Intimidation And Violence By Males In High School Athletics

Adolescence, Fall, 1999 by Edgar W. Shields, Jr.

A preliminary concern was whether variables that were possibly antecedent to VI, PI, and PV formed coherent, relatively independent subsets. No theoretical or empirical rationale has been established for specifying a particular structure for this type of data. Therefore, an exploratory PCA was undertaken to identify the structure having the greatest interpretability.

PCA is the method of choice if the goal is an empirical summary of the data with the specific objective of reducing the number of variables to a smaller number of components. In PCA, all the observed variance is analyzed (contrary to factor analysis, in which only shared variance is analyzed), and is distributed to components, including error and unique variance for each observed variable (duplicating exactly the observed correlation matrix and the standard scores of the observed variables--if all components are retained). PCA extracts maximum variance from a data set with a few orthogonal components and produces a unique mathematical solution, whereas most forms of factor analysis are not unique. Each component is orthogonal to all other components, a characteristic that greatly facilitates interpretation of results and their use in other analyses; for example, as predictor variables in regression models (Tabachnick & Fidell, 1989). For this study, identification of subsets (or components) of the antecedent items was considered to be of greater importance than was parsimony.

Use of PCA for the data on antecedents was supported. Bartlett's test of sphericity was significant ([X.sup.2] = 195.277, p [less than].0001), indicating that the coefficients in the correlation matrix were different from zero and did not likely occur as a function of chance. Further, the total matrix sampling adequacy of all antecedent items, an index of the extent to which the matrix of partial and multiple correlation coefficients conforms to zero partials and large multiple rs, was found to be satisfactory (.633). A common standard is for this value to be greater than .500; however, Tabachnick and Fidell (1989) specify a higher standard of .600. In addition, sampling adequacy for individual antecedent items exceeded .500 for two variables, with all remaining variables exceeding .600.

Initially, components extracted and retained for further analysis were selected based on the larger of the numbers determined by a 75% variance rule and root curve analysis. The 75% variance criterion dictates that when the sum of the proportionate contributions of the eigenvalues exceeds 75%, components are no longer retained in the final solution. The roots-greater-than-one criterion retains as many components as there are eigenvalues greater than or equal to one. This procedure extracted four components from ten antecedent variables. The next step in the analysis was to compare the utility of one-, two-, three-, and four-component models in accounting for the interrelations of the ten antecedent items.

Models consisting of one, two, and three components were developed. Both orthogonal and oblique rotations were performed with each model. Variables jeopardizing the pursuit of a relatively simple structure by substantial loadings on more than one factor were closely examined; subsequently, two (antecedent items 5 and 8) were dropped from the PCA. A four-component solution (eight variables) was selected after considering several psychometric criteria, including correlation matrices, partial correlation matrices, test for sphericity, sampling adequacy, differences in adjacent eigenvalues, differences in the amount of variance accounted for by adjacent components, the internal consistency of the components, component loadings, the scree plot of eigenvalues, both orthogonal and oblique component plots, and perhaps most important, the identification and interpretability of logical subgroupings. Internal consistency, as estimated by Cronbach's alpha for the eight variables retained in the four-component solution, was .65, which was above the lower limits of acceptability (generally considered to be around .50 to .60).