Bivariate Distributions Underlying Responses to Ordinal Variables

The association between two ordinal variables can be expressed with a polychoric correlation coefficient. This coefficient is conventionally based on the assumption that responses to ordinal variables are generated by two underlying continuous latent variables with a bivariate normal distribution. When the underlying bivariate normality assumption is violated, the estimated polychoric correlation coefficient may be biased. In such a case, we may consider other distributions. In this paper, we aimed to provide an illustration of fitting various bivariate distributions to empirical ordinal data and examining how estimates of the polychoric correlation may vary under different distributional assumptions. Results suggested that the bivariate normal and skew-normal distributions rarely hold in the empirical datasets. In contrast, mixtures of bivariate normal distributions were often not rejected.

Evidence-based decision making and covid-19: what a posteriori probability distributions speak

Background: In the absence of any pharmaceutical interventions, the management of the COVID-19 pandemic is based on public health measures. The present study fosters evidence-based decision making by estimating various “a posteriori probability distributions" from COVID-19 patients. Methods: In this retrospective observational study, 987 RT-PCR positive COVID-19 patients from SMS Medical College, Jaipur, India, were enrolled after approval of the institutional ethics committee. The data regarding age, gender, and outcome were collected. The univariate and bivariate distributions of COVID-19 cases with respect to age, gender, and outcome were estimated. The age distribution of COVID-19 cases was compared with the general population's age distribution using the goodness of fit c2 test. The independence of attributes in bivariate distributions was evaluated using the chi-square test for independence. Results: The age group ‘25-29’ has shown highest probability of COVID-19 cases (P [25-29] = 0.14, 95% CI: 0.12- 0.16). The men (P [Male] = 0.62, 95%CI: 0.59-0.65) were dominant sufferers. The most common outcome was recovery (P [Recovered] = 0.79, 95%CI: 0.76-0.81) followed by admitted cases (P [Active]= 0.13, 95%CI: 0.11-0.15) and death (P [Death] = 0.08, 95%CI: 0.06-0.10). The age distribution of COVID-19 cases differs significantly from the age distribution of the general population (c2 =399.04, P < 0.001). The bivariate distribution of COVID-19 across age and outcome was not independent (c2 =106.21, df = 32, P < 0.001). Conclusion: The knowledge of disease frequency patterns helps in the optimum allocation of limited resources and manpower. The study provides information to various epidemiological models for further analysis.

Tree Bounds for Sums of Bernoulli Random Variables: A Linear Optimization Approach

We study the problem of computing the tightest upper and lower bounds on the probability that the sum of n dependent Bernoulli random variables exceeds an integer k. Under knowledge of all pairs of bivariate distributions denoted by a complete graph, the bounds are NP-hard to compute. When the bivariate distributions are specified on a tree graph, we show that tight bounds are computable in polynomial time using a compact linear program. These bounds provide robust probability estimates when the assumption of conditional independence in a tree-structured graphical model is violated. We demonstrate, through numericals, the computational advantage of our compact linear program over alternate approaches. A comparison of bounds under various knowledge assumptions, such as univariate information and conditional independence, is provided. An application is illustrated in the context of Chow–Liu trees, wherein our bounds distinguish between various trees that encode the maximum possible mutual information.