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)     

Multivariate group comparisons of variable systems: MANOVA and structural equation modeling.

Multivariate analysis of variance (MANOVA) and structural equation modeling (SEM) are compared as alternative approaches to the detection of multivariate mean differences in between-groups designs. MANOVA remains the most commonly implemented multivariate test of between-groups mean differences. When the set of outcome variables reflects a latent-variable system rather than an emergent-variable system, however, SEM provides an elegant alternative to MANOVA. Three generic types of latent-variable systems were considered. In the simplest case, SEM and MANOVA led to very similar substantive interpretations. In the 2 more complex variable systems, SEM provided parameter estimates and collateral information that was at least as informative as those provided by MANOVA. Researchers are encouraged to consider more carefully the nature of their variable systems before choosing too casually MANOVA over SEM. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Simultaneous test procedures and the choice of a test statistic in MANOVA.

Suggests that problems associated with the use of a preliminary MANOVA test in conjunction with univariate follow-up tests can be avoided if follow-up tests are carried out with a simultaneous test procedure (STP) derived from the MANOVA test statistic used for the overall test. It is argued that the choice of a MANOVA test statistic for such analyses should be based on the power and robustness of MANOVA STP rather than on the properties of the corresponding overall tests. Monte Carlo data are presented that show that the STP based on the trace statistic V can be extremely conservative relative to the STP based on the largest root statistic R. Data suggest that the lack of robustness of the R statistic is unlikely to produce problems if the R STP is used to evaluate interpretable contrasts on linear combinations of variates of interest to the experimenter as opposed to contrasts for which coefficients referring to groups and variates are determined by the data. (24 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Reexamination of a plea for multivariate analyses.

Re-examines 2 issues raised in an article by M. R. Leary and E. M. Altmaier (see record 1981-02539-001) proposing MANOVA as a solution to the problem of inflated Type I error. Guidelines are discussed for choosing an overall MANOVA test statistic and post hoc tests that determine the dependent variable or variables responsible for any significant effects. It is concluded that guidelines based on recent comparisons of the various test statistics be used by researchers rather than advice found in basic textbooks. A discussion of the power of MANOVA concludes that although MANOVA may under some conditions reduce the chances of detecting significance, it is powerful for detecting legitimate experimental effects that are spread across more than one dependent variable. (37 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Nonparametric one-way multivariate analysis of variance: A computational approach based on the Pillai-Bartlett trace.

Demonstrates a method for computing the omnibus test statistic, which illustrates the relation of nonparametric methods to parametric MANOVA, based on K. Pillai's (1955) and M. Bartlett's (1939) trace statistic. The required test statistic is equal to (N?–?1)V, where V is the trace statistic computed on the transformed data. The test statistic can be obtained by submitting the data, already converted to ranks or normal scores, to a packaged MANOVA program. (25 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Contrast tests of interaction hypothesis.

This article argues for the use of contrasts to test a priori interaction hypotheses in 2-way analysis of variance designs. It focuses on 3 underused types of interaction contrast tests: a "matching" pattern for cognate levels of row and column factors; the "qualitative quadratic" for monotonic profiles of means in the same direction but with opposed concavities; and a "differential curvilinearity" test for differences in the curvature of two profiles with the same direction of concavity. The circumstances that best capitalize on the potential advantages of a priori contrast tests are indicated and an effect size measure for contrasts is presented. Investigators are urged to examine residuals after accounting for the variation attributable to the chosen contrast for patterns that may provide hints for more textured hypotheses in further research. If a posteriori contrasts are used, their effect sizes should be noted. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Approximate degrees of freedom tests: A unified perspective on testing for mean equality.

This article presents a statistic for tests of mean equality in between-subjects and within-subjects designs when variances are heterogeneous. The approximate degrees of freedom statistic of S. Johansen (1980) can be used to test main and interaction effects, as well as multiple comparison hypotheses related to these effects. Thus, researchers need only be familiar with a single statistic, rather than the many statistics that have been defined in the literature, to perform these tests of significance. Also included is a computer program to obtain a numerical solution. (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)     

How the power of MANOVA can both increase and decrease as a function of the intercorrelations among the dependent variables.

This article directly addresses explicit contradictions in the literature regarding the relation between the power of multivariate analysis of variance (MANOVA) and the intercorrelations among the dependent variables. Artificial data sets, as well as analytical methods, revealed that (1) power increases as correlations between dependent variables with large consistent effect sizes (that are in the same direction) move from near 1.0 toward –2.0, (2) power increases as correlations become more positive or more negative between dependent variables that have very different effect sizes (i.e., one large and one negligible), and (3) power increases as correlations between dependent variables with negligible effect sizes shift from positive to negative (assuming that there are dependent variables with large effect sizes still in the design). Implications for the reliability of dependent variables and strategies for selecting these variables in MANOVA designs are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Multivariate statistics for family psychology research.

The application of selected multivariate statistics is illustrated for use in family psychology research. The use of multivariate analysis of variance (MANOVA) and discriminant analysis in factorial designs and profile analysis is discussed. Profile analysis provides a method for dealing with unit of analysis issues in family psychology research. Applications of confirmatory factor analysis are also discussed as useful methods for researchers examining multiple components of families and handling multiple perspectives of various family members. Limitations and applications of these methods in family psychology research are reviewed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Under the Skin: On the Impartial Treatment of Genetic and Environmental Hypotheses of Racial Differences.

Environmental and genetic explanations have been given for Black-White racial differences in intelligence and other traits. In science, viable, alternative hypotheses are ideally given equal Bayesian prior weights; but this has not been true in the study of racial differences. This article advocates testing environmental and genetic hypotheses of racial differences as competing hypotheses. Two methods are described: (a) fitting means within structural equation models and (b) predicting means of interracial children. These methods have limitations that call for improved research designs of racial differences. One improvement capitalizes on biotechnology. Genetic admixture estimates--the percentage of genes of European origin that a Black individual possesses (independent of genes related to skin coloration)--can represent genetic influences. The study of interracial children can be improved by increasing sample size and by choosing family members who are most informative for a research question. Eventually, individual-admixture estimates will be replaced by molecular genetic tests of alleles of those genes that influence traits. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Testing hypotheses on specific environmental causal effects on behavior.

There have been strong critiques of the notion that environmental influences can have an important effect on psychological functioning. The substance of these criticisms is considered in order to infer the methodological challenges that have to be met. Concepts of cause and of the testing of causal effects are discussed with a particular focus on the need to consider sample selection and the value (and limitations) of longitudinal data. The designs that may be used to test hypotheses on specific environmental risk mechanisms for psychopathology are discussed in relation to a range of adoption strategies, twin designs, various types of "natural experiments," migration designs, the study of secular change, and intervention designs. In each case, consideration is given to the need for samples that "pull-apart" variables that ordinarily go together, specific hypotheses on possible causal processes, and the specification and testing of key assumptions. It is concluded that environmental risk hypotheses can be (and have been) put to the test but that it is usually necessary to use a combination of research strategies. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A three-factor model encompassing many possible statistical tests on independent groups.

Combined 3 factors, parameter used, technique used, and method of control of Type I errors, into a model that includes 100 different statistical tests, of which 64 are defensible. Tests on complex hypotheses about correlations, ρ, proportions, P, and variances, ?–2, comparable to tests on means, μ, are available. For the equal n case, the statistics needed can all be formulated either as t statistics or as omnibus F statistics. The technique factor with 5 levels includes 3 variations whereby a t is contrasted with 1 of 3 critical values appropriate for a given set of contrasts. The F statistic may be used on 1-way or multifactor designs on any of the above parameters. The experiment's design and experimental hypotheses dictate which cells of the crossing of these 2 factors are appropriate. The experimenter's major choice is the method of control of Type I errors. A simultaneous and 4 stepwise methods are discussed as general methods that could be used with most statistics. Setting alpha as the familywise rate of Type I errors and the use of simultaneous methods are recommended. (38 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Generalized Eta and Omega Squared Statistics: Measures of Effect Size for Some Common Research Designs.

The editorial policies of several prominent educational and psychological journals require that researchers report some measure of effect size along with tests for statistical significance. In analysis of variance contexts, this requirement might be met by using eta squared or omega squared statistics. Current procedures for computing these measures of effect often do not consider the effect that design features of the study have on the size of these statistics. Because research-design features can have a large effect on the estimated proportion of explained variance, the use of partial eta or omega squared can be misleading. The present article provides formulas for computing generalized eta and omega squared statistics, which provide estimates of effect size that are comparable across a variety of research designs. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Pupillographic assessment of sleepiness in sleep-deprived healthy subjects

Medical research frequently involves the statistical comparison of >2 groups, often using data obtained through the application of complex experimental designs. Fortunately, inferential statistical methodologies exist to address these situations. Analysis of variance (ANOVA) in its many forms is used to simultaneously test the equality of all groups in a study. One-way (with 1 independent variable), 2-way (with 2 independent variables), and repeated-measures (patients serve as their own controls) ANOVAs are forms of this technique. Each form has been developed to analyze data from a specific experimental design. Analysis of covariance (ANCOVA) allows the researcher to control for confounding variables that may influence the response of the dependent variable. Finally, multivariate analysis of variance (MANOVA) evaluates the simultaneous responses of multiple dependent variables to > or = 1 independent variable. Whereas ANOVA is the correct alternative to statistically inappropriate multiple t-tests, MANOVA is the correct alternative to statistically inappropriate multiple univariate ANOVA calculations. Use of each of these statistical methods requires an appropriate experimental design and data meeting a number of assumptions. When used properly, each of these methods provides a powerful statistical analysis technique.     

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)     

Hypothesis testing and the comparison of imprecise hypotheses.

Suggests that hypotheses cannot be tested in isolation; to evaluate a hypothesis, it shoud be compared with some alternative hypothesis. The most satisfactory way of doing this is considered to be by means of a likelihood ratio (LR), which indicates the degree to which observed data support one or the other hypothesis. The present article deals with hypotheses about the mean (μ) or standard deviation (?) of a normal process, especially those hypotheses that are imprecise. An imprecise hypothesis is one that does not lead to the prediction of exact values for μ and ?. A technique is described for defining imprecise hypotheses by probability distributions, and a method for estimating parameters of these distributions is given. Equations are presented for computing LR when comparing 2 such imprecise hypotheses. The procedure is also applicable to the testing of precise null hypotheses. A comparison of LR with traditional significance levels illustrates the unsuitability of significance levels for evaluating null hypotheses. (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)     

Bayesian methods for assessing importance of effects.

In experimental data analysis when it comes to assessing the importance of effects of interest, 2 situations are commonly met. In Situation 1, asserting largeness is sought: "The effect is large in the population." In Situation 2, asserting smallness is sought: "The effect is small in the population." In both situations, as is well known, conventional significance testing is far from satisfactory. The claim of this article is that Bayesian inference is ideally suited to making adequate inferences. Specifically, Bayesian techniques based on "noninformative" priors provide intuitive interpretations and extensions of familiar significance tests. The use of Bayesian inference for assessing importance is discussed elementarily by comparing 2 treatments, then by addressing hypotheses in complex analysis of variance designs. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Enhancing a sense of agency through career planning.

Tested the effectiveness of a program for parents to help their adolescent sons and daughters develop a greater sense of agency regarding a career. In a pretest–posttest control group design involving 39 families in the experimental group and 25 families in a wait-listed control group, a multivariate analysis of variance (MANOVA) with repeated measures was used to test improvement on 6 variables concerned with career and a sense of agency. The MANOVA yielded a significant effect for groups over time. ANOVA with repeated measures showed that 4 of 6 univariate tests were also significant. Through career planning with a parent, adolescents in Grade 12 showed a greater career certainty, less indecision, more career salience, and stronger ego identity. (PsycINFO Database Record (c) 2010 APA, all rights reserved)