Defining and testing hypotheses in multidimensional contingency tables.

Describes the log-linear model as a framework for analyzing effects in multidimensional contingency tables, i.e., tables of frequencies formed by 2 or more variables of classification. Variables are considered to have nominal categories. A general purpose analysis is proposed for such tables, similar to analysis of variance. 2 test procedures are considered: (a) maximum likelihood estimation of expected cell frequencies and associated chi-square tests, and (b) chi-square tests based on logarithms of adjusted cell frequencies. In addition, 2 multiple-comparison methods, related to the latter approach, are considered as supplementary or alternative procedures. (28 ref.) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

On the partitioning of contingency tables.

The chi-square test is presented in general. If a contingency table is greater than 2 X 2 the usual chi-square test of independence fails to determine the source of interaction if interaction is present. Methods for extending the chi-square analysis of r X c contingency tables are presented. The procedures outlined consist of partitioning an r X c contingency table into (r-1) (c-1) independent components, each with 1 degree of freedom. A special partition of a 2 X c table into 3 independent components is also given. The results of partitioning permit precise inference about independence or interaction in a contingency table and allow specific hypotheses to be tested without loss of generality. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Loglinear models: A way to study main effects and interactions for multidimensional contingency tables with categorical data.

Until recently, behavioral researchers failed to examine categorical data based on multidimensional data in a unified whole because of the complex arithmetic such procedures entail. With the appearance of the loglinear model and user-friendly computer software, the problems associated with the analysis of multifaceted contingency tables no longer exist. In this article, we describe the loglinear model and apply it to categorical data cross-tabulated on four dimensions. We define contrasts similar to those of the analysis of variance and describe post hoc and planned comparison strategies. We illustrate hypothesis testing and model building for categorical data, and provide guidelines for performing an analysis on frequency data by using the loglinear model. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Analysis of contingency tables with between-subjects variability.

When separate contingency tables are collected from each S or experimental unit, 2 null hypotheses concerning the association between the attributes can be tested: (1) No S shows any association, and (2) there is no mean association over the population of Ss. In almost all psychological data, the association varies among Ss, and the population hypothesis (Hypothesis 2) is appropriate. Common contingency-table methods (e.g., log-linear models) test Hypothesis 1 and have a positive bias. To test Hypothesis 2, the association should be compared with an empirical estimate of its variability, as in the ANOVA, not with the multinomial sampling variability of the observations. Nine test statistics were examined by simulation. For dichotomous classifications, the best test was a t test on the log odds ratio. In larger tables and multiway designs, a Hotelling statistic was appropriate. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A comment on the use of canonical correlation in the analysis of contingency tables.

T. Holland et al (see record 1980-08743-001) proposed the use of canonical correlation for analyzing contingency tables. The present authors contend that Holland et al failed to discuss several problems with the canonical correlation test procedure, and they committed interpretational errors when using the technique. Furthermore, in dismissing the chi-square approach, they implied that once a significant Pearson chi-square test has been obtained, no further analysis is possible. Such a position is completely false and ignores a great deal of the literature dealing with categorical data. Both prior and posterior comparison procedures for contingency tables are referenced in the present article. (35 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Estimates of contingency between two dichotomous variables.

Seven experiments with 292 undergraduates determined what heuristics people use in estimating the contingency between 2 dichotomous variables. It was hypothesized that people assess contingency solely based on the number of instances in which both Factors 1 and 2 (the 2 variables) are present. By manipulating column and row labels, various values could be placed in this "present/present" cell, also called Cell A. If Ss do base their contingency estimate on Cell A, a monotonic relation would be expected between Cell A frequency and the contingency estimate. This test of the Cell A heuristic led to the conclusion that it could not represent a complete explanation of contingency estimation. Although Exp II resulted in a rejection of one possible explanation of the results, Exps II and III indicated that low cell frequencies were greatly overestimated. Ss in a contingency estimation task involving no memory demands used complex heuristics in judging contingency. When demands were increased, the Cell A heuristic emerged as the modal strategy. Two factors, the use of simple heuristics by most Ss and the overestimation of small cell frequencies, combined to explain the results. In a contingency estimation task, salience augmented the impact of one type of data but not another. The method by which the contingency estimate was assessed, as data were received vs at the end of the stream, influenced the final estimate. It is concluded that the "framing" of the task affects the contingency estimate. (41 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Bayesian evaluation of inequality and equality constrained hypotheses for contingency tables.

In this article, a Bayesian model selection approach is introduced that can select the best of a set of inequality and equality constrained hypotheses for contingency tables. The hypotheses are presented in terms of cell probabilities allowing researchers to test (in)equality constrained hypotheses in a format that is directly related to the data. The proposed method is investigated by several simulation studies and shows good performance. Software that allows researchers to apply the Bayesian approach to their own data is also provided. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Graphic determination of significance of 2 X 2 contingency tables.

There are several techniques available for determining the significance of differences in proportions, or frequencies in the 2 X 2 contingency tables. All of these techniques have their limitations. A method is proposed which overcomes some of these limitations. The method is applicable for the situation where the two samples being compared are independently selected and contain the same number of Ss. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Taylor-Russell tables for dichotomous criterion variables.

Considers that the widely used tables of H. Taylor and T. Russell (see pa, vol. 14:1111) for evaluating the usefulness of selection tests are inappropriate for evaluating predictors used to discriminate discrete normally distributed criterion groups where a point-biserial correlation coefficient is the proper index of validity. A set of tables is presented, similar to the taylor-russell tables, for evaluating the usefulness of selection tests where the assumptions require point-biserial validity coefficients. A graphic illustration of the rationale for developing the tables is provided. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Extension of multiple-range tests to interaction tables in the analysis of variance: A rapid approximate solution.

Discusses the application of multiple-range tests to means derived from the 1-way analysis of variance, in which none of the possible contrasts is confounded. Often, however, it is ncessary to compare Ms in an interaction table derived from a factorial analysis of variance. In this case, the only unconfounded comparisons are those made within rows and columns. An approximate solution is described that adjusts the number of treatments by basing the q statistic upon the number of unconfounded comparisons only. The solution is then applied, using actual data, (a) when only the K(K-1)/2 contrasts are desired (the method of J. Tukey), and (b) when all possible contracts are desired (the method of H. Sheffe). The Duncan and Newman-Keuls tests are deemphasized, since research demonstrates these tests fail to control adequately for Type I error. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A general equation and technique for the exact partitioning of chi-square contingency tables.

Considers the technique of the exact partitioning of chi-square contingency tables. Methods are presented for partitioning contingency tables into components. A general equation for chi-square is derived. The equation may be used for the calculation of exact values for (1) non-exhaustive sets of categories, and (2) situations in which some cells have small expected frequencies. (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)     

Small-sample tests for homogeneity of response probabilities in 2?×?2 contingency tables.

Classical tests of significance for homogeneity of response probabilities in 2?×?2 tables tend to have actual Type I error probabilities that differ substantially from the theoretical alpha level. Fisher's exact test and chi-square with Yates's (1934) correction for continuity have severely conservative biases, whereas the Pearson chi-square is frequently nonconservative. A z test proposed by Haber (1986) has the advantage of never producing nonconservative results while evidencing only modest conservative bias in most cases. We show several other new tests to have actual Type I error probabilities that vary more closely about theoretical alpha levels. They avoid the conservative biases of the Fisher, Yates, and Haber tests and are less often nonconservative than is the Pearson chi-square. We present computational formulas. (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)     

Assessing the impact of common method variance on higher order multidimensional constructs.

Researchers are often concerned with common method variance (CMV) in cases where it is believed to bias relationships of predictors with criteria. However, CMV may also bias relationships within sets of predictors; this is cause for concern, given the rising popularity of higher order multidimensional constructs. The authors examined the extent to which CMV inflates interrelationships among indicators of higher order constructs and the relationships of those constructs with criteria. To do so, they examined core self-evaluation, a higher order construct comprising self-esteem, generalized self-efficacy, emotional stability, and locus of control. Across 2 studies, the authors systematically applied statistical (Study 1) and procedural (Study 2) CMV remedies to core self-evaluation data collected from multiple samples. Results revealed that the nature of the higher order construct and its relationship with job satisfaction were altered when the CMV remedies were applied. Implications of these findings for higher order constructs are discussed. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

A sample size correction for Pearson chi-square in 2?×?2 contingency tables.

A test statistic that is asymptotically equivalent to the Pearson chi-square, but which is more conservative by a factor of (N?–?1)/N, is proposed for use with small samples when testing for homogeneity of response probabilities in 2?×?2 tables. Evaluation of this test statistic indicates that it produces actual Type I error probabilities that are less often and less seriously nonconservative than the Pearson chi-square, although it does not result in the severely conservative bias previously reported for Yates's correction or Fisher's exact probability test. Other recent evaluations of bias in the Pearson chi-square test for 2?×?2 tables have resulted in conflicting recommendations, primarily because the evaluations have been based on different assumptions concerning the degree of experimenter control over marginal frequencies. Three conditions of control over the marginal frequencies are identified, and the results of exact probability evaluations for the Pearson chi-square under different assumptions are discussed. (9 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)     

Diagnostic validity of tests--binary tables or multidimensional statistical methods?

The authors criticize the use of concepts of sensitivity and specificity calculated by means of a binary table in differential diagnostic reflections pertaining to more than one disease or several, i.e. more than one diagnostic test. On examples they present possibilities of different ways for calculating these concepts with different not correlating results. They indicate the justification of using multidimensional statistical methods in those instances and their better diagnostic yield.     

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)