Sequential Bayesian Ability Estimation Applied to Mixed-Format Item Tests

Large-scale tests often contain mixed-format items, such as when multiple-choice (MC) items and constructed-response (CR) items are both contained in the same test. Although previous research has analyzed both types of items simultaneously, this may not always provide the best estimate of ability. In this paper, a two-step sequential Bayesian (SB) analytic method under the concept of empirical Bayes is explored for mixed item response models. This method integrates ability estimates from different item formats. Unlike the empirical Bayes method, the SB method estimates examinees' posterior ability parameters with individual-level sample-dependent prior distributions estimated from the MC items. Simulations were used to evaluate the accuracy of recovery of ability and item parameters over four factors: the type of the ability distribution, sample size, test length (number of items for each item type), and person/item parameter estimation method. The SB method was compared with a traditional concurrent Bayesian (CB) calibration method, EAPsum, that uses scaled scores for summed scores to estimate parameters from the MC and CR items simultaneously in one estimation step. From the simulation results, the SB method showed more accurate and reliable ability estimation than the CB method, especially when the sample size was small (150 and 500). Both methods presented similar recovery results for MC item parameters, but the CB method yielded a bit better recovery of the CR item parameters. The empirical example suggested that posterior ability estimated by the proposed SB method had higher reliability than the CB method.