On Corrado Gini's 1932 paper "Intorno alle curve di concentrazione". A selection of translated excerpts

The main focus of the paper is the study of the concentration curve, with special emphasis on its fundamental features and properties and on the relationship with other relevant curves. One of the most innovative contributions (rediscovered forty years later) is the alternative analytical representation of the concentration curves in a coordinate system which assumes the so-called equidistribution line as x-axis and its perpendicular line as y-axis. Furthermore, the impact of the presence of a superior and/or inferior limit in the variable of interest on the maximum concentration triangle is examined. Suitable correction coefficients are derived for computing the corresponding concentration ratio, that take into account these restrictions

Bayesian Set Estimation with Alternative Loss Functions: Optimality and Regret Analysis

Decision-theoretic interval estimation requires the use of loss functions that, typically, take into account the size and the coverage of the sets. We here consider the class of monotone loss functions that, under quite general conditions, guarantee Bayesian optimality of highest posterior probability sets. We focus on three specific families of monotone losses, namely the linear, the exponential and the rational losses whose difference consists in the way the sizes of the sets are penalized. Within the standard yet important set-up of a normal model we propose: 1) an optimality analysis, to compare the solutions yielded by the alternative classes of losses; 2) a regret analysis, to evaluate the additional loss of standard non-optimal intervals of fixed credibility. The article uses an application to a clinical trial as an illustrative example

Sample Size Requirements for Calibrated Approximate Credible Intervals for Proportions in Clinical Trials

In Bayesian analysis of clinical trials data, credible intervals are widely used for inference on unknown parameters of interest, such as treatment effects or differences in treatments effects. Highest Posterior Density (HPD) sets are often used because they guarantee the shortest length. In most of standard problems, closed-form expressions for exact HPD intervals do not exist, but they are available for intervals based on the normal approximation of the posterior distribution. For small sample sizes, approximate intervals may be not calibrated in terms of posterior probability, but for increasing sample sizes their posterior probability tends to the correct credible level and they become closer and closer to exact sets. The article proposes a predictive analysis to select appropriate sample sizes needed to have approximate intervals calibrated at a pre-specified level. Examples are given for interval estimation of proportions and log-odds

On the predictive performance of a non-optimal action in hypothesis testing

In Bayesian decision theory, the performance of an action is measured by its pos- terior expected loss. In some cases it may be convenient/necessary to use a non- optimal decision instead of the optimal one. In these cases it is important to quantify the additional loss we incur and evaluate whether to use the non-optimal decision or not. In this article we study the predictive probability distribution of a relative measure of the additional loss and its use to define sample size determination criteria in a general testing set-up

Spunti di didattica della statistica utilizzando il linguaggio R

In questo periodo di pandemia è diventato ancora più chiaro quanto sia urgente migliorare le competenze di base nel campo del ragionamento probabilistico e statistico. Per realizzare questo obiettivo può essere utile affiancare agli strumenti più tradizionali, quelli derivanti dall’uso di software o linguaggi di programmazione. In particolare strumenti come i fogli di calcolo e Geogebra sono entrati nella pratica didattica di molti docenti. Se è vero che grazie a un foglio di calcolo è possibile produrre analisi di dataset (anche) di grandi dimensioni, che sarebbero impensabili ‘con carta e penna’, l’uso di un linguaggio di programmazione offre opportunità ancor più ampie

Borrowing historical information for non-inferiority trials on Covid-19 vaccines

Non-inferiority vaccine trials compare new candidates to active controls that provide clinically significant protection against a disease. Bayesian statistics allows to exploit pre-experimental information available from previous studies to increase precision and reduce costs. Here, historical knowledge is incorporated into the analysis through a power prior that dynamically regulates the degree of information-borrowing. We examine non-inferiority tests based on credible intervals for the unknown effects-difference between two vaccines on the log odds ratio scale, with an application to new Covid-19 vaccines. We explore the frequentist properties of the method and we address the sample size determination problem

Optimal Sample Size for Evidence and Consensus in Phase III Clinical Trials

Design and analysis of clinical trials imply decisions that often involve multiple parties. We focus here on one of the main design issues in phase III trials, that is the choice of the sample size, that influences the final probability of success of the experiment, i.e. showing evidence of superiority of a new treatment over the standard one. Bayesian Statistics allows one to exploit pre-experimental information and uncertainty that can be translated into probability distributions for the effects-difference parameter. Sometimes sources of prior knowledge can be in striking contrast (skeptimism vs optimism), possibly leading to divergent final post-experimental conclusions. We propose a sample size criterion that controls not only the achievement of minimal evidence of superiority but also posterior consensus. The method is illustrated for trials involving binary outcomes with normal approximation for the log odds ratio with application to a comparative study of two interventions for diabetic patients with coronary artery disease

Joint control of consensus and evidence in Bayesian design of clinical trials

In Bayesian inference, prior distributions formalize preexperimental information and uncertainty on model parameters. Sometimes different sources of knowledge are available, possibly leading to divergent posterior distributions and inferences. Research has been recently devoted to the development of sample size criteria that guarantee agreement of posterior information in terms of credible intervals when multiple priors are available. In these articles, the goals of reaching consensus and evidence are typically kept separated. Adopting a Bayesian performance-based approach, the present article proposes new sample size criteria for superiority trials that jointly control the achievement of both minimal evidence and consensus, measured by appropriate functions of the posterior distributions. We develop both an average criterion and a more stringent criterion that accounts for the entire predictive distributions of the selected measures of minimal evidence and consensus. Methods are developed and illustrated via simulation for trials involving binary outcomes. A real clinical trial example on Covid-19 vaccine data is presented