The BLIM, the DINA, and their polytomous extensions. Rejoinder to the Commentary by Chiu, Köhn, and Ma

The basic local independence model (BLIM) is a probabilistic model developed in knowledge space theory (KST). Recently, Stefanutti, de Chiusole, et al. (2020, Psychometrika 85, 684-715) proposed the polytomous local independence model (PoLIM), which is an extension of the BLIM to items with more than two response alternatives (polytomous items). In a Commentary to this paper, Chiu et al. (2023, Psychometrika 88, 656-671) claimed that (i) the BLIM is just a deterministic input noisy AND-gate (DINA) model where every item has a single skill and, as a consequence of this, (ii) the PoLIM is simply a paraphrase of a DINA model in cognitive diagnosis (CD) for polytomous items (p. 656). This rejoinder shows that such statements are invalid and totally misleading. Its aim is to clarify the nature of the relationship between the BLIM and the DINA, as well as that between the PoLIM and the Polytomous DINA. It builds upon formal results by Heller, et al. (2015, Psychometrika 80(4), 995-1019) on the intimate relation between KST and CD notions, and shows that the BLIM/PoLIM may be conceived as marginal models for whole classes of CD models.