Michael Escobar

Bayesian Density Estimation and Inference Using Mixtures

Abstract We describe and illustrate Bayesian inference in models for density estimation using mixtures of Dirichlet processes. These models provide natural settings for density estimation and are exemplified by special cases where data are modeled as a sample from mixtures of normal distributions. Efficient simulation methods are used to approximate various prior, posterior, and predictive distributions. This allows for direct inference on a variety of practical issues, including problems of local versus global smoothing, uncertainty about density estimates, assessment of modality, and the inference on the numbers of components. Also, convergence results are established for a general class of normal mixture models.

Evidence that dyslexia may represent the lower tail of a normal distribution of reading ability.

BACKGROUND Dyslexia is now widely believed to be a biologically based disorder that is distinct from other, less specific reading problems. According to this view, reading ability is considered to follow a bimodal distribution, with dyslexia as the lower mode. We hypothesized that, instead, reading ability follows a normal distribution, with dyslexia at the lower end of the continuum. METHODS AND RESULTS We used data from the Connecticut Longitudinal Study, a sample survey of 414 Connecticut children who entered kindergarten in 1983 and were followed as a longitudinal cohort. Dyslexia was defined in terms of a discrepancy score, which represents the difference between actual reading achievement and achievement predicted on the basis of measures of intelligence. Data were available from intelligence tests administered in grades 1, 3, and 5 and achievement tests administered yearly in grades 1 through 6. For each child there were 108 possible discrepancy scores ([3 x 3 years] x [2 x 6 years]) based on combinations of the ability scores (full-scale, verbal, and performance IQ) in each of three years and two achievement scores (reading and mathematics) in each of six years. We demonstrated that each of the discrepancy scores followed a univariate normal distribution and that the interrelation of two different discrepancy scores followed a bivariate normal distribution. At most, only 9 of 108 discrepancy scores (8.3 percent) and 171 of 3402 pairs of discrepancy scores (5.0 percent) were significantly different (at the 5 percent level) from the expected scores--well within the expected values for data with univariate and bivariate normal distributions, respectively. We also examined the stability of dyslexia over time. The normal-distribution model predicted (and the data indicated) that only 7 of the 25 children (28 percent) classified as having dyslexia in grade 1 would also be classified as having dyslexia in grade 3. CONCLUSIONS Reading difficulties, including dyslexia, occur as part of a continuum that also includes normal reading ability. Dyslexia is not an all-or-none phenomenon, but like hypertension, occurs in degrees. The variability inherent in the diagnosis of dyslexia can be both quantified and predicted with use of the normal-distribution model.

Estimating Normal Means with a Dirichlet Process Prior

Abstract In this article, the Dirichlet process prior is used to provide a nonparametric Bayesian estimate of a vector of normal means. In the past there have been computational difficulties with this model. This article solves the computational difficulties by developing a “Gibbs sampler” algorithm. The estimator developed in this article is then compared to parametric empirical Bayes estimators (PEB) and nonparametric empirical Bayes estimators (NPEB) in a Monte Carlo study. The Monte Carlo study demonstrates that in some conditions the PEB is better than the NPEB and in other conditions the NPEB is better than the PEB. The Monte Carlo study also shows that the estimator developed in this article produces estimates that are about as good as the PEB when the PEB is better and produces estimates that are as good as the NPEB estimator when that method is better.

Long term outcomes after moderate to severe traumatic brain injury

Objective: This research examined the long-term outcomes of rehabilitation patients with moderate to severe traumatic brain injury (TBI). Design: Retrospective cohort study. Setting and subjects: We examined consecutive records of persons with moderate to severe traumatic brain injury who were discharged from a large rehabilitation hospital in Pennsylvania from 1973 to 1989. We interviewed consenting participants (n = 306) up to 24 years post-injury. Main outcome measures: Self-rated health, activity limitations, employment, living arrangements, marital status, Community Integration Questionnaire, and use of rehabilitation services. Results: Participants were most limited in activities such as managing money and shopping. Twenty-nine per cent of our participants were working full time. There were significant relationships between activity limitations and residual cognitive impairment at follow-up. Self-rated health was correlated with most instrumental activities of daily living. Conclusion: Our findings document health and function in a large post acute TBI population and implications for rehabilitation are discussed.

The use of the Tobit model for analyzing measures of health status.

Self-reported health status is often measured using psychometric or utility indices that provide a score intended to summarize an individuals health. Measurements of health status can be subject to a ceiling effect. Frequently, researchers want to examine relationships between determinants of health and measures of health status. Regression methods that ignore the presence of a ceiling effect, or of censoring in the health status measurements can produce biased coefficient estimates. The Tobit regression model is a frequently used tool for modeling censored variables in econometrics research. The authors carried out a Monte-Carlo simulation study to contrast the performance of the Tobit model for censored data with that of ordinary least squares (OLS) regression. It was demonstrated that in the presence of a ceiling effect, if the conditional distribution of the measure of health status had uniform variance, then the coefficient estimates from the Tobit model have superior performance compared with estimates from OLS regression. However, if the conditional distribution had non-uniform variance, then the Tobit model performed at least as poorly as the OLS model.

Early Language Impairment and Young Adult Delinquent and Aggressive Behavior.

Clinic and forensic studies have reported high rates of language impairments in conduct disordered and incarcerated youth. In community samples followed to early adolescence, speech and language impairments have been linked to attention deficits and internalizing problems, rather than conduct problems, delinquency, or aggression. This study examines the young adult antisocial outcomes of speech or language impaired children. Language impaired boys had higher levels of parent-rated delinquency symptoms by age 19 than boys without language impairment, controlled for verbal IQ and for demographic and family variables. Language impaired boys did not differ from controls in self-reported delinquency or aggression symptoms on a standardized checklist; however, language impaired boys reported higher rates of arrests and convictions than controls. Language impairment was not related to aggression or delinquency in girls. We examine alternate models of the interrelationships between language, academics, and behavior, at ages 5, 12, and 19.

Computing Nonparametric Hierarchical Models

Bayesian models involving Dirichlet process mixtures are at the heart of the modern nonparametric Bayesian movement. Much of the rapid development of these models in the last decade has been a direct result of advances in simulation-based computational methods. Some of the very early work in this area, circa 1988-1991, focused on the use of such nonparametric ideas and models in applications of otherwise standard hierarchical models. This chapter provides some historical review and perspective on these developments, with a prime focus on the use and integration of such nonparametric ideas in hierarchical models. We illustrate the ease with which the strict parametric assumptions common to most standard Bayesian hierarchical models can be relaxed to incorporate uncertainties about functional forms using Dirichlet process components, partly enabled by the approach to computation using MCMC methods. The resulting methodology is illustrated with two examples taken from an unpublished 1992 report on the topic.

Eye tracking dysfunction in schizophrenia: characterization of component eye movement abnormalities, diagnostic specificity, and the role of attention.

To characterize oculomotor components and diagnostic specificity of eye tracking abnormalities in schizophrenia, we examined a large consecutively admitted series of psychotic patients and matched controls. The most common abnormality in schizophrenic patients was low gain (slow) pursuit eye movements (47% of cases). Pursuit and saccadic eye movement abnormalities were no more severe in schizophrenic Ss than in those with affective psychoses, except that high rates of catch-up saccades were unique to schizophrenic Ss (17% of cases). These findings indicate that impaired pursuit eye movements are a major cause of eye tracking impairments in schizophrenia, that tracking dysfunctions commonly occur in affective psychoses, and that markedly high rates of catch-up saccades during eye tracking may be specific to schizophrenia.

Development of chlorine dioxide-related by-product models for drinking water treatment

Factorial experiments were conducted using source waters from seven drinking water treatment plants in Ontario, Canada to develop statistically based model equations capable of predicting chlorine dioxide consumption and chlorite and chlorate formation upon chlorine dioxide application. The equations address raw water quality and operational parameters including pH, temperature, chlorine dioxide concentration, reaction time and water organic content (as described by non-purgeable organic carbon x ultraviolet absorbance measured at 254 nm, NPOC x UV254). Terms describing two-factor interaction effects were also included, improving the accuracy of the predictive equations in fitting measured response concentrations as evaluated through internal and external validations. Nearly 80% of the predictions for chlorine dioxide consumption and chlorite formation were observed to be within 20% of the measured levels. Over 90% of the predicted chlorate levels were within +/- 0.1 mg/L of the measured levels. Chlorine dioxide concentration and NPOC x UV254 were key parameters when developing the predictive models.

Models and determinants of vocabulary growth from kindergarten to adulthood.

BACKGROUND Increasing evidence suggests that childhood language problems persist into early adulthood. Nevertheless, little is known about how individual and environmental characteristics influence the language growth of individuals identified with speech/language problems. METHOD Individual growth curve models were utilised to examine how speech/language impairment and environmental variables (socioeconomic status, family separation, and maternal factors) influence vocabulary development from age 5 to 25. Participants were taken from a community sample of children initially diagnosed with speech/language problems at age 5 and their sex- and age-matched controls. RESULTS The language impaired group had significantly poorer receptive vocabulary than the speech impaired and control groups throughout the 20-year period. Family income was a significant predictor of vocabulary growth when considered separately, but ceased to be a predictor when language impairment status was taken into account. Maternal education and family separation were determinants of vocabulary at age 5, over and above language impairment status. CONCLUSION Language impairment is a significant risk factor for vocabulary development from childhood to adulthood. Individuals with speech impairment were less impaired on receptive vocabulary than individuals with language impairment. Further investigation into maternal and familial risk factors may provide targets for early intervention with children at risk for language impairment.