Maximum-Likelihood Estimation of Parameters Subject to Restraints

Aitchison, J.; Silvey, S. D.

doi:10.1214/aoms/1177706538

Cited by 504 publications

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“…It is well-known that minus twice the LR statistic has a limiting central chi-square distribution under the null hypothesis (Wilks (193 8) ), and a limiting non-central chisquare distribution under a sequence of local alternatives (Wald (1943)) with a non-centrality parameter equal to that of the Wald statistic (Wald (1943)) and Lagrange Multiplier statistic (Aitchinson and Sil vey (1958) , Silvey (1959) ). However , as Foutz and Srivastana 4 (1977) , Kent (1982) , and White (1982a) pointed out, when the largest model is misspecified, the LR statistic is no longer ne cessarily chi square distributed under the null hypothesis where the null hypothesis must be appropriately redefined in terms of the pseudo-true values satisfying the specified restrictions .…”

Section: Introduction Quang H Vuong California Institute Of Technologymentioning

confidence: 99%

Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses

Vuong¹

1989

Econometrica

4,924

3,637

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Section: Introduction Quang H Vuong California Institute Of Technologymentioning

confidence: 99%

Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses

Vuong¹

1989

Econometrica

4,924

3,637

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“…In the present study, the focus is on country-specific CDIF. CDIF can be detected using Lagrange multiplier (LM) test statistics (Rao 1947; see also, Aitchison and Silvey 1958) and CDIF can be modeled using country-specific item parameters. Glas and Jehangir (2014) already showed the feasibility of the method using PISA data, although in the slightly simpler framework of one-dimensional IRT models.…”

Section: Detection and Modeling Of Differential Item Functioningmentioning

confidence: 99%

Modeling Parental Involvement

Punter

Glas

Meelissen

2016

IEA Research for Education

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In this chapter, we first outline the models used and the estimation and testing procedures employed, and then summarize the results revealed by these models. Estimation and Testing ProceduresThe procedures we used for parameter estimation and evaluation of model fit are based on marginal maximum likelihood (MML). Most of the procedures we discuss are documented in more detail elsewhere (see Bock and Aitkin 1981;Bock et al. 1988;Gibbons and Hedeker 1992;Glas 1999;Adams and Wu 2006;De Jong et al. 2007;Jennrich and Bentler 2011;Glas and Jehangir 2014). We used the public domain software package MIRT (Glas 2010) in the calculations. Additional estimation and testing procedures were used for the bi-factor model, with unidimensional models as special cases, and random item parameters as a generalization. MML EstimationThe bi-factor model used in this study was in two parts: a measurement model (i.e., an IRT model) and a structural model. The measurement model pertains to a polytomously-scored response of a student n to an item i. The possible item scores range from 0 to m i and the score of student n on item i is denoted by the variables x nij (j = 1, …, m i ) where x nij = 1 if the response is in category 1 and zero otherwise. Note that m i has an index i, which indicates that the maximum score of items can differ.We describe the procedure for the bi-factor model, combined with the partial credit model (PCM;Masters 1982) and generalized partial credit model (GPCM; Muraki 1992) as IRT models, since these two models were the ones we selected for the present study. However, the theory also applies to other IRT models, such as the unidimensional PCM and GPCM, the graded response model (Samejima 1969), the

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“…The calculation of the expected information matrix remains relatively simple, but does include the Jacobian of the identifying constraints (e.g., 1 3 i 1 a i ). We refer the reader to Aitchison and Silvey (1958) for the relevant details.…”

Section: Other Parameterizationsmentioning

confidence: 99%

Parameter identification in multinomial processing tree models

Schmittmann

Dolan

Raijmakers

et al. 2010

Behavior Research Methods

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Multinomial processing tree (MPT) models form an increasingly popular class of stochastic models for categorical data that have applications in a variety of research areas in cognitive, differential, and social psychology (e.g., Batchelder & Riefer, 1999Erdfelder et al., 2009;Stahl, 2006). For example, Batchelder and Riefer (1999) discussed over 80 applications of MPT models in various areas of psychology, including memory, perception, and reasoning. The statistical theory of MPT models, including maximum likelihood (ML) parameter estimation, overall model testing, and tests of specific hypotheses within models, has been discussed by Hu and Batchelder (1994) and by Riefer and Batchelder (1988). Flexible software to fit MPT models by means of ML estimation has been developed by Hu and Phillips (1999), Moshagen (2010), Rothkegel (1999), and Stahl and Klauer (2007. MPT models entail a reparameterization of the cell probabilities of the multinomial or product-multinomial distribution (Andersen, 1980;Bishop, Fienberg, & Holland, 1975) in terms of parameters assumed to represent the probabilities of underlying cognitive processes. The underlying cognitive architecture is represented as a rooted tree, in which each branch represents a processing sequence leading to an observable categorical response. More than one branch may terminate in a given response.In MPT models, as in all statistical models, global model identification (the ability to infer unique parameter values from data) is an important issue. A common cause of nonidentifiability of a model is parameter redundancy, in which case the likelihood of the model can be expressed as a function of fewer than the original number of parameters (Catchpole & Morgan, 1997). If a model is not parameter redundant, it is locally, and possibly even globally, identified; we go into the details of local and global identification in terms of MPT models below.In an MPT model, there is a function f ( ) that maps the model's parameter space into the set of possible multinomial probability distributions. Global identification can be established by proving that inequality of two parameter vectors implies inequality of the modeled multi nomial cell probability vectors f ( ) f ( ) for all parameter vectors and in the parameter space Meiser, 2005;Riefer & Batchelder, 1988; see also Bishop et al., 1975, p. 510). The required calculations for establishing global identification can become tedious, and even intractable, in complex models. In this case, the next best thing is to establish local identification; that is, implies f ( ) f ( ) for all in an open neighborhood around . One can empirically investigate local identification at a specific point G in the parameter space by fitting the generating MPT model to artificial (simulated) data generated by G , or to exact summary statistics-that is, the expected cell counts, derived from the probabilities f ( G ) implied by the generating MPT model. Obtaining parameter estimates sufficiently close to the true values G (in the case of simulated...

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Maximum-Likelihood Estimation of Parameters Subject to Restraints

Cited by 504 publications

References 0 publications

Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses

Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses

Modeling Parental Involvement

Parameter identification in multinomial processing tree models

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