Alphas, betas and skewy distributions: two ways of getting the wrong answer

Although many parametric statistical tests are considered to be robust, as recently shown in Methodologist’s Corner, it still pays to be circumspect about the assumptions underlying statistical tests. In this paper I show that robustness mainly refers to α, the type-I error. If the underlying distribution of data is ignored there can be a major penalty in terms of the β, the type-II error, representing a large increase in false negative rate or, equivalently, a severe loss of power of the test

Long-term nutrient reductions lead to the unprecedented recovery of a temperate coastal region

Humans strongly impact the dynamics of coastal systems, yet surprisingly few studies mechanistically link management of anthropogenic stressors and successful restoration of nearshore habitats over large spatial and temporal scales. Such examples are sorely needed to ensure the success of ecosystem restoration efforts worldwide. Here, we unite 30 consecutive years of watershed modeling, biogeochemical data, and comprehensive aerial surveys of Chesapeake Bay, United States to quantify the cascading effects of anthropogenic impacts on submersed aquatic vegetation (SAV), an ecologically and economically valuable habitat. We employ structural equation models to link land use change to higher nutrient loads, which in turn reduce SAV cover through multiple, independent pathways. We also show through our models that high biodiversity of SAV consistently promotes cover, an unexpected finding that corroborates emerging evidence from other terrestrial and marine systems. Due to sustained management actions that have reduced nitrogen concentrations in Chesapeake Bay by 23% since 1984, SAV has regained 17,000 ha to achieve its highest cover in almost half a century. Our study empirically demonstrates that nutrient reductions and biodiversity conservation are effective strategies to aid the successful recovery of degraded systems at regional scales, a finding which is highly relevant to the utility of environmental management programs worldwide

Simpson's Paradox, Lord's Paradox, and Suppression Effects are the same phenomenon – the reversal paradox

This article discusses three statistical paradoxes that pervade epidemiological research: Simpson's paradox, Lord's paradox, and suppression. These paradoxes have important implications for the interpretation of evidence from observational studies. This article uses hypothetical scenarios to illustrate how the three paradoxes are different manifestations of one phenomenon – the reversal paradox – depending on whether the outcome and explanatory variables are categorical, continuous or a combination of both; this renders the issues and remedies for any one to be similar for all three. Although the three statistical paradoxes occur in different types of variables, they share the same characteristic: the association between two variables can be reversed, diminished, or enhanced when another variable is statistically controlled for. Understanding the concepts and theory behind these paradoxes provides insights into some controversial or contradictory research findings. These paradoxes show that prior knowledge and underlying causal theory play an important role in the statistical modelling of epidemiological data, where incorrect use of statistical models might produce consistent, replicable, yet erroneous results

A gap in the spectrum of the Neumann-Laplacian on a periodic waveguide

We will study the spectral problem related to the Laplace operator in a singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder with contains periodic arrangement of inclusions. On the boundary of the waveguide we consider both Neumann and Dirichlet conditions. We will prove that provided the diameter of the inclusion is small enough in the spectrum of Laplacian opens spectral gaps, i.e. frequencies that does not propagate through the waveguide. The existence of the band gaps will verified using the asymptotic analysis of elliptic operators.Comment: 26 pages, 6 figure

Parametric methods outperformed non-parametric methods in comparisons of discrete numerical variables

<p>Abstract</p> <p>Background</p> <p>The number of events per individual is a widely reported variable in medical research papers. Such variables are the most common representation of the general variable type called discrete numerical. There is currently no consensus on how to compare and present such variables, and recommendations are lacking. The objective of this paper is to present recommendations for analysis and presentation of results for discrete numerical variables.</p> <p>Methods</p> <p>Two simulation studies were used to investigate the performance of hypothesis tests and confidence interval methods for variables with outcomes {0, 1, 2}, {0, 1, 2, 3}, {0, 1, 2, 3, 4}, and {0, 1, 2, 3, 4, 5}, using the difference between the means as an effect measure.</p> <p>Results</p> <p>The Welch U test (the T test with adjustment for unequal variances) and its associated confidence interval performed well for almost all situations considered. The Brunner-Munzel test also performed well, except for small sample sizes (10 in each group). The ordinary T test, the Wilcoxon-Mann-Whitney test, the percentile bootstrap interval, and the bootstrap-<it>t </it>interval did not perform satisfactorily.</p> <p>Conclusions</p> <p>The difference between the means is an appropriate effect measure for comparing two independent discrete numerical variables that has both lower and upper bounds. To analyze this problem, we encourage more frequent use of parametric hypothesis tests and confidence intervals.</p

Parametric study of EEG sensitivity to phase noise during face processing

&lt;b&gt;Background: &lt;/b&gt; The present paper examines the visual processing speed of complex objects, here faces, by mapping the relationship between object physical properties and single-trial brain responses. Measuring visual processing speed is challenging because uncontrolled physical differences that co-vary with object categories might affect brain measurements, thus biasing our speed estimates. Recently, we demonstrated that early event-related potential (ERP) differences between faces and objects are preserved even when images differ only in phase information, and amplitude spectra are equated across image categories. Here, we use a parametric design to study how early ERP to faces are shaped by phase information. Subjects performed a two-alternative force choice discrimination between two faces (Experiment 1) or textures (two control experiments). All stimuli had the same amplitude spectrum and were presented at 11 phase noise levels, varying from 0% to 100% in 10% increments, using a linear phase interpolation technique. Single-trial ERP data from each subject were analysed using a multiple linear regression model. &lt;b&gt;Results: &lt;/b&gt; Our results show that sensitivity to phase noise in faces emerges progressively in a short time window between the P1 and the N170 ERP visual components. The sensitivity to phase noise starts at about 120–130 ms after stimulus onset and continues for another 25–40 ms. This result was robust both within and across subjects. A control experiment using pink noise textures, which had the same second-order statistics as the faces used in Experiment 1, demonstrated that the sensitivity to phase noise observed for faces cannot be explained by the presence of global image structure alone. A second control experiment used wavelet textures that were matched to the face stimuli in terms of second- and higher-order image statistics. Results from this experiment suggest that higher-order statistics of faces are necessary but not sufficient to obtain the sensitivity to phase noise function observed in response to faces. &lt;b&gt;Conclusion: &lt;/b&gt; Our results constitute the first quantitative assessment of the time course of phase information processing by the human visual brain. We interpret our results in a framework that focuses on image statistics and single-trial analyses

Age-related delay in information accrual for faces: Evidence from a parametric, single-trial EEG approach

Background: In this study, we quantified age-related changes in the time-course of face processing by means of an innovative single-trial ERP approach. Unlike analyses used in previous studies, our approach does not rely on peak measurements and can provide a more sensitive measure of processing delays. Young and old adults (mean ages 22 and 70 years) performed a non-speeded discrimination task between two faces. The phase spectrum of these faces was manipulated parametrically to create pictures that ranged between pure noise (0% phase information) and the undistorted signal (100% phase information), with five intermediate steps. Results: Behavioural 75% correct thresholds were on average lower, and maximum accuracy was higher, in younger than older observers. ERPs from each subject were entered into a single-trial general linear regression model to identify variations in neural activity statistically associated with changes in image structure. The earliest age-related ERP differences occurred in the time window of the N170. Older observers had a significantly stronger N170 in response to noise, but this age difference decreased with increasing phase information. Overall, manipulating image phase information had a greater effect on ERPs from younger observers, which was quantified using a hierarchical modelling approach. Importantly, visual activity was modulated by the same stimulus parameters in younger and older subjects. The fit of the model, indexed by R2, was computed at multiple post-stimulus time points. The time-course of the R2 function showed a significantly slower processing in older observers starting around 120 ms after stimulus onset. This age-related delay increased over time to reach a maximum around 190 ms, at which latency younger observers had around 50 ms time lead over older observers. Conclusion: Using a component-free ERP analysis that provides a precise timing of the visual system sensitivity to image structure, the current study demonstrates that older observers accumulate face information more slowly than younger subjects. Additionally, the N170 appears to be less face-sensitive in older observers