No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Analysis of the Generalized Certain Nested Repeated Measures Models
Abstract
Previously a general method to analyze a nested repeated measure model was developed when the covariance matrix had a certain pattern. A case of being a number of sub-individuals of a particular individual, such as sub-field or other types of offsprings, receives several treatments. As a consequence, the observations are correlated with certain covariance matrix pattern and such a model is known as nested repeated measures model (NRMM). In this paper, a weaker assumption is used when the covariance matrix is arbitrary and has no specific pattern. Independent normally distributed individuals are taken with their own mean and common positive definite covariance matrix. It is aimed to test hypotheses about the mean. Two techniques are used for testing. The first is based on the multivariate one sample model (MOSM), when each individual receives the same treatments and hence has the same mean vector, whilst the second is based on the multivariate linear model (MLM). Different individuals receive different treatments and hence have different mean vectors. For each technique a uniformly most powerful (UMP) invariant size test is found.
ResearchGate has not been able to resolve any citations for this publication.
Enterprise value is the result of interaction between financial and nonfinancial factors. Financial factors represent important resources in production of goods but their contribution on enterprise has decreased with the development of knowledge-based economy. Acceptance of nonfinancial factors as elements generating future benefits imposed the application of European politicies concerning nonfinancial reporting for multinationals.
In this paper, we investigate tests of linear hypotheses in heteroscedastic one-way MANOVA via proposing a modified Bartlett (MB) test. The MB test is easy to conduct via using the usual χ2-table. It is shown to be invariant under affine transformations, different choices of the contrast matrix used to define the same hypothesis and different labeling schemes of the mean vectors. Simulation studies and real data applications demonstrate that the MB test performs well and is generally comparable to Krishnamoorthy and Lu’s (J Statist Comput Simul 80(8):873–887, 2010) parametric bootstrap test in terms of size controlling and power.
For heteroscedastic two-way MANOVA, the so-called modified MANOVA tests proposed recently are too conservative, not powerful and not affine-invariant. In this note, we show how they can be improved and can be made affine-invariant. A real data example and some simulation studies are used to illustrate and demonstrate the methodologies.
The test procedures, invariant under certain groups of transformations [4], for testing a set of multivariate linear hypotheses in the linear normal model depend on the characteristic roots of a random matrix. The power function of such a test depends on the characteristic roots of a corresponding population matrix as parameters; these roots may be regarded as measures of deviation from the hypothesis tested. In this paper sufficient conditions on the procedure for the power function to be a monotonically increasing function of each of the parameters are obtained. The likelihood-ratio test [1], Lawley-Hotelling trace test [1], and Roy's maximum root test [6] satisfy these conditions. The monotonicity of the power function of Roy's test has been shown by Roy and Mikhail [5] using a geometrical method.
Analysis of variance (ANOVA) is a fundamental procedure for event-related potential (ERP) research and yet there is very little guidance for best practices. It is important for the field to develop evidence-based best practices: 1) to minimize the Type II error rate by maximizing statistical power, 2) to minimize the Type I error rate by reducing the latitude for varying procedures, and 3) to identify areas for further methodological improvements. While generic treatments of ANOVA methodology are available, ERP datasets have many unique characteristics that must be considered. In the present report, a novelty oddball dataset was utilized as a test case to determine whether three aspects of ANOVA procedures as applied to ERPs make a real-world difference: the effects of reference site, regional channels, and robust ANOVAs. Recommendations are provided for best practices in each of these areas.
Repeated measurements and multimodal data are common in neuroimaging research. Despite this, conventional approaches to group level analysis ignore these repeated measurements in favour of multiple between-subject models using contrasts of interest. This approach has a number of drawbacks as certain designs and comparisons of interest are either not possible or complex to implement. Unfortunately, even when attempting to analyse group level data within a repeated-measures framework the methods implemented in popular software packages make potentially unrealistic assumptions about the covariance structure across the brain. In this paper, we describe how this issue can be addressed in a simple and efficient manner using the multivariate form of the familiar general linear model (GLM), as implemented in a new MATLAB toolbox. This multivariate framework is discussed, paying particular attention to methods of inference by permutation. Comparisons with existing approaches and software packages for dependent group-level neuroimaging data are made. We also demonstrate how this method is easily adapted for dependency at the group level when multiple modalities of imaging are collected from the same individuals. Follow-up of these multimodal models using linear discriminant functions (LDA) is also discussed, with applications to future studies wishing to integrate multiple scanning techniques into investigating populations of interest.
We develop parametric and nonparametric bootstrap methods for multi-factor multivariate data, without assuming normality, and allowing for covariance matrices that are heterogeneous between groups. The newly proposed, general procedure includes several situations as special cases, such as the multivariate Behrens–Fisher problem, the multivariate one-way layout, as well as crossed and hierarchically nested two-way layouts. We derive the asymptotic distribution of the bootstrap tests for general factorial designs and evaluate their performance in an extensive comparative simulation study. For moderate sample sizes, the bootstrap approach provides an improvement to existing methods in particular for situations with nonnormal data and heterogeneous covariance matrices in unbalanced designs. For balanced designs, less computationally intensive alternatives based on approximate sampling distributions of multivariate tests can be recommended.
In the complete balanced model for the analysis of variance, the equivalence of sums of squares and quadratic forms is seen to imply well-fitting patterns involving Kronecker products of identity matrices and scalar multiples of matrices with all elements equal to 1. The questions of symmetry, idempotency, and orthogonality so central to this topic are answered by simple multiplications; ranks are determined from simple traces. The associations between the forms of the two-factor model are presented here in a way that is accessible to first-year students and makes generalizations to higher order models transparent. The lack of patterns in incomplete or unbalanced models is noted. Additional steps in design and analysis are suggested in the references.
In this article, we propose a parametric bootstrap (PB) test for testing main, simple and interaction effects in heteroscedastic two-way MANOVA models under multivariate normality. The PB test is shown to be invariant under permutation-transformations, and affine-transformations, respectively. Moreover, the PB test is independent of the choice of weights used to define the parameters uniquely. The proposed test is compared with existing Lawley-Hotelling trace (LHT) and approximate Hotelling T2T2 (AHT) tests by the invariance and the intensive simulations. Simulation results indicate that the PB test performs satisfactorily for various cell sizes and parameter configurations when the homogeneity assumption is seriously violated, and tends to outperform the LHT and AHT tests for moderate or larger samples in terms of power and controlling size. In addition, simulation results also indicate that the PB test does not lose too much power when the homogeneity assumption is actually valid or the model assumptions are approximately correct.
We introduce nonparametric versions for many of the hypotheses tested in analysis of variance and repeated measures models, such as the hypotheses of no main effects, no interaction effects, and no factor effects. These natural extensions of the nonparametric hypothesis of equality of the k distributions in the k sample problem have appealing practical interpretations. We concentrate on multivariate repeated measures designs and obtain simple rank statistics for testing these hypotheses. These statistics are the rank transform (RT) versions of the classical statistics for testing hypotheses in repeated measures designs. We emphasize that even though recent research has demonstrated the inappropriateness of the RT method for many parametric hypotheses, the RT procedure is always valid for testing our nonparametric hypotheses. We show that the rank statistics converge in distribution to central chi-squared distributions under their respective nonparametric null hypotheses. The noncentrality parameters under nonparametric contiguous alternatives are obtained. In addition, we present an interpretation of simultaneous confidence intervals and multiple comparison procedures based on rank statistics. Finally, we illustrate the proposed rank tests with a real data set from the statistical literature.
We show that if overall sample size and effect size are held constant, the power of theF test for a one-way analysis of variance decreases dramatically as the number of groups increases. This reduction in power
is even greater when the groups added to the design do not produce treatment effects. If a second independent variable is
added to the design, either a split-plot or a completely randomized design may be employed. For the split-plot design, we
show that the power of theF test on the betweengroups factor decreases as the correlation across the levels of the within-groups factor increases. The
attenuation in between-groups power becomes more pronounced as the number of levels of the withingroups factor increases.
Sample size and total cost calculations are required to determine whether the split-plot or completely randomized design is
more efficient in a particular application. The outcome hinges on the cost of obtaining (or recruiting) a single subject relative
to the cost of obtaining a single observation: We call this thesubject-to-observation cost (SOC) ratio. Split-plot designs are less costly than completely randomized designs only when the SOC ratio is high, the correlation
across the levels of the within-groups factor is low, and the number of such levels is small.
A dictionary between operator-based and matrix-based languages in multivariate statistical analysis is proposed. Then this formulary is applied to asymptotic factorial analyses, especially for giving asymptotic covariance matrices and operators in an explicit form. Finally, we present the mathematical foundations on which are based the functional tools, i.e. tensor products of linear spaces, of vectors, and of operators.
A repeated measures design occurs in analysis of variance when a particular individual receives several treatments. Let be the vector of observations on the ith individual. It is assumed that the are independently normally distributed with mean and common covariance . The researcher wants to test hypotheses about the . Let . For this paper, in order to get powerful tests, the simplifying assumption that the are exchangeable is made. We assume that the design is given and use a coordinate-free approach to find optimal (i.e., UMP invariant, UMP unbiased, most stringent, etc.) procedures for testing a large class of hypotheses about the .
Thesis (Ph. D.)--Pennsylvania State University. Microfilm (positive).
Simulated data for a two-group repeated measurements design were generated with different numbers of equally-spaced measurements interposed between baseline and the end of the study. A standard repeated measurements ANOVA for a split-pilot design was used to test the significance of the between-groups main effect, the Geisser-Greenhouse corrected groups x times interaction, and the difference in linear trends across time. The analyses were repeated with and without baseline measurements entered as a covariate in the model. Monte Carlo results confirmed that increasing the number of repeated measurements across a fixed treatment period generally had negative or neutral implications for power of the tests of significance in the presence of serial dependencies that produced heterogeneous correlations among the repeated measurements.
Statistical power of completely randomized and split-plot factorial designs
- D R Bradley
- R A Orfaly
Bradley, D. R., & Orfaly, R. A. (1999). Statistical power of completely randomized and split-plot
factorial designs. Behavioral Neuroscience.