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31.
This paper analyzes the sum score based (SSB) formulation of the Rasch model, where items and sum scores of persons are considered
as factors in a logit model. After reviewing the evolution leading to the equality between their maximum likelihood estimates,
the SSB model is then discussed from the point of view of pseudo-likelihood and of misspecified models. This is then employed
to provide new insights into the origin of the known inconsistency of the difficulty parameter estimates in the Rasch model.
The main results consist of exact relationships between the estimated standard errors for both models; and, for the ability
parameters, an upper bound for the estimated standard errors of the Rasch model in terms of those for the SSB model, which
are more easily available.
The authors acknowledge partial financial support from the FONDECYT Project No. 1060722 from the Chilean Government, and the
BIL05/03 grant to P. De Boeck, E. Lesaffre and G. Molenberghs (Flanders) for a collaboration with G. del Pino, E. San Martín,
F. Quintana and J. Manzi (Chile). 相似文献
32.
Francesco Bartolucci 《Psychometrika》2007,72(2):141-157
We illustrate a class of multidimensional item response theory models in which the items are allowed to have different discriminating
power and the latent traits are represented through a vector having a discrete distribution. We also show how the hypothesis
of unidimensionality may be tested against a specific bidimensional alternative by using a likelihood ratio statistic between
two nested models in this class. For this aim, we also derive an asymptotically equivalent Wald test statistic which is faster
to compute. Moreover, we propose a hierarchical clustering algorithm which can be used, when the dimensionality of the latent
structure is completely unknown, for dividing items into groups referred to different latent traits. The approach is illustrated
through a simulation study and an application to a dataset collected within the National Assessment of Educational Progress,
1996.
The author would like to thank the Editor, an Associate Editor and three anonymous referees for stimulating comments. I also
thank L. Scaccia, F. Pennoni and M. Lupparelli for having done part of the simulations. 相似文献
33.
Robert J. Jannarone 《Psychometrika》1986,51(3):357-373
Conjunctive item response models are introduced such that (a) sufficient statistics for latent traits are not necessarily additive in item scores; (b) items are not necessarily locally independent; and (c) existing compensatory (additive) item response models including the binomial, Rasch, logistic, and general locally independent model are special cases. Simple estimates and hypothesis tests for conjunctive models are introduced and evaluated as well. Conjunctive models are also identified with cognitive models that assume the existence of several individually necessary component processes for a global ability. It is concluded that conjunctive models and methods may show promise for constructing improved tests and uncovering conjunctive cognitive structure. It is also concluded that conjunctive item response theory may help to clarify the relationships between local dependence, multidimensionality, and item response function form.I appreciate the many helpful suggestions that were given by the reviewers and Ivo Molenaar. 相似文献
34.
Consideration will be given to a model developed by Rasch that assumes scores observed on some types of attainment tests can
be regarded as realizations of a Poisson process. The parameter of the Poisson distribution is assumed to be a product of
two other parameters, one pertaining to the ability of the subject and a second pertaining to the difficulty of the test.
Rasch's model is expanded by assuming a prior distribution, with fixed but unknown parameters, for the subject parameters.
The test parameters are considered fixed. Secondly, it will be shown how additional between- and within-subjects factors can
be incorporated. Methods for testing the fit and estimating the parameters of the model will be discussed, and illustrated
by empirical examples. 相似文献
35.
36.
David Andrich 《Psychometrika》1995,60(1):7-26
It is common in educational, psychological, and social measurement in general, to collect data in the form of graded responses and then to combine adjacent categories. It has been argued that because the division of the continuum into categories is arbitrary, any model used for analyzing graded responses should accommodate such action. Specifically, Jansen and Roskam (1986) enunciate ajoining assumption which specifies that if two categoriesj andk are combined to form categoryh, then the probability of a response inh should equal the sum of the probabilities of responses inj andk. As a result, they question the use of the Rasch model for graded responses which explicitly prohibits the combining of categories after the data are collected except in more or less degenerate cases. However, the Rasch model is derived from requirements of invariance of comparisons of entities with respect to different instruments, which might include different partitions of the continuum, and is consistent with fundamental measurement. Therefore, there is a strong case that the mathematical implication of the Rasch model should be studied further in order to understand how and why it conflicts with the joining assumption. This paper pursues the mathematics of the Rasch model and establishes, through a special case when the sizes of the categories are equal and when the model is expressed in the multiplicative metric, that its probability distribution reflects the precision with which the data are collected, and that if a pair of categories is collapsed after the data are collected, it no longer reflects the original precision. As a consequence, and not because of a qualitative change in the variable, the joining assumption is destroyed when categories are combined. Implications of the choice between a model which satisfies the joining assumption or one which reflects on the precision of the data collection considered are discussed. 相似文献
37.
Paul W. Holland 《Psychometrika》1990,55(1):5-18
The Dutch Identity is a useful way to reexpress the basic equations of item response models that relate the manifest probabilities to the item response functions (IRFs) and the latent trait distribution. The identity may be exploited in several ways. For example: (a) to suggest how item response models behave for large numbers of items—they are approximate submodels of second-order loglinear models for 2
J
tables; (b) to suggest new ways to assess the dimensionality of the latent trait—principle components analysis of matrices composed of second-order interactions from loglinear models; (c) to give insight into the structure of latent class models; and (d) to illuminate the problem of identifying the IRFs and the latent trait distribution from sample data.This research was supported in part by contract number N00014-87-K-0730 from the Cognitive Science Program of the Office of Naval Research. I realized the usefulness of the identity in Theorem 1 while lecturing in the Netherlands during October, 1986. Because this was in no small part due to the stimulating psychometric atmosphere there, I call the result the Dutch Identity. 相似文献
38.
The problem of characterizing the manifest probabilities of a latent trait model is considered. The item characteristic curve
is transformed to the item passing-odds curve and a corresponding transformation is made on the distribution of ability. This
results in a useful expression for the manifest probabilities of any latent trait model. The result is then applied to give
a characterization of the Rasch model as a log-linear model for a 2
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-contingency table. Partial results are also obtained for other models. The question of the identifiability of “guessing”
parameters is also discussed.
The research reported here is collaborative in every respect and the order of authorship is alphabetical. Dr. Cressie was
a Visiting Research Scientist at ETS during the Fall of 1980. His current address is: School of Mathematical Sciences, The
Flinders University of South Australia, Bedford Park SA, 5042, AUSTRALIA. The preparation of this paper was supported, in
part, by the Program Statistics Research Project in the Research Statistics Group at ETS. 相似文献
39.
Ivo W. Molenaar 《Psychometrika》1983,48(1):49-72
Although several goodness of fit tests have been developed for the Rasch model for dichotomous items, most of them are of
a global, asymptotic, and confirmatory type. This paper, based on ideas from a recent thesis by Van den Wollenberg, offers
some suggestions for local, small sample, and exploratory techniques: difficulty plots for person groups scoring right and
wrong on a specific item, a slope test per item based on a binomial distribution per score group, and a unidimensionality
check based on an extended hypergeometric distribution per score group.
This paper owes much to the inspiring and pioneering work of Arnold Van den Wollenberg, of which only minor aspects are criticized.
Thanks go to Charles Lewis for stimulating discussions and for solutions to some programming problems. 相似文献
40.
Dean Follmann 《Psychometrika》1988,53(4):553-562
Consider the class of two parameter marginal logistic (Rasch) models, for a test ofm True-False items, where the latent ability is assumed to be bounded. Using results of Karlin and Studen, we show that this class of nonparametric marginal logistic (NML) models is equivalent to the class of marginal logistic models where the latent ability assumes at most (m + 2)/2 values. This equivalence has two implications. First, estimation for the NML model is accomplished by estimating the parameters of a discrete marginal logistic model. Second, consistency for the maximum likelihood estimates of the NML model can be shown (whenm is odd) using the results of Kiefer and Wolfowitz. An example is presented which demonstrates the estimation strategy and contrasts the NML model with a normal marginal logistic model.This research was supported by NIMH traning grant, 2 T32 MH 15758-06 and by ONR contract N00014-84-K-0588. The author would like to thank Diane Lambert, John Rolph, and Stephen Fienberg for their assistance. Also, the comments of the referees helped to substantially improve the final version of this paper. 相似文献