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In this work we obtain boundedness results for fractional operators associated with Schr\"odinger operators $\ \mathcal{L}=-\Delta+V$ on weighted variable Lebesgue spaces. These operators include fractional integrals and their respective commutators. Particularly, we obtain weighted inequalities of the type $L^{p(\cdot)}$-$L^{q(\cdot)}$ and estimates of the type $L^{p(\cdot)}$-Lipschitz variable integral spaces. For this purpose, we developed extrapolation results that allow us to obtain boundedness results of the type described above in the variable setting by starting from analogous inequalities in the classical context.

In social science research there are a number of instruments that use a rating scale such as a Likert response scale. For a number of reasons, a respondent's response vector may not contain responses to each item. This study investigated the effect on a respondent's location estimate when a respondent is presented an item, has ample time to answer the item, but decides not to respond to the item. For these situations, different strategies have been developed for handling missing data. In this study, four different approaches for handling missing data were investigated for their capability to mitigate the effect of omitted responses on person location estimation. These methods included ignoring the omitted response, selecting the midpoint response category, Hot-decking, and a likelihood-based approach. A Monte Carlo study was performed and the effect of different levels of omission on the simulees location estimates was determined. Results show that the Hot-decking procedure performed the best of the methods examined. Implications for practitioners were discussed. (Contains 6 figures and 10 references.) (Author/SLD)

This study investigated the effect on examinees' ability estimate under item response theory (IRT) when they are presented an item, have ample time to answer the item, but decide not to respond to the item. Simulation data were modeled on an empirical data set of 25,546 examinees that was calibrated using the 3-parameter logistic model. The study was conducted in three phases. Phase 1 was an exploratory study comparing the various estimation methods under different conditions. Methods compared were the Biweight, expected a posteriori (EAP), and maximum likelihood estimation (MLE) methods. Phase 2, based on Phase 1 results, examined a modified EAP approach. The third phase examined the use of R. Bock's 1972 nominal response (NR) model for handling omission data. Results seem to indicate that omits should not be treated as incorrect. It also appears that ignoring omits can have a greater impact on thetas using certain estimation approaches (e.g., MLE) than with others. To the extent that a model with a pseudo-guessing parameter more accurately describes the data than a model without one, then the use of the NR model may not produce results comparable to those seen here. (Contains 5 tables, 10 figures, and 14 references.) (SLD)

Differential item functioning (DIF) may be defined as an item that displays different statistical properties for different groups after the groups are matched on an ability measure. For instance, with binary data, DIF exists when there is a difference in the conditional probabilities of a correct response for two manifest groups. This paper suggests that the occurrence of DIF can be explained by recognizing that the observed data do not reflect a homogeneous population of individuals, but are a mixture of data from multiple latent populations or classes. This conceptualization of DIF hypothesizes that when one observes DIF using the current conceptualization of DIF, it is only to the degree that the manifest groups are represented in the latent classes in different proportions. A Monte Carlo study was conducted to compare various approaches to detecting DIF under this formulation of DIF. Results show that as the latent class proportions became more equal, the DIF detection methods identification rates approached null condition levels. (Contains 6 tables, 3 figures, and 27 references.) (Author/SLD)

This study extended item parameter recovery studies in item response theory to the nominal response model (NRM). The NRM may be used with computerized adaptive testing, testlets, demographic items, and items whose alternatives provide educational diagnostic information. Moreover, with the increasing popularity of performance-based assessment, the use of polytomous item response theory models, in general, and the NRM in particular, will more than likely see increased application. Establishing guidelines for reasonable item parameter estimation was seen as fundamental to the use of the NRM. Factors studied through simulation were the sample size ratio, the latent ability distribution, and item information level. Results showed that as the latent ability distribution departs from a uniform distribution the accuracy of estimating the slope parameter decreased. This decrease in accuracy may be compensated for, in part, by increasing the sample size. Moreover, more informative items tended not to be as well estimated as less informative items. The results appear to indicate that if one is interested in estimating ability, a sample size ratio of 5:1 can produce reasonably accurate item parameter estimates for this purpose. (Contains 7 figures, 7 tables, and 26 references.) (Author/SLD)

Expected a posteriori has a number of advantages over maximum likelihood estimation or maximum a posteriori (MAP) estimation methods. These include ability estimates (thetas) for all response patterns, less regression towards the mean than MAP ability estimates, and a lower average squared error. R. D. Bock and R. J. Mislevy (1982) state that the posterior standard deviation (PSD theta) is virtually interchangeable with the standard error (SEE). A typical criterion for terminating an adaptive test is when the theta's SEE is equal to or less than a predetermined value. However, if there are conditions in which the PSD theta is not interchangeable with the SEE, then the adaptive test may not be validly terminated. The use of the PSD theta in situations where an examinee is classified on the basis of his or her ability may lead to incorrect classifications if the PSD theta does not agree with the SEE. Results of this Monte Carlo study showed that the use of 10 quadrature points tends to result in PSD thetas which underestimate the observed standard error. The use of 80 quadrature points, given the test's length, is recommended where accurate PSD thetas are required. (Contains 8 figures, 14 tables, and 14 references.) (Author/SLD)

Previous work on the effects of dimensionality on parameter estimation was extended from dichotomous models to the polytomous graded response (GR) model. A multidimensional GR model was developed to generate data in one-, two-, and three-dimensions, with two- and three-dimensional conditions varying in their interdimensional associations. Test length (15 and 30 items) and the ratio of sample size to the number of item parameters to estimate were also investigated, using sample sizes of 375 and 750 for the short test and 750 and 1,500 for the longer test. Results show that for unidimensional data a sample size ratio of 5:1 provided reasonably accurate estimation, and that increasing the sample size did not have a significant impact on the accuracy of item parameter estimation. Regardless of data dimensionality, the difficulty parameters were well-estimated, and for the multidimensional data the correlations between estimated item discrimination and the average and the sum of the dimensional discrimination were greater than the correlations between the estimated item discrimination and individual dimensional discriminations. Fidelity coefficients between the mean ability and the ability estimate were greater than those between the ability estimate and the latent traits. The impact of equating on accuracy indices in a multidimensional context was discussed. Seven tables and 16 graphs present analysis data. (Author/SLD)

One important and promising application of item response theory (IRT) is computerized adaptive testing (CAT). The implementation of a nominal response model-based CAT (NRCAT) was studied. Item pool characteristics for the NRCAT as well as the comparative performance of the NRCAT and a CAT based on the three-parameter logistic (3PL) model were examined. Ability estimates were generated at test lengths of 10, 15, 20, 25, and 30 items from item pools of 90 items. Abilities were generated for 1,300 examinees in 1 study and for 900 examinees in the other study. Results show that for 2-, 3-, and 4-category items, items with maximum information of at least 0.16 produced reasonably accurate ability estimation for tests with a minimum test length of about 15 to 20 items. Moreover, the NRCAT was able to produce ability estimates comparable to those of the 3PL CAT. Implications of these results were discussed. Eight tables and six graphs illustrate the discussion. (SLD)

The robustness of a partial credit (PC) model-based computerized adaptive test's (CAT's) ability estimation to items that did not fit the PC model was investigated. A CAT program was written based on the PC model. The program used maximum likelihood estimation of ability. Item selection was on the basis of information. The simulation terminated when a maximum of 30 items was reached or when a predetermined standard error of estimate (SEE) was obtained. SEE termination criteria of 0.20, 0.25, and 0.30 were used. Responses to 150 5-alternative items generated according to a linear factor analytic model were simulated for 1,000 examinees. Results indicate that reasonably accurate ability estimation could be obtained despite the adaptive tests, which, on the average, contained up to 45% misfitting items. The inclusion of the misfitting items did not appear to increase the PC CAT test lengths. The benefits of polytomous model-based CATs were discussed. Three data tables, five figures, and a 30-item list of references are included. (SLD)

The effect of dimensionality on an adaptive test's ability estimation was examined. Two-dimensional data sets, which differed from one another in the interdimensional ability association, the correlation among the difficulty parameters, and whether the item discriminations were or were not confounded with item difficulty, were generated for 1,600 simulated examinees. The generated data were used for Bayesian computerized adaptive testing (CAT) simulations (three-parameter logistic model), and the CAT ability estimates were compared with the simulated examinees' known abilities. The dimensionality of response data shifted the focus for the minimization of measurement errors from known abilities (with unidimensional data) to the average of the latent abilities (with bidimensional data). Three tables and 24 graphs summarize the study results. (Author/SLD)

Computerized adaptive testing procedures (CATPs) based on the graded response method (GRM) of F. Samejima (1969) and the partial credit model (PCM) of G. Masters (1982) were developed and compared. Both programs used maximum likelihood estimation of ability, and item selection was conducted on the basis of information. Two simulated data sets, one with 1,000 simulated examinees and one with 500 simulated examinees, were generated according to a linear analytic model. Both contained responses to 180 five-alternative items, of which 55 were retained for 997 simulated examinees with infit statistics between -3.0 and 3.0. The MULTILOG calibration program of D. J. Thissen (1988) was used to obtain item parameter estimates for both models from the data set. It was expected that using a data set fitted to the PCM model would result in no differences between PCM and GRM CATPs. However, the GRM CATPs provided more accurate information than did the PCM CATPs and the estimation was considered adequate. Reasons for difficulties with the PCM model are discussed. Two tables and five graphs present study data. (SLD)