Appropriately centering Level 1 predictors is vital to the interpretation of intercept and slope parameters in multilevel models (MLMs). The issue of centering has been discussed in the literature, but it is still widely misunderstood. The purpose of this article is to provide a detailed overview of grand mean centering and group mean centering in the context of 2-level MLMs. The authors begin with a basic overview of centering and explore the differences between grand and group mean centering in the context of some prototypical research questions. Empirical analyses of artificial data sets are used to illustrate key points throughout. The article provides a number of practical recommendations designed to facilitate centering decisions in MLM applications.

Centering predictor variables in cross-sectional multilevel models: a new look at an old issue.

The idea that parental involvement has positive influence on students' academic achievement is so intuitively appealing that society in general, and educators in particular, have considered parental involvement an important ingredient for the remedy for many problems in education. The vast proportion of the literature in this area, however, is qualitative and nonempirical. Among the empirical studies that have investigated the issue quantitatively, there appear to be considerable inconsistencies. A meta-analysis was conducted to synthesize the quantitative literature about the relationship between parental involvement and students' academic achievement. The findings reveal a small to moderate, and practically meaningful, relationship between parental involvement and academic achievement. Through moderator analysis, it was revealed that parental aspiration/expectation for children's education achievement has the strongest relationship, whereas parental home supervision has the weakest relationship, with students' academic achievement. In addition, the relationship is stronger when academic achievement is represented by a global indicator (e.g., GPA) than by a subject-specific indicator (e.g., math grade). Limitations of the study are noted, and suggestions are made for future studies.

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Parental involvement and students' academic achievement: A meta-analysis.

The following paper discusses exploratory factor analysis and gives an overview of the statistical technique and how it is used in various research designs and applications. A basic outline of how the technique works and its criteria, including its main assumptions are discussed as well as when it should be used. Mathematical theories are explored to enlighten students on how exploratory factor analysis works, an example of how to run an exploratory factor analysis on SPSS is given, and finally a section on how to write up the results is provided. This will allow readers to develop a better understanding of when to employ factor analysis and how to interpret the tables and graphs in the output.

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A Beginner's Guide to Factor Analysis: Focusing on Exploratory Factor Analysis

Rationale, aims and objectives The diversity of the population worldwide suggests a great need for cross-culturally validated research instruments or scales. Researchers and clinicians must have access to reliable and valid measures of concepts of interest in their own cultures and languages to conduct cross-cultural research and/or provide quality patient care. Although there are well-established methodological approaches for translating, adapting and validating instruments or scales for use in cross-cultural health care research, a great variation in the use of these approaches continues to prevail in the health care literature. Therefore, the objectives of this scholarly paper were to review published recommendations of cross-cultural validation of instruments and scales, and to propose and present a clear and user-friendly guideline for the translation, adaptation and validation of instruments or scales for cross-cultural health care research. Methods A review of highly recommended methodological approaches to translation, adaptation and cross-cultural validation of research instruments or scales was performed. Recommendations were summarized and incorporated into a seven-step guideline. Each one of the steps was described and key points were highlighted. Example of a project using the proposed steps of the guideline was fully described. Conclusions Translation, adaptation and validation of instruments or scales for crosscultural research is very time-consuming and requires careful planning and the adoption of rigorous methodological approaches to derive a reliable and valid measure of the concept of interest in the target population.

Translation, adaptation and validation of instruments or scales for use in cross‐cultural health care research: a clear and user‐friendly guideline

ANOSIM, PERMANOVA, and the Mantel test are all resemblance-based permutation methods widely used in ecology. Here, we report the results of the first simulation study, to our knowledge, specifically designed to examine the effects of heterogeneity of multivariate dispersions on the rejection rates of these tests and on a classical MANOVA test (Pillai's trace). Increasing differences in dispersion among groups were simulated under scenarios of changing sample sizes, correlation structures, error distributions, numbers of variables, and numbers of groups for balanced and unbalanced one-way designs. The power of these tests to detect environmental impacts or natural large-scale biogeographic gradients was also compared empirically under simulations based on parameters derived from real ecological data sets.

Overall, ANOSIM and the Mantel test were very sensitive to heterogeneity in dispersions, with ANOSIM generally being more sensitive than the Mantel test. In contrast, PERMANOVA and Pillai's trace were largely unaffected by heterogeneity for balanced designs. PERMANOVA was also unaffected by differences in correlation structure, unlike Pillai's trace. For unbalanced designs, however, all of the tests were (1) too liberal when the smaller group had greater dispersion and (2) overly conservative when the larger group had greater dispersion, especially ANOSIM and the Mantel test. For simulations based on real ecological data sets, PERMANOVA was generally, but not always, more powerful than the others to detect changes in community structure, and the Mantel test was usually more powerful than ANOSIM. Both the error distributions and the resemblance measure affected results concerning power.

Differences in the underlying construction of these test statistics result in important differences in the nature of the null hypothesis they are testing, their sensitivity to heterogeneity, and their power to detect important changes in ecological communities. For balanced designs, PERMANOVA and PERMDISP can be used to rigorously identify location vs. dispersion effects, respectively, in the space of the chosen resemblance measure. ANOSIM and the Mantel test can be used as more “omnibus” tests, being sensitive to differences in location, dispersion or correlation structure among groups. Unfortunately, none of the tests (PERMANOVA, Mantel, or ANOSIM) behaved reliably for unbalanced designs in the face of heterogeneity.

PERMANOVA, ANOSIM, and the Mantel test in the face of heterogeneous dispersions: What null hypothesis are you testing?

Applied multivariate statistics for the social sciences, 4th ed.

One Way Analysis of Variance Power Analysis Factorial Analysis of Variance Repeated Measures Analysis Simple and Multiple Regression Analysis of Covariance. Appendices: Data Sets Statistical Tables Power Tables.

Intermediate Statistics: A Modern Approach

Because the results of a regression analysis can be quite sensitive to outliers (either on y or in the space of the predictors), it is important to be able to detect such points. This article discusses and interrelates the following four diagnostics that are useful in identifying outliers: studentized residuals, the hat elements, Cook's distance, and Mahalanobis distance. Guidelines are given for interpretation of the diagnostics. Outliers will not necessarily be influential in affecting the regression coefficients. This important fact is illustrated and emphasized.

Outliers and influential data points in regression analysis.

http://psycnet.apa.org/journals/bul/88/3/728.pdf

James P. Stevens

Papers

Applied multivariate statistics for the social sciences, 4th ed.

Intermediate Statistics: A Modern Approach

Outliers and influential data points in regression analysis.

Power of the multivariate analysis of variance tests.

Comment on Olson: Choosing a test statistic in multivariate analysis of variance.