Relationships Between Full and Layer Models with Applications to Level Merging

doi:10.1080/03610920903470909

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Relationships Between Full and Layer Models with Applications to Level Merging

Journal Article•DOI•

Relationships Between Full and Layer Models with Applications to Level Merging

Vivek Vijay¹•Institutions (1)

Birla Institute of Technology and Science¹

24 Jan 2011-Communications in Statistics-theory and Methods (Taylor & Francis Group)-Vol. 40, Iss: 4, pp 745-761

TL;DR: Relationships between the interaction parameters of the full log-linear model and that of its corresponding layer models are obtained and the concept of merging of factor levels based on these interaction parameters is discussed.

Abstract: Analysis of a large dimensional contingency table is quite involved. Models corresponding to layers of a contingency table are easier to analyze than the full model. Relationships between the interaction parameters of the full log-linear model and that of its corresponding layer models are obtained. These relationships are not only useful to reduce the analysis but also useful to interpret various hierarchical models. We obtain these relationships for layers of one variable, and extend the results for the case when layers of more than one variable are considered. We also establish, under conditional independence, relationships between the interaction parameters of the full model and that of the corresponding marginal models. We discuss the concept of merging of factor levels based on these interaction parameters. Finally, we use the relationships between layer models and full model to obtain conditions for level merging based on layer interaction parameters. Several examples are discussed to illustrate th...

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Journal Article•DOI•

The Statistical Analysis of Discrete Data

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Douglas G. Simpson

01 May 1991-Technometrics

TL;DR: In this article, the authors proposed a large sample theory for linear algebra and applied it to large-scale data sets, including linear algebra, loglinear models, and univariate discrete responses with covariates.

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Abstract: 1 Introduction.- 2 Univariate Discrete Responses.- 3 Loglinear Models.- 4 Cross-Classified Data.- 5 Univariate Discrete Data with Covariates.- Appendix 1. Some Results from Linear Algebra.- Appendix 2. Maximization of Concave Functions.- Appendix 3. Proof of Proposition 3.3.1 (ii) and (iii).- Appendix 4. Elements of Large Sample Theory.- Problems.- References.- List of Notation.- Index to Data Sets.- Author Index.

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12 citations

Journal Article•DOI•

Analyzing Returns and Pattern of Financial Data Using Log-linear Modeling

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Vivek Vijay, Parmod Kumar Paul

01 Jan 2016

TL;DR: In this article, the authors use log-linear modeling for the analysis of the contingency table and test various hypotheses of association for these variables by using Chi-square test for contingency tables.

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Abstract: Technical analysis is useful for forecasting the price movement through the analysis of historic data. This sort of movement has Turn of the year effect also and useful for short term prediction. If the direction of price of two or more assets is same, it becomes necessary to analyze the returns also. We first use optimal band to predict the direction of price and create a contingency table of the data to analyze the pattern (movement) against returns. We use log-linear modeling for the analysis of the contingency table. We next include the volume of transactions as one more variable in the contingency table. The table consisting of three variables, Pattern, Returns and Volume is further analyzed by using log-linear modeling. We test various hypotheses of association for these variables by using Chi-square test for contingency tables.

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1 citations

References

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Book•

Discrete multivariate analysis: theory and practice

[...]

Yvonne M. M. Bishop, Paul W. Holland, Stephen E. Fienberg

01 Jan 1975

TL;DR: Discrete Multivariate Analysis is a comprehensive text and general reference on the analysis of discrete multivariate data, particularly in the form of multidimensional tables, and contains a wealth of material on important topics.

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Abstract: "At last, after a decade of mounting interest in log-linear and related models for the analysis of discrete multivariate data, particularly in the form of multidimensional tables, we now have a comprehensive text and general reference on the subject. Even a mediocre attempt to organize the extensive and widely scattered literature on discrete multivariate analysis would be welcome; happily, this is an excellent such effort, but a group of Harvard statisticians taht has contributed much to the field. Their book ought to serve as a basic guide to the analysis of quantitative data for years to come." --James R. Beninger, Contemporary Sociology "A welcome addition to multivariate analysis. The discussion is lucid and very leisurely, excellently illustrated with applications drawn from a wide variety of fields. A good part of the book can be understood without very specialized statistical knowledge. It is a most welcome contribution to an interesting and lively subject." --D.R. Cox, Nature "Discrete Multivariate Analysis is an ambitious attempt to present log-linear models to a broad audience. Exposition is quite discursive, and the mathematical level, except in Chapters 12 and 14, is very elementary. To illustrate possible applications, some 60 different sets of data have been gathered together from diverse fields. To aid the reader, an index of these examples has been provided. ...the book contains a wealth of material on important topics. Its numerous examples are especially valuable." --Shelby J. Haberman, The Annals of Statistics

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5,249 citations

"Relationships Between Full and Laye..." refers background in this paper

...It is well known, from (4.9), that (see Bishop et al., 1975) 12 1 j = ln p 1 j − 1 I ∑ i ln p i j − 1 J ∑ r ln p 1 r + 1 IJ ∑ i r ln p i r (4.10) and similarly, 12 2 j = ln p 2 j − 1 I ∑ i ln p i j − 1 J ∑ r ln p 2 r + 1 IJ ∑ i r ln p i r (4.11) Using (4.10) and (4.11), 12 1 j = 12 2 j implies ln p…...
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...When, for example, Z = 1 r , the interaction parameter Z satisfies the following zero-sum property (for more details, see Bishop et al., 1975; Vellaisamy and Vijay, 2007):∑ ij Z iZ = 0 for all j ∈ 1 r (2.2) Similarly, let the log-linear model for the (n− 1)-dimensional table corresponding to Xn =…...
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Book•

Graphical Models in Applied Multivariate Statistics

[...]

Joe Whittaker

01 Apr 1990

TL;DR: This introduction to the use of graphical models in the description and modeling of multivariate systems covers conditional independence, several types of independence graphs, Gaussian models, issues in model selection, regression and decomposition.

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Abstract: The Wiley Paperback Series makes valuable content more accessible to a new generation of statisticians, mathematicians and scientists.Graphical models--a subset of log-linear models--reveal the interrelationships between multiple variables and features of the underlying conditional independence. This introduction to the use of graphical models in the description and modeling of multivariate systems covers conditional independence, several types of independence graphs, Gaussian models, issues in model selection, regression and decomposition. Many numerical examples and exercises with solutions are included.This book is aimed at students who require a course on applied multivariate statistics unified by the concept of conditional independence and researchers concerned with applying graphical modelling techniques.

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1,717 citations

Journal Article•DOI•

The Interpretation of Interaction in Contingency Tables

[...]

E. H. Simpson

01 Jul 1951-Journal of the royal statistical society series b-methodological

1,657 citations

"Relationships Between Full and Laye..." refers background or methods in this paper

...The reduction depends on whether the direction and/or strength of interaction among the factors remains the same for both the full table and the reduced table (Simpson, 1951). For log-linear models, Whittemore (1978) and Vellaisamy and Vijay (2007) discussed this concept of collapsibility. If this reduction is not possible, that is, if Simpson’s paradox occurs, then we need to analyze the full (large) contingency table. When reduction of dimension is not possible, that is, when collapsibility conditions do not hold, and the number of levels of variables is large, then merging of levels also may reduce the analysis. Although, the merging of factor levels does not lead to reduction of dimension, it reduces the number of factor levels. Wermuth and Cox (1998) discussed factor level merging for ordinal variables based on interaction parameters of a new parametrization. For nominal data, Dellaportas and Tarantola (2005) extended their results for graphical log-linear models....
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...The reduction depends on whether the direction and/or strength of interaction among the factors remains the same for both the full table and the reduced table (Simpson, 1951). For log-linear models, Whittemore (1978) and Vellaisamy and Vijay (2007) discussed this concept of collapsibility....
[...]
...The reduction depends on whether the direction and/or strength of interaction among the factors remains the same for both the full table and the reduced table (Simpson, 1951)....
[...]
...The reduction depends on whether the direction and/or strength of interaction among the factors remains the same for both the full table and the reduced table (Simpson, 1951). For log-linear models, Whittemore (1978) and Vellaisamy and Vijay (2007) discussed this concept of collapsibility. If this reduction is not possible, that is, if Simpson’s paradox occurs, then we need to analyze the full (large) contingency table. When reduction of dimension is not possible, that is, when collapsibility conditions do not hold, and the number of levels of variables is large, then merging of levels also may reduce the analysis. Although, the merging of factor levels does not lead to reduction of dimension, it reduces the number of factor levels. Wermuth and Cox (1998) discussed factor level merging for ordinal variables based on interaction parameters of a new parametrization....
[...]
...The reduction depends on whether the direction and/or strength of interaction among the factors remains the same for both the full table and the reduced table (Simpson, 1951). For log-linear models, Whittemore (1978) and Vellaisamy and Vijay (2007) discussed this concept of collapsibility. If this reduction is not possible, that is, if Simpson’s paradox occurs, then we need to analyze the full (large) contingency table. When reduction of dimension is not possible, that is, when collapsibility conditions do not hold, and the number of levels of variables is large, then merging of levels also may reduce the analysis. Although, the merging of factor levels does not lead to reduction of dimension, it reduces the number of factor levels. Wermuth and Cox (1998) discussed factor level merging for ordinal variables based on interaction parameters of a new parametrization. For nominal data, Dellaportas and Tarantola (2005) extended their results for graphical log-linear models. The results are again based on the same parametrization. We first note that the results obtained by Wermuth and Cox (1998) are valid for nominal variables if the chain independence condition is satisfied....
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Book•

The Statistical Analysis of Discrete Data

[...]

Thomas J. Santner, Diane E. Duffy

16 Oct 1989

TL;DR: This paper presents a meta-analysis of large sample theory of univariate Discrete Responses and some results from Linear Algebra suggest that the model chosen may be biased towards linear models.

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253 citations