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Abstract This paper develops a mathematical model and statistical methods to quantify trends in presence/absence observations of snow cover (not depths) and applies these in an analysis of Northern Hemispheric observations extracted from satellite flyovers during 1967–2021. A two-state Markov chain model with periodic dynamics is introduced to analyze changes in the data in a cell by cell fashion. Trends, converted to the number of weeks of snow cover lost/gained per century, are estimated for each study cell. Uncertainty margins for these trends are developed from the model and used to assess the significance of the trend estimates. Cells with questionable data quality are explicitly identified. Among trustworthy cells, snow presence is seen to be declining in almost twice as many cells as it is advancing. While Arctic and southern latitude snow presence is found to be rapidly receding, other locations, such as eastern Canada, are experiencing advancing snow cover. Significance StatementThis project quantifies how the Northern Hemisphere’s snow cover has recently changed. Snow cover plays a critical role in the global energy balance due to its high albedo and insulating characteristics and is therefore a prominent indicator of climate change. On a regional scale, the spatial consistency of snow cover influences surface temperatures via variations in absorbed solar radiation, while continental-scale snow cover acts to maintain thermal stability in the Arctic and subarctic regions, leading to spatial and temporal impacts on global circulation patterns. Changing snow presence in Arctic regions could influence large-scale releases of carbon and methane gas. Given the importance of snow cover, understanding its trends enhances our understanding of climate change. 

Abstract The use of likelihood ratios for quantifying the strength of forensic evidence in criminal cases is gaining widespread acceptance in many forensic disciplines. Although some forensic scientists feel that subjective likelihood ratios are a reasonable way of expressing expert opinion regarding strength of evidence in criminal trials, legal requirements of reliability of expert evidence in the United Kingdom, United States and some other countries have encouraged researchers to develop likelihood ratio systems based on statistical modelling using relevant empirical data. Many such systems exhibit exceptional power to discriminate between the scenario presented by the prosecution and an alternate scenario implying the innocence of the defendant. However, such systems are not necessarily well calibrated. Consequently, verbal explanations to triers of fact, by forensic experts, of the meaning of the offered likelihood ratio may be misleading. In this article, we put forth a statistical approach for testing the calibration discrepancy of likelihood ratio systems using ground truth known empirical data. We provide point estimates as well as confidence intervals for the calibration discrepancy. Several examples, previously discussed in the literature, are used to illustrate our method. Results from a limited simulation study concerning the performance of the proposed approach are also provided. 

High intratumoral heterogeneity is thought to be a poor prognostic indicator. However, the source of heterogeneity may also be important, as genomic heterogeneity is not always reflected in histologic or ‘visual’ heterogeneity. We aimed to develop a predictor of histologic heterogeneity and evaluate its association with outcomes and molecular heterogeneity. We used VGG16 to train an image classifier to identify unique, patient-specific visual features in 1655 breast tumors (5907 core images) from the Carolina Breast Cancer Study (CBCS). Extracted features for images, as well as the epithelial and stromal image components, were hierarchically clustered, and visual heterogeneity was defined as a greater distance between images from the same patient. We assessed the association between visual heterogeneity, clinical features, and DNA-based molecular heterogeneity using generalized linear models, and we used Cox models to estimate the association between visual heterogeneity and tumor recurrence. Basal-like and ER-negative tumors were more likely to have low visual heterogeneity, as were the tumors from younger and Black women. Less heterogeneous tumors had a higher risk of recurrence (hazard ratio = 1.62, 95% confidence interval = 1.22–2.16), and were more likely to come from patients whose tumors were comprised of only one subclone or had a TP53 mutation. Associations were similar regardless of whether the image was based on stroma, epithelium, or both. Histologic heterogeneity adds complementary information to commonly used molecular indicators, with low heterogeneity predicting worse outcomes. Future work integrating multiple sources of heterogeneity may provide a more comprehensive understanding of tumor progression.