A square root method of selecting a minimum set of variables in multiple regression: I. The method

An extension of Dwyer's “square root” method has been made to the problem of selecting a minimum set of variables in a multiple regression problem. The square root method of selection differs from the Wherry-Doolittle method primarily in that (1) the computations required are more compact, (2) anF ratio criterion is used which leads to the selection of fewer variables. The method provides solutions for the problems of test selection, item analysis, analysis of variance with disproportionate frequencies, and other problems requiring the rejection of superfluous variables. In a subsequent article a worked example will be given, and the square root and Wherry-Doolittle methods compared.     

A criterion for selecting variables in a regression analysis

Methods are given for deciding whether to use some or no predictor variables in a regression analysis. Previously obtained results on the more general problem, whether to usek ork — r predictor variables are reviewed with emphasis on applications.     

Some uses of the square root method as a tool in multiple prediction

It is shown how the square root method can be used in the derivation of formulas in multiple prediction and in the computations necessary for their application. A formula derived by Horst indicates the validity a test must possess in order to contribute a specified amount to the multiple correlation of a battery. Then a formula is derived which relates the length of one test in a battery to the multiple correlation of the battery as a whole. The formulas hold for population parameters; in this paper, sampling theory is not discussed. A numerical example is presented; it is shown that one square root analysis provides all the data needed for the formulas.     

The square root method and multiple group methods of factor analysis

The square root method for the solution of a set of simultaneous linear equations or the reduction of a matrix has been known for some time under a variety of names. Because of its usefulness in statistical work, especially in factor analysis, the square root method is presented in general terms and an example given. Several independently developed multiple group methods for factor analysis are compared and synthesized. Their fundamental concepts are set forth and an appropriate system of notation developed. Detailed computational procedures are outlined, and the square root method is emphasized as a computing aid in multiple group analysis.The opinions expressed in this paper are those of the author and do not necessarily reflect official Department of the Army policy. The author is now with The RAND Corporation, Santa Monica, Calif.     

Multivariate multiple regression analyses: a permutation method for linear models

A multivariate extension of a univariate procedure for the analysis of experimental designs is presented. A Euclidean-distance permutation procedure is used to evaluate multivariate residuals obtained from a regression algorithm, also based on Euclidean distances. Applications include various completely randomized and randomized block experimental designs such as one-way, Latin square, factorial, nested, and split-plot designs, with and without covariates. Unlike parametric procedures, the only required assumption is the randomization of subjects to treatments.     

Effect size and power in assessing moderating effects of categorical variables using multiple regression: a 30-year review

The authors conducted a 30-year review (1969-1998) of the size of moderating effects of categorical variables as assessed using multiple regression. The median observed effect size (f(2)) is only .002, but 72% of the moderator tests reviewed had power of .80 or greater to detect a targeted effect conventionally defined as small. Results suggest the need to minimize the influence of artifacts that produce a downward bias in the observed effect size and put into question the use of conventional definitions of moderating effect sizes. As long as an effect has a meaningful impact, the authors advise researchers to conduct a power analysis and plan future research designs on the basis of smaller and more realistic targeted effect sizes.     

Moderated multiple regression for interactions involving categorical variables: a statistical control for heterogeneous variance across two groups

Moderated multiple regression (MMR) arguably is the most popular statistical technique for investigating regression slope differences (interactions) across groups (e.g., aptitude-treatment interactions in training and differential test score-job performance prediction in selection testing). However, heterogeneous error variances can greatly bias the typical MMR analysis, and the conditions that cause heterogeneity are not uncommon. Statistical corrections that have been developed require special calculations and are not conducive to follow-up analyses that describe an interaction effect in depth. A weighted least squares (WLS) approach is recommended for 2-group studies. For 2-group studies, WLS is statistically accurate, is readily executed through popular software packages (e.g., SAS Institute, 1999; SPSS, 1999), and allows follow-up tests.     

The statistical regression phenomenon: II. Application of a metamodel

The relevance of the traditional regression concept to measurement of change and to causal research is investigated by applying a metamodel for the concept as a frame of reference in a logical analysis of these two research fields. It is concluded that the concept is considerably less important than ordinarily maintained in the methodological research literature. As for measurement of change, only a special error case of the concept often constitutes a biasing factor, while all special cases are of little interest for describing true changes. Concerning causal research, the regression phenomenon will probably seldom constitute any threat to internal validity. The unjustified attention given to the concept in psychological research is probably mainly due to the fact that central assumptions have been overlooked and that a thorough logical analysis of the relevance issue has not been attempted previously.     

Current misuses of multiple regression for investigating bivariate hypotheses: an example from the organizational domain

By definition, multiple regression (MR) considers more than one predictor variable, and each variable’s beta will depend on both its correlation with the criterion and its correlation with the other predictor(s). Despite ad nauseam coverage of this characteristic in organizational psychology and statistical texts, researchers’ applications of MR in bivariate hypothesis testing has been the subject of recent and renewed interest. Accordingly, we conducted a targeted survey of the literature by coding articles, covering a five-year span from two top-tier organizational journals, that employed MR for testing bivariate relations. The results suggest that MR coefficients, rather than correlation coefficients, were most common for testing hypotheses of bivariate relations, yet supporting theoretical rationales were rarely offered. Regarding the potential impact on scientific advancement, in almost half of the articles reviewed (44 %), at least one conclusion of each study (i.e., that the hypothesis was or was not supported) would have been different, depending on the author’s use of correlation or beta to test the bivariate hypothesis. It follows that inappropriate decisions to interpret the correlation versus the beta will affect the accumulation of consistent and replicable scientific evidence. We conclude with recommendations for improving bivariate hypothesis testing.     

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.     

A Monte Carlo study of the effects of correlated method variance in moderated multiple regression analysis

It has been suggested that hierarchical regression analysis provides an unambiguous conclusion with regard to the existence of moderator effects (Arnold & Evans, 1979). This paper examines the impact of correlated error among the dependent and independent variables in order to explore whether or not artificial interaction terms can be generated. A Monte Carlo study was performed to investigate the effects of correlated error on noninteraction and interaction models. The results are clear-cut. Artifactual interaction cannot be created; true interactions can be attentuated. Some practical suggestions are provided for drawing inferences from hierarchical regression analysis.     

A woman under the influence: An example of multiple victimization within a family

Abstract

The interactional processes depicted in rich detail in A WOMAN UNDER THE INFLUENCE are examined from a family therapist's viewpoint. Rather than portraying the persecution of a single individual by other members of the family, the film's drama demonstrates that the complex interactions within the extended family victimizes all the participants. When events are examined closely, it becomes clear that even the overt victimizers in the family system suffer as a result of the very processes in which they engage. The discussion of the film underscores the need for the therapist to empathize with all members of a dysfunctional system, especially when one member appears to be the victim.     

A comparison of three predictor selection techniques in multiple regression

Three methods for selecting a few predictors from the many available are described and compared with respect to shrinkage in cross-validation. From 2 to 6 predictors were selected from the 15 available in 100 samples ranging in size from 25 to 200. An iterative method was found to select predictors with slightly, but consistently, higher cross-validities than the popularly used stepwise method. A gradient method was found to equal the performance of the stepwise method only in the larger samples and for the largest predictor subsets.     

A method of determining unbiased distribution in the latin square

The paper describes a method of constructing Latin-square designs in which treatment sequences are unbiased with regard to serial order as well as position of treatment. The procedure is useful for those Latin squares in which the number of cells in each column (or row) is an even number, which, when written in ascending and descending series, contain corresponding ordered numbers prime to each other. Such numbers are 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, etc.     

Sample size determination for confidence intervals of interaction effects in moderated multiple regression with continuous predictor and moderator variables

Moderated multiple regression (MMR) has been widely employed to analyze the interaction or moderating effects in behavior and related disciplines of social science. Much of the methodological literature in the context of MMR concerns statistical power and sample size calculations of hypothesis tests for detecting moderator variables. Notably, interval estimation is a distinct and more informative alternative to significance testing for inference purposes. To facilitate the practice of reporting confidence intervals in MMR analyses, the present article presents two approaches to sample size determinations for precise interval estimation of interaction effects between continuous moderator and predictor variables. One approach provides the necessary sample size so that the designated interval for the least squares estimator of moderating effects attains the specified coverage probability. The other gives the sample size required to ensure, with a given tolerance probability, that a confidence interval of moderating effects with a desired confidence coefficient will be within a specified range. Numerical examples and simulation results are presented to illustrate the usefulness and advantages of the proposed methods that account for the embedded randomness and distributional characteristic of the moderator and predictor variables.     

Coordinating the effects of multiple variables: A skill fundamental to scientific thinking

The skill of predicting outcomes based on simultaneous effects of multiple factors was examined. Over five sessions, 91 sixth graders engaged this task either individually or in pairs and either preceded or followed by six sessions on the more widely studied inquiry task that requires designing and interpreting experiments to identify individual effects. Final assessment, while indicating a high level of mastery on the inquiry task, showed progress but continuing conceptual challenges on the multivariable prediction task having to do with understanding of variables, variable levels, and consistency of a variable’s operation across occasions. Task order had a significant but limited effect, and social collaboration conferred only a temporary benefit that disappeared in a final individual assessment. In a follow-up study, the lack of effect of social collaboration was confirmed, as was that of feedback on incorrect answers. Although fundamental to science, the concept that variables operate jointly and, under equivalent conditions, consistently across occasions is one that children appear to acquire only gradually and, therefore, one that cannot be assumed to be in place.     

Psychotherapists' attitudes towards and observations of nonverbal communication in a greeting situation: II. Relationships to background variables.

An interview study of 79 Swedish psychotherapists was performed to investigate their attitudes towards and observation of nonverbal communication in a reference situation in relation to background factors such as gender, length of professional experience, and theoretical approach. The greeting situation, i.e., the first time a therapist and patient meet in a waiting-room, was chosen as the reference situation. Face communication in that greeting situation was significantly less achieved by male psychotherapists with relatively short experience than by female therapists. However, this sex difference was not significant among therapists whose psychotherapeutic experience was longer than 13 years. Attaching great importance to nonverbal communication in psychotherapy was related to an intention to be well-kept and to perform correctly in front of the patient, to establish direct communication in the greeting situation, and to background factors such as frequent work in psychotherapy and an eclectic psychotherapeutic approach.     

A comparison of the psychological profiles of teenage mothers and their nonmother peers: II. Responses to a set of TAT cards.

In Part I of this investigation (Oz, Tari, & Fine, 1992), it was found that, in comparison with nonmother peers, teenage mothers demonstrated a higher degree of ego development. This was in contradiction to most of the literature on the subject. In the second part of the study, the Thematic Apperception Test was employed to examine defenses and motivations which may impact upon the young girls' functioning within family and heterosexual relationships. Results confirmed reports in the literature which have indicated that adolescent mothers attempt to resolve autonomy-dependence conflicts in heterosexual relationships, whereas nonmothers do so within the family of origin. Consistent with their enhanced ego maturity, teenage mothers were more likely to confront their conflicts than were nonmothers. However, whereas nonmothers reacted to environmental stressors by rejecting the source of the stress, teenage mothers tended to react with self-abasement. It is recommended that further research be directed toward uncovering the emotional and cognitive resources of teenage mothers rather than focusing on their more obvious weaknesses.