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where Ui is the response variable for item i and Pr Ui 1 j  is the probability of a correct response on the item by an examinee with ability level  2 1; 1 In addition, parameter bi 2 1; 1 can be interpreted as the dif culty item i, parameter ai 2 0; 1 as its discriminating power, and ci 2 0; 1 as a parameter that represents the probability of success for an examinee who guesses without any knowledge Even for this relatively simple dichotomous response model, not too long ago the scoring of examinees by estimating the value of the ability parameter  from their responses was a timeintensive activity The same held for the use of its item parameters in assembling a test from a pool of items Nowadays, however, we are able to execute such procedures in real time As a consequence, we can produce interim estimates of an examinee s ability at any time during the test, use these estimates in re ned methods of item selection, diagnose the response behavior of an examinee while (s)he is on line, and immediately update item-parameter estimates after each item in continuous computer-based testing (van der Linden & Pashley, 2000) This is not only possible for the model in (21) but also for models for more involved response formats and models for response times The message in this chapter is that, though the new stimulus and response formats made possible by the computer have been breathtaking, the true revolution in CBT may very well be this marriage between the current computing power of PCs and recent developments in psychometric modeling In a fortunate twist of history, computing power became plentiful and cheap in the 1990s, just when fundamental models in psychometrics had matured and become statistically tractable for use on a routine basis Another fortunate coincidence is that the same type of modeling now helps us solve various practical problems that did not exist in the pre-computer era in testing, for example, the need to minimize the number of items that are stolen and to eliminate other forms of security breaches In this chapter, we will illustrate our claim by reviewing a few applications of psychometric modeling to problems in CBT We begin with an application for computer-based tests with an adaptive format For a description of computerized adaptive tests (CATs), see, for instance, Wainer (2000) The problem we address is how to build tests on line that are not only based on the principle of adaptation but also meet the numerous content and other statistical speci cations we may have in mind The next two applications deal with threats to test integrity and are for any type of individualized computer-based testing In one application, we show how to build individual tests without overexposing some of the items in the pool In the other, we use response times recorded during testing to diagnose examinee behavior on line for possible aberrances due to aws in the test design or cheating by the examinee The nal application deals with the problem of item shortage in adaptive testing One of the proposed solutions is item cloning, that is, generating a family of new items from an existing item or an item structure We will discuss a psychometric model for calibration and adaptive testing from a pool with such item families
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A fundamental dilemma in testing is that if we knew the examinee s ability level, , we could easily assemble a test that would be statistically optimal, but the very reason we need a test is that the examinee s ability level is unknown How to assemble a test if  were known follows directly from quantities known in IRT as the item and test information function For the model in (21), Figure 21 gives the information functions for ve items Their curves show
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