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The present research compares the ANOVA F-test, the Kruskal-Wallis test, and the normal scores test in terms of empirical alpha and empirical power with samples from the normal distribution and two exponential distributions. Empirical evidence supports the use of the ANOVA F-test even under violation of assumptions when testing hypotheses about means. If the researcher is willing to test hypotheses about medians, the Kruskal-Wallis test was found to be competitive to the F-test. However, in the cases investigated, the normal scores test was not consistently better than the F-test or the Kruskal-Wallis test and could not be recommended on the basis of this research.

A procedure for estimating the mean and variance components and their variances in a-stage nested designs with composited samples are discussed. The estimation procedure allows for the class or sub-class means or observations to be weighted in different ways. Four different estimation procedures are numerically compared to determine the conditions under which any one of them is best.

Consider the standard one-way analysis of variance model with random effects (variance components model). The customary assumptions of zero means, zero correlations, and equal variances are made for the random terms of this balanced model. The customary normality assumptions are also made when other than point estimates are desired. This standard model is generalized, in several ways, by addition of one or two types of “error” terms. The extended models are applicable to much more general situations than the standard model. However, exact results are developed for investigating the mean and both variance components (of the standard model) and for investigating subsets of these parameters. The generality level for an extended model depends on which parameters are investigated. Customary procedures for the standard model remain usable for some investigations and extensions, but more general models are applicable when use of customary results is not required. A method for rejection of outliers by use of the standard model is described and shown to be applicable for some of the extensions.

This program for three-factor analysis of variance with repeated measures on two factors is based on an unweighted means analysis, permitting the use of unequal sample sizes. Each factor may be fixed or random since the program will select an appropriate error term or will calculate a quasi- F ratio. The number of possible levels of each factor is extended over those in other programs. Rounding errors are minimized.

Considered are balanced factorial models for one-way analysis of variance with continuous data, fixed effects, and at least four replications. Special definitions, with a median basis, are given for the constant term and for contrasts in the effects. For normality models and many other important kinds of models, however, the resulting parameters are the same as those occurring for these models. The only assumption necessarily made is that of independence among replications. Then, approximate tests, confidence intervals, and median estimates can be obtained for the constant term and for specified contrasts in effects. Exact tests, intervals, and estimates can be obtained for special cases which are themselves of a moderately general nature. Some of the exact results are of a sign test type while others can use methodology developed for investigating symmetry about zero. These models and methods are a generalization of median analysis of variance through use of concepts developed for the linearized nonlinear regression approach.

Uniformity analysis of a genetically heterogeneous plantation has been used as a technique for separating total variance into genetic and environmental components. By superimposing a plot structure on a plantation of randomly allocated genotypes and calculating an analysis of variance between plots and within plots, Shrikhande (1957) showed that application of H.F. Smith's empirical variance law to the ’between plots‘ mean square results in identifiability of the two variance components. An alternative procedure consists of exploiting the regularity of the autocorrelation function within rows; various empirical studies have shown that in uniformity trials the correlation ρ k between the first and the kth plant in a row decreases as ρ k = ρ k . Such a model for environmental correlation (with no correlation between the first and kth genotype) results in identifiability of genetic and environmental components of variance and covariance.

A procedure is presented for obtaining estimates of components of variance for the main effects and interaction in a two-way factorial arrangement of treatments with only partially cross-classified data (sometimes referred to as unbalanced data in a two-way crossed classification model with interaction). The procedure entails the use of two hierarchical analyses of variance in which each main effect is first treated as classes and then as subclasses. The estimation of the variance component for each main effect results from the solution of two simultaneous equations with values used obtained directly from the hierarchical analyses. The interaction component of variance can be obtained from the variance components for main effects. An example is presented for the purpose of illustrating the procedure. Interpretation of the interaction variance component estimated by the method is limited due to the absence of some of the subclasses.

In a contingency table one classification often corresponds to samples from different populations and the other classification to ordered categories. Methods for analyzing data of this type may be compared using recent extensions in the theory of rank tests by Conover (1973) and Vorlickova (1970) to non-continuous distributions. In this paper the Kruskal-Wallis Test and the Median Test are compared in those situations where the data are given in a contingency table with ordered categoriesf and where one test or the other is optimal. Some comparisons with the Pearson chi-square statistic are given.

The basic ideas for the Shine Combined Analysis of Variance are extended to the case where some factors do not have repeated measures. Schematic calculation procedures are presented for the two-way case with repeated measures on one experimental factor. Some useful modifications of the basic Shine-Bower error term, which is used to test subject sources of variation, are also presented.

It is known that in the random effect model, exact F-tests are not obtainable from the analysis of variance table for testing the main effects in a three-way layout. Similarly exact F-tests are not obtainable for the main effects and certain interactions in higher-way layouts. In this paper level-α: tests obtainable from the analysis of variance table are given for testing these effects. Tables are provided to illustrate the size function and the power function of the tests.

The analysis of variance is a technique for comparing treatment means or for estimating components of variation. The former is called fixed effects analysis of variance and the latter is called random effects analysis of variance. Many nonstatisticians ..

Models used for investigation of variance components in analysis of variance are moderately restrictive for point estimation and quite restrictive, assuming normality, when tests and confidence regions are developed. However, the same results as occur for these models can be obtained for generalizations that are much less restrictive. In particular, results based on normality for the random variables of the model, and on their independence, remain valid for situations where each observation can have a different nonnormal distribution and dependence between each two observations can be different. The level of generalization for a model depends on what is being investigated and decreases as the number of variance components being investigated increases. Considered are the N-way classification models of various kinds and some specific examples. Similar extensions can be made for virtually all types of variance components models. The verification consists in showing that the statistics involved have the same joint distribution for the extended model as they do for the original model. Some ways of investigating outliers of nested designs, which also apply to the generalizations, are discussed.

Forecasts may be combined using a minimum variance criterion to yield a composite forecast of smaller error variance than any of the components. This paper considers the sampling distributions of the weights to be attached to the components and of the error variance of the combined forecast. Confidence limits are derived for the estimates of the weights and of the variance of a composite forecast with two components. The theoretical analysis reveals that, in practice, it is doubtful whether combined forecasts offer much improvement because of the unreliability of the weight estimates.

(1971). A New Analysis of Variance Model for Non-additive Data. Technometrics: Vol. 13, No. 1, pp. 1-18.

A formula for estimating the reliability of a test, based on the analysis of variance theory, is developed and illustrated. The data needed for the required computation are the number of correct responses to each item and the score for each subject. The results obtained from this formula are identical with those from one of the special cases of the Kuder-Richardson formulation. The relationships of the new procedure to other approaches to the problem are indicated.

A development of the mathematical structure of nested designs is presented using, for a general case, a four-stage nested design. For the casea where the variables at each stage are normally distributed, analytical results can be obtained. This is done for the unbalanced “staggered” and “inverted” designns. Empirical estimattas of variance for the non-normal case are obtained. These are compared with the analytical solutions. Also considered is the probability of negative variance estimates. It ia interesting to note that for these alternatives to balanced nested designs, one can decrease the probabilities of negative estimates of some variances at the cost of increasing them for others.

Discusses the relationship between 2 basic models for analysis of variance for repeated measures: the “traditional” model, which involves the assumptions of equal correlations between all variables both within and between S groups, and a less restrictive model based on a multivariate analysis of variance technique which is considered as an alternative to the traditional model. Suggestions are made for efficient computer programing for the traditional model. (PsycINFO Database Record (c) 2006 APA, all rights reserved)

Subjects estimated average value of angle pairs using magnitude estimations or graphic ratings. These numerical response data follow a simple averaging model. Functional scaling yields a linear relation between subjective and objective angle. The numerical data are then reduced to rank orders, and J. B. Kruskal's monotone analysis of variance (MONANOVA) procedure is applied. This allows a reconstruction of the original metric information from the strictly ordinal information, illustrating the power of MONANOVA in scaling. Limitations of MONANOVA in testing the underlying model are discussed.

Minimum variance unbiased quadratic order estimates for the variance is derived. An example of a rectangular distribution is given. For a normal distribution, certain relationships among the moments of order statistics are obtained and may be used to simplify the determination of the said estimates.

SynopsisA method is presented by which the sampling variance of all heritability estimates derived from ratios involving variance components may be found.

Procedures are developed and illustrated by application to real usage data of analysis of variance techniques for measuring components of variance in the data due to trend and arbitrary random variability in usage of equipment in a group as a whole. The usage data are obtained by periodic scanning of the equipment group at fixed intervals. Procedures for fitting the Polya distribution to the data are presented. Fitted Polya distributions are used to estimate the effects of trend on blocking. The procedures are extended to measuring and estimating the effects on blocking of day-to-day traffic variation and are illustrated by application to real usage data. A procedure is presented for adjusting estimates of variance for autocorrelation due to the usage measurement procedure and call holding times. The adjustment procedure is based on incomplete research and is, therefore, tentative. The effects of using a variance-stabilizing transformation of the data are shown. Other areas of application of the theory and procedures are discussed.

Summary.-The analysis of factor variance is a multivariate approach particularly applicable in those instances when the number of variables is large, when the reliabilities of some measures are questionable, when no reasonable a prior; combinations of measures exist, and when a search for structure among the measures is a partial objective of the study. The methods of analysis of factor variance, however, allow for a number of important options. Two of these options or cases were investigated in this study. The first required that the "matrix co be factored" be centered around rhe grand means for rhe total dam while the second required that the "matrix to be factored" be centered around individual group means. The results showed that the choice of method is crucial to the interpretaeon of results. Suppose that p response measures have been observed on each of the subjects in k groups. What Is the best approach to answering the question, "Are the groups different on these measures?" An over-all answer can be provided by multivariate analysis of variance, but there are at least three objections to the multivariate analysis. First, if k = 2, such an analysis can be done by collapsing the p response measures into a single variable and performing a simple analysis of variance on it. Since the single variable generated by this multivariate analysis of variance is chosen solely to maximize the difference between the groups and not for its intrinsic "meaning," it is hard to interpret, is unlikely to fit well into anyone's theory and usually leaves the experimenter without a satisfactory interpretation of results. For example, the groups may be different, but different on what dimensions? If nore than two groups are involved, the interpretation is even more elusive. Secondly, if the study is replicated and another multivariate analysis performed, the variable generated in the second analysis may besit little relationship to that generated in the first. Thus, it can happen easily that two identical studies each yield significant results, but on quite a different combination of variables. How then can they be compared? Thirdly, multivariate analysis of variance is limited by practical considerations to a relatively small value of p and to variables which are not too highly correlated. Thus, it is not recommended for exploratory studies on a large number of outcome measures whose individual characteristics and relationships are not well understood. In smaller studies, with fewer outcome measures, a priori combinations of measures can be formed and tested in follow-up tests to the multivariate analysis. But then, it is theory and not the data which must provide the rationale for grouping, combining and reducing the number of outcome measures. Hence

This paper presents procedures for multivariate analysis of variance which simultaneously test for significance all variables together as well as all subsets of variables, or the differences between all samples as well as between the samples of any subgroup. These procedures ensure a predetermined probability of making no type I error on any of the tests and are coherent in preserving the implication relations which exist between hypotheses on different groups of samples. The procedure using Roy's maximum characteristic root statistics is found to be preferable to all other procedures of this kind.

To err is human, and now that behavioral researchers are engaging in inferential activity with increasing frequency, it is more than likely that the number of erroneous inferences is also increasing. Familiar to all seasoned hypothesis testers are those twin gremlins, Type I and Type II errors. The first faux pas refers, of course, to the rejection of a true hypothesis, while the second is the non-rejection of a false hypothesis. The rightful originator of Type III errors is disputable, as is the commonly accepted definition of them: "In 1947, F. N. David, perhaps not entirely seriously, suggested that there was a third kind of error which might be committed in

The purpose of this note is to derive a design criterion that can be used by investigators Who wish to perform experiments to discriminate among rival models when the variance σ2 is unknown. As would be expected, when the number of experiments is large, the expression for this design criterion is almost equivalent to replacing σ2 in the corresponding equation for the case where the variance is known with a suitable estimate s 2.