The concept of -equivalence has especially been developed for tests as the units of measurement. However, the concept also applies for items (within tests) as the units of measurement. In the realm of tests, -equivalence is defined in terms of classical test theory. However, in the realm of items within tests, -equivalence is defined in terms of the specific item response model concerned. In the case of intelligence test, item response models can be subdivided into models for binary data (correct vs. incorrect) and models for raection time data. First, the binary response models will be discussed.
Analogous to factor analytic modelling, first unifactorial models were developed, such as the normal ogive model proposed by Lawley (1954 & 1944) and the logistic model proposed by Birnbaum (1968). Deterministic models, such as the Guttman scale, will be left out of consideration, because it seems more appropriate to consider test performance in intelligence tests as a random process. A special case of the logistic model is the Rasch model (Rasch, 1960). This model is of special interest because it has the property of separate sufficiency, a statistical term coined by van Breukelen (1995, p. 96). Separate sufficiency implies the existence of sufficient statistics for the subject parameters that are independent of the item parameters and of sufficient statistics for the item parameters that are independent of the subject parameters. The Rasch model uses separate sufficiency of the marginal totals.
In the case of the Rasch model, the logit of the probability , that subject i gives a correct response to item j, is equal to the difference between the subject parameter and the item parameter , that is
This property implies that the items are essentially -equivalent as far as the logit of the probability of a correct response is concerned.
The logistic model proposed by Birnbaum (1969) can be considered as an extension of the Rasch model. In the logistic model, the logit of the probability , that subject i gives a correct response to item j, is the summation of a weighted subject parameter and an unweighted item parameter . The weight is item dependent. In formula:
In the cease of the Birnbaum model the items are linearly -equivalent as far as the logit of the probability of a correct response is concerned. This model is completely isomorphic to the uni-factorial model proposed by Spearman (1904). The item weights correspond to the factor loadings in Spearman's model.
Similar to the history of factor analysis, multidimensional extensions of the Rasch model, or better of the logistic model, have also been proposed. McKinley and Reckase (1985) have proposed a model, in which the logit of the probability is equal to the weighted sum of p subject parameters , and the item parameter . In formula:
This model is completely isomorphic to the well-known multiple factor model.
The normal ogive model has the same rationale as the logistic model. But, instead of the logit function, in this model the normit function is used. Therefore, in the case of the normal ogive model the items are linearly -equivalent as far as the probit of the probability of a correct response is concerned.
Recently, van Breukelen (1995) discussed five item response models
for RT data:
Micko's logistic model (1969, 1970), Scheiblechner's exponential model
(1979), Thissen's lognormal model (1983), and two models which are
proposed by himself. These models will be referred to as van Breukelen's
gamma model (1995, p. 103) and van Breukelen's gaussian model (1995,
p. 104). Only three of these models have the property of separate
sufficiency. Therefore these models will be discussed in more detail.
In the sequal F(t) denotes the distribution function of the random
variable T and f(t) the probability density function. The expectation
of T will be denoted as E(T). Note, that f(t) = F'(t).
Scheiblechner (1979) proposed the exponential distribution and a generalization for the RT of subject i on item j:
where , and b(t) is some monotone function of the time t that satisfies b(0) = 0 and for . The exponential distribution is described in full detail in Johnson & Kotz (1970, p. 207). Scheiblechner considered the cases b(t) = t and , yielding the exponential and the Weibull distribution for the RT, respectively. The Weibull distribution is described in full detail in Johnson & Kotz (1970, p. 250). He chose an additive parameter structure, , to obtain separete sufficiency of the marginal totals and for and , respectively (Scheiblechner, 1997, pp. 33-34). The expectation is equal to . Consequently, the inverse of the expectation of time, which is needed by subject i to solve item j, is equal to the difference between the subject parameter and the item parameter , that is:
In Scheiblechner's exponential model the items are essentially -equivalent as far as the inverse of the expectation of the reaction time is concerned.
Thissen (1983) proposed the lognormal distribution with parameters and for the RT of subject i on item j. According to the lognormal distribution, has a normal distribution with mean en standard deviation . In a simple version of his model, the assumption is made, that
Sufficient statistics for and are respectively and . In Thissen's lognormal model the items are essentially -equivalent as far as the expectation of the natural logarithm of the reaction time is concerned.
Van Breukelen (1995, p. 103) proposed the gamma distribution with parameters and for the RT of subject i on item j. Van Breukelen assumes "... that an item is solved by completing all component processes in a serial-additive way, and each process has an exponential distributed duration with rate ." If the component processes are independent, then the distribution of the total solution time has a gamma distribution, which, since is an integer > 1, equals
If is a positive integer, the gamma distribution is an Erlang distribution. A more appropriate name would, therefore, be van Breukelen's Erlang model. Note, that in this model the parameters and , respectively, correspond to (but are not equal to) the parameters and . Sufficient statistics for and are, respectively, and . The expectation of the gamma distribution reads as follows:
Consequently, . If and , then . In van Breukelen's gamma model the items are essentially -equivalent as far as the logarithm of the expectation of the reaction time is concerned.