Consistent Estimation of the Structural Distribution Function

Motivated by problems in linguistics we consider a multinomial random vector for which the number of cells N is not much smaller than the sum of the cell frequencies, i.e. the sample size n . The distribution function of the uniform distribution on the set of all cell probabilities multiplied by N is called the structural distribution function of the cell probabilities. Conditions are given that guarantee that the structural distribution function can be estimated consistently as n increases indefinitely although n / N does not. The natural estimator is inconsistent and we prove consistency of essentially two alternative estimators.     

Bayes linear analysis for graphical models: The geometric approach to local computation and interpretive graphics

This paper concerns the geometric treatment of graphical models using Bayes linear methods. We introduce Bayes linear separation as a second order generalised conditional independence relation, and Bayes linear graphical models are constructed using this property. A system of interpretive and diagnostic shadings are given, which summarise the analysis over the associated moral graph. Principles of local computation are outlined for the graphical models, and an algorithm for implementing such computation over the junction tree is described. The approach is illustrated with two examples. The first concerns sales forecasting using a multivariate dynamic linear model. The second concerns inference for the error variance matrices of the model for sales, and illustrates the generality of our geometric approach by treating the matrices directly as random objects. The examples are implemented using a freely available set of object-oriented programming tools for Bayes linear local computation and graphical diagnostic display.     

On optimal testing for the equality of equicorrelation: An example of loss in power

Sen Gupta (1988) considered a locally most powerful (LMP) test for testing nonzero values of the equicorrelation coefficient of a standard symmetric multivariate normal distribution. This paper constructs analogous tests for the symmetric multivariate normal distribution. It shows that the new test is uniformly most powerful invariant even in the presence of a nuisance parameter, σ². Further applications of LMP invariant tests to several equicorrelated populations have been considered and an extension to panel data modeling has been suggested.     

Importance of interpolation when constructing double-bootstrap confidence intervals

We show that, in the context of double-bootstrap confidence intervals, linear interpolation at the second level of the double bootstrap can reduce the simulation error component of coverage error by an order of magnitude. Intervals that are indistinguishable in terms of coverage error with theoretical, infinite simulation, double-bootstrap confidence intervals may be obtained at substantially less computational expense than by using the standard Monte Carlo approximation method. The intervals retain the simplicity of uniform bootstrap sampling and require no special analysis or computational techniques. Interpolation at the first level of the double bootstrap is shown to have a relatively minor effect on the simulation error.     

Standard errors for EM estimation

The EM algorithm is a popular method for computing maximum likelihood estimates. One of its drawbacks is that it does not produce standard errors as a by-product. We consider obtaining standard errors by numerical differentiation. Two approaches are considered. The first differentiates the Fisher score vector to yield the Hessian of the log-likelihood. The second differentiates the EM operator and uses an identity that relates its derivative to the Hessian of the log-likelihood. The well-known SEM algorithm uses the second approach. We consider three additional algorithms: one that uses the first approach and two that use the second. We evaluate the complexity and precision of these three and the SEM in algorithm seven examples. The first is a single-parameter example used to give insight. The others are three examples in each of two areas of EM application: Poisson mixture models and the estimation of covariance from incomplete data. The examples show that there are algorithms that are much simpler and more accurate than the SEM algorithm. Hopefully their simplicity will increase the availability of standard error estimates in EM applications. It is shown that, as previously conjectured, a symmetry diagnostic can accurately estimate errors arising from numerical differentiation. Some issues related to the speed of the EM algorithm and algorithms that differentiate the EM operator are identified.     

The exponentiated power exponential regression model with different regression structures: application in nursing data

We define the exponentiated power exponential distribution and propose a regression model with different systematic structures based on the new distribution. We show that the new regression model can be applied to dispersion data since it represents a parametric family of models that includes as sub-models some widely-known regression models. It then can be used more effectively in the analysis of real data. We use maximum likelihood estimation and derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes. Some global-influence measurements are also investigated and simulation studies are performed to evaluate the accuracy of the estimates. We provide an application of the regression model with four systematic structures to nursing activities score data in the Unit of the Medical Clinic of University of São Paulo (USP) Hospital.     

Learning from a lot: Empirical Bayes for high‐dimensional model‐based prediction

Empirical Bayes is a versatile approach to “learn from a lot” in two ways: first, from a large number of variables and, second, from a potentially large amount of prior information, for example, stored in public repositories. We review applications of a variety of empirical Bayes methods to several well‐known model‐based prediction methods, including penalized regression, linear discriminant analysis, and Bayesian models with sparse or dense priors. We discuss “formal” empirical Bayes methods that maximize the marginal likelihood but also more informal approaches based on other data summaries. We contrast empirical Bayes to cross‐validation and full Bayes and discuss hybrid approaches. To study the relation between the quality of an empirical Bayes estimator and p, the number of variables, we consider a simple empirical Bayes estimator in a linear model setting. We argue that empirical Bayes is particularly useful when the prior contains multiple parameters, which model a priori information on variables termed “co‐data”. In particular, we present two novel examples that allow for co‐data: first, a Bayesian spike‐and‐slab setting that facilitates inclusion of multiple co‐data sources and types and, second, a hybrid empirical Bayes–full Bayes ridge regression approach for estimation of the posterior predictive interval.     

Maternal Postpartum Depression Increases Vulnerability for Toddler Behavior Problems through Infant Cortisol Reactivity

The current study examined the role of hypothalamic‐pituitary‐adrenal reactivity (a physiological indicator of stress) in early infancy as a mediator of the relationship between maternal postpartum depression and toddler behavior problems. Participants were 137 at‐risk mothers and their children participating in a longitudinal study of intergenerational transmission of risk. Mothers’ depression was measured five times during the infants’ first 18 months. Infant cortisol was collected during a social stressor (the still‐face paradigm) when infants were 6 months old, and mothers reported on toddlers’ internalizing and externalizing symptoms at 18 months. Among this sample of high‐risk mother–infant dyads, early postpartum depression predicted atypical infant cortisol reactivity at 6 months, which mediated the effect of maternal depression on increased toddler behavior problems. Clinical implications are discussed.     

Changing Opportunities for Learning in Everyday Life: Infant Body Position Over the First Year

Developmental theories depend on characterizing the input to potential learning mechanisms—infants’ everyday experiences. The current study employed a novel ecological momentary assessment (EMA) to measure two aspects of the physical context of those experiences: body position and location. Infant body position was selected because it relates to the development of a variety of other skills. Caregivers of 3‐, 6‐, 9‐, and 12‐month‐olds reported infants’ body position—held, supine, reclined, prone, sitting, or upright—in response to text message notifications over a week to capture infants’ experiences across the entire range of their daily activities. Findings revealed a tremendous disparity in the distribution of body position experiences over the first year. Younger infants spend more time held, supine, and reclined, whereas older infants spend more time sitting and upright. Body position experiences differed substantially between same‐age infants who possess a motor skill (e.g., ability to sit or walk) compared with those who did not, suggesting that developing motor skills change infants’ everyday experiences. Finally, the success of the methodology suggests that similar EMAs might be used to study a wide range of infants’ naturalistic experiences.     

Issues in the integration of religious education and worldviews education in an intercultural context

ABSTRACT

The Council of Europe’s 2008 ‘Recommendation’ advocates the study of ‘non-religious convictions’ in schools in addition to religions. In 2018, there is evidence of growing academic interest in the inclusion of non-religious worldviews in the school curriculum, but few European countries include such a study within religious education. The guidance document, Signposts, recognises that the integration of religions and non-religious worldviews is problematic for policy makers, teacher trainers and schools and that there is a need for further research. Norway and England are at significant but different stages in the process of integration. This article presents a comparative study which draws on the findings of research in both countries which has investigated the process at classroom and policy levels, and issues raised by this. Four issues are explored: practical challenges; differing understandings of the concept of worldviews; the inclusion of non-religious worldviews as a political issue and influences on the selection of worldviews. The article concludes with a discussion of issues and findings within a supranational context and makes recommendations which highlight the value of integration and the need for both contextual studies and further comparative research.