COMPUTER-BASED TESTING AND THE INTERNET in .NET

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COMPUTER-BASED TESTING AND THE INTERNET
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Item Pool
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Figure 26 Venn diagram of a pool of cloned items, with the size of the sets representing the variability of the item parameters
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Glas and van der Linden (2003) show that the second part of this conclusion is not necessary, provided we model the situation appropriately Their model is based on two-stage (or strati ed) selection of a test from a pool of cloned items The steps are (1) selection of the families of items in the pool from which an item is to be administered, and (2) random sampling of an items from each selected family A graphical representation of the selection process is given by the Venn diagram in Figure 26 The selection of the families is assumed to be based on a statistical criterion; how this can be done is discussed below However, it is important that the selection of the items from the families is random; the model in the next section is based on this assumption In the hypothetical case depicted in Figure 26, we have much more variability between than within the families Intuitively, the proportion of the total variability that is between the families is an upper bound to the gain in the precision of ability estimation that can be made by optimizing the rst step in the item selection procedure using a statistical criterion The proportion that is within the families determines the loss in precision due to random item sampling in the second step
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Two-Level Model
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The proposed item-selection procedure suggests a two-level IRT model At the rst level, each of the individual items in each family is represented by a regular response model, such as the 3PL model in (21) At the second level, the distribution of the item parameters in each family is modeled These distributions are allowed to vary across families Speci cally, suppose we have a pool with families p 1; ; P and individual items ip 1; ; IP for family p The rst-level model is the 3PL model in (21), which describes the probability of success on item ip as a function of ability level  and the item parameters For brevity, we denote the parameters as a vector nip aip ; bip ; cip The distribution of the responses of examinee j to
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MODEL-BASED INNOVATIONS
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item ip from family is thus determined by the probabilities Pr Uip j 1 p j ; nip 2:23
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given in (21) The second-level model describes the distribution of the parameters nip by a separate trivariate normal distribution for each family p nip $ MVN lp ; Rp ; 2:24
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where lp and Rp are the vector with the mean values of the item parameters and their covariance matrix in family p, respectively Observe again that we have a different normal distribution of the item parameters for each family This property allows us to deal with the facts that some item families are more dif cult or discriminating on average than others, have less variability in their values for the dif culty or guessing parameter, and so on Item-pool calibration now takes the form of estimating the parameters lp and Rp of each family in the pool These parameters are estimated from the response data of examinees to items randomly sampled from the families Procedures for marginal maximum likelihood estimation of the parameters are given by Glas and van der Linden (2003); for more informative Bayesian estimation procedures, see Glas and van der Linden (in press)
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Selection of Families of Items
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It is obvious that the selection of an entire item family for a test cannot be based on the information function for a single item de ned in (22), but because the actual items in the test are drawn randomly from the families selected it makes sense to select the family using the average of the information functions of the items in it Adapting our notation to the case of families of items, the information function for item ip in family p, with vector of item parameters nip , is denoted as Iip ; nip : 2:25
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The expectation or average of this information function over the item parameters in family p is Ip  Z Iip ; nip p np jlp ; Rp dnp : 2:26
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If the test administered by the computer is xed, it can be assembled optimally from the pool using a test assembly model such as the one in (23) (29) The model then has to be formulated at the level of the families rather than the individual items That is, we then de ne decision variables xp , which take
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