Corrigendum: Data Analytic Methods for Latent Partially Ordered Classification Models

Abstract

Inadvertently, an incorrect data set was used for the paper. The correct data set contains only the item responses of 536 students, as opposed to 2144, and the data that were used in the paper were actually a quadrupled copy of the correct data set. Also, the computer software has since been improved, and it is believed that the estimates that were obtained below are derived from a more accurate implementation of Gibbs sampling. Although the point estimates (and estimated posterior means) from the correct data set are for the most part similar to those found previously, the estimates of the posterior standard deviations are larger than previously reported (see the amended Table 2). This is of course also due in part to the smaller sample size. The models also have been modified and are more accurate in that they now contain exactly all the identifiable states (see the amended 1, 2, which contain Hasse diagrams for models I and II respectively). Model I Model II Given the smaller sample size, a more parsimonious approach to the estimation and classi-fication analysis was employed. Previously, 214 sets of student responses were set aside for the analysis of classification results based on parameter estimates obtained from 1940 student responses. In this subsequent analysis, the same 214 students are classified. However, to obtain estimates from a larger pool of data, the 214 student responses were split into two equal-sized subsets. For the first subset of size 107, the remaining 429 student responses data were used to estimate item parameters, and classification of the 107 students is based on the set of corres-ponding estimates. For the second subset of size 107, a second set of estimates and classification results were also obtained in this manner. The same Bayesian estimation scheme and techniques were employed as originally for both sets. Gibbs sampling was implemented with 100000 iterations, with uniform prior distributions on [0,1] for all the parameters. The stopping criteria for convergence appear to be satisfied, and both sets of estimates appear to be stable across a range of priors. In the amended Table 2, the mean values of the two sets of estimates are given for model II. The maximum range of values between corresponding posterior mean estimates was 0.05423, with a median range of 0.01560. Between the corresponding estimated posterior standard deviations, the maximum range was 0.00529, and the median range was 0.00210. The estimates thus appear to be stable across the two sets. For items 5 and 18 with respect to model II, multiple strategies were again specified. The two specified possible strategies for correctly answering item 5 actually should be stated as involving either attributes B, D and G, or attributes B, C and G. State 23 is associated with mastery of attributes B, C and G, whereas state 14 is associated with mastery of attributes B, D and G. In the amended Fig. 2, state 14 is greater than state 23 because state 14 can be viewed as the image of {B,C,D,F} and {B,D,F} in the order-induced mapping of the power set lattice of attributes to identifiable states, whereas {B,C,F} is mapped into state 23 (see Tatsuoka (1996)). Item 5 is thus associated with only one up-set. Finally, the overall classification results are similar to those which were found previously.