Stability of nonlinear principal components analysis: an empirical study using the balanced bootstrap

Principal components analysis (PCA) is used to explore the structure of data sets containing linearly related numeric variables. Alternatively, nonlinear PCA can handle possibly nonlinearly related numeric as well as nonnumeric variables. For linear PCA, the stability of its solution can be established under the assumption of multivariate normality. For nonlinear PCA, however, standard options for establishing stability are not provided. The authors use the nonparametric bootstrap procedure to assess the stability of nonlinear PCA results, applied to empirical data. They use confidence intervals for the variable transformations and confidence ellipses for the eigenvalues, the component loadings, and the person scores. They discuss the balanced version of the bootstrap, bias estimation, and Procrustes rotation. To provide a benchmark, the same bootstrap procedure is applied to linear PCA on the same data. On the basis of the results, the authors advise using at least 1,000 bootstrap samples, using Procrustes rotation on the bootstrap results, examining the bootstrap distributions along with the confidence regions, and merging categories with small marginal frequencies to reduce the variance of the bootstrap results.     

Stand Tall to Protect Students: School Counselors Strengthening School Connectedness

In an urban sample of 5,595 students in Grades 7–12, the authors found that (a) personalized counseling services that were responsive to student needs enhanced school connectedness, (b) risk factors (e.g., poverty, mobility rates, limited English proficiency) were associated with lower school connectedness scores, and (c) responsive counseling services acted as a protective factor that reduced some of the negative effects of risk factors on school connectedness, and consequently, student success in academic and nonacademic areas.