An R package to compute commonality coefficients in the multiple regression case: An introduction to the package and a practical example

Multiple regression is a widely used technique for data analysis in social and behavioral research. The complexity of interpreting such results increases when correlated predictor variables are involved. Commonality analysis provides a method of determining the variance accounted for by respective predictor variables and is especially useful in the presence of correlated predictors. However, computing commonality coefficients is laborious. To make commonality analysis accessible to more researchers, a program was developed to automate the calculation of unique and common elements in commonality analysis, using the statistical package R. The program is described, and a heuristic example using data from the Holzinger and Swineford (1939) study, readily available in the MBESS R package, is presented.     

The asymptotic distribution of commonality components

Commonality components have been defined as a method of partitioning squared multiple correlations. In this paper, the asymptotic joint distribution of all 2 ^k – 1 squared multiple correlations is derived. The asymptotic joint distribution of linear combinations of squared multiple correlations is obtained as a corollary. In particular, the asymptotic joint distribution of commonality components are derived as a special case. Simultaneous and nonsimultaneous asymptotic confidence intervals for commonality components can be obtained from this distribution.This work was supported in part by the Spencer Foundation and the National Science Foundation.The authors are grateful to Bryna Siegel-Gorlick for her help in obtaining the data used in Example 4.3, and to the referees for their comments and suggestions.     

Visual-manual commonality in size judgments

Visual-manual commonality was studied in terms of commonality of process and commonality of achievement. Fifty-four Ss madelarge-small categorical judgments about a set of stimuli presented in counterbalanced order through both the visual modality and the haptic, touch, or kinesthetic modality. Using a linear discriminant function model, analyses of the several visual-manual pairings were made on the basis of intra-S absolute differences in the weighting of the physical dimensions (process) and the degree of category separation (achievement). A correlational analysis revealed the independence of these measures, and analyses of variance revealed greater visual-haptic commonality for weighting systems and similar visual-haptic and visual-touch commonality for category separation. The results were interpreted in terms of similarities and differences in information gathering and processing characteristics.     

Confusing developmental and individual differences

T. Globerson (1985, Developmental Review, 5, 261–273) compared the relative effects of an individual difference variable field dependency (FD/I) with a developmental variable (age) on a measure of M-capacity, in an analysis of variance (ANOVA) design. Age is a highly significant effect and the sum of squares associated with this factor accounts for 66% of the variance in M-capacity. FD/I is not significant and accounts for less than 1% of the variance in M-capacity. This is interpreted as indicating that M-capacity is a purely developmental phenomenon showing negligible individual differences. This paper argues that this result is an inevitable artifact of the experimental design. Further, the design is a common one in developmental psychology, and its dangers are largely unrecognized. The main problems concern (1) dichotomizing a continuous variable and treating the dichotomy as two levels of a factorial variable, (2) drawing inferences about the size of relationship between the continuous variable and the dependent variable on the basis of the variance accounted for by the factorial variable in the ANOVA, (3) differential reliability of measures of the independent variables, and (4) selective control of the sampling of the independent variables. These issues are explored in a simulation of Globerson's data and in an analogous analysis of data relating age and height, as independent variables, with weight as a dependent variable.     

3～5岁儿童概念比较过程研究

本研究以两个实验探讨了儿童的一个认知过程:人优先考虑共同性,其次才是组合差异性与非组合差异性。3~5岁三个年龄组儿童分别对日常生活的基本概念进行类似性判断与差异性判断。实验1的结果表明,3岁年龄组的儿童还没有表现出组合差异性,4岁年龄组的儿童初步表现出明显的组合差异性,5岁年龄组的儿童表现出明显的组合差异性。他们首先考虑共同性,其次才是组合差异性与非组合差异性。实验2的结果表明,5岁儿童比4岁儿童更多地表现了组合差异性的判断能力。最后以结构组合的观点讨论了这些结果。     

Examining the unique and combined effects of grit,trait self-control,and conscientiousness in predicting motivation for academic goals: A commonality analysis

The purpose of the present research was to examine the predicative ability of both the unique and combined components of grit, trait self-control, and conscientiousness in the context of academic goal pursuit. Participants (n₁ = 163, n₂ = 551) were asked to complete assessments of each self-regulatory trait and reported their motivation for an academic goal. Together, grit, trait self-control, and conscientiousness explained 9.9% of the variance in academic goal motivation across both samples. Using commonality analysis, we found that the overlapping components of grit, trait self-control, and conscientiousness accounted for 49.6% of the explained variance (4.9% of the total variance), with the individual components each accounting for less than 20% (2% of the total variance). Implications for research on self-regulatory traits are discussed.     

Revisiting Interpretation of Canonical Correlation Analysis: A Tutorial and Demonstration of Canonical Commonality Analysis

In the face of multicollinearity, researchers face challenges interpreting canonical correlation analysis (CCA) results. Although standardized function and structure coefficients provide insight into the canonical variates produced, they fall short when researchers want to fully report canonical effects. This article revisits the interpretation of CCA results, providing a tutorial and demonstrating canonical commonalty analysis. Commonality analysis fully explains the canonical effects produced by using the variables in a given canonical set to partition the variance of canonical variates produced from the other canonical set. Conducting canonical commonality analysis without the aid of software is laborious and may be untenable, depending on the number of noteworthy canonical functions and variables in either canonical set. Commonality analysis software is identified for the canonical correlation case and we demonstrate its use in facilitating model interpretation. Data from Holzinger and Swineford (1939) Holzinger, K. J. and Swineford, F. 1939. “A study in factor analysis: The stability of a bi-factor solution.”. In Supplementary Educational Monographs Vol. 48, 81–91. Chicago: Department of Education, University of Chicago.  [Google Scholar] are employed to test a hypothetical theory that problem-solving skills are predicted by fundamental math ability.     

Children's Classification Skills as Predictors of Their Referential Communicative Encodings

Younger and older children's classification skills were assessed, and their classificatory performance was used as a predictor variable in a multiple-linear-regression analysis to determine whether their classificatory skills accounted for a significant proportion of the variance in their performance on referential communication tasks. We tested 10 boys and 10 girls at each of six age levels between 5 and 10 years individually on a vocabulary test, four classification tasks, and six referential communication tasks. Order of task presentations was counterbalanced; reliabilities for all tasks were high. A series of related multiple-regression analyses showed that grade and vocabulary accounted for a significant proportion of the variance on the referential communication tasks; the series also showed a large decrease in the F value for grade and vocabulary when the contribution of the classification variables was partialled out. No evidence for an independent contribution of the classification variables was found.     

The reasoning skills and thinking dispositions of problem gamblers: a dual‐process taxonomy

We present a taxonomy that categorizes the types of cognitive failure that might result in persistent gambling. The taxonomy subsumes most previous theories of gambling behavior, and it defines three categories of cognitive difficulties that might lead to gambling problems: The autonomous set of systems (TASS) override failure, missing TASS output, and mindware problems. TASS refers to the autonomous set of systems in the brain (which are executed rapidly and without volition, are not under conscious control, and are not dependent on analytic system output). Mindware consists of rules, procedures, and strategies available for explicit retrieval. Seven of the eight tasks administered to pathological gamblers, gamblers with subclinical symptoms, and control participants were associated with problem gambling, and five of the eight were significant predictors in analyses that statistically controlled for age and cognitive competence. In a commonality analysis, an indicator from each of the three categories of cognitive difficulties explained significant unique variance in problem gambling, indicating that each of the categories had some degree of predictive specificity. Copyright © 2006 John Wiley & Sons, Ltd.     

Statistical mediation analysis with a multicategorical independent variable

Virtually all discussions and applications of statistical mediation analysis have been based on the condition that the independent variable is dichotomous or continuous, even though investigators frequently are interested in testing mediation hypotheses involving a multicategorical independent variable (such as two or more experimental conditions relative to a control group). We provide a tutorial illustrating an approach to estimation of and inference about direct, indirect, and total effects in statistical mediation analysis with a multicategorical independent variable. The approach is mathematically equivalent to analysis of (co)variance and reproduces the observed and adjusted group means while also generating effects having simple interpretations. Supplementary material available online includes extensions to this approach and Mplus, SPSS, and SAS code that implements it.     

The perils of partialling: cautionary tales from aggression and psychopathy

Although a powerful technique, the partialling of independent variables from one another in the context of multiple regression analysis poses certain perils. The present article argues that the most important and underappreciated peril is the difficulty in knowing what construct an independent variable represents once the variance shared with other independent variables is removed. The present article presents illustrative analyses in a large sample of inmates (n=696) using three measures from the psychopathy and aggression fields. Results indicate that in terms of relations among items on a single scale and relations between scales, the raw and residualized scores bore little resemblance to one another. It is argued that researchers must decide to which construct-the one represented by the original scale or the one represented by the residualized scale-conclusions are meant to apply. Difficulties in applying the conclusions to the residualized scale are highlighted and best practices suggested.     

The analysis of multifactor experiments designed to contrast the within- and between-subjects manipulation of the independent variables

Experiments are frequently performed in which the major question of interest is whether the effect of the substantive independent variable depends on its manipulation within as opposed to between subjects. Erlebacher (1977) discussed the design of such experiments in which one independent variable is manipulated, and presented a method of analysis by an analysis of variance technique. This paper discusses the analysis of three different experimental designs in which the within-subjects/between-subjects design type is one factor and two substantive independent variables are manipulated. A rule is also presented that can be used to generate statistical tests in a generalized multifactor design.     

Two classes of models for magnitude estimation

One class of models assumes that presentation of a signal results in an internal representation as a random variable. Depending on whether the signal is close to or far from the preceding signal, the variance of the representation is smaller or larger. Responses are determined largely by this random variable; however, when the signal is close to the preceding one, the response is generated by modifying the representation multiplicatively by some function of the ratio of the previous response to its representation. Power and linear functions are explored. The form of the random variable is assumed to be that arising from either the timing or the counting model operating on a Poisson process. Detailed analyses are carried out successfully only for the timing model with neural sample sizes independent of intensity; however, the data require the sample to increase with intensity. The linear response function coupled with the constant sample size counting model appears somewhat viable, but detailed calculations are very difficult to carry out. The second class of models postulates a power function relation between magnitude estimates and signals intensity for which the exponent is a Gaussian distributed random variable and the unit is the product of two log normal random variables. Again we assume an attention band such that succesive stimuli that are widely separated in intensity lead to independent samples of the random variables while a variety of assumptions is explored for successive stimuli that are near each other in intensity. Although they each give rise to the qualitative features of the data, estimates of parameters are sufficiently inconsistent that we are led to reject all of the submodels studied.     

Analysis of trend: a permutation alternative to the F test

When the categories of the independent variable in an analysis of variance are quantitative, it is more informative to evaluate the trends in the treatment means than to simply compare differences among the treatment means. A permutation alternative to the conventional F test is shown to possess significant advantages when analyzing trend among quantitative treatments in a one-way analysis of variance. An example with and without an extreme data point illustrates the effectiveness of the permutation alternative for the analysis of trend when homogeneity of variance is compromised.     

Posttraumatic Stress Disorder Symptom Clusters and Acquired Capability for Suicide: A Reexamination Using DSM-5 Criteria

This study used the interpersonal–psychological theory of suicide to explore the relationships among DSM-5 posttraumatic stress disorder (PTSD) symptom clusters derived from the six-factor anhedonia model and facets of acquired capability for suicide (ACS). In a sample of 373 trauma-exposed undergraduates, most PTSD symptom clusters were negatively associated with facets of ACS in bivariate correlations, but the anhedonia cluster was positively associated with ACS in regression models. Structure coefficients and commonality analysis indicated that anhedonia served as a suppressor variable for the other symptom clusters. Our findings further elucidate the complex relationship between specific PTSD symptom clusters and ACS.     

The Correlation of General Cognitive Ability and Psychomotor Tracking Tests

A study was conducted to investigate the nexus of cognitive and psychomotor tests as might be used for personnel selection and assessment. These two domains are frequently seen as independent. A multiple aptitude cognitive test battery and a psychomotor test battery were administered to 354 United States Air Force recruits. The average multiple correlation of the cognitive tests and each psychomotor score as a criterion was 0.34, corrected for range restriction. Confirmatory factor analyses disclosed general cognitive and general psychomotor factors, three lower-order psychomotor factors, and two lower-order cognitive factors. The general cognitive factor accounted for 39% of the variance and the general psychomotor factor accounted for 29% of the variance. Residualized, the lower-order factors accounted for between 10% and 3% of the variance. The average g saturations (loadings) of the cognitive and psychomotor tests were 0.82 and 0.34 respectively. An implication for personnel selection is that the incremental validity of psychomotor tracking tests beyond the validity of cognitive tests will be small due to the commonality of measurement. A further implication of findings is the need to study the validity of the general and specific psychomotor factors.     

Analyzing aphasia data in a multidimensional symptom space

The utility of single-case vs. group studies has been debated in neuropsychology for many years. The purpose of the present study is to illustrate an alternative approach to group studies of aphasia, in which the same symptom dimensions that are commonly used to assign patients to classical taxonomies (fluency, naming, repetition, and comprehension) are used as independent and continuous predictors in a multivariate design, without assigning patients to syndromes. One hundred twenty-six Italian-speaking patients with aphasia were first classified into seven classic aphasia categories, based on fluency, naming, auditory comprehension, and repetition scales. There were two goals: (1) compare group analyses based on aphasia types with multivariate analyses that sidestep classification and treat aphasic symptoms as continuous variables; (2) present correlation-based outlier analyses that can be used to identify individuals who occupy unusual positions in the multivariate "symptom space." In the service of the first goal, group performance on an external validation measure (the Token Test) was assessed in three steps: analyses of variance based on aphasia type, regressions using the same cut-offs for fluency, naming, comprehension and repetition as independent but dichotomous predictors, and regressions using the same subscales as continuous predictors (with no cut-offs). More variance in Token Test performance was accounted for when symptoms were treated as continuous predictors than with the other two methods, though use of independent but dichotomous predictors accounted for more variance than aphasia taxonomies. Thus, if we by-pass classical taxonomies and treat patients as points in a multidimensional symptom space, better predictions are obtained. Outlier analyses show that group results depend on heterogeneity among patients, which can be used as a search tool to identify potentially interesting dissociations. Hence this multivariate group approach is complementary to and compatible with single-case methods.     

Bias in Estimation of Misclassification Rates

When a simple random sample of size n is employed to establish a classification rule for prediction of a polytomous variable by an independent variable, the best achievable rate of misclassification is higher than the corresponding best achievable rate if the conditional probability distribution is known for the predicted variable given the independent variable. In typical cases, this increased misclassification rate due to sampling is remarkably small relative to other increases in expected measures of prediction accuracy due to samplings that are typically encountered in statistical analysis. This issue is particularly striking if a polytomous variable predicts a polytomous variable, for the excess misclassification rate due to estimation approaches 0 at an exponential rate as n increases. Even with a continuous real predictor and with simple nonparametric methods, it is typically not difficult to achieve an excess misclassification rate on the order of n ⁻¹. Although reduced excess error is normally desirable, it may reasonably be argued that, in the case of classification, the reduction in bias is related to a more fundamental lack of sensitivity of misclassification error to the quality of the prediction. This lack of sensitivity is not an issue if criteria based on probability prediction such as logarithmic penalty or least squares are employed, but the latter measures typically involve more substantial issues of bias. With polytomous predictors, excess expected errors due to sampling are typically of order n ⁻¹. For a continuous real predictor, the increase in expected error is typically of order n ^−2/3     

Common genetic influences on impulsivity facets are related to goal management,psychopathology, and personality

In this study, we characterized the genetic/environmental commonality and heterogeneity of impulsivity facets and tested the hypothesis that goal-management is central to their common variance. 764 young-adult twins completed the UPPS-P Impulsive Behavior Scale and measures of goal management, personality, and psychopathology. We found common genetic influences across all impulsivity facets except sensation seeking. These impulsivity genetic influences explained 40.0% of variance in goal-management ability. Other results supported three hypotheses concerning heterogeneity: that sensation seeking is independent of other facets, that urgency is more related to psychopathology than lack of premeditation, and that lack of perseverance is more similar to urgency than lack of premeditation. Thus, impulsivity facets show considerable heterogeneity in addition to common variation related to goal-management abilities.     

Predictors of Health Self-Appraisal: What's Involved in Feeling Healthy?

Increased interest in the prevention of illness underscores the importance of understanding how people evaluate their health. This study investigated self-report measures of health status, medical complaints, and psychosocial factors in 455 undergraduates. After controlling for neuroticism, regression analyses indicated that positive mood, negative mood, perceived vulnerability to illness, and vitality were significant contributors to self-assessed health independent of physical symptoms and diseases. Together, psychosocial factors accounted for more variance in health assessments (R² = .44) than did disease and symptom reports alone (R² = .25; p. < .01 for all betas). These results demonstrate the strong association of psychosocial stares with evaluations of health and suggest that physical discomfort may play a relatively minor role in how "healthy" we feel.