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)     

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)     

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)     

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)     

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)     

A Monte Carlo study of three approaches to nonorthogonal analysis of variance.

Several alternative procedures have been advocated for analyzing nonorthogonal ANOVA data. Two in particular, J. E. Overall and D. K. Spiegel's (see record 1970-01534-001) Methods 1 and 2, have been the focus of controversy. A Monte Carlo study was undertaken to explore the relative sensitivity and error rates of these 2 methods, in addition to M. I. Applebaum and E. M. Cramer's (see record 1974-28956-001) procedure. Results of 2,250 3?×?3 ANOVAs conducted with each method and involving 3 underlying groups of population effects supported 3 hypotheses raised in the study: (a) Method 2 was more powerful than Method 1 in the absence of interaction; (b) Method 2 was biased upwards in the presence of interaction; and (c) Methods 1 and 2 both had Type I error rates close to those expected in the absence of interaction. In addition, it was found that in the absence of interaction, the Appelbaum and Cramer procedure was more powerful than Method 2 but slightly increased the Type I error rate. (16 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Comment on "Some problems in the nonorthogonal analysis of variance."

M. I. Appelbaum and E. M. Cramer (see record 1974-28956-001) described ignoring tests of main effects as "irrelevant" when 1 eliminating test is significant in a 2-way nonorthogonal analysis of variance. It is stated by the present author that such tests are not irrelevant, however, because there are situations when A eliminating B is significant and A ignoring B is nonsignificant, making it reasonable to include B in the model, even though the eliminating test for B is nonsignificant. An example is given, and the necessary modifications to the Appelbaum and Cramer procedure are proposed. In addition, another ignoring-eliminating significance pattern is shown to be impossible. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Using nonorthogonal analysis of variance designs in psychological research: Comment on Donahue and Costar.

Argues that there are a number of potential problems with the methodological employed in the study of T. J. Donahue and J. W. Costar (see record 1978-24142-001) and that these issues were not appropriately addressed in their article in order to provide the reader with a means of evaluating the correctness of the measurement instrument, the hypotheses tested, and the data analysis procedures. The problems are discussed. (5 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

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)     

Analysis of nonorthogonal fixed-effects designs.

Discusses nonorthogonal fixed-effects experimental designs using both the full-rank and reduction in error sums of squares conceptualizations of data analysis. The hypotheses tested by several commonly used methods of analysis are clarified, and suggestions for choice of the most appropriate procedure are proposed. (17 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)     

The overparameterized analysis of variance model.

Analyses of variance (ANOVA) with the general linear model (OLM) in many standard statistical packages use an overparameterized model, a model unfamiliar to most behavioral science researchers. Estimates and significance tests with GLM procedures are calculated by computing generalized inverses and estimates of estimable functions. Using simple examples, the authors discuss the concepts that underlie the solutions for 1-way and 2-way ANOVAs with overparameterized models and illustrate how these models allow one to evaluate the research hypotheses. The authors also extend the discussion of overparameterized models to a more general modeling approach than GLM, the general linear mixed model. (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)     

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)     

Robustness considerations in Bayesian analysis

All statistical analyses demand uncertain inputs or assumptions. This is especially true of Bayesian analyses. In addition to the usual concerns about the agreement of the data and model, a Bayesian must contemplate the effect of an uncertain prior specification. The degree to which inferences are robust to changes in the prior is of primary interest. This article discusses some robust techniques that have been suggested in the literature. One goal is to make apparent the relevance of some of these techniques to biostatistical work.     

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)     

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)