# Relationships Between Full and Layer Models with Applications to Level Merging

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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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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1,717 citations

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....

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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)....

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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. 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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253 citations

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

...Secondly, these are useful to interpret various hierarchical log-linear models (Santner and Duffy, 1989, p. 154)....

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150 citations

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

...For log-linear models, Whittemore (1978) and Vellaisamy and Vijay (2007) discussed this concept of collapsibility....

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