Sample Size Planning for the Squared Multiple Correlation Coefficient: Accuracy in Parameter Estimation via Narrow Confidence Intervals

Methods of sample size planning are developed from the accuracy in parameter approach in the multiple regression context in order to obtain a sufficiently narrow confidence interval for the population squared multiple correlation coefficient when regressors are random. Approximate and exact methods are developed that provide necessary sample size so that the expected width of the confidence interval will be sufficiently narrow. Modifications of these methods are then developed so that necessary sample size will lead to sufficiently narrow confidence intervals with no less than some desired degree of assurance. Computer routines have been developed and are included within the MBESS R package so that the methods discussed in the article can be implemented. The methods and computer routines are demonstrated using an empirical example linking innovation in the health services industry with previous innovation, personality factors, and group climate characteristics.     

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.     

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.     

The second order approximation to sample influence curve in canonical correlation analysis

A second order approximation to the sample influence curve (SIC) in canonical correlation analysis has been derived in the literature. However, it does not seem satisfactory for some cases. In this paper, we present a more accurate second order approximation. As a particular case, the proposed method is exact for the SIC of the squared multiple correlation coefficient. An example is given. The authors are most grateful to the associate editor and three reviewers for valuable comments and suggestions which improved the presentation of the paper considerably. The first author was partly supported by a RGC earmarked research grant of Hong Kong.     

COST-BENEFIT CONSIDERATIONS IN CHOOSING AMONG CROSS-VALIDATION METHODS

There are two general methods of cross-validation: (a) empirical estimation, and (b) formula estimation. In choosing a specific cross-validation procedure, one should consider both costs (eg. inefficient use of available data in estimating regression parameters) and benefits (eg. accuracy in estimating population cross-validity). Empirical cross-validation methods involve significant costs, since they are typically laborious and wasteful of data, but under conditions represented in Monte Carlo studies, they are generally not more accurate than formula estimates. Consideration of costs and benefits suggests that empirical estimation methods are typically not worth the cost, except in a limited number of cases in which Monte Carlo sampling assumptions are not met in the derivation sample. Designs which use multiple samples to estimate the cross-validity of a single regression equation are clearly preferable to single-sample designs; the latter are never expected to be more accurate than formula estimates and thus are never worth the cost. Multi-equation designs are more accurate than single equation designs, but they appear to estimate the wrong parameter, and thus are difficult to interpret.     

Properties of the maximum likelihood solution in factor analysis regression

Algebraic properties of the normal theory maximum likelihood solution in factor analysis regression are investigated. Two commonly employed measures of the within sample predictive accuracy of the factor analysis regression function are considered: the variance of the regression residuals and the squared correlation coefficient between the criterion variable and the regression function. It is shown that this within sample residual variance and within sample squared correlation may be obtained directly from the factor loading and unique variance estimates, without use of the original observations or the sample covariance matrix.     

The Interactive Influence of Prosocial Places,Youth Population,and Poverty on Incidents of Violent Assault

The article builds on previous work by Roman and colleagues in order to specify the potential causes for increased incidents of violent assault in areas featuring a high number of prosocial institutions (i.e., schools, recreation centers). Utilizing aspects of routine activity theory and crime pattern theory, the argument is presented that both poverty and youth population will have a conditioning impact on the relationship between the presence of prosocial places and incidence of violent assault. Least squares regression was utilized in order to show that such institutions can serve as generators for offending and/or victimization, and that their location in lower income areas may magnify the problem.     

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.     

Approximate uniqueness estimates for singular correlation matrices

The residual variance (one minus the squared multiple correlation) is often used as an approximation to the uniqueness in factor analysis. An upper bound approximation to the residual variance is given for the case when the correlation matrix is singular. The approximation is computationally simpler than the exact solution, especially since it can be applied routinely without prior knowledge as to the singularity or nonsingularity of the correlation matrix.     

Some developments in multivariate generalizability

This article is concerned with estimation of components of maximum generalizability in multifacet experimental designs involving multiple dependent measures. Within a Type II multivariate analysis of variance framework, components of maximum generalizability are defined as those composites of the dependent measures that maximize universe score variance for persons relative to observed score variance. The coefficient of maximum generalizability, expressed as a function of variance component matrices, is shown to equal the squared canonical correlation between true and observed scores. Emphasis is placed on estimation of variance component matrices, on the distinction between generalizability- and decision-studies, and on extension to multifacet designs involving crossed and nested facets. An example of a two-facet partially nested design is provided.Appreciation is expressed to the Office of Research in Medical Education, University of Texas Medical Branch, for permitting use of their data.     

Influence in canonical correlation analysis

The perturbation theory of the generalized eigenproblem is used to derive influence functions of each squared canonical correlation coefficient and the corresponding canonical vector pair. Three sample versions of these functions are described and some properties are noted. As particular applications, the influence function of the squared multiple correlation coefficient and influence functions of eigenvalues and eigenvectors in correspondence analysis are obtained. Three numerical examples are briefly discussed.We thank the Editor and the anonymous reviewers for their helpful comments. This research was carried out with the financial support of the Italian Ministry of the University and the National Research Council.     

Sample size requirements for the design of reliability studies: precision consideration

In multilevel modeling, the intraclass correlation coefficient based on the one-way random-effects model is routinely employed to measure the reliability or degree of resemblance among group members. To facilitate the advocated practice of reporting confidence intervals in future reliability studies, this article presents exact sample size procedures for precise interval estimation of the intraclass correlation coefficient under various allocation and cost structures. Although the suggested approaches do not admit explicit sample size formulas and require special algorithms for carrying out iterative computations, they are more accurate than the closed-form formulas constructed from large-sample approximations with respect to the expected width and assurance probability criteria. This investigation notes the deficiency of existing methods and expands the sample size methodology for the design of reliability studies that have not previously been discussed in the literature.     

Fungible Weights in Multiple Regression

Every set of alternate weights (i.e., nonleast squares weights) in a multiple regression analysis with three or more predictors is associated with an infinite class of weights. All members of a given class can be deemed fungiblebecause they yield identical SSE (sum of squared errors) and R ² values. Equations for generating fungible weights are reviewed and an example is given that illustrates how fungible weights can be profitably used to evaluate parameter sensitivity in multiple regression. The author wishes to thank Drs. Robyn Dawes, William Grove, Markus Keel, Leslie Yonce, Joe Rausch, the editor, and three anonymous reviewers for helpful comments on earlier versions of this article.     

Estimation of the simple correlation coefficient

This article investigates some unfamiliar properties of the Pearson product—moment correlation coefficient for the estimation of simple correlation coefficient. Although Pearson’s r is biased, except for limited situations, and the minimum variance unbiased estimator has been proposed in the literature, researchers routinely employ the sample correlation coefficient in their practical applications, because of its simplicity and popularity. In order to support such practice, this study examines the mean squared errors of r and several prominent formulas. The results reveal specific situations in which the sample correlation coefficient performs better than the unbiased and nearly unbiased estimators, facilitating recommendation of r as an effect size index for the strength of linear association between two variables. In addition, related issues of estimating the squared simple correlation coefficient are also considered.     

Sample Size Calculation for Estimating or Testing a Nonzero Squared Multiple Correlation Coefficient

The problems of hypothesis testing and interval estimation of the squared multiple correlation coefficient of a multivariate normal distribution are considered. It is shown that available one-sided tests are uniformly most powerful, and the one-sided confidence intervals are uniformly most accurate. An exact method of calculating sample size to carry out one-sided tests (null hypothesis may involve a nonzero value for the multiple correlation coefficient) to attain a specified power is given. Sample size calculation for computing confidence intervals for the squared multiple correlation coefficient with a specified expected width is also provided. Sample sizes for powers and confidence intervals are tabulated for a wide range of parameter configurations and dimensions. The results are illustrated using the empirical data from Timm (1975) Timm, N. H. 1975. Multivariate analysis with applications in education and psychology, Belmont, CA: Wadsworth.  [Google Scholar] that related scores from the Peabody Picture Vocabulary Test to four proficiency measures.     

α系数的区间估计方法比较

多数情况下,α系数可以用来评价测验信度。诸多研究建议,在报告测验信度的时候应当包括其置信区间。通过蒙特卡洛模拟研究,比较了7种α系数区间估计方法,包括Fisher法、Bonett-02法、Bonett-10法、精确Koning-Franses法、渐近ID法、渐近Koning-Franses法和ADF法。结果发现Bonett-10法和精确Koning-Franses法较好,它们的结果相差很小。这两种方法都比较简单,只需要样本的α值、测验题数、被试人数及F临界值,通过简单的运算便可得到α系数的置信区间。     

A measure of interrelationship for overlapping groups

A coefficient of interrelationship between overlapping groups that avoids indeterminacies inherent in the construction of fourfold tables for such purposes and, at the same time, is relatively insensitive to the absolute magnitude of marginal totals of fourfold tables, is developed. Under assumptions that are consistent with the objectives of organizational analysis, this coefficient is shown to be equivalent to a product-moment correlation coefficient. The advantages and limitations of this coefficient are pointed out. A numerical example giving computational procedures is presented.This measure was developed in connection with a study made by Dorothy C. Adkins (1). Her influence in the development was felt in many ways. The article was prepared while the author was employed at The University of North Carolina.     

A Note on the Reliability Coefficients for Item Response Model-Based Ability Estimates

Assuming item parameters on a test are known constants, the reliability coefficient for item response theory (IRT) ability estimates is defined for a population of examinees in two different ways: as (a) the product-moment correlation between ability estimates on two parallel forms of a test and (b) the squared correlation between the true abilities and estimates. Due to the bias of IRT ability estimates, the parallel-forms reliability coefficient is not generally equal to the squared-correlation reliability coefficient. It is shown algebraically that the parallel-forms reliability coefficient is expected to be greater than the squared-correlation reliability coefficient, but the difference would be negligible in a practical sense.     

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.