Multi-Validity Process and Factor-Invariance. Perceived Self-Efficacy-Scale for the Prevention of Obesity in Preteens

Given the lack of scales with a robust psychometric assessment of self-efficacy related to obesity in early adolescence, we aimed to obtain an instrument with high-quality validity and reliability items. Nonrandom samples (N = 2371) classified boys (1174, M = 12.83, SD = 0.84) and girls (1197, M = 12.68, SD = 0.78) from Mexico City and some cities of the Mexican Republic with obesity rates near to the national level mean. A multi-validity process and structural invariance analysis using the Perceived Self-efficacy Scale for Obesity Prevention were performed. A two-factor—physical activity and healthy eating—model with high effect-sized values—girls R2 (0.88, p < 0.01) and boys R2 (0.87, p < 0.01)—were obtained. Each factor explained more than half of the variance with high-reliability coefficients in each group and acceptable adjustment rates. The self-efficacy scale proved to have only girls, an invariant factor structure, or a psychometric equivalence between the groups. The obtained scale showed that a two-factor structure is feasible and appropriate, according to the highest quality of validity and reliability.

Finding clusters of groups with measurement invariance: Unraveling intercept non-invariance with mixture multigroup factor analysis

Comparisons of latent constructs across groups are ubiquitous in behavioral research and, nowadays, often numerous groups are involved. Measurement invariance of the constructs across the groups is imperative for valid comparisons and can be tested by multigroup factor analysis. For many groups, metric invariance (invariant factor loadings) often holds, whereas scalar invariance (invariant intercepts) is rarely supported. Scalar invariance is a prerequisite for comparing latent means, however. One may inspect group-specific intercepts to pinpoint non-invariances, but this is a daunting task in case of many groups. This paper presents mixture multigroup factor analysis (MMG-FA) for clustering groups based on their intercepts. Clusters of groups with scalar invariance are obtained by imposing cluster-specific intercepts and invariant loadings whereas unique variances, factor means and factor (co)variances can differ between groups. Thus, MMG-FA ties down the number of intercepts to inspect and generates clusters of groups wherein latent means can be validly compared.

Finding clusters of groups with measurement invariance: Unraveling intercept non-invariance with mixture multigroup factor analysis

Comparisons of latent constructs across groups are ubiquitous in behavioral research and, nowadays, often numerous groups are involved. Measurement invariance of the constructs across the groups is imperative for valid comparisons and can be tested by multigroup factor analysis. For many groups, metric invariance (invariant factor loadings) often holds, whereas scalar invariance (invariant intercepts) is rarely supported. Scalar invariance is a prerequisite for comparing latent means, however. One may inspect group-specific intercepts to pinpoint non-invariances, but this is a daunting task in case of many groups. This paper presents mixture multigroup factor analysis (MMG-FA) for clustering groups based on their intercepts. Clusters of groups with scalar invariance are obtained by imposing cluster-specific intercepts and invariant loadings whereas unique variances, factor means and factor (co)variances can differ between groups. Thus, MMG-FA ties down the number of intercepts to inspect and generates clusters of groups wherein latent means can be validly compared.

The smallest invariant factor of the multiplicative group

Let [Formula: see text] denote the least invariant factor in the invariant factor decomposition of the multiplicative group [Formula: see text]. We give an asymptotic formula, with order of magnitude [Formula: see text], for the counting function of those integers [Formula: see text] for which [Formula: see text]. We also give an asymptotic formula, for any even [Formula: see text], for the counting function of those integers [Formula: see text] for which [Formula: see text]. These results require a version of the Selberg–Delange method whose dependence on certain parameters is made explicit, which we provide in Appendix A. As an application, we give an asymptotic formula for the counting function of those integers [Formula: see text] all of whose prime factors lie in an arbitrary fixed set of reduced residue classes, with implicit constants uniform over all moduli and sets of residue classes.

TESTING FOR STRUCTURAL CHANGES IN FACTOR MODELS VIA A NONPARAMETRIC REGRESSION

We propose a model-free test for structural changes in factor models. The basic idea is to regress the data on commonly estimated factors by local smoothing and compare the fitted values of time-varying factor loadings with those of time-invariant factor loadings estimated via principal component analysis. By construction, the test is designed to be powerful against both smooth structural changes and sudden structural breaks with a possibly unknown number of breaks and unknown break dates in the factor loadings. No restrictions on the form of alternatives or trimming of boundary regions near the beginning or end of the sample period is required for the test. The test has power to detect the usual nonparametric rate of local alternatives. Monte Carlo studies demonstrate excellent power of the test in detecting both smooth and sudden structural changes in the factor loadings. In an application using U.S. asset returns, we find significant evidence against time-invariant factor loadings.

Raters Interpret Positively and Negatively Worded Items Similarly in a Quality of Life Instrument for Children

Measurement invariance is an important assumption to meaningfully compare children’s quality of life (QoL) between different raters (eg, children and parents) and across genders. Moreover, QoL instruments may combine using negatively and positively worded items—a common method to reduce response bias. However, the wording effects may have different levels of impact on different raters and genders. Our aim was to investigate the measurement invariance of Kid-KINDL, a commonly used QoL instrument, across genders and raters and to consider the wording effects simultaneously. Third to sixth graders (208 boys and 235 girls) completed the self-rated Kid-KINDL, and 1 parent each of 241 children completed the parent-rated Kid-KINDL. The wording effects were accounted for by correlated traits-uncorrelated methods model. The measurement invariance was examined using multigroup confirmatory factor analysis. Item loadings and item intercepts were invariant across gender and rater when we simultaneously accounted for the wording effects of Kid-KINDL. Our results suggest that Kid-KINDL could be used to compare QoL across gender and that parent-rated Kid-KINDL could be used to measure children’s QoL. Specifically, the invariant factor loadings across child-rated and parent-rated Kid-KINDL suggest that the score weights in each item were the same for both children and parents (ie, the important items identified by the children are the same items identified by the parents). The invariant item intercepts suggest that both children and parents share the same threshold for each item. Based on the results, we tentatively recommend that each score of a parent-rated Kid-KINDL can stand for each child’s QoL.

Average of the first invariant factor of the reductions of abelian varieties of CM type

For a field of definition [Formula: see text] of an abelian variety [Formula: see text] and prime ideal [Formula: see text] of [Formula: see text] which is of a good reduction for [Formula: see text], the structure of [Formula: see text] as abelian group is: [Formula: see text] where [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text]. We are interested in finding an asymptotic formula for the number of prime ideals [Formula: see text] with [Formula: see text], [Formula: see text] has a good reduction at [Formula: see text], [Formula: see text]. We succeed in proving this under the assumption of the Generalized Riemann Hypothesis (GRH). Unconditionally, we achieve a short range asymptotic for abelian varieties of CM type, and the full cyclicity theorem for elliptic curves over a number field containing the CM field.