Multivariate analysis of variance.

We provide an expository presentation of multivariate analysis of variance (MANOVA) for both consumers of research and investigators by capitalizing on its relation to univariate analysis of variance models. We address several questions: (a) Why should one use MANOVA? (b) What is the structure of MANOVA? (c) How are MANOVA test statistics obtained and interpreted? (d) How are MANOVA follow-up tests obtained and interpreted? (e) How is strength of association assessed in MANOVA? (f) How should the results of MANOVA be presented? (g) Are there any alternatives to MANOVA? We use an example data set throughout the article to illustrate these points. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Nonorthogonal two-way analysis of variance.

Examined the advantage of disadvantages of 5 different exact analyses of variance for nonorthogonal 2-way designs with respect to orthogonality of the analyses, parametric hypotheses tested, and model comparisons made by the analyses. It is proposed that experimenters, when faced with the necessity of performing a 2-way ANOVA, carefully consider these analyses with regard to the a priori information they have about the data, the questions they expect the analysis to help answer, and the questions each analysis is best equipped to answer. It is also suggested that experimenters choose the analysis that best fits their needs rather than depend on one for all situations. (16 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Calculating power in analysis of variance.

Presents an overview of procedures for calculating power of the F test under 3 models of ANOVA (fixed effects, random effects, and mixed effects). A comparison of power of tests on fixed and random factors shows the latter to have substantially lower power. Consequences for designing experiments and for interpreting experimental results are discussed, and the simplicity with which power calculations are done is emphasized. (15 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Multiple comparisons in analysis of variance.

Of the many multiple comparisons techniques described by Ryan (see 34: 1416), the procedure, derived from analysis of variance, of partitioning the degrees of freedom attributable to the main effect into n orthogonal components was omitted. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Note on nonorthogonal analysis of variance.

Criticizes J. E. Overall and D. K. Spiegel's article (see record 1970-01534-001) discussing 3 methods for performing nonorthogonal analysis of variance (ANOVA). It is observed that the statistics obtained do not provide exact tests for main effects when one is assuming an interaction model. An alternative method is presented for treating nonorthogonal ANOVA which uses an existing general linear model program. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A note on variance explained in the mixed analysis of variance models.

Examines assumptions about the general linear model for interaction terms in the mixed analysis of variance. Some well-known results of S. R. Searle (1971) demonstrate that the inconsistencies between J. H. Dwyer's (see record 1975-02166-001) technique and that of G. M. Vaughn and M. C. Corballis (see record 1969-16617-001) in estimating the magnitude of effect for a mixed interaction are the direct result of specific assumptions made. If it is assumed that the interaction source of variance is a random variable, then the equations obtained by Vaughn and Corballis are correct; however, if an alternative assumption is made (i.e., that the iteraction term is fixed in one direction), then Dwyer's equations are correct. Researchers are called on to be cognizant of these two sets of assumptions and to be aware of the dramatic effects they may have on estimates of magnitude of effect for mixed interactions. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Robustness properties of nonorthogonal analysis of variance.

Nonorthogonal analysis of variance (ANOVA) is defined as a factorial ANOVA with differing numbers of subjects in the cells. Despite the volume of literature in the psychological journals dealing with the problems of nonorthogonal ANOVA, little attention has been focused on the robustness properties of the corresponding hypothesis tests. Monte Carlo results are presented for the two-way ANOVA design indicating that all of the standard computational routines for the unequal cell size case are nonrobust. This occurs when the assumptions of homogeneity of variance or normality are violated. The user is cautioned against collecting such data. If such data must be analyzed, then alternative or supplemental analysis strategies should be used. Alternative approaches would include simulation, rank transformation, modified ANOVA procedures, and alternative developments in linear models, such as nonparametric factorial ANOVA. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Comments on Rawlings' nonorthogonal analysis of variance.

In J. Overall and D. Spiegel's reply to R. Rawlings's (see record 1972-26084-001) criticism of their previous article, the authors state that Rawlings's alternative nonorthogonal analysis of variance is equivalent to their method, which Rawlings criticized as incorrect. In 2 separate articles (a) Rawlings replies to Overall and Spiegel's present article, and (b) I. Smith contends that there is a statistical error in G. Joe's (see record 1971-25969-001) attempt to clarify the original Overall and Speigel article. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Multivariate analysis of variance and confidence levels.

Describes how a 2?×?2?×?2 factorial analysis of variance (ANOVA) affects confidence levels of results. Discussion focuses on uses and limitations of statistical tools such as ANOVAs, as well as the appropriate times to use them. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A distribution-free test of analysis of variance hypotheses.

Rao's technique of decomposing chi-square into components is modified to derive a distribution-free or nonparametric test of hypotheses involving main effects and interaction examined customarily by the analysis of variance of the two-factor or two-way variety. The proposed nonparametric test of analysis of variance hypotheses is described in terms of six principal steps, illustrated with a computational example, discussed with regard to small expected frequencies, compared with Mood's tests which appear to be disadvantageous in treating interaction effects, and is possible to extend for designs of three or more factors. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Equivalence of orthogonal and nonorthogonal analysis of variance.

Considers that the controversy surrounding dummy variate multiple regression approaches to nonorthogonal analysis of variance would be cleared up if a criterion could be accepted for deciding what constitutes a proper generalization of the classical analysis of variance for orthogonal factorial designs. It is proposed that a general multiple regression solution be interpreted as testing analysis of variance effects only if it results in an estimation of the same parameters and tests of the same hypotheses that might otherwise be estimated and tested in an orthogonal design involving the same factors. A method which satisfies this criterion is identified, and a simple procedure for examining equivalence in orthogonal and nonorthogonal cases is suggested. (19 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Multivariate analysis of variance by multiple regression methods.

Discusses multivariate analysis of variance as a general case of familiar multiple regression analysis. A consequence of this approach is a unified treatment of multivariate analysis of variance which can be used by psychologists who are generally familiar with multiple regression approaches to univariate analysis of variance. It is suggested that the generality of the approach permits solutions consistent with any of the several available strategies for dealing with problems of unequal and disproportionate cell frequencies. Inherent in the multiple regression formulation is the otherwise not so obvious fact that univariate analysis of variance results are an integral part of the multivariate solution and that both are important for understanding complex data. Methods of interpreting multivariate analysis of variance results in complex factorial experimental designs are discussed. (32 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Factorial analysis of variance with unequal cell frequencies.

Unequal cell frequencies in a factorial design create the problem of nonorthogonality: an intercorrelation among the main and interaction effects. This article presents a nontechnical discussion of the problem introduced by nonorthogonality. Four least-squares approaches to the solution of this problem are presented (the unadjusted main effects, hierarchical, fitting constants, and simultaneous solutions). Each is discussed with reference to its method of dealing with nonorthogonality and the conclusions it permits. Recommendations are provided as to the circumstances under which each solution is appropriate. (French abstract) (14 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Some problems in the nonorthogonal analysis of variance.

Contends that nonorthogonal analysis of variance has been much misunderstood by psychologists, and as a result there has been considerable controversy as to the appropriate methods of analysis. These problems traditionally associated with the nonorthogonal multifactor analysis of variance are rather easily resolved by viewing the analysis of variance (either orthogonal or nonorthogonal) as a series of model comparisons. From this point of view, the analysis of highly confounded designs is seen to yield results that correspond to those that a purely logical analysis would suggest. A logical flow of comparisons and decisions is developed for both the 2- and 3-factor designs that, although more complicated than procedures previously proposed, seems necessary for drawing proper inferences. It is further shown that there is no logical difference between orthogonal and nonorthogonal analysis of variance. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

On Wilson's distribution-free test of analysis of variance hypotheses.

The power of Wilson's proposed chi-square test for testing n-way analysis of variance designs is very low in comparison with the F test. Several illustrative problems are presented. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Response variable hypotheses in the multivariate analysis of variance.

Es using multivariate analysis of variance (MANOVA) usually want to analyze effects separately for each response variable after rejecting a null hypothesis of multivariate dispersion. From the standpoint of the multivariate general linear model, 4 measures of importance for response variables are discussed: univariate F statistic for each response, standardized canonical coefficient for each response, contribution to the MANOVA test criterion by each response, and simultaneous confidence intervals on estimates of treatment effects on each response. Artificial data are presented to illustrate problems in using these measures. (17 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Consequences of violating the independence assumption in analysis of variance.

Discusses the biasing effects of nonindependence of observations on the mean squares used to test the effect of some discrete independent variable. Nonindependence of observations is defined, and 3 commonly assumed patterns of nonindependence are identified: nonindependence due to groups, nonindependence due to sequence, and nonindependence due to space. How the bias in both the mean square for treatment and the mean square for error can be derived when each of the 3 patterns of nonindependence is ignored in analyzing the effect of a discrete independent variable is demonstrated. Ways to eliminate the sometimes considerable biases, either by including the source of nonindependence in the analysis, transforming the data to remove it, or modeling it, are discussed. It is concluded that nonindependence of observations should be viewed as a substantive issue central to many areas by psychological research. (28 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

On choosing a test statistic in multivariate analysis of variance.

Users of multivariate analysis of variance must choose which of several available test statistics to employ as a generalization of the usual univariate F statistic. A review of statistical literature concerning the power and robustness of the 4 most promising tests leads to the recommendation of K. C. Pillai's (1955) and M. S. Bartlett's (1939) trace statistic for general use. A survey of recent experimental reports revealed that psychologists have been using a 2nd best statistic and that they have frequently failed to specify their statistic to let readers judge its appropriateness. To facilitate increased use of the Pillai-Bartlett statistic, information is given concerning computation, the availability of significance tables, and a convenient F approximation. (45 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Comparison of two strategies for analysis of variance in nonorthogonal designs.

Compared the M. I. Appelbaum and E. M. Cramer (see record 1974-28956-001) comparison of models strategy for analysis of data from nonorthogonal designs with the J. E. Overall and D. K. Spiegal (see record 1970-01534-001) Method 1 general linear model analysis. Data were generated by Monte Carlo methods to include known true ANOVA main and interaction effects. In the presence of a true but nonsignificant interaction, estimates of main effect parameters derived from the Method 1 general linear model analysis were significantly closer to the true values. Greater accuracy in estimation of main effects in the presence of a significant interaction was also observed. The danger of letting observed data determine the ANOVA model and the hypotheses to be tested is emphasized. (12 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Reorganization of variables in analysis of variance and multidimensional contingency tables.

Appropriate reorganization of variables in some analysis of variance designs may make the obtained results more easily interpretable and may also expand the range of experimental designs that can easily be analyzed by standard procedures. A rule is given for determining equivalences of effects in terms of original and reorganized variables, and an example illustrates the usefulness for theoretical purposes of such restructuring. The potential of reorganization for broadening the range of readily applicable experimental designs, and the implications of the possibility of reorganization for nonorthogonal analysis of variance, are explored. Applications of restructuring in multidimensional contingency table analysis are noted. (19 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)