Statistical stability and linear response for random hyperbolic dynamics

We consider families of random products of close-by Anosov diffeomorphisms, and show that statistical stability and linear response hold for the associated families of equivariant and stationary measures. Our analysis relies on the study of the top Oseledets space of a parametrized transfer operator cocycle, as well as ad-hoc abstract perturbation statements. As an application, we show that, when the quenched central limit theorem (CLT) holds, under the conditions that ensure linear response for our cocycle, the variance in the CLT depends differentiably on the parameter.

Anxiety attenuates learning advantages conferred by statistical stability and induces loss of volatility-attuning in brain activity

Anxiety can alter an individual's perception of their external sensory environment. Previous studies suggest that anxiety can increase the magnitude of neural responses to unexpected (or surprising) stimuli. Additionally, surprise responses are reported to be boosted during stable compared to volatile environments. Few studies, however, have examined how learning is impacted by both threat and volatility. To investigate these effects, we used threat-of-shock to transiently increase subjective anxiety in healthy adults during an auditory oddball task, in which the regularity could be stable or volatile, while undergoing functional Magnetic Resonance Imaging (fMRI) scanning. We then used Bayesian Model Selection (BMS) mapping to pinpoint the brain areas where different models of anxiety displayed the highest evidence. Behaviourally, we found that threat-of-shock eliminated the accuracy advantage conferred by environmental stability over volatility in the task at hand. Neurally, we found that threat-of-shock led to both attenuation and loss of volatility-attuning of neural activity evoked by surprising sounds across most subcortical and limbic brain regions including the thalamus, basal ganglia, claustrum, insula, anterior cingulate, hippocampal gyrus and also the superior temporal gyrus. Conversely, within two small clusters in the left medial frontal gyrus and extrastriate area, threat-of-shock boosted the neural activity (relative to the safe and volatile condition) to the levels observed during the safe and stable condition, while also inducing a loss of volatility-attuning. Taken together, our findings suggest that threat eliminates the learning advantage conferred by statistical stability compared to volatility. Thus, we propose that anxiety disrupts behavioural adaptation to environmental statistics, and that multiple subcortical and limbic regions are implicated in this process.

The Statistical Fragility of Platelet-Rich Plasma in Rotator Cuff Surgery: A Systematic Review and Meta-analysis

Background: The practice of evidence-based medicine relies on objective data to guide clinical decision-making with specific statistical thresholds conveying study significance. Purpose: To determine the utility of applying the fragility index (FI) and the fragility quotient (FQ) analysis to randomized controlled trials (RCTs) evaluating the utilization of platelet-rich plasma (PRP) in rotator cuff repairs (RCRs). Study Design: Systematic review and meta-analysis. Methods: RCTs pertaining to the utilization of PRP in surgical RCRs published in 13 peer-reviewed journals from 2000 to 2020 were evaluated. The FI was determined by manipulating each reported outcome event until a reversal of significance was appreciated. The associated FQ was determined by dividing the FI by the sample size. Results: Of the 9746 studies screened, 19 RCTs were ultimately included for analysis. The overall FI incorporating all 19 RCTs was only 4, suggesting that the reversal of only 4 events is required to change study significance. The associated FQ was determined as 0.092. Of the 43 outcome events reporting lost to follow-up data, 13 (30.2%) represented lost to follow-up >4. Conclusion: Our analysis suggests that RCTs evaluating PRP for surgical RCRs may lack statistical stability with only a few outcome events required to alter trial significance. Therefore, we recommend the reporting of an FI and an FQ in conjunction with P value analysis to carefully interpret the integrity of statistical stability in future comparative trials. Clinical Relevance: Clinical decisions are often informed by statistically significant results. Thus, a true understanding of the robustness of the statistical findings informing clinical decision-making is of critical importance.

Quantitative statistical stability and linear response for irrational rotations and diffeomorphisms of the circle

<p style='text-indent:20px;'>We prove quantitative statistical stability results for a large class of small <inline-formula><tex-math id="M1">\begin{document}$ C^{0} $\end{document}</tex-math></inline-formula> perturbations of circle diffeomorphisms with irrational rotation numbers. We show that if the rotation number is Diophantine the invariant measure varies in a Hölder way under perturbation of the map and the Hölder exponent depends on the Diophantine type of the rotation number. The set of admissible perturbations includes the ones coming from spatial discretization and hence numerical truncation. We also show linear response for smooth perturbations that preserve the rotation number, as well as for more general ones. This is done by means of classical tools from KAM theory, while the quantitative stability results are obtained by transfer operator techniques applied to suitable spaces of measures with a weak topology.</p>