General formulae for expectations, variances and covariances of the mean squares for staggered nested designs

Staggered nested experimental designs are the most popular class of unbalanced nested designs. Using a special notation which covers the particular structure of the staggered nested design, this paper systematically derives the canonical form for the arbitrary m-factors. Under the normality assumption for every random variable, a vector comprising m canonical variables from each experimental unit is normally independently and identically distributed. Every sum of squares used in the analysis of variance (ANOVA) can be expressed as the sum of squares of the corresponding canonical variables. Hence, general formulae for the expectations, variances and covariances of the mean squares are directly obtained from the canonical form. Applying the formulae, the explicit forms of the ANOVA estimators of the variance components and unbiased estimators of the ratios of the variance components are introduced in this paper. The formulae are easily applied to obtain the variances and covariances of any linear combinations of the mean squares, especially the ANOVA estimators of the variance components. These results are eff ectively applied for the standardization of measurement methods.     

Experimental designs with dependent observations: New results and comments on previous results

We present a new sufficient condition on the covariance matrix of the normality distributed observations of an ANOVA model (with orthogonal decomposition of the total sum of squares) under which the F-statictics are distributed proportionally to Fisher F-random variables variables. A new proof of a previous result, a necessary and sufficient condition for applicability of Barlett’s test to the observations of a one-way ANOVA models, and comments on recent results are also given.     

Review: Reversed low-rank ANOVA model for transforming high dimensional genetic data into low dimension

A general modeling procedure for analyzing genetic data is reviewed. We review ANOVA type model that can handle both the continuous and discrete genetic variables in one modeling framework. Unlike the regression type models which typically set the phenotype variable as a response, this ANOVA model treats the phenotype variable as an explanatory variable. By reversely treating the phenotype variable, usual high dimensional problem is turned into low dimension. Instead, the ANOVA model always includes interaction term between the genetic locations and phenotype variable to find potential association between them. The interaction term is designed to be low rank with the multiplication of bilinear terms so that the required number of parameters is kept in a manageable degree. We compare the performance of the reviewed ANOVA model to the other popular methods via microarray and SNP data sets.     

The L2 Rate of Convergence for Event History Regression with Time-dependent Covariates

Consider repeated events of multiple kinds that occur according to a right-continuous semi-Markov process whose transition rates are influenced by one or more time-dependent covariates. The logarithms of the intensities of the transitions from one state to another are modelled as members of a linear function space, which may be finite- or infinite-dimensional. Maximum likelihood estimates are used, where the maximizations are taken over suitably chosen finite-dimensional approximating spaces. It is shown that the L ₂ rates of convergence of the maximum likelihood estimates are determined by the approximation power and dimension of the approximating spaces. The theory is applied to a functional ANOVA model, where the logarithms of the intensities are approximated by functions having the form of a specified sum of a constant term, main effects (functions of one variable), and interaction terms (functions of two or more variables). It is shown that the curse of dimensionality can be ameliorated if only main effects and low-order interactions are considered in functional ANOVA models.     

Bivariate distributions of some ratios of independent noncentral chi-square random variables

The bivariate distributions of three pairs of ratios of in¬dependent noncentral chi-square random variables are considered. These ratios arise in the problem of computing the joint power function of simultaneous F-tests in balanced ANOVA and ANCOVA. The distributions obtained are generalizations to the noncentral case of existing results in the literature. Of particular note is the bivariate noncentral F distribution, which generalizes a special case of Krishnaiah*s (1964,1965) bivariate central F distribution. Explicit formulae for the cdf's of these distribu¬tions are given, along with computational procedures     

基于Fisher变换的Bayes判别方法探索

判别分析是三大多元统计分析方法之一,在许多领域都有广泛的应用。通常认为距离判别、Fisher判别和Bayes判别是三种不同的判别分析方法,本文的研究表明,距离判别与Bayes判别是两种实质的判别方法,前者实际依据的是百分位点或置信区间,后者实际依据的是概率。而著名的Fisher判别,只是依据方差分析的思想,对判别变量进行线性变换,然后用于距离判别,其实不能算是一种实质的判别方法。本文将Fisher变换与Bayes判别结合起来,即先做Fisher变换,再利用概率最大原则做Bayes判别,得到一种新的判别途径,可进一步提高判别效率。理论与实证分析表明,基于Fisher变换的Bayes判别,适用场合广泛,判别效率最高。     

Power of analysis of covariance tests under conditional specification

The procedure-wise power functions of two strategies for balanced single-factor analysis of covariance in the presence of possibly unequal regression slopes are evaluated and illustrated. The strategies differ in the action to be taken following a re-^ jection by the preliminary test for equal slopes. The first strategy simply discards the covariate and respecifies the model as the one-way ANOVA model for testing factor effects. The second leaves the unequal slopes covariance model intact, but respecifies the factor effects hypothesis to address the factor level means adjusted to the sample average of the covariate. One additional strategy, that of testing factor effects only if the preliminary slopes test does not reject, is included for comparison purposes. Computation of the power functions requires extensive use of the results obtained in Hawkins and Han (1986) concerning the bivariate distributions of certain ratios of independent noncentral chi-square random variables.     

Choice of the primary analysis in longitudinal clinical trials

Missing data, and the bias they can cause, are an almost ever‐present concern in clinical trials. The last observation carried forward (LOCF) approach has been frequently utilized to handle missing data in clinical trials, and is often specified in conjunction with analysis of variance (LOCF ANOVA) for the primary analysis. Considerable advances in statistical methodology, and in our ability to implement these methods, have been made in recent years. Likelihood‐based, mixed‐effects model approaches implemented under the missing at random (MAR) framework are now easy to implement, and are commonly used to analyse clinical trial data. Furthermore, such approaches are more robust to the biases from missing data, and provide better control of Type I and Type II errors than LOCF ANOVA. Empirical research and analytic proof have demonstrated that the behaviour of LOCF is uncertain, and in many situations it has not been conservative. Using LOCF as a composite measure of safety, tolerability and efficacy can lead to erroneous conclusions regarding the effectiveness of a drug. This approach also violates the fundamental basis of statistics as it involves testing an outcome that is not a physical parameter of the population, but rather a quantity that can be influenced by investigator behaviour, trial design, etc. Practice should shift away from using LOCF ANOVA as the primary analysis and focus on likelihood‐based, mixed‐effects model approaches developed under the MAR framework, with missing not at random methods used to assess robustness of the primary analysis. Copyright © 2004 John Wiley & Sons, Ltd.     

Interpretation of the coefficient of determination of an ANOVA model as a measure of polarization

In this paper, it is demonstrated that coefficient of determination of an ANOVA linear model provides a measure of polarization. Taking as the starting point the link between polarization and dispersion, we reformulate the measure of polarization of Zhang and Kanbur using the decomposition of the variance instead of the decomposition of the Theil index. We show that the proposed measure is equivalent to the coefficient of determination of an ANOVA linear model that explains, for example, the income of the households as a function of any population characteristic such as education, gender, occupation, etc. This result provides an alternative way to analyse polarization by sub-populations characteristics and at the same time allows us to compare sub-populations via the estimated coefficients of the ANOVA model.     

Analysis of ranked data in randomized blocks when there are missing values

Data consisting of ranks within blocks are considered for randomized block designs when there are missing values. Tied ranks are possible. Such data can be analysed using the Skillings–Mack test. Here we suggest a new approach based on carrying out an ANOVA on the ranks using the general linear model platform available in many statistical packages. Such a platform allows an ANOVA to be calculated when there are missing values. Indicative sizes and powers show the ANOVA approach performs better than the Skillings–Mack test.     

The effect of interaction and rounding error in two-way ANOVA: example of impact on testing for normality

A key issue in various applications of analysis of variance (ANOVA) is testing for the interaction and the interpretation of resulting ANOVA tables. In this note it is demonstrated that for a two-way ANOVA, whether interactions are incorporated or not may have a dramatic influence when considering the usual statistical tests for normality of residuals. The effect of numerical rounding is also discussed.     

Simulation-based model comparison methodology with application to road accident models

This article assumes the goal of proposing a simulation-based theoretical model comparison methodology with application to two time series road accident models. The model comparison exercise helps to quantify the main differences and similarities between the two models and comprises of three main stages: (1) simulation of time series through a true model with predefined properties; (2) estimation of the alternative model using the simulated data; (3) sensitivity analysis to quantify the effect of changes in the true model parameters on alternative model parameter estimates through analysis of variance, ANOVA. The proposed methodology is applied to two time series road accident models: UCM (unobserved components model) and DRAG (Demand for Road Use, Accidents and their Severity). Assuming that the real data-generating process is the UCM, new datasets approximating the road accident data are generated, and DRAG models are estimated using the simulated data. Since these two methodologies are usually assumed to be equivalent, in a sense that both models accurately capture the true effects of the regressors, we are specifically addressing the modeling of the stochastic trend, through the alternative model. Stochastic trend is the time-varying component and is one of the crucial factors in time series road accident data. Theoretically, it can be easily modeled through UCM, given its modeling properties. However, properly capturing the effect of a non-stationary component such as stochastic trend in a stationary explanatory model such as DRAG is challenging. After obtaining the parameter estimates of the alternative model (DRAG), the estimates of both true and alternative models are compared and the differences are quantified through experimental design and ANOVA techniques. It is observed that the effects of the explanatory variables used in the UCM simulation are only partially captured by the respective DRAG coefficients. This a priori, could be due to multicollinearity but the results of both simulation of UCM data and estimating of DRAG models reveal that there is no significant static correlation among regressors. Moreover, in fact, using ANOVA, it is determined that this regression coefficient estimation bias is caused by the presence of the stochastic trend present in the simulated data. Thus, the results of the methodological development suggest that the stochastic component present in the data should be treated accordingly through a preliminary, exploratory data analysis.     

Tests for homogeneity of variance

Six procedures which convert tests of homogeneity of variance into tests for mean equality for independent groups are compared. The tests are the analysis of variance (ANOVA) and Welch F statistics. The Welch statistics are included since it was anticipated that ANOVA would not provide a robust test when samples of unequal sizes are obtained from non-normal populations. However, the Welch tests are not found to be uniformly preferrable. In addition, a prior recommendation for Miller's jackknife procedure is not supported for the unequal sample size case. The data indicates that the current tests for variance heterogeneity are either sensitive to non-normality or, if robust, lacking in power. Therefore, these tests cannot be recommended for the purpose of testing the validity of the ANOVA homogeneity assumption.     

An empirical comparison of several methods for testing the equality of dependent proportions

Three methods for testing the equality of nonindependent proportions were compared with, the use of Monte Carlo techniques. The three methods included Cochran's test, an ANOVA F test, and Hotelling's T² test. With respect to empirical significance levels, the ANOVA F test is recommended as the preferred method of analysis.

Oftentimes an experimenter is interested in testing the equality of several proportions. When the proportions are independent Kemp and Butcher (1972) and Butcher and Kemp (1974) compared several methods for analysing large sample binomial data for the case of a 3 x 3 factorial design without replication. In addition, Levy and Narula (1977) compared many of the same methods for analyzing binomial data; however, Levy and Narula investigated the relative utility of the methods for small sample sizes.     

An extension of the median test to analysis of variance

In a k-way analysis of variance model, the major concern is testing for main effects and for the presence of interaction between the factors. When the assumptions of normality and equal variances are satisfied, the appropriate test to use is the usual F-test for ANOVA. However, when the normality assumption is not satisfied then a robust or nonparametric test is needed to conduct the analysis. In this paper a nonparametric method based on cell counts is proposed. Each cell is divided into L subcells based on predetermined outpoints and the resulting frequencies are laid out in a contingency table. Then the Pearson x² and tne likelihood ratio tests are performed. A comparison with the classical ANOVA F-test indicates that the proposed method is preferable when the data comes from a thick-tailed highly skewed distribution.     

Comparison of designs for the three-fold nested random model

The quality of estimation of variance components depends on the design used as well as on the unknown values of the variance components. In this article, three designs are compared, namely, the balanced, staggered, and inverted nested designs for the three-fold nested random model. The comparison is based on the so-called quantile dispersion graphs using analysis of variance (ANOVA) and maximum likelihood (ML) estimates of the variance components. It is demonstrated that the staggered nested design gives more stable estimates of the variance component for the highest nesting factor than the balanced design. The reverse, however, is true in case of lower nested factors. A comparison between ANOVA and ML estimation of the variance components is also made using each of the aforementioned designs.     

Comparison of the estimators of the intra-cluster correlation for the nested error regression model

The intra-cluster correlation is insisted on nested error regression model that, in practice, is rarely known. This article demonstrates the size in generalized least squares (GLS) F-test using Fuller–Battese transformation and modification F-test. For the balanced case, the former using strictly positive, analysis of covariance (ANCOVA) and analysis of variance (ANOVA) estimators of intra-cluster correlation can control the size for moderate intra-cluster correlations. For small intra-cluster correlation, they perform well when the numbers of cluster are large. The latter using the ANOVA estimator performs well except for small numbers of cluster. When intra-cluster correlation is large, it cannot control the size. For the unbalanced case, the GLS F-test using the Fuller–Battese transformation and the modification F-test using the strictly positive, the ANCOVA and the ANOVA estimators maintain the significance level for small total sample size and small intra-cluster correlations when there is a large variation in cluster sizes, but they perform well in controlling the size for large total sample size and small different variation in cluster sizes. Besides, Henderson’s method 3 estimator maintains the significance level for a few situations.     

Interpretation of the four types of analysis of variance tables in sas

Various computational methods exist for generating sums of squares in an analysis of variance table. When the ANOVA design is balanced, most of these computational methods will produce equivalent sums of squares for testing the significance of the ANOVA model parameters. However, when the design is unbalanced, as is frequently the case in practice, these sums of squares depend on the computational method used.- The basic reason for the difference in these sums of squares is that different hypotheses are being tested. The purpose of this paper is to describe these hypotheses in terms of population or cell means. A numerical example is given for the two factor model with interaction. The hypotheses that are tested by the four computational methods of the SAS general linear model procedure are specified.

Although the ultimate choice of hypotheses should be made by the researcher before conducting the experiment, this paper

PENDLETON,VON TRESS,AND BREMER

presents the following guidelines in selecting these hypotheses:

When the design is balanced, all of the SAS procedures will agree.

In unbalanced ANOVA designs when there are no missing cells. SAS Type III should be used. SAS Type III tests an unweighted hypothesis about cell means. SAS Types I and II test hypotheses that are functions of the ceil frequencies. These frequencies are often merely arti¬facts of the experimental process and not reflective of any underlying frequencies in the population.

When there are missing cells, i.e. no observations for some factor level combinations. Type IV should be used with caution. SAS Type IV tests hypotheses which depend     

Confidence intervals and tests on the variance ratio in random models with two variance components

The LM test is modified to test any value of the ratio of two variance components in a mixed effects linear model with two variance components. The test is exact, so it can be used to construct exact confidence intervals on this ratio.Exact Neyman-Pearson (NP) tests on the variance ratio are described.Their powers provide attainable upper bounds on powers of tests on the variance ratio.Efficiencies of LM tests, which include ANOVA tests, and NP tests are compared for unbalanced, random, one-way ANOVA models.Confidence intervals corresponding to LM tests and NP tests are described.     

Umvu estimators for the population mean and variance based on random effects models for lognormal data

Cross-classified data are often obtained in controlled experimental situations and in epidemiologic studies. As an example of the latter, occupational health studies sometimes require personal exposure measurements on a random sample of workers from one or more job groups, in one or more plant locations, on several different sampling dates. Because the marginal distributions of exposure data from such studies are generally right-skewed and well-approximated as lognormal, researchers in this area often consider the use of ANOVA models after a logarithmic transformation. While it is then of interest to estimate original-scale population parameters (e.g., the overall mean and variance), standard candidates such as maximum likelihood estimators (MLEs) can be unstable and highly biased. Uniformly minimum variance unbiased (UMVU) cstiniators offer a viable alternative, and are adaptable to sampling schemes that are typiral of experimental or epidemiologic studies. In this paper, we provide UMVU estimators for the mean and variance under two random effects ANOVA models for logtransformed data. We illustrate substantial mean squared error gains relative to the MLE when estimating the mean under a one-way classification. We illustrate that the results can readily be extended to encompass a useful class of purely random effects models, provided that the study data are balanced.