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.
