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.     

Identifying Children at Risk of Problematic Development: Latent Clusters Among Childhood Arrestees

The presence of clusters characterized by distinct profiles of individual, family and peer characteristics among childhood arrestees was investigated and cluster membership stability after 2 years was determined. Identification of such clusters in this heterogeneous at-risk group can extend insight into the presence and severity of children’s co-occurring problems and guide intervention and prevention efforts. Latent class analysis (LCA) was used to detect clusters among 308 childhood arrestees (mean age 10.7 years), based on dichotomous dynamic correlates of offending present at the time of first arrest. Correlates in the individual, peer and family domains were assessed at baseline and 2-year follow-up, using standardized instruments. This resulted in identification of a low problem group characterized by few problems across all domains (40.2 %), an externalizing intermediate problem group characterized by mainly externalizing problems on the individual and peer domains (39.4 %), and a pervasive high problem group characterized by numerous problems across all domains (20.4 %). Cluster membership was most stable for the low problem group (71.4 %), followed by the externalizing intermediate problem group (49.5 %). Transition was highest in the pervasive high problem group (63.0 %), with the majority of children progressing to the externalizing intermediate problem group. The identification of such distinct clusters among childhood arrestees, differing in the presence of co-occurring problems, stresses the importance of a first police arrest as an opportunity for early recognition of children in need of care. As problems present at the time of first arrest do not persist in every child, careful periodic monitoring is needed.