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)     

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)     

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)     

Longitudinal designs in randomized group comparisons: When will intermediate observations increase statistical power?

A variety of experimental designs have been developed to estimate the effect of an intervention. This article compared analyses in 2 standard randomized designs, the posttest-only and pretest-posttest designs, with an analysis in the newly suggested intensive design in terms of the statistical power and precision of estimated intervention effect afforded by each design. These comparisons are especially interesting because the suggested analysis of data from the intensive design is equivalent to a possible use of hierarchical linear modeling when there are no missing data. Results show that an analysis of variance (ANOVA) of slopes from the intensive design is almost always more powerful than an ANOVA in the posttest-only design, and can also be more powerful than an analysis of covariance in the pretest-posttest design, but typically only when the number of measurement waves is 5 or more. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Comments on the Overall and Spiegel Paper.

Responds to the comments by J. F. Overall and D. K. Spiegel (see record 1973-20070-001) on the current author's original article "Note on nonorthogonal analysis of variance" (see record 1972-26084-001). The purpose of this note is the clarification of certain erroneous interpretations concerning the nonorthogonal analysis of variance which appear in the comment by Overall and Spiegel. This note demonstrates that the method proposed by Overall and Spiegel and the method proposed by the current author are not equivalent. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Rules for coding dummy variables in multiple regression.

Describes how an apparent contradiction between the methods of coding dummy variables proposed by J. Cohen (see record 1969-06106-001) and those by J. Overall and D. Spiegel (see record 1970-01534-001) led to the discovery of a general formula for such coding, based on demonstrating a theoretical connection between multiple comparison and dummy multiple regression. Examples are given for various cases of orthogonal and nonorthogonal designs, which explicitly include assumptions about sample size. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Cancellous bone architecture: advantages of nonorthogonal trabecular alignment under multidirectional joint loading

Wolff proposed that trabeculae align at 90 degrees angles (orthogonal). However, nonorthogonal alignment of trabeculae has been observed near many joints, including the proximal femur. We propose that nonorthogonal alignment is an adaptation to multidirectional joint loads. When the loading direction does not correspond with the trabecular alignment, warping or shear coupling occurs leading to large shear strains within the cancellous structure. Using a simplified continuum model for trabecular bone, we demonstrate that shear coupling caused by multidirectional joint loads is reduced 33-75% when trabeculae are aligned 60 degrees from one another (as is observed in regions of the proximal femur), as opposed to 90 degrees from one another (as was predicted by Wolff). The results suggest that an optimal cancellous structure may appear differently under multidirectional joint loads than the 'trajectorial' organization proposed by Wolff, which was based upon assumptions drawn from unidirectional loading.     

Errata to Carlson and Timm.

Reports an error in the original article by J. E. Carlson and N. H. Timm (Psychological Bulletin, 1974[Sep], Vol No.[81], 563-570). Errors made in some equations and estimations of the analysis of variance parameters are corrected. (The following abstract of this article originally appeared in record 1975-02163-001.) 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. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A comment on coding in nonorthogonal designs.

Briefly discusses the controversial issue of dummy variable coding in nonorthogonal designs. It is argued that there is no one single coding method that is "correct." Different coding methods carry different implicit assumptions. The decision as to which coding method should be used depends upon these assumptions. (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)     

Regression analyses of repeated measures data in cognitive research.

Repeated measures designs involving nonorthogonal variables are being used with increasing frequency in cognitive psychology. Researchers usually analyze the data from such designs inappropriately, probably because the designs are not discussed in standard textbooks on regression. Two commonly used approaches to analyzing repeated measures designs are considered in this article. It is argued that both approaches use inappropriate error terms for testing the effects of independent variables. A more appropriate analysis is presented, and two alternative computational procedures for the analysis are illustrated. (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)     

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)     

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)     

Type IV errors and interactions.

Extends L. A. Marascuilo and J. R. Levin's notion of Type IV errors, emphasizing the interpretation of interactions in factorial analysis of variance (ANOVA) designs. To help clarify what an interaction is and what it is not in terms of the ANOVA model, it is emphasized that (a) comparisons designed to explain an interaction are acceptable only if they reduce to comparisons involving interaction parameters exclusively; and (b) such comparisons may be both specified and directional, i.e., they may be defined to test an E's a priori hypotheses. (17 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)     

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)     

Lack of method variance in self-reported affect and perceptions at work: Reality or artifact?

Spector (1987) recently concluded that there is little evidence of method variance in multitrait-multimethod studies of self-reported affect and perceptions at work. In this article we propose that this conclusion was incorrect and was the result of improper analytical procedures. Spector's data were reanalyzed by using a more powerful approach: confirmatory factor analysis. Model comparisons and variance partitioning indicated that method variance is present and accounts for approximately 25% of the variance in the measures examined by Spector. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Orthogonality of patterns does not predict their discriminability.

The Lie algebra of continuous transformation groups proposes a theoretical basis for predicting the discriminability of patterns. Three pairs of orthogonal patterns generated by 1st-order Lie operators were used to test the predictions that (1) pair mates that are orthogonal would be more discriminable than nonorthogonal pair mates and (2) the ease of discrimination for the 3 pairs of orthogonal patterns would be predictable from the relative complexity of the Lie operators that generate them. 18 Mongolian gerbils (Meriones unguiculatus) were trained in a jumping stand to discriminate 3 successive pairs of patterns; training continued to a criterion of 2 consecutive days at 90% correct or better. The 1st prediction was not confirmed, and the 2nd was only partially confirmed. Discriminations, using a grating as the positive or negative stimulus or both, were learned significantly faster than those in which no grating appeared. Results offer no strong support for a model of pattern analysis based on the theory of Lie operators. It is concluded that simple orientation of linear contours, rather than contour orthogonality, is the most salient feature of simple patterns. (French abstract) (20 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Defining phonological awareness and its relationship to early reading.

Phonological awareness (PA) has been operationally defined by many different tasks, and task comparisons have been confounded by differing levels of linguistic complexity among items. A sample of 113 kindergartners and first graders completed PA tasks designed to separate task difficulty from linguistic complexity. These measures were, in turn, compared with measures of early literacy. Results indicated that the measures loaded on a single factor and that PA measured by differences in linguistic complexity, rather than by task differences, seemed to be more closely related to the factor. A logical analysis suggested that alphabet knowledge is necessary for children to separate onsets from rimes and that awareness of onsets and rimes is necessary both for word reading and for more complex levels of phonemic analysis. (PsycINFO Database Record (c) 2010 APA, all rights reserved)