Analysis of Two-Way Layout of Count Data Involving Multiple Counts in Each Cell
01 Dec 1998-Journal of the American Statistical Association (Taylor & Francis Group)-Vol. 93, Iss: 444, pp 1419-1429
TL;DR: In this paper, the authors developed C(α) tests for interaction and main effects assuming data to be Poisson distributed and also assuming that data within the cells have extra (over/under) dispersion beyond that explained by a Poisson distribution.
Abstract: Multiple counts may occur in each cell of an a × b two-way layout (balanced or unbalanced) of two fixed factors A and B. Standard log-linear model analysis based on a Poisson distribution assumption of the cell counts is not applicable here, because of the unbalanced nature of the table or because the Poisson distribution assumption is not valid. We develop C(α) tests for interaction and main effects assuming data to be Poisson distributed and also assuming that data within the cells have extra (over/under) dispersion beyond that explained by a Poisson distribution. For this we consider an extended negative binominal distribution and a semiparametric model using the quasi-likelihood. We show that in all situations the C(α) tests for interaction are of very simple forms. For C(α) tests for the main effect in presence of no interaction, such simplification is possible only under certain conditions. A score test for detecting extra dispersion in presence of interaction is also obtained and is of sim...
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1,275 citations
TL;DR: A first-order bias-corrected maximum likelihood estimator for the negative binomial dispersion parameter is derived and has superior bias and efficiency properties in most instances.
Abstract: Nous decrivons un estimateur du maximum de vraisemblance avec correction du premier ordre pour le biais sur le parametre de dispersion d'une distribution binomiale negative. Cet estimateur est compare, en terme debiais et d'efficacite, avec l'estimateur du maximum de vraisemblance etudie par Piegorsch (1990, Biometrics 46, 863-867), les estimateurs de moment et du maximum de quasi-vraisemblance etendue de Clark et Perry (1989, Biometrics 45, 309-316) et un estimateur de double quasi-vraisemblance etendue. L'estimateur du maximum de vraisemblance avec correction du biais a des proprietes de biais et d'efficacite superieures dans la plupart des cas. Pour faciliter la comparaison, nous donnons les resultats pour le modele binomial negatif a deux parametres. Cependant un exemple impliquant la regression binomiale negative est donne.
116 citations
Cites methods from "Analysis of Two-Way Layout of Count..."
...Different authors have used different parameterizations for the negative binomial distribution (see, for example, Paul and Plackett, 1978; Barnwal and Paul, 1988; Piegorsch, 1990; Paul and Banerjee, 1998)....
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01 Jan 2003
TL;DR: For any generalized linear model, the Pearson goodness of fit statistic is the score test statistic for testing the current model against the saturated model, and the relationship between the Pearson statistic and the residual deviance is therefore explained in this paper.
Abstract: For any generalized linear model, the Pearson goodness of fit statistic is the score test statistic for testing the current model against the saturated model. The relationship between the Pearson statistic and the residual deviance is therefore the relationship between the score test and the likelihood ratio test statistics, and this clarifies the role of the Pearson statistic in generalized linear models. The result is extended to cases in which there are multiple reponse observations for the same combination of explanatory variables.
62 citations
Cites background from "Analysis of Two-Way Layout of Count..."
...Paul and Banerjee [19] derive the score test for interaction in a two-way contingency table with multiple counts in each cell....
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...For this reason score tests have been proposed frequently in generalized linear model contexts to test for various sorts of model complications such as overdispersion [3, 5, 7, 13, 19, 24], zero inflation [8], adequacy of the link function [9, 20], or extra terms in the fitted model [1, 2,4, 19, 21, 26]....
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...This setup arises frequently when extra parameters are introduced to accommodate overdispersion in generalized linear models [1, 2, 7, 19]....
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...Corollary 1 includes Paul and Banerjee's Theorem 1 as a special case....
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...It includes for example as special cases the results on tests for independence in two-way contingency tables of Thall [26] and Paul and Banerjee [19]....
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TL;DR: In this paper, the authors developed score tests of goodness of fit for discrete generalized linear models against zero inflation against binomial and Poisson models, and in the latter case the proposed test reduces to that of Broek (1995).
Abstract: The authors develop score tests of goodness of fit for discrete generalized linear models against zero inflation. The binomial and Poisson models are treated as examples, and in the latter case the proposed test reduces to that of Broek (1995). Some simulation results and an illustrative example are presented.
Les auteurs developpent des procedures scores permettant de tester l'adequation de modeles lineaires generalises discrets lorsque la valeur zero est en surnombre dans l'echantillon. Les modeles binomial et de Poisson font l'objet d'une attention particuliere et, dans ce dernier cas, le test obtenu se ramene a celui de Broek (1995). Des simulations et un exemple sont egalement presentes.
49 citations
Cites background from "Analysis of Two-Way Layout of Count..."
...For additional discussion on the choice of C(α) tests, cf. Breslow (1990) and Paul & Banerjee (1998)....
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01 Jan 2005
TL;DR: In this article, the class of zero-inated over-dispersed generalized linear models and score tests for selecting a model that can handle such data were proposed, and the power properties of the tests were examined by a limited simulation study.
Abstract: Discrete data in the form of counts often exhibit extra variation that cannot be explained by a simple model, such as the binomial or the Poisson. Also, these data sometimes show more zero counts than what can be predicted by a simple model. Therefore, a discrete generalized linear model (Poisson or binomial) may fail to t a set of discrete data either because of zero-ination, because of over- dispersion, or because there is zero-ination as well as over-dispersion in the data. Previous published work deals with goodness of t tests of the generalized linear model against zero-ination and against over-dispersion separately. In this paper we deal with the class of zero-inated over-dispersed generalized linear models and propose procedures based on score tests for selecting a model that ts such data. For over-dispersion we consider a general over-dispersion model and specic over- dispersion models. We show that in certain cases and under certain conditions, the score tests derived using the general over-dispersion model and those devel- oped under specic over-dispersion models are identical. Empirical level and power properties of the tests are examined by a limited simulation study. Simulations show that the score tests, in general, hold nominal levels well and have good power properties. Two illustrative examples and a discussion are presented.
46 citations
Cites background from "Analysis of Two-Way Layout of Count..."
...However, both of these tests require estimates of the parameters under the alternative hypotheses and often show liberal or conser- vative behaviour in small samples (See, for example, Barnwal and Paul (1988), Thall (1992) and Paul and Banerjee (1998))....
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References
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Abstract: JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Blackwell Publishing and Royal Statistical Society are collaborating with JSTOR to digitize, preserve and extend access to Journal of the Royal Statistical Society. Series A (General). SUMMARY The technique of iterative weighted linear regression can be used to obtain maximum likelihood estimates of the parameters with observations distributed according to some exponential family and systematic effects that can be made linear by a suitable transformation. A generalization of the analysis of variance is given for these models using log-likelihoods. These generalized linear models are illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables) and gamma (variance components). The implications of the approach in designing statistics courses are discussed.
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