## 1. Introduction

The purpose of this paper is to show that the ‘deterministic-input noisy-AND’ (DINA) model (Haertel, 1989; Junker & Sijtsma, 2001; Macready & Dayton, 1977) is equivalent to a special case of a more general compensatory family of diagnostic models.

Two variants of this model equivalency are presented in this paper. An empirical illustration of the result is given using example data that comes with the Ox (Doornik, 2002) implementation of the DINA model (de la Torre, 2009). The equivalencies are based on defining two mappings of the DINA skill space and the design **Q**-matrix of item by skills associations. This (re)mapping produces an instance of the compensatory general diagnostic model (GDM; von Davier, 2005) that is mathematically equivalent to the (conjunctive) DINA model.

Whenever it can be shown that several equivalent model variants exist or one model can be viewed as a special case of another more general one, conclusions derived from any particular model-based estimates are called into question.

It is a widely known that multidimensional models can often be specified in multiple ways while the model-based probabilities of observed variables stay the same. Maris and Bechger (2004, 2009) have shown for MIRID (‘model with internal restrictions on the item difficulties’; Butter *et al*., 1998) models and diagnostic models that there are often multiple design (**Q**-matrix) and skill-space definitions for the same diagnostic model, and that these produce the same model-based probabilities of observed quantities. For the testlet model (Bradlow *et al*., 1999) and the (constrained) bifactor model (Gibbons & Hedeker, 1992), this was shown, for example, by Rijmen (2010). For structural equation models (SEMs), it is a widely known result that the covariance matrix can be reproduced by rather different SEMs in identical or almost identical fashion. Finally, the equivalency between higher-order factor models and the hierarchical factor model has been established by Yung *et al*. (1999).

This paper goes beyond this type of equivalency by showing that a *conjunctive* diagnostic classification model can be expressed as a special case embedded in a general *compensatory* diagnostic modelling framework. More specifically, this paper looks at a different type of equivalency between cognitive diagnosis models. While previous research was mainly concerned with transformational or rotational invariance, this paper shows that a conjunctive diagnostic classification model can be expressed as a constrained model in a compensatory modelling framework – the GDM (von Davier, 2005, 2008, 2010).

The equivalencies presented in this paper hold for all DINA models with any **Q**-matrix, not only for trivial (simple-structure) cases. Also, these equivalencies are in no way implied by the recently suggested log-linear cognitive diagnosis model (L-CDM; Henson *et al*., 2009) – derived on the basis of the GDM (Rupp *et al*., 2010) – or the generalized DINA (G-DINA; de la Torre, 2011) approaches. The skill interaction features that these models add to the linear GDM are neither used nor approximated in the results presented here. These equivalencies solely require a linear – compensatory – GDM without any skill interaction terms. That is, in the result presented here, no additional structures are introduced, and both the DINA and the GDM are taken in their original form. An equivalency between the two is established purely by remapping the skill space of the DINA into an alternative skill space of a DINA-equivalent GDM.

At some level it may seem trivial to show that the vast majority of models considered ‘diagnostic’ can be considered latent class, or better, latent structure models (von Davier, 2009; von Davier *et al*., 2008; von Davier & Yamamoto, 2004; Rupp *et al*., 2010). But it is less obvious that a *conjunctive* model (i.e., a model that does not allow for compensatory functioning of skills) can be re-expressed using a mapping of the DINA skill space onto a new set of skills so that a *compensatory* diagnostic model can be used to define a DINA-equivalent model.

The long-standing knowledge that all diagnostic models can be viewed as latent structure models or generalized latent variable models and that therefore, at a very high level, these models are ‘all the same’ is not of much help to gain a deeper understanding of them. At this level of generality, it is also true for the mixture item response theory (IRT) and multidimensional IRT models, but it is still necessary to develop equivalency results with regard to these models in order to facilitate understanding of the similarities and differences between these approaches (Rijmen & De Boeck, 2005).

Another example is the seminal paper by Takane and de Leeuw (1987): while factor analysis for discrete variables and IRT are both examples of a general class of latent variable models, it is important to develop a deeper understanding of the relationship between the specific instances of this general class.

The point of the results presented here is related to these types of published research on model equivalencies: within this class of latent structure models, different model assumptions can be made that lead to very different interpretations of diagnostic models, as in the case of assumed conjunctive or compensatory skills. If an equivalency is proven between these two apparently incompatible assumptions, this is new knowledge not explained by the fact that at a high level, both are instances of general latent variable models. To make an analogy, it is new knowledge if a researcher finds out that a rodent and a bear are genetically closer than their differences in appearance suggest even though it has been known for a long time that both are mammals.

As an added value gained en passant, we show how the equivalent-DINA model contains parameter constraints as well as constraints on the distribution of skills, and these results facilitate decisions as to whether the DINA model or some other, more general, diagnostic model is appropriate to fit the data at hand.

The selection of a particular model should be based on an examination of how well model assumptions relate to the theoretical consideration that served as the basis for test construction. By selecting and committing to use the DINA model early on, without considering whether other models may be more appropriate, the researcher skips this important step in determining whether one or more models are suitable representation(s) of the construct of interest. With this early decision comes an inability to examine whether the restrictions used in the DINA model are suitable or whether a more general model should have been used.

Embedding the DINA model – or, better, embedding the equivalent-DINA model variants into a larger modelling framework – allows comparisons to other models. The famous adage that all models are wrong while some may be useful (Box & Draper, 1987) also holds true for diagnostic models. Therefore, any (diagnostic) model is at best an approximation of reality, and a more general basis upon which different models can be compared is useful in determining whether the assumed skill requirements for each item, or even more so, whether the assumed functioning of skills as compensatory or non-compensatory/conjunctive, are indeed appropriate. It is expected that the result presented in this paper facilitates these types of model comparisons.