Abstract
In a recent paper, Hessen (Psychometrika 70(3):497–516, 2005) introduces the class of constant latent oddsratios models as an extension of the Rasch model for which the sum score is still the sufficient statistic for ability. In this paper the relation between both the general and the general parametric constant latent oddsratios model and the Rasch model is considered.
Hessen (2005) introduces the class of constant latent oddsratios models (CLOR, for short) as a generalization of the Rasch model (Rasch, 1960), and shows how differential item functioning by means of a MantelHaenszel procedure can be studied. In this paper we focus on the class of CLOR models and clarify their relation to the Rasch model. In the Rasch model, the probability with which a person with ability t gives a correct response to item i is the following:
where b _{ i } (c _{ i }) is the easiness (difficulty) of the ith item. This function is usually called the item response function (IRF, for short). In the definition of the class of CLOR models, the oddsratio,
and the ratio of oddsratios, for items i and j,
play a crucial role. For the Rasch model it holds that ω _{ ij }(t) does not depend on the ability t.
The class of CLOR models is characterized by three properties:

1.
Unidimensionality.

2.
Local independence (LI, for short): \(\Pr ({\rm{X}} = xt) = {\Pi _i}\Pr ({X_i} = {x_i}t)\)

3.
Constant ratio of oddsratios ω _{ ij }(t) = ω _{ ij }.
CLOR models may have both monotone and nonmonotone IRFs. Hessen (2005) claims that the class of CLOR models is an extension of the Rasch model for which the property that the sum score is sufficient for the latent trait is retained. This claim is illustrated with some parametric and nonparametric IRT models. In this paper we show that whenever unidimensionality and LI are assumed, the assumption of constant ratio of oddsratios is equivalent to the assumption that the IRF can be expressed as in (1). For nonmonotone CLOR models we show that the latent trait is not identifiable from X, and that the same distribution can be specified with the IRF in (1). Furthermore, we show that the general parametric CLOR model is equivalent to the Rasch model together with the assumption that the range of the latent trait is constrained.
1. CLOR Models in Relation to the Rasch Model
From the assumptions of CLOR models we readily obtain that
and hence
.
If, without loss of generality, we choose a parametrization for the latent trait t such that it equals the oddsratio for an arbitrary item (denoted as item 1 in what follows):
we obtain that the IRF for item 1 can be written in the following form:
, whereas the IRFs for the other items can be obtained from (5),
.
We see that for any CLOR model the IRF can be expressed, as in (1), with item easiness equal to the ratio of oddsratios ω _{ i1}, and ability as defined by the oddsratio V _{1}(t). Observe that both the item easiness parameters as well as the ability parameters can be uniquely determined from X. In other words, these parameters are identifiable. Observe, furthermore, that specific CLOR models may impose additional constraints beyond those characterizing the class of CLOR models, as illustrated below. In a CLOR model we not only have that the sum score is the sufficient statistic for the latent trait, but also that the number of persons giving a correct answer to item i is the sufficient statistic for the item easiness parameter ω _{ ij } (j fixed).
For future reference we state our result as a theorem.
Theorem 1. Whenever unidimensionality and local independence are assumed, the assumption of a constant ratio of oddsratios is equivalent to the assumption that, for every item, the IRF can be expressed as in (1) with item easiness equal to ω _{ ij } (j fixed).
Observe that unless the range of the latent trait is explicitly constrained, as in the general parametric CLOR model considered in the next section, the IRF for every item satisfies the following limits: lim_{ t→∞}π_{ i }(t) = 1 and lim_{ t→0} π_{ i }(t) = 0.
Hessen (2005) also considers CLOR models where the IRF is not a monotone function of the latent trait. From Theorem 1 it follows that for any CLOR model the IRF can be written as in (1), and hence the same applies to every nonmonotone CLOR model. An example may clarify the strange phenomenon that the same distribution can at the same time have monotone and nonmonotone IRFs. Let the IRFs for a set of items be defined as follows:
, where b _{1} equals one and θ can assume both positive and negative values. A routine derivation shows that this model is indeed a CLOR model with nonmonotone IRFs. However, the same distribution can be written in terms of a latent trait t = exp(θ ^{2}) as follows:
which is recognized as (1), with the additional assumption that the latent trait t is larger than or equal to one. This means that the latent trait θ is not identifiable from X, whereas the latent trait t is identifiable from X. In other words, all that is identifiable from X about θ is its magnitude ‖θ‖, whereas its sign is not identifiable from X. The model in (9) gives an example of a specific CLOR model that imposes an additional assumption beyond those characterizing the class of CLOR models.
As a special case of a nonmonotone IRF we consider an IRF which remains constant over some part of the domain of the latent trait t ^{*}. First, if the IRF of one item is constant over the range from t ^{*} equal to a to t ^{*} equal to b, the IRFs of all other items are constant over this same range. This follows readily from the constant latent oddsratio property. First, if for t _{1} ^{*} ≠ t _{2} ^{*} it holds that the IRF for item 1 has the same value, then also the oddsratio has the same value. Second, because the ratio of latent oddsratios is constant we have that
and hence
.
It is readily seen that also in this case the latent trait t ^{*} cannot be identifiable from X.
2. The General Parametric CLOR Model is Equivalent to the Rasch Model with a Constraint on the Range of the Latent Trait
Hessen (2005) introduces the general parametric CLOR model, which is a special case of the fourparameter logistic model of Barton and Lord (1981), and has the following IRF:
, where 0 ≤δ < c _{ i } ^{*} and 0 ≤γ < 1/c _{ i } ^{*} for all i. The general parametric CLOR model defines nonintersecting item response functions for which the lower (upper) asymptotes may differ from zero (one) depending on the value of δ and γ .
A routine derivation shows that the IRF in (13) can also be expressed in the form of (1):
, where t _{ γ,δ } and c _{ γ,δ } are the following monotone functions:
and
.
Notice that in order to obtain a positive result in (15) and (16) the parameters δ and γ have to be constrained. Specifically, the constraints imposed by Hessen (2005) (0 ≤ γ < c _{ i } ^{*} and 0 ≤ γ < 1/c _{ i } ^{*} ) meet these requirements. Moreover, the transformed latent trait satisfies the following constraints:
which means that the general parametric CLOR model in (13) is equivalent to the Rasch model with a constraint on the range of the latent trait. By restricting the range of the latent trait, we obtain that at the lowest possible level of the latent trait (i.e., t _{ γ,δ }(t ^{*}) = δ) the probability to give a correct response can be larger than zero and, similarly, that at the highest possible level of the canonical latent trait (i.e., t _{γ,δ}(t ^{*}) = 1/γ ) the probability to give a correct response can be smaller than one.
3. Discussion
We have shown for the general CLOR model that the IRF can be expressed as in (1). Whether this result can be interpreted as saying that the general CLOR model is equivalent to the Rasch model, or not, depends on how one defines the Rasch model. Specifically, it depends on whether the range of the latent trait is part of the definition or not. If the Rasch model is defined as (1) together with local independence, the result states that the general CLOR model is equivalent to the Rasch model. If the Rasch model is defined as (1) together with local independence and the assumption that t ranges over the nonnegative real number, the result states that the general CLOR model is not equivalent to the Rasch model. We have also shown that, for nonmonotone CLOR models, the latent trait parameter is not identifiable from X. Different CLOR models correspond to different ways to parametrize the latent trait. If the latent trait t in (1) is transformed monotonically we obtain a monotone CLOR model, whereas if the latent trait is transformed in a nonmonotone way we obtain a nonmonotone CLOR model. Observe that in order to retain the sufficiency property of the sum score the same transformation is to be used for the latent trait with all items. Specific instances of a CLOR model, such as the general parametric CLOR model, may add additional assumptions to those of Theorem 1. In the case of the general parametric CLOR model the assumption is added that the range of the latent trait is restricted. In this way, it becomes possible that the IRFs have lower and upper asymptotes different from zero and one, respectively.
References
Barton, M., & Lord, F. (1981). An upper asymptote for the threeparameter logistic itemresponse model (Technical Report No. 8120). New York: Educational Testing Service.
Hessen, D.J. (2005). Constant latent oddsratios models and the Mantel–Haenszel null hypothesis. Psychometrika 70(3), 497–516.
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: The Danish Institute of Educational Research. (Expanded edition, 1980. Chicago: The University of Chicago Press.)
Author information
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section for details of this license and what reuse is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and reuse information, please contact the Rights and Permissions team.
About this article
Cite this article
Maris, G. A Note on “Constant Latent OddsRatios Models and the Mantel–Haenszel Null Hypothesis” Hessen, 2005. Psychometrika 73, 153 (2008). https://doi.org/10.1007/s1133600790330
Received:
Revised:
Published:
Keywords
 fourparameter logistic model
 constant latent oddsratios model
 Rasch model