b_i)}}{1+e^{D{a}_i({\theta }_j-b_i)}},$$

where Pi(θj) indicates the probability of a correct response to item i from examinee j with ability θj, ai is the item discrimination, bi is the item difficulty, and ci is the lower asymptote. D is an optional scaling parameter which either equals 1 (in which case it can be dropped) or 1.7; if D = 1.7 the a parameters will be approximately on the normal (probit) metric. The lower asymptote is the probability of correct response for examinees with very low ability. For example, if low-ability examinees perceive all of the distractors to be about as likely as the correct answer, the c parameter might be equal to random guessing. The subscripts are often omitted.

For the two-parameter logistic model, all lower asymptotes are zero, so the c parameter can be dropped from the model. For the one-parameter logistic (1PL) model, all item discrimination parameters are equal. The Rasch model is mathematically equivalent to the 1PL model; the a parameter is dropped from the model (all a = 1), and the item discrimination is displaced onto the variance of θ: Tests composed of more discriminating items have greater θ variance.

The peak of the item information function occurs at or just above the item difficulty, depending on the model. In Figure 1, Items 3 and 4 are more difficult than Items 1 and 2. The information function steepens as the a parameter increases. Comparing two items with the same b and c parameters but different a parameters, the item with the higher a parameter will have more information but in a narrower range of θ. In Figure 1, for example, b = −1.5 and c = .05 for items 1 and 2, but a = 1.4 for Item 1 compared to 0.8 for Item 2. Thus, Item 1 has more information for low θ values, but slightly less information than Item 2 for very high θ values. The c parameter dampens the information function, especially for hard items. Items 3 and 4 have the same a and b parameters (a = 1.2, b = 1.5), but the c parameter = .05 [Page 900]for Item 3 compared to .25 for Item 4. Thus, Item 4 provides less information.

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