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Even if a CBT program uses an effective exposure-control method, it may still be prudent to have routine checks for possible remaining irregularities during
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COMPUTER-BASED TESTING AND THE INTERNET
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testing For an IRT model such as the one in (21), these checks take the form of person- t analysis, that is, checks for individual examinees with unexpected responses in their score vector For example, a correct response on a dif cult item for an individual of low ability may point to an item known before the test, or to copying of the response from someone else For a review of the available statistical techniques for detecting unexpected responses, see Meijer and Sijtsma (1995) A common feature of all these techniques, however, is the lower statistical power of detecting cheating than desirable due to the binary nature of the responses For a well-designed test with the dif culties of the items near the true ability of the examinee, the probability of a correct response approximates 050 As a consequence, correct and incorrect responses become equally probable, and any statistical test of cheating tends to lose its power entirely This result is particularly likely for adaptive testing, where the dif culty of the items is adapted to the examinee s ability level It seems therefore attractive to look for other potential sources of information on examinee behavior that could be diagnosed for possible irregularities An obvious source are the response times by the examinees, which are recorded automatically recorded in CBT These times are continuous and not binary as are the responses themselves Also, if the test is assembled to have maximum information at the examinee s estimated or anticipated ability level, this selection criterion does not involve any detriment to the power of a test for diagnosing response behavior based on response times
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Lognormal Model for Response Times
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The approach we follow is analogous to the one typical of a statistical test based on the responses only That is, analogous to (21) we model the response times by an examinee on an item by a probability distribution with examinee and item parameters, estimate the item parameters during pretesting and check the model for its validity, estimate the examinee s speed, predict the times we expect the examinee to spend on the items, and use a statistical test to determine whether the actual times deviate too much from the prediction Of course, to have a convincing proof of cheating, we must have such additional evidence as reports by test proctors, seating plan or material con scated during the test As an example, suppose that we have evidence that an examinee might be part of a network that tries to memorize items in a testing program A statistical test on item preknowledge would then check this examinee for the combination of correct responses on items for which this person has low probability of success and response times that are much shorted than expected given this person s ability and speed estimates A useful response-time model is the lognormal model, which models the logarithm of the response time as a normal distribution The logarithmic transformation allows for the typically skewness in response-time distributions;
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MODEL-BASED INNOVATIONS
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because time is on a non-negative scale, its distribution tends to have a short lower tail and a longer upper tail Lognormal distributions for response times have been applied used successfully by Schnipke and Scrams (1997), Thissen (1982), van der Linden, Scrams, and Schnipke (1999), van der Linden and van Krimpen-Stoop (2003) We will use a parameterization for this distribution similar to the one for the two-parameter logistic version of the response model in (21) proposed by van der Linden (in press) Let Tij denote the response time by examinee j 1; ; J on item i 1; ; I The proposed model for the distribution of Tij is   i 1 2 f tij ; j ; i ; i p exp i ln tij i j 2 tij 2 2:20
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which is the probability density for the normal distribution, with ln tij substituted for tij on the original scale (eg time measured in seconds) The model has an parameter j which represents the speed at which this examinee j responds to the items in the test A larger value of j implies a larger probability for a shorter time In addition, the model has two parameters, i and i , which parallel the item dif culty and discrimination parameters in (21) Parameter i represents the amount of time item i tends to require from the examinees The larger this parameter, the larger the amount of time each examinee tends to spend on the item It is therefore appropriate to call i the time-intensity parameter for item i Parameter i can be interpreted as a discrimination parameter A larger value for i means less dispersion for the response-time distribution of each examinee on item i, and hence better discrimination by the item between the distributions of two persons with levels of speed slightly above and below i The model in (220) does not have the equivalent of a guessing parameter; it does not need to have one because response times have a natural lower limit at zero The scale for parameters j and i in the model is not yet determined; an increase in the former can always be compensated by an increase in the latter We remove this indeterminacy by imposing the following identi ability constraint:
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