Underestimation of Standard Errors in Multi-Site Time Series Studies

Multi-site time series studies of air pollution and mortality and morbidity have figured prominently in the literature as comprehensive approaches for estimating acute effects of air pollution on health. Hierarchical models are generally used to combine site-specific information and estimate pooled air pollution effects taking into account both within-site statistical uncertainty, and across-site heterogeneity. Within a site, characteristics of time series data of air pollution and health (small pollution effects, missing data, highly correlated predictors, non linear confounding etc.) make modelling all sources of uncertainty challenging. One potential consequence is underestimation of the statistical variance of the site-specific effects to be combined. In this paper we investigate the impact of variance underestimation on the pooled relative rate estimate. We focus on two-stage normal-normal hierarchical models and on under- estimation of the statistical variance at the first stage. By mathematical considerations and simulation studies, we found that variance underestimation does not affect the pooled estimate substantially. However, some sensitivity of the pooled estimate to variance underestimation is observed when the number of sites is small and underestimation is severe. These simulation results are applicable to any two-stage normal-normal hierarchical model for combining information of site-specific results, and they can be easily extended to more general hierarchical formulations. We also examined the impact of variance underestimation on the national average relative rate estimate from the National Morbidity Mortality Air Pollution Study and we found that variance underestimation as much as 40% has little effect on the national average

Constraints on the Clustering, Biasing and Redshift Distribution of Radio Sources

We discuss how different theoretical predictions for the variance $\sigma^2$ of the distribution of radio sources can be matched to measurements from the FIRST survey at different flux limits. The predictions are given by the integration of models for the angular correlation function $w(\theta)$ for three different functional forms of the redshift distribution $N(z)$, different spatial correlation functions and by different evolutions of the bias $b(z)$ with redshift. We also consider the two cases of open and flat Universes. Although the predicted $w(\theta)$ show substantial differences due to differences in the $N(z)$'s, these differences are not significant compared to the uncertainties in the current observations. It turns out that the best fit is provided by models with constant biasing at all times, although the difference between models with epoch-independent bias and models with bias that evolves linearly with redshift is not very large. All models with strong evolution of bias with epoch are ruled out. As a further step we directly calculated $w_{obs}(\theta)$ at 3mJy from the catalogue and matched it with our models for the angular correlation function in the hypothesis that the clustering signal comes from two different populations, namely AGN-powered sources and starbursting galaxies. The results are consistent with a scenario for hierarchical clustering where the fainter starbursting galaxies trace the mass at all epochs, while brighter AGN's are strongly biased, with $b(z)$ evolving linearly with redshift, as suggested by some theories of galaxy formation and evolution.Comment: 14 pages, 12 figures, version to appear on MNRA

From neural PCA to deep unsupervised learning

A network supporting deep unsupervised learning is presented. The network is an autoencoder with lateral shortcut connections from the encoder to decoder at each level of the hierarchy. The lateral shortcut connections allow the higher levels of the hierarchy to focus on abstract invariant features. While standard autoencoders are analogous to latent variable models with a single layer of stochastic variables, the proposed network is analogous to hierarchical latent variables models. Learning combines denoising autoencoder and denoising sources separation frameworks. Each layer of the network contributes to the cost function a term which measures the distance of the representations produced by the encoder and the decoder. Since training signals originate from all levels of the network, all layers can learn efficiently even in deep networks. The speedup offered by cost terms from higher levels of the hierarchy and the ability to learn invariant features are demonstrated in experiments.Comment: A revised version of an article that has been accepted for publication in Advances in Independent Component Analysis and Learning Machines (2015), edited by Ella Bingham, Samuel Kaski, Jorma Laaksonen and Jouko Lampine

Modeling general, specific, and method variance in personality measures: Results for ZKA-PQ and NEO-PI-R

Reprinted by permission of SAGE PublicationsContemporary models of personality assume a hierarchical structure in which broader traits contain narrower traits. Individual differences in response styles also constitute a source of score variance. In this study, the bifactor model is applied to separate these sources of variance for personality subscores. The procedure is illustrated using data for two personality inventories—NEO Personality Inventory–Revised and Zuckerman–Kuhlman–Aluja Personality Questionnaire. The inclusion of the acquiescence method factor generally improved the fit to acceptable levels for the Zuckerman–Kuhlman–Aluja Personality Questionnaire, but not for the NEO Personality Inventory–Revised. This effect was higher in subscales where the number of direct and reverse items is not balanced. Loadings on the specific factors were usually smaller than the loadings on the general factor. In some cases, part of the variance was due to domains being different from the main one. This information is of particular interest to researchers as they can identify which subscale scores have more potential to increase predictive validit

Analysis of variance--why it is more important than ever

Analysis of variance (ANOVA) is an extremely important method in exploratory and confirmatory data analysis. Unfortunately, in complex problems (e.g., split-plot designs), it is not always easy to set up an appropriate ANOVA. We propose a hierarchical analysis that automatically gives the correct ANOVA comparisons even in complex scenarios. The inferences for all means and variances are performed under a model with a separate batch of effects for each row of the ANOVA table. We connect to classical ANOVA by working with finite-sample variance components: fixed and random effects models are characterized by inferences about existing levels of a factor and new levels, respectively. We also introduce a new graphical display showing inferences about the standard deviations of each batch of effects. We illustrate with two examples from our applied data analysis, first illustrating the usefulness of our hierarchical computations and displays, and second showing how the ideas of ANOVA are helpful in understanding a previously fit hierarchical model.Comment: This paper discussed in: [math.ST/0508526], [math.ST/0508527], [math.ST/0508528], [math.ST/0508529]. Rejoinder in [math.ST/0508530

The relation between school leadership from a distributed perspective and teachers' organizational commitment: examining the source of the leadership function

Purpose: In this study the relationship between school leadership and teachers’ organizational commitment is examined by taking into account a distributed leadership perspective. The relation between teachers’ organizational commitment and contextual variables of teachers’ perceptions of the quality and the source of the supportive and supervisory leadership function, participative decision making, and cooperation within the leadership team are examined. Research Design: A survey was set up involving 1,522 teachers from 46 large secondary schools in Flanders (Belgium). Because the data in the present study have an inherent hierarchical structure, that is, teachers are nested into schools, hierarchical linear modeling techniques are applied. Findings: The analyses reveal that 9% of the variance in teachers’ organizational commitment is attributable to differences between schools. Teachers’ organizational commitment is mainly related to quality of the supportive leadership, cooperation within the leadership team, and participative decision making. Who performed the supportive leadership function plays only a marginally significant positive role. The quality of the supervisory leadership function and the role of the leadership team members in this function were not significantly related to teachers’ organizational commitment. Conclusions: The implications of the findings are that to promote teachers’ organizational commitment teachers should feel supported by their leadership team and that this leadership team should be characterized by group cohesion, role clarity, and goal orientedness. Recommendations for further research are provided