On normal theory based inference for multilevel models with distributional violations

Data in social and behavioral sciences are often hierarchically organized though seldom normal, yet normal theory based inference procedures are routinely used for analyzing multilevel models. Based on this observation, simple adjustments to normal theory based results are proposed to minimize the consequences of violating normality assumptions. For characterizing the distribution of parameter estimates, sandwich-type covariance matrices are derived. Standard errors based on these covariance matrices remain consistent under distributional violations. Implications of various covariance estimators are also discussed. For evaluating the quality of a multilevel model, a rescaled statistic is given for both the hierarchical linear model and the hierarchical structural equation model. The rescaled statistic, improving the likelihood ratio statistic by estimating one extra parameter, approaches the same mean as its reference distribution. A simulation study with a 2-level factor model implies that the rescaled statistic is preferable.This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of North Texas faculty research grant. We would like to thank the Associate Editor and two reviewers for suggestions that helped to improve the paper.     

The Process Through Which an Advocacy Intervention Resulted in Positive Change for Battered Women Over Time

A prior experimental evaluation of a community-based advocacy program for women with abusive partners demonstrated positive change in the lives of women even 2 years postintervention (C. M. Sullivan & D. I. Bybee, 1999). The current study explored the complex mediational process through which this change occurred, using longitudinal structural equation modeling and formal tests of mediation. As hypothesized, the advocacy intervention first resulted in women successfully obtaining desired community resources and increasing their social support, which enhanced their overall quality of life. This improvement in well-being appeared to serve as a protective factor from subsequent abuse, as women who received the intervention were significantly less likely to be abused at 2-year follow-up compared with women in the control condition. Increased quality of life completely mediated the impact of the advocacy intervention on later reabuse. Discussion places advocacy for women in the context of other efforts that are needed to build an effective community response to preventing intimate violence against women.     

Beyond bruises and broken bones: the joint effects of stress and injuries on battered women's health

We investigated the joint mediating effects of injuries and stress on the relationship between abuse and women's health. A community sample of 397 women, half of whom had been assaulted by an intimate partner within the prior 6 months, was interviewed about their experience of intimate partner violence, injuries, stress, income, depression, and physical health problems. Structural equation modeling techniques confirmed the complex model of hypothesized indirect effects of abuse on women's physical health problems through injuries, stress, and depression. Stress accounted for 80% of the indirect effect of abuse on women's physical health. Its direct effect on physical health was somewhat larger than its indirect effect through depression, but both processes played a key role in determining the effect of abuse on women's physical health problems. Furthermore, abuse was a stronger predictor of women's stress than was poverty. Implication and future research are discussed.     

Sea lions and equivalence: expanding classes by exclusion

Experiments have shown that human and nonhuman subjects are capable of performing new arbitrary stimulus-stimulus relations without error. When subjects that are experienced with matching-to-sample procedures are presented with a novel sample, a novel comparison, and a familiar comparison, most respond by correctly selecting the novel comparison in the presence of the new sample. This exclusion paradigm was expanded with two California sea lions that had previously formed two 10-member equivalence classes in a matching-to-sample procedure. Rather than being presented with a novel sample on a given trial, the sea lions were presented with a randomly selected familiar member of one class as the sample. One of the comparisons was a randomly selected familiar member of the alternative class, and the other was a novel stimulus. When required to choose which comparison matched the sample, the subjects reliably rejected the familiar comparison, and instead selected the unfamiliar one. Next, the sea lions were presented with transfer problems that could not be solved by exclusion; they immediately grouped the new stimuli into the appropriate classes. These findings show that exclusion procedures can rapidly generate new stimulus relations that can be used to expand stimulus classes.     

Testing for symmetry in the conditional discriminations of language-trained chimpanzees

If subjects are taught to match Stimulus A to B and then, without further training, match B to A, they have passed a test of symmetry. It has been suggested that non-humans' lack of success on symmetry tests might be overcome by giving them a history of symmetry exemplar training, that is, by directly teaching a large number of conditional relations (e.g., AB, CD, EF,...) and also directly training the "reverse" of these relations (e.g., BA, DC, FE,...). The chimpanzee subjects of the present study, Sherman, Austin, and Lana, had already received extensive symmetry exemplar training as a result of attempts to teach a selection-based language system of lexigrams. The present study systematically subjected 2 of these chimps (Sherman and Lana), for the first time, to standard symmetry tests in controlled conditions. Both chimps failed the tests, even when their correct responses on test trials were reinforced. The findings do not support the exemplar training hypothesis, and cast doubt upon whether the chimps can pass tests of stimulus equivalence.     

Cognitive processing in literate and illiterate subjects: a review of some recent behavioral and functional neuroimaging data

The study of illiterate subjects, which for specific socio-cultural reasons did not have the opportunity to acquire basic reading and writing skills, represents one approach to study the interaction between neurobiological and cultural factors in cognitive development and the functional organization of the human brain. In addition the naturally occurring illiteracy may serve as a model for studying the influence of alphabetic orthography on auditory-verbal language. In this paper we have reviewed some recent behavioral and functional neuroimaging data indicating that learning an alphabetic written language modulates the auditory-verbal language system in a non-trivial way and provided support for the hypothesis that the functional architecture of the brain is modulated by literacy. We have also indicated that the effects of literacy and formal schooling is not limited to language related skills but appears to affect also other cognitive domains. In particular, we indicate that formal schooling influences 2D but not 3D visual naming skills. We have also pointed to the importance of using ecologically relevant tasks when comparing literate and illiterate subjects. We also demonstrate the applicability of a network approach in elucidating differences in the functional organization of the brain between groups. The strength of such an approach is the ability to study patterns of interactions between functionally specialized brain regions and the possibility to compare such patterns of brain interactions between groups or functional states. This complements the more commonly used activation approach to functional neuroimaging data, which characterize functionally specialized regions, and provides important data characterizing the functional interactions between these regions.     

Bayesian estimation and testing of structural equation models

The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those based on the maximum likelihood solution, for example, output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters underidentified models, as we illustrate on a simple errors-in-variables model.We thank David Spiegelhalter for suggesting applying the Gibbs sampler to structural equation models to the first author at a 1994 workshop in Wiesbaden. We thank Ulf Böckenholt, Chris Meek, Marijtje van Duijn, Clark Glymour, Ivo Molenaar, Steve Klepper, Thomas Richardson, Teddy Seidenfeld, and Tom Snijders for helpful discussions, mathematical advice, and critiques of earlier drafts of this paper.     

Latent variables should remain as such: Evidence from a Monte Carlo study

Use of subject scores as manifest variables to assess the relationship between latent variables produces attenuated estimates. This has been demonstrated for raw scores from classical test theory (CTT) and factor scores derived from factor analysis. Conclusions on scores have not been sufficiently extended to item response theory (IRT) theta estimates, which are still recommended for estimation of relationships between latent variables. This is because IRT estimates appear to have preferable properties compared to CTT, while structural equation modeling (SEM) is often advised as an alternative to scores for estimation of the relationship between latent variables. The present research evaluates the consequences of using subject scores as manifest variables in regression models to test the relationship between latent variables. Raw scores and three methods for obtaining theta estimates were used and compared to latent variable SEM modeling. A Monte Carlo study was designed by manipulating sample size, number of items, type of test, and magnitude of the correlation between latent variables. Results show that, despite the advantage of IRT models in other areas, estimates of the relationship between latent variables are always more accurate when SEM models are used. Recommendations are offered for applied researchers.     

Symmetry and transitivity of conditional relations in monkeys (Cebus apella) and pigeons (Columba livia)

In Experiment 1 six monkeys were tested with discriminative relations that were backward relative to their training in a 0-second conditional (“symbolic”) matching procedure. Although there was some indication of backward associations, the evidence was generally weak, and statistical evaluations did not reach conventional significance levels. Unlike children, who show backward associations to the point of symmetry, monkeys and pigeons display at best only weak and transient backward associations. In Experiment 2 associative transitivity was assessed across two sets of conditional matching tasks. All four monkeys tested demonstrated strong transitivity. In contrast, in Experiment 3 there was no evidence of transitivity in three pigeons tested under conditions closely comparable to those of Experiment 2. These results may identify some key features of interspecies differences and contribute to analyses of serial learning in animals.