Usual Analysis Of Variance Research Articles

Resampling methods in ANOVA for data from the von Mises-Fisher distribution

An important problem in directional statistics is to test the null hypothesis of a common mean direction for several populations. The Analysis of Variance (ANOVA) test for vectorial data may be used to test the hypothesis of the equality of the mean directions for several von Mises-Fisher populations. As this test is valid only for large concentrations, we propose in this paper to apply the resampling techniques of bootstrap and permutation to the ANOVA test. We carried out an extensive simulation study in order to evaluate the performance of the ANOVA test with the resampling techniques, for several sphere dimensions and different sample sizes and we compare with the usual ANOVA test for data from von Mises-Fisher populations. The purpose of this simulation study is also to investigate whether the proposed tests are preferable to the ANOVA test, for low concentrations and small samples. Finally, we present an example with spherical data.

Evaluation and Interpretation of Interactions

Understanding the factors that define a given interaction is important in agricultural, agronomic, and plant breeding research, where agronomic treatments or genotypes are evaluated under several environmental conditions and where interactions usually complicate a researcher's decisions. We give examples of how interactions, in common agricultural experiments, can be examined and studied to make use of the rich information available on the interaction term of the model. Examples with different levels of interaction complexity are used to illustrate how to analyze and interpret interactions and how interaction components can be partitioned into comparisons with sensible biological interpretations. It will offer researchers a greater understanding of how to exploit interaction information beyond the standard statistical tests performed in the usual analysis of variance. Simple SAS codes for performing standard interaction contrasts and defining interaction covariables are provided.

Robustified MANOVA with applications in detecting differentially expressed genes from oligonucleotide arrays

Oligonucleotide arrays such as Affymetrix GeneChips use multiple probes, or a probe set, to measure the abundance of mRNA of every gene of interest. Some analysis methods attempt to summarize the multiple observations into one single score before conducting further analysis such as detecting differentially expressed genes (DEG), clustering and classification. However, there is a risk of losing a significant amount of information and consequently reaching inaccurate or even incorrect conclusions during this data reduction. We developed a novel statistical method called robustified multivariate analysis of variance (MANOVA) based on the traditional MANOVA model and permutation test to detect DEG for both one-way and two-way cases. It can be extended to detect some special patterns of gene expression through profile analysis across k (>or=2) populations. The method utilizes probe-level data and requires no assumptions about the distribution of the dataset. We also propose a method of estimating the null distribution using quantile normalization in contrast to the 'pooling' method (Section 3.1). Monte Carlo simulation and real data analysis are conducted to demonstrate the performance of the proposed method comparing with the 'pooling' method and the usual Analysis of Variance (ANOVA) test based on the summarized scores. It is found that the new method successfully detects DEG under desired false discovery rate and is more powerful than the competing method especially when the number of groups is small. The package of robustified MANOVA can be downloaded from http://faculty.ucr.edu/~xpcui/software

Combining multiple microarrays in the presence of controlling variables

Microarray technology enables the monitoring of expression levels for thousands of genes simultaneously. When the magnitude of the experiment increases, it becomes common to use the same type of microarrays from different laboratories or hospitals. Thus, it is important to analyze microarray data together to derive a combined conclusion after accounting for the differences. One of the main objectives of the microarray experiment is to identify differentially expressed genes among the different experimental groups. The analysis of variance (ANOVA) model has been commonly used to detect differentially expressed genes after accounting for the sources of variation commonly observed in the microarray experiment. We extended the usual ANOVA model to account for an additional variability resulting from many confounding variables such as the effect of different hospitals. The proposed model is a two-stage ANOVA model. The first stage is the adjustment for the effects of no interests. The second stage is the detection of differentially expressed genes among the experimental groups using the residuals obtained from the first stage. Based on these residuals, we propose a permutation test to detect the differentially expressed genes. The proposed model is illustrated using the data from 133 microarrays collected at three different hospitals. The proposed approach is more flexible to use, and it is easier to accommodate the individual covariates in this model than using the meta-analysis approach. A set of programs written in R will be electronically sent upon request.

Analysis of Data with Imbalance in the Baseline Outcome Variable for Randomized Clinical Trials

In randomized clinical trials, substantial imbalance in the baseline outcome variable may occur by chance. Inference on treatment effect may be confounded by such imbalance if not properly accounted for. The usual unadjusted analysis may be conditionally biased with an inflated Type I error rate or reduced power conditioning on the baseline imbalance in the observed sample. This paper reviews methods for baseline adjustment with emphasis on the analysis of co-variance (ANCOVA) model with the baseline outcome as a covariate. Many issues on the ANCOVA model, including the nature of the adjustment, unconditional and conditional properties of treatment effect estimate from the ANCOVA, and baseline with measurement errors are discussed. In summary, the ANCOVA model with the baseline outcome as a covariate is more efficient unconditionally, and has better statistical properties conditionally, than the usual analysis of variance (ANOVA) analysis based on either posttreat-ment outcome or change from baseline and, therefore, is recommended. A real-life data set is analyzed and used to illustrate the unconditional and conditional properties.

Testing dispersion effects from general unreplicated fractional factorial designs

Continuous improvement of the quality of industrial products is an essential factor in modern-day manufacturing. The investigation of those factors that affect process mean and process dispersion (standard deviation) is an important step in such improvements. Most often, experiments are executed for such investigations. To detect mean factors, I use the usual analysis of variance on the experimental data. However, there is no unified method to identify dispersion factors. In recent years several methods have been proposed for identifying such factors with two levels. Multilevel factors, especially three-level factors, are common in industrial experiments, but we lack methods for identifying dispersion effects in multilevel factors. In this paper, I develop a method for identifying dispersion effects from general fractional factorial experiments. This method consists of two stages. The first stage involves the identification of mean factors using the performance characteristic as the response. The second stage involves the computation of a dispersion measure and the identification of dispersion factors using the dispersion measure as the response. The sequence for identifying dispersion factors is first to test the significance of the total dispersion effect of a factor, then to test the dispersion contrasts of interest, which is a method similar to the typical post hoc testing procedure in the ANOVA analysis. This familiar approach should be appealing to practitioners. Copyright © 2001 John Wiley & Sons, Ltd.

Analysis of factorial experiments for survival data with long-tailed distributions

Two left-truncated survival data sets are collected in one-way factorial designs to examine the quality of products. We cannot specify our survival function completely, and only know that the tail has a power functional form of its argument. Thus, our problem is a left-truncated one with incomplete survivor functions. One of our data sets is the case where the usual analysis of variance (ANOVA) may be adapted. The other is a repeated measurement case. We note that the likelihood function is expressed as a product of conditional and marginal likelihood functions. Estimates of power parameters are always obtained by the conditional likelihood. Location parameters describing treatment eff ects are included in the marginal likelihood only, and their estimates are undetermined, because of missing values resulting from left truncation. However, in the ANOVA case, we show that a common structure of power parameters and some simple assumptions about the missing values enable us to construct an approximate F test for treatment effects through the marginal likelihood. This result is extended to a regression case. With the data in repeated measurements, a systematic variation of the power parameters and an apparent deviation from our presupposed model make an application of the ANOVA mentioned impossible, and compel us to generalize our model. By using the ratio of those generalized models, we show that a descriptive model for evaluating treatment effects can be constructed.

Tests for the Analysis of Variance of Crossover Designs with Correlated Errors

The usual analysis of variance based on ordinary least squares (OLS) may be inappropriate to analyze the crossover designs because of correlations within subjects arising from the repeated measurements. The tests used with OLS are compared with three alternative tests that take into account the structure of the covariance matrix for testing the treatment and carryover effects under the assumption of multivariate normality. A modified F-test approximation and a test derived from a Pearson curve fit are shown to give adequate control over Type I error compared to the OLS tests and to tests following an empirical generalized least squares analysis.

Predictive and postdictive success of statistical analyses of yield trials

The accuracy of a yield trial can be increased by improved experimental techniques, more replicates, or more efficient statistical analyses. The third option involves nominal fixed costs, and is therefore very attractive. The statistical analysis recommended here combines the Additive main effects and multiplicative interaction (AMMI) model with a predictive assessment of accuracy. AMMI begins with the usual analysis of variance (ANOVA) to compute genotype and environment additive effects. It then applies principal components analysis (PCA) to analyze non-additive interaction effects. Tests with a New York soybean yield trial show that the predictive accuracy of AMMI with only two replicates is equal to the predictive accuracy of means based on five replicates. The effectiveness of AMMI increases with the size of the yield trial and with the noisiness of the data. Statistical analysis of yield trials with the AMMI model has a number of promising implications for agronomy and plant breeding research programs.

Statistical Analysis of a Yield Trial

AbstractYield trials frequently have both significant main effects and a significant genotype X environment (GE) interaction. Traditional statistical analyses are not always effective with this data structure: the usual analysis of variance (ANOVA), having a merely additive model, identifies the GE interaction as a source but does not analyze it; principal components analysis (PCA), on the other hand is a multiplicative model and hence contains no sources for additive genotype or environment main effects; and linear regression (LR) analysis is able to effectively analyze interaction terms only where the pattern fits a specific regression model. The consequence of fitting inappropriate statistical models to yield trial data is that the interaction may be declared nonsignificant, although a more appropriate analysis would find agronomically important and statistically significant patterns in the interaction. Therefore, agronomists and plant breeders may fail to perceive important interaction effects. This paper compares the above three traditional models with the additive main effects and multiplicative interaction (AMMI) Model, in an analysis of a soybean [Glycine max (L.) Merr.] yield trial. ANOVA fails to detect a significant interaction component, PCA fails to identify and separate the significant genotype and environment main effects, and LR accounts for only a small portion of the interaction sum of squares. On the other hand, AMMI analysis reveals a highly significant interaction component that has clear agronomic meaning. Since ANOVA, PCA, and LR are sub‐cases of the more complete AMMI model, AMMI offers a more appropriate first statistical analysis of yield trials that may have a genotype X environment interaction. AMMI analysis can then be used to diagnose whether or not a specific sub‐case provides a more appropriate analysis. AMMI has no specific experimental design requirements, except for a two‐way data structure.

A critical appraisal of the analysis of variance for two-way classifications

This paper discusses two reasons for invalidity of analysis of variance, namely heterogeneous variance and heterogeneous correlation. Textbook examples of data in two classifications are presented in which these failures of validating assumptions were not recognised and which led to analyses which were seriously in error. Both forms of heterogeneity result in a residual mean square in the usual analysis of variance being made up of heterogeneous components; the difficulty is overcome by partitioning the mean square and using the component appropriate to the hypothesis to be tested and the contrast to be estimated.

Genotype by environment interactions and genetic correlations involving two environmental factors

The concept of the genetic correlation for one trait across environments was extended to two environmental factors B and C. Three additive genetic correlations for the same trait were defined: rG, across both environmental factors; rG(B), across C and within B; and rG(C), across B and within C. As genotype x environment variances increase, the genetic correlations across environments decrease. These three genetic correlations are biased downward in the presence of heterogeneity of genetic variances within environments when they are calculated from the usual analysis of variance (r*G, r*G(B), r*G(C)). Correction factors were derived to remove the bias. The two-way genotype by environment interaction variances can be biased upward or downward by the heterogeneity, but the three-way interaction variance is always biased upward. Correction factors for the interaction variances were also derived. Four additive genetic correlations between two traits (X and Y) were derived: rG xy, across both B and C; rG(B) xy, and rG(C) xy, across one environmental factor and within the other; and rG(BC) xy, within both B and C. These concepts were extended to genetic correlations for dominance and maternal effects. Paternal half-sib and factorial mating designs were used to obtain the various genetic correlations. An example of a paternal half-sib design with beef cattle was used to illustrate the methodology.

Some extensions of anova techniques to location-scale models

In the usual analysis of variance (ANOVA) framework, the different distributions being compared are assumed to differ only in location so that the various measures of comparison are based on these location parameters only. The shift functions, introduced by Doksum (1974, provide a natural basis for extending some of the ANOVA techniques to nonlinear model so We consider a location-scale model and discuss several measures for comparing the various populations. These measures have intuitive interpretations in control-treatments situations. We develop various estimation procedures and discuss their large sample properties. Asymptotically efficient multiple comparison procedures are also considered.

A review of statistical methods in the analysis of data arising from observer reliability studies (Part I)*

Summary This paper reviews research situations in medicine, epidemiology and psychiatry, in psychological measurement and testing, and in sample surveys in which the observer(rater or interviewer) can be an important source of measurement error. Moreover, most of the statistical literature in observer variability is surveyed with attention given to a notational unification of the various models proposed. In the continuous data case, the usual analysis of variance (ANOVA) components of variance models are presented with an emphasis on the intraclass correlation coefficient as a measure of reliability. Other modified ANOVA models, response error models in sample surveys, and related multivariate extensions are also discussed. For the categorical data case, special attention is given to measures of agreement and tests of hypotheses when the data consist of dichotomous responses. In addition, similarities between the dichotomous and continous cases are illustrated in terms of intraclass correlation coefficients. Finally, measures of agreement, such as kappa and weighted‐kappa, are discussed in the context of nominal and ordinal data. A proposed unifying framework for the categorical data case is given in the form of concluding remarks.

A Comparison of Designs and Estimators For the Two-Stage Nested Random Model

The problem of selecting a design and estimating procedure for estimating with minimum mean squared error (MSE) the variance components in a two-stage nested random model is considered. For balanced designs a modified maximum likelihood (ML) estimator is superior, and for this estimator the optimal design is less sensitive to the intra-class correlation τ for τ ≤ .5 than those designs based on minimizing the variance of the usual analysis of variance (AOV) estimator. For τ > .5, where an unbalanced design is preferable, asymptotic results were derived to indicate optimal designs for ML and AOV estimators; ML estimators have smaller MSE's than truncated AOV estimators or iterated least squares estimators. The optimal number of classes is somewhat less than the number needed for minimizing the variance of the usual AOV estimator. Large sample results for unbalanced designs were compared with small sample results obtained by simulation for a wide range of intra-class correlation and several selected designs.

Estimation of Heritability from Experiments with Inbred and Related Individuals

SUMMARY Assuming an arbitrary amount of inbreeding and certain types of relationships among individuals in the parent population, it is shown how for nested and diallel mating designs the genetic and environmental variance components can be estimated from the mean squares in the usual analyses of variance. The systems of equations are given explicitly for the case of equal numbers of offspring per mating. A method for dealing with unequal numbers of offspring is outlined. In a previous paper (Hinkelmann [1969]) modifications of the usual procedures for estimating heritability from nested and diallel mating designs were given for the case that the dams are related. It was shown how this relationship induces a certain covariance structure for some random effects in the underlying linear model. However, just as the variance components can be expressed in terms of genetic and environmental variance components, these covariance components are simply functions of the genetic variance components and the coefficients of relationship. Since the coefficients of relationship can be evaluated from the pedigree, it is then easy to estimate the genetic and environmental variance components from the mean squares in the usual analysis of variance, and hence estimate heritability. In the present paper this method is extended to allow for an arbitrary amount of inbreeding and certain types of relationships in the parental population. The most general situation would be to allow for all types of relationships, i.e., among sires, among dams, and among sires and dams. This, however, may lead to inbreeding in the offspring generation which in turn leads to problems in evaluatiing covariances between relatives, an approach on which the results in this paper are based. Harris [1964] has developed formulae for the covariance between inbred relatives which involve additional genetic parameters not present in the usual covariance formulae for non-inbred relatives (e.g. Kempthorne [1957]). These parameters are not estimable from the usual mating designs. Also, Cockerham [1963] and Harris [19641 give special conditions under which the covariance between inbred

On a Class of Aligned Rank Order Tests in Two-way Layouts

The present investigation is concerned with the formulation of a multivariate approach for the construction of a class of aligned rank order tests for the analysis of variance (ANOVA) problem relating to two-way layouts. The problems of simultaneous testing and testing for ordered alternatives based on aligned rank order statistics are also considered. Various efficiency results pertaining to the proposed tests are studied. Let us consider a two factor experiment comprising $n$ blocks, each block containing $p(\geqq 2)$ plots receiving $p$ different treatments. In accordance with the two-way ANOVA model, we express the yield $X_{ij}$ of the plot receiving the $j$th treatment in the $i$th block as \begin{equation*}\tag{1.1}X_{ij} = \mu + \alpha_i + \tau_j + \epsilon_{ij}, \quad i = 1, \cdots, n, j = 1, \cdots p;\end{equation*} where $\mu$ stands for the mean effect, $\alpha_1, \cdots, \alpha_n$ for the block effects (may or may not be stochastic), $\tau_1, \cdots, \tau_p$ for the treatment effects (assumed to be non-stochastic), and $\epsilon_{ij}$'s are the residual error components. It is assumed that $\mathscr{\epsilon}_i = (\epsilon_{i1}, \cdots, \epsilon_{ip}), i = 1, \cdots, n$ are independent and identically distributed stochastic vectors having a continuous (joint) cumulative distribution function (cdf) $G(x_1, \cdots, x_p)$ which is symmetric in its $p$ arguments; (this includes the conventional situation of independence and identity of distributions of all the $np$ error components as a special case). We may set without any loss of generality $\sum^p_1 \tau_j = 0$, and frame the null hypothesis of no treatment effect as \begin{equation*}\tag{1.2}H_0 : \tau_1 = \cdots = \tau_p = 0.\end{equation*} The usual ANOVA test based on the variance-ratio criterion is valid only when $G$ is a $p$-variate (totally symmetric) multinormal cdf. For arbitrarily continuous cdf $G(x_1, \cdots, x_p)$, intra-block rank tests are due to Friedman [7], Brown and Mood [3], and Sen [21]; generalizations of these tests to incomplete layouts are due to Durbin [6], Benard and Elteren [1], and Bhapkar [2]. Hodges and Lehmann [9] have pointed out that intra-block rank tests do not utilize the information contained in the interblock comparisons, and hence, are comparatively less efficient. They have suggested the use of ranking after alignment and also considered the Wilcoxon's and Kruskal-Wallis tests based on aligned observations whose asymptotic efficiency have been studied very recently by Mehra and Sarangi [14]: (After the first draft of the paper was submitted, the author came to know through the editor about the paper by Mehra and Sarangi [14], submitted earlier to the Annals. In view of this, the overlapping part is not considered here.) The object of the present investigation is to formulate briefly the theory of rank order tests based on observations after alignment and through a multivariate approach to justify the validity and efficacy of the proposed tests (which include the earlier works as special cases).

Transformações dos dados experimentais

Neste artigo apresentamos a análise estatística de um experimento de sistema - época de plantio de milho, fatorial 2x4, em que a heterogeneidade das variâncias dos tratamentos foi eliminada pelo uso da transformação logarítmica dos dados de produção. Os dados originais representam a produção, em pêso (quilos), de grãos, por canteiro. A transformação logarítmica trouxe a necessária independência entre as médias dos tratamentos e os desvios padrão respectivos, assegurando, dessa forma, a validez da análise da variância.

Biases encountered in large-scale yield tests

Abstract does not appear. First page follows. Introduction Certain difficulties in interpreting the results of the ordinary analysis of variance when applied to yield tests of genetically similar wheat strains, derived through backcrossing, prompted a critical examination of the data. These data indicated a bias inherent in any large-scale experiment that imposes an inflexible design upon a soil whose fertility may fluctuate markedly within short distances. Because of this bias, the usual analysis of variance tends to overestimate significance grossly when the number of varieties is large and the productivity levels of the soil change rapidly and erratically (as at Davis). One may briefly describe the bias by saying that the inflexible design of the experiment tends to subtract too little from the naturally high-yielding plots and too much from the naturally low-yielding plots in attempting to correct for differences in soil productivity. This has a spreading effect on the part of the variation that is labeled “varietal differences” and thus causes a serious overestimation of significance. In a conventional analysis of variance, the natural variation of an experiment is arbitrarily partitioned into categories according to a preconceived mathematical model. These categories are labeled “variation due to varieties,” “variation due to soil productivity,” and so on. If the experiment is in exact accord with the model, the labels are accurate. If the experiment is not as called for by the model, the labels are misleading: for instance, the category labeled “variation due to varieties” may contain some of the variation due to soil-productivity. A mathematical model may be pleasing and beautiful to its creator or users. If, then, nature does not conform to the model, workers having limited firsthand experience with the vagaries of biological material may even feel that nature has erred and should be corrected.