A New Analysis of Variance Model for Non-additive Data

Mandel, John

doi:10.1080/00401706.1971.10488751

Cited by 270 publications

(139 citation statements)

References 8 publications

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“…Degrees of freedom for the secondary factors have been discussed by Mandel,9 who states that a number of degrees of freedom can be defined by recognizing that the expected value of a secondary eigenvalue, divided by the appropriate number of degrees of freedom, should be an unbiased estimate of the error variance σ 2 :…”

Section: Mandel's Degrees Of Freedom For Secondary Factorsmentioning

confidence: 99%

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Modification of Malinowski'sF-test for abstract factor analysis applied to the Quail Roost II data sets

Faber

Kowalski

1997

J. Chemometrics

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Section: Mandel's Degrees Of Freedom For Secondary Factorsmentioning

confidence: 99%

“…More elaborate simulations 8 showed that this is only approximately true. Building on the pioneering work of Mandel 9 we developed a modification of Malinowski's F-test that yields sharper significance levels for the significant factors. 10 This advantage is due to the larger number of degrees of freedom used in the modified test.…”

mentioning

confidence: 99%

Modification of Malinowski'sF-test for abstract factor analysis applied to the Quail Roost II data sets

Faber

Kowalski

1997

J. Chemometrics

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“…Such restrictions are used in the analysis of variance with non-additive fits (Mandel, 1969(Mandel, , 1971 and in factor analysis (Gollob, 1968). With help of the relations (18)- (21), we represent (17) as a conditional objective…”

Section: Generalized Svdmentioning

confidence: 99%

Generalized Singular Value Decomposition with Additive Components

Lipovetsky

2016

J. Mod. App. Stat. Meth.

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The singular value decomposition (SVD) technique is extended to incorporate the additive components for approximation of a rectangular matrix by the outer products of vectors. While dual vectors of the regular SVD can be expressed one via linear transformation of the other, the modified SVD corresponds to the general linear transformation with the additive part. The method obtained can be related to the family of principal component and correspondence analyses, and can be reduced to an eigenproblem of a specific transformation of a data matrix. This technique is applied to constructing dual eigenvectors for data visualizing in a two dimensional space.

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“…Given a mean and covariance structure, the vector of observations per individual were generated from a multivariate normal distribution. In addition to Tukey’s and saturated models, we considered simulations under additive main effects and multiplicative interaction (AMMI) models [27, 28]. AMMI models are a class of interaction models that have a flexible structure, which essentially entails a singular value decomposition (SVD) of the cell residual matrix after removing the additive main effects.…”

Section: Simulation Studymentioning

confidence: 99%

Testing departure from additivity in Tukey's model using shrinkage: application to a longitudinal setting

Mukherjee

Smith

et al. 2014

Statistics in Medicine

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While there has been extensive research developing gene-environment interaction (GEI) methods in case-control studies, little attention has been given to sparse and efficient modeling of GEI in longitudinal studies. In a two-way table for GEI with rows and columns as categorical variables, a conventional saturated interaction model involves estimation of a specific parameter for each cell, with constraints ensuring identifiability. The estimates are unbiased but are potentially inefficient because the number of parameters to be estimated can grow quickly with increasing categories of row/column factors. On the other hand, Tukey’s one degree of freedom (df) model for non-additivity treats the interaction term as a scaled product of row and column main effects. Due to the parsimonious form of interaction, the interaction estimate leads to enhanced efficiency and the corresponding test could lead to increased power. Unfortunately, Tukey’s model gives biased estimates and low power if the model is misspecified. When screening multiple GEIs where each genetic and environmental marker may exhibit a distinct interaction pattern, a robust estimator for interaction is important for GEI detection. We propose a shrinkage estimator for interaction effects that combines estimates from both Tukey’s and saturated interaction models and use the corresponding Wald test for testing interaction in a longitudinal setting. The proposed estimator is robust to misspecification of interaction structure. We illustrate the proposed methods using two longitudinal studies — the Normative Aging Study and the Multi-Ethnic Study of Atherosclerosis.

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A New Analysis of Variance Model for Non-additive Data

Cited by 270 publications

References 8 publications

Modification of Malinowski'sF-test for abstract factor analysis applied to the Quail Roost II data sets

Modification of Malinowski'sF-test for abstract factor analysis applied to the Quail Roost II data sets

Generalized Singular Value Decomposition with Additive Components

Testing departure from additivity in Tukey's model using shrinkage: application to a longitudinal setting

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