The Bivariate Normal Copula

We collect well known and less known facts about the bivariate normal distribution and translate them into copula language. In addition, we prove a very general formula for the bivariate normal copula, we compute Gini's gamma, and we provide improved bounds and approximations on the diagonal.Comment: 24 page

Deep Learning Using Preoperative AS-OCT Predicts Graft Detachment in DMEK

Purpose: To evaluate a novel deep learning algorithm to distinguish between eyes that may or may not have a graft detachment based on pre–Descemet membrane endothelial keratoplasty (DMEK) anterior segment optical coherence tomography (AS-OCT) images. Methods: Retrospective cohort study. A multiple-instance learning artificial intelligence (MIL-AI) model using a ResNet-101 backbone was designed. AS-OCT images were split into training and testing sets. The MIL-AI model was trained and validated on the training set. Model performance and heatmaps were calculated from the testing set. Classification performance metrics included F1 score (harmonic mean of recall and precision), specificity, sensitivity, and area under curve (AUC). Finally, MIL-AI performance was compared to manual classification by an experienced ophthalmologist. Results: In total, 9466 images of 74 eyes (128 images per eye) were included in the study. Images from 50 eyes were used to train and validate the MIL-AI system, while the remaining 24 eyes were used as the test set to determine its performance and generate heatmaps for visualization. The performance metrics on the test set (95% confidence interval) were as follows: F1 score, 0.77 (0.57–0.91); precision, 0.67 (0.44–0.88); specificity, 0.45 (0.15–0.75); sensitivity, 0.92 (0.73–1.00); and AUC, 0.63 (0.52–0.86). MIL-AI performance was more sensitive (92% vs. 31%) but less specific (45% vs. 64%) than the ophthal-mologist’s performance. Conclusions: The MIL-AI predicts with high sensitivity the eyes that may have post-DMEK graft detachment requiring rebubbling. Larger-scale clinical trials are warranted to validate the model. Translational Relevance: MIL-AI models represent an opportunity for implementation in routine DMEK suitability screening

Phi-divergence-type test for positive dependence alternatives in 2 x k contingency tables

In this chapter, we consider 2 x k contingency tables and derive a new family of test statistics for detecting positive dependence in them. The family of test statistics introduced here is based on the phi-divergence measures of which the likelihood ratio test is a special case