Abstract:

Existing literature has shown that dual-objective CD-CAT testing can facilitate the achievement of measurement objectives for both formative and summative assessments. And the Gini Index can be used as a measurement for the degree of uncertainty of random variables since a smaller Gini value indicates a lower degree of uncertainty. Hence, this paper proposed a Gini-Index-based selection method for dual-objective CD-CAT, and it measured the changes in the posterior probability of knowledge state and confidence interval for latent traits estimation. By adopting the Bayesian Decision Theory, the potential information of participants could be detected based on participants’ responses and changes in posterior probability distribution of two the random variables.
Monte Carlo Simulation was used to test the performances of the selection method based on Gini, ASI, IPA, and JSD, respectively. The item banks measured 5 attributes consisting of 250 items in total, and each item measured 3 attributes at most. The true knowledge state of each participant was generated by HO-CDM and Multivariate Normal Models (both means were 0 and covariance coefficient was 0.8 and 0.2, respectively). G-DINA, DINA, and R-RUM were adopted as the cognitive diagnostic models, and the item bank of each of these three models included both CDM and 2PL parameters. Specifically, CDM parameters were generated by a G-DINA package in R software with the slipping and guessing parameters randomly selected from a uniform distribution in a range from 0.05 to 0.25. The 2PL parameters were estimated by factoring in the responses elicited from 3,000 participants’ responses to all items in item banks using the mirt package. Four indexes, namely the pattern match ratio, root mean square error of latent trait, chi-square value, and time needed for item selection, were adopted in comparing the efficiency of different item selection methods. The value for each index was the mean of 10 repeated simulations of 1,000 participants’ responses to all item bank.
The results showed that (1) The Gini and IPA selection methods had similar performance in terms of pattern match ratio, root mean square error of latent trait, and chi-square value. Both methods were high in precision measurement and low in sensitivity to CDM and the distribution of participants’ cognitive patterns, making both methods applicable to the item banks featuring a mixture of cognitive diagnosis models. By comparison, the Gini method outperformed the IPA method slightly in pattern match ratio and time needed for item selection in which the Gini method was only one-tenth that of the IPA method; (2) Both the Gini and ASI selection methods were weighted linear combination approaches. The performances of the two methods were very close in the short test. In the long test, however, although the time needed for item selection using the ASI method was only one-third that of the Gini method, the latter was superior to the former in terms of measurement accuracy and chi-square value; (3) Although the JSD method outperformed the Gini method in terms of uniformity of item bank usage and time needed for item selection, its measurement accuracy was far less than the latter.
To summarize, the Gini, IPA, and ASI selection methods all have good measurement accuracy and hence are all recommended for short tests. For medium and long tests with a limited number of attributes and a smaller item bank, the Gini and IPA selection methods are recommended. As the number of attributes and item bank size grow, the Gini method is recommended. When there are high correlations among different attributes, as well as a large number of attributes and big item bank size, the ASI and JSD selection methods are recommended, with the ASI method slightly outperforming the JSD method in measurement accuracy.

Key words: cognitive diagnostic, items response theory, Gini index, dual objective CD-CAT, selection method