Eta Squared,Partial Eta Squared,and Misreporting of Effect Size in Communication Research

Communication researchers, along with social scientists from a variety of disciplines, are increasingly recognizing the importance of reporting effect sizes to augment significance tests. Serious errors in the reporting of effect sizes, however, have appeared in recently published articles. This article calls for accurate reporting of estimates of effect size. Eta squared (η²) is the most commonly reported estimate of effect sized for the ANOVA. The classical formulation of eta squared (Pearson, 1911; Fisher, 1928) is distinguished from the lesser known partial eta squared (Cohen, 1973), and a mislabeling problem in the statistical software SPSS (1998) is identified. What SPSS reports as eta squared is really partial eta squared. Hence, researchers obtaining estimates of eta squared from SPSS are at risk of reporting incorrect values. Several simulations are reported to demonstrate critical issues. The strengths and limitations of several estimates of effect size used in ANOVA are discussed, as are the implications of the reporting errors. A list of suggestions for researchers is then offered.     

方差分析效果大小报告的新指标

心理学期刊论文中可重复性不高的现象,原因之一是研究结果的效果普遍较小。并且在报告效果大小的文章中,效果大小指标使用不当。在方差分析中最经常报告的是η2和η2_p,但是在不同的研究设计中,这些效果大小是无法直接进行比较的。广义eta方(η²_G)是近年来新出现的一种新的效果大小指标,可克服的η2和η2_p不足,灵活处理重复测量等多种研究设计下个体差异的计算问题,实现跨研究设计效果大小的可比性。论文结合实例介绍了η²_G的原理和计算方法,并对其优缺点、使用和报告等问题进行了讨论。研究人员在报告效果大小时要考虑到不同的研究设计和研究假设,并选择恰当的指标防止过高估计效果大小。     

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.     

Recommended effect size statistics for repeated measures designs

Investigators, who are increasingly implored to present and discuss effect size statistics, might comply more often if they understood more clearly what is required. When investigators wish to report effect sizes derived from analyses of variance that include repeated measures, past advice has been problematic. Only recently has a generally useful effect size statistic been proposed for such designs: generalized eta squared (η_G²; Olejnik & Algina, 2003). Here, we present this method, explain that η_G² is preferred to eta squared and partial eta squared because it provides comparability across between-subjects and within-subjects designs, show that it can easily be computed from information provided by standard statistical packages, and recommend that investigators provide it routinely in their research reports when appropriate.     

Exact Analysis of Squared Cross-Validity Coefficient in Predictive Regression Models

In regression analysis, the notion of population validity is of theoretical interest for describing the usefulness of the underlying regression model, whereas the presumably more important concept of population cross-validity represents the predictive effectiveness for the regression equation in future research. It appears that the inference procedures of the squared multiple correlation coefficient have been extensively developed. In contrast, a full range of statistical methods for the analysis of the squared cross-validity coefficient is considerably far from complete. This article considers a distinct expression for the definition of the squared cross-validity coefficient as the direct connection and monotone transformation to the squared multiple correlation coefficient. Therefore, all the currently available exact methods for interval estimation, power calculation, and sample size determination of the squared multiple correlation coefficient are naturally modified and extended to the analysis of the squared cross-validity coefficient. The adequacies of the existing approximate procedures and the suggested exact method are evaluated through a Monte Carlo study. Furthermore, practical applications in areas of psychology and management are presented to illustrate the essential features of the proposed methodologies. The first empirical example uses 6 control variables related to driver characteristics and traffic congestion and their relation to stress in bus drivers, and the second example relates skills, cognitive performance, and personality to team performance measures. The results in this article can facilitate the recommended practice of cross-validation in psychological and other areas of social science research.     

Note on the squared multiple correlation as a lower bound to communality

A simple proof that the squared multiple correlation of a variable with the remaining variables in the set of variables is a lower bound to the communality of that variable is presented.     

Effects of prenatal X-irradiation on open-field behavior in rats: application of randomized fostering technique and mapping results

Sprague-Dawley (SD) rats were given X-irradiation (150 R) on Day 17 of gestation. After birth, all male pups were pooled once and then assigned randomly to irradiated mothers and control mothers. Offspring were administered an open-field test at about 7 weeks of age. The analysis was performed on the basis of two approaches: In the per subject approach, individual subject data (aggregation across Day 2 through Day 4) were treated as the basic unit of statistical analysis. In the per litter approach, double aggregation (aggregation across Day 2 through Day 4 for each subject and aggregation across subjects within each litter) was used. The per subject approach was slightly more sensitive as to the treatment effect, but it induced a reduction in the magnitude of eta squared. A principal component analysis was performed using eta squared together with those of several reference groups. Results were plotted on a map constructed from component scores. The characteristics of behavior in X-irradiated rats were very similar to those of the earlier stage of trials in terms of the location on the map. The postnatal maternal effect on open-field behavior was not serious and was adequately negligible in practice. A new fostering procedure was proposed and its advantages discussed.     

Effects of Prenatal X-irradiation on Open-Field Behavior in Rats: Application of Randomized Fostering Technique and Mapping Results

Sprague-Dawley (SD) rats were given X-irradiation (150 R) on Day 17 of gestation. After birth, all male pups were pooled once and then assigned randomly to irradiated mothers and control mothers. Offspring were administered an open-field test at about 7 weeks of age. The analysis was performed on the basis of two approaches: In the per subject approach, individual subject data (aggregation across Day 2 through Day 4) were treated as the basic unit of statistical analysis. In the per litter approach, double aggregation (aggregation across Day 2 through Day 4 for each subject and aggregation across subjects within each litter) was used. The per subject approach was slightly more sensitive as to the treatment effect, but it induced a reduction in the magnitude of eta squared. A principal component analysis was performed using eta squared together with those of several reference groups. Results were plotted on a map constructed from component scores. The characteristics of behavior in X-irradiated rats were very similar to those of the earlier stage of trials in terms of the location on the map. The postnatal maternal effect on open-field behavior was not serious and was adequately negligible in practice. A new fostering procedure was proposed and its advantages discussed.     

Note on the squared multiple correlation as a lower bound to communality

It is demonstrated that the squared multiple correlation of a variable with the remaining variables in a set of variables is a function of the communalities and the squared canonical correlations between the observed variables and common factors. This equation is shown to imply a strict inequality between the squared multiple correlation and communality.     

Correlation coefficients for more than one scale type: An alternative to the Janson and Vegelius approach

Janson and Vegelius have recently suggested a family of correlations for variables of mixed scale types, including nominal scales. The resulting correlations areE-coefficients, which means that they are unity if the variables involved are identical up to permissible transformations, and that they can be considered as inner products in a Euclidian space. Some of the coefficients of the correlation family suggested by Janson and Vegelius are generalized squared product-moment correlations and some are not. In the present paper, a family of correlations for variables of mixed scale types is advocated all members of which are generalized squared product-moment correlations. Some practical advantages of the latter family are explained. The authors are obliged to Klaas Nevels and Henk Kiers for helpful comments.     

Some relationships between factors and components

The asymptotic correlations between the estimates of factor and component loadings are obtained for the exploratory factor analysis model with the assumption of a multivariate normal distribution for manifest variables. The asymptotic correlations are derived for the cases of unstandardized and standardized manifest variables with orthogonal and oblique rotations. Based on the above results, the asymptotic standard errors for estimated correlations between factors and components are derived. Further, the asymptotic standard error of the mean squared canonical correlation for factors and components, which is an overall index for the closeness of factors and components, is derived. The results of a Monte Carlo simulation are presented to show the usefulness of the asymptotic results in the data with a finite sample size.The author is indebted to anonymous referees for their comments, corrections and suggestions which have led to the improvement of this article.     

Obtaining squared multiple correlations from a correlation matrix which may be singular

A theorem is presented relating the squared multiple correlation of each measure in a battery with the other measures to the unique generalized inverse of the correlation matrix. This theorem is independent of the rank of the correlation matrix and may be utilized for singular correlation matrices. A coefficient is presented which indicates whether the squared multiple correlation is unity or not. Note that not all measures necessarily have unit squared multiple correlations with the other measures when the correlation matrix is singular. Some suggestions for computations are given for simultaneous determination of squared multiple correlations for all measures.The research reported in this paper was supported by the Personnel and Training Branch of the Office of Naval Research under Contract Number 00014-67-A-0305-0003 with the University of Illinois.     

The effect of affine transformation on redundancy analysis

Canonical redundancy analysis provides an estimate of the amount of shared variance between two sets of variables and provides an alternative to canonical correlation. The proof that the total redundancy is equal to the average squared multiple correlation coefficient obtained by regressing each variable in the criterion set on all variables in the predictor set is generalized to the case in which there are a larger number of criterion than predictor variables. It is then shown that the redundancy for the criterion set of variables is invariant under affine transformation of the predictor variables, but not invariant under transformation of the criterion variables.     

Differences of Mean Scores on the Preliminary Scholastic Aptitude Test (PSAT) for Classical Christian Schools Compared to Non-Classical Christian Schools

Abstract

This causal-comparative quantitative study compares mean scores on the Preliminary Scholastic Aptitude Test (PSAT) between Classical Christian schools and non-Classical Christian schools using data randomly selected from survey responses. The sample consisted of 4,486 mean scores from the 2003–2004 school year through 2012–2013. Welch’s t-tests for unequal variances was used with an alpha set at 0.05 and 0.017 for Bonferroni correction and returned statistically significant results for all three academic areas: reading, math, and writing. Effect size measured by Cohen’s d and eta squared indicated Classical Christian methodology should have a large, positive effect on PSAT scores. Demographic and between school comparison limitations noted. Future research and associated correlational studies are recommended.     

The Influence of Differing the Paths to an Incentive on Third Graders’ Reading Achievement and Attitudes

The purpose of this study was to examine the impact of four incentive paths on third graders’ reading vocabulary and comprehension achievement and recreational and academic reading attitude. One hundred and twenty third-grade students were assigned to one of four incentive path treatment conditions. Data were analyzed using analysis of covariance (ANCOVA) with Fisher's least significant difference (LSD) post hoc comparisons. An effect size of .09 was calculated using a partial η² (eta squared) statistic. Findings indicated that the four incentive path treatment conditions did not differentially impact student vocabulary, comprehension, and recreational reading. A significant difference for students’ academic reading attitudes was identified across the four incentive path treatment conditions.     

What spelling tells us about the orthographic development and word study instruction with emergent bilingual secondary students

Educators need ways to assess orthographic knowledge and differentiate word study instruction for secondary, emergent bilingual learners. In this study, the spelling of 199 students in grades 7–12 across eight features and four spelling stages was examined to understand students' orthographic development; all but two were learning Spanish and English. Spelling data were collected three times over a nine-month period. There were significant changes across the time periods that show students progressing across four stages of spelling development, with a large effect size (partial eta squared = .498). Data from a standardized assessment of language proficiency were examined to study the relationship between students' orthographic knowledge and their language scores. This study suggests ways to use spelling assessments to plan differentiated vocabulary and word study instruction.     

On the partitioning of squared Euclidean distance and its applications in cluster analysis

The partitioning of squared Eucliean distance between two vectors in M-dimensional space into the sum of squared lengths of vectors in mutually orthogonal subspaces is discussed and applications given to specific cluster analysis problems. Examples of how the partitioning idea can be used to help describe and interpret derived clusters, derive similarity measures for use in cluster analysis, and to design Monte Carlo studies with carefully specified types and magnitudes of differences between the underlying population mean vectors are presented. Most of the example applications presented in this paper involve the clustering of longitudinal data, but their use in cluster analysis need not be limited to this arena.     

A Comparative Investigation of Confidence Intervals for IndependentVariables in Linear Regression

In linear regression, the most appropriate standardized effect size for individual independent variables having an arbitrary metric remains open to debate, despite researchers typically reporting a standardized regression coefficient. Alternative standardized measures include the semipartial correlation, the improvement in the squared multiple correlation, and the squared partial correlation. No arguments based on either theoretical or statistical grounds for preferring one of these standardized measures have been mounted in the literature. Using a Monte Carlo simulation, the performance of interval estimators for these effect-size measures was compared in a 5-way factorial design. Formal statistical design methods assessed both the accuracy and robustness of the four interval estimators. The coverage probability of a large-sample confidence interval for the semipartial correlation coefficient derived from Aloe and Becker was highly accurate and robust in 98% of instances. It was better in small samples than the Yuan-Chan large-sample confidence interval for a standardized regression coefficient. It was also consistently better than both a bootstrap confidence interval for the improvement in the squared multiple correlation and a noncentral interval for the squared partial correlation.     

Robust regression: Testing global hypotheses about the slopes when there is multicollinearity or heteroscedasticity

A well-known concern regarding the usual linear regression model is multicollinearity. As the strength of the association among the independent variables increases, the squared standard error of regression estimators tends to increase, which can seriously impact power. This paper examines heteroscedastic methods for dealing with this issue when testing the hypothesis that all of the slope parameters are equal to zero via a robust ridge estimator that guards against outliers among the dependent variable. Included are results related to leverage points, meaning outliers among the independent variables. In various situations, the proposed method increases power substantially.     

Methodological and computational considerations for multiple correlation analysis

The squared multiple correlation coefficient has been widely employed to assess the goodness-of-fit of linear regression models in many applications. Although there are numerous published sources that present inferential issues and computing algorithms for multinormal correlation models, the statistical procedure for testing substantive significance by specifying the nonzero-effect null hypothesis has received little attention. This article emphasizes the importance of determining whether the squared multiple correlation coefficient is small or large in comparison with some prescribed standard and develops corresponding Excel worksheets that facilitate the implementation of various aspects of the suggested significance tests. In view of the extensive accessibility of Microsoft Excel software and the ultimate convenience of general-purpose statistical packages, the associated computer routines for interval estimation, power calculation, a nd samplesize determination are alsoprovided for completeness. The statistical methods and available programs of multiple correlation analysis described in this article purport to enhance pedagogical presentation in academic curricula and practical application in psychological research.