Multidimensional models

6 Multidimensional models

All models discussed thus far have one very important feature in common that has not been discussed explicitly thus far: they all assume that the items in the test are indicators of one latent variable. The assumption of conditional indepen- dence amounts to claiming that any pair of items do not share any variance beyond the variance that can be attributed to that latent variable. However, this is precisely what is assumed in the one-factor model. In that sense, unidimensional IRT models are just variations of the factor analytical model for a single factor. This relation has been discussed in detail by Takane and De Leeuw (1987), and

a formal proof is given that the two parameter normal ogive model as discussed above is exactly equivalent with the one-factor model. The generalization to multidimensional models is then straightforward and can be conceptualized as consisting of three components:

• For each item the performance is dependent on a linear combination of abilities:

(27) where the coefficients ␣ i1 , . . ., ␣ ip have exactly the same meaning as in factor

␣⬘ i ␪=␣ i1 ␪ 1 +␣ i2 ␪ 2 + ... + ␣ ip ␪ p ,

analysis and are commonly referred to as factor loadings. • The assumption of a latent response z i , which is a random draw from some hypothesized distribution, as exemplified in Figure 8.5. The mean of this distribution is the linear combination given by (25). In the tradition of factor analysis, the standard deviation of this distribution can vary across items, and the distribution itself is assumed to be normal, but there is no objection in principle to assuming that this distribution is logistic, or might have yet another form. In most applications, it is assumed that the latent responses are conditionally independent. The possibility provided in SEM for ‘correlated errors’ is not to be conceived as a new feature of these models. It fits in the general model described here, but in a concrete application a specification error might occur, because ‘in reality’ there are p factors and the model assumes that there are less. The unexplained covariance caused by this underspecification may then show up as correlated residuals.

• If the latent response variables z i were observable, the preceding model assumptions would be nothing else than the theory on factor analysis. However, the variables z i are continuous, and the data delivered from a test administration are highly discrete. So, one needs a mechanism to convert the continuous latent responses to discrete overt responses. This mechanism

180 Different methodological orientations is the model of a Thurstonian threshold, exemplified in Figure 8.5. General -

izations to polytomous observations are obvious: the number of thresholds equals the number of observable categories minus one (Bock and Lieberman 1970; Christofferson 1975: Muthén 1978). It is even possible to conceive of these thresholds as linear combinations of dimension-wise thresholds (Glas and Verhelst 1995, Kelderman and Rijkes 1994).

A model where the performance depends on a linear combination of latent abilities is a compensatory model: a low ability on one dimension or factor may be compensated by a high value on one or more other dimensions; the probability of a correct answer only depends on the result of the linear combination, and not on its components.

There may be situations in educational assessment where such a compensatory rule is not realistic. To find the solution to a mathematics problem that is embedded in a more or less complicated description of a real-life situation, one might hypothesize that a successful performance will depend on a sufficient reading ability and a sufficient mathematics ability, but that deficiency in either of these abilities cannot be compensated by an excess in the other. Models that require such a multiple requirement are known as ‘conjunctive’ models (Hendrickson and Mislevy 2005; Maris 1995; van Leeuwe and Roskam 1991).

A model that is logically tightly related to the conjunctive model is the disjunctive model, where it is assumed that an item or a task can be successfully solved by appealing to one of several abilities. In fact, both models are logically equivalent: requiring sufficient reading and mathematics ability to solve an item, is logically equivalent to requiring insufficient reading or mathematics ability to fail the item. This means that using the complement of the observed data (exchanging ones and zeros) and reversing the direction of the latent dimensions turns a conjunctive model into a disjunctive one and vice versa. An example is

a test where the use of different strategies may lead to the correct solution. The multidimensional ability then refers to the ability to successfully handle these strategies. This might be reflected, for example, in mathematics problems where different strategies may be used to find the solution to a task, for example, by using mainly algebraic arguments or mainly geometric ones.