Statistics

This course provides an introduction to probability and statistics with a view toward applications. It includes topics on mathematical models for random phenomena, random variables, expectation, and discrete & continuous distributions. This course also covers laws of large numbers, central limit theorem, and basic techniques of inferential statistics. Students are expected to be familiar with statistical thinking and the basic concepts of descriptive statistics, probability distribution, and inferential statistics through this course.

This course introduces the foundational and applied concepts of probability and statistical modelling for data science in engineering. Strong emphasis is placed on using the material covered to solve engineering problems, with a focus on the R statistical computing software. The main sections of the course are descriptive statistics; laws of probability; random variables; statistical inference; simple linear regression; and statistical methods for quality control. In addition, students are required to complete a sequence of computer laboratory sessions using the R software package. Students learn to perform exploratory data analyses using graphical and numerical descriptive statistics, calculate probabilities and simulate from common probability distributions, calculate confidence intervals and perform hypothesis tests, and fit linear regression models.

This is an advanced course in linear and logistic regression, which expounds on the knowledge gained in introductory mathematical statistics courses. It covers matrix formulation of multivariate regression, methods for model validation, residuals, outliers, influential observations, construction and use of F- and t- tests, likelihood-ratio-test, confidence intervals and prediction, and applied implementation of various techniques in R software. Students also consider correlated errors, Poisson regression, multinominal and ordinal logistic regression. The first part of the course expands on previous study of linear regression to consider how to check if the model fits the data, what to do if it does not fit, how uncertain it is, and how to use it to draw conclusions about reality. The second part of the course explores logistic regression, which is used in surveys where the answers follow a categorical alternative pattern such as “yes/no,” “little/just fine/much,” or “car/bicycle/bus.” Students describe differences between continuous and discrete data, and the resulting consequences for the choice of statistical model. Students learn to give an account of the principles behind different estimation principles, and describe the statistical properties of such estimates as they appear in regression analysis. The interpretation of regression relations in terms of conditional distributions is studied. Odds and odds ration are presented, and students describe their relation to probabilities and to logistic regression. Students formulate both linear and logistic regression models for concrete problems, estimate and interpret the parameters, examine the validity of the model and make suitable modifications, use the model for prediction, utilize a statistical computer program for analysis, and present the analysis and conclusions of a practical problem in a written report and oral presentation. The course makes use of lectures, exercises, computer exercises, and project work.

This course presents some basic techniques for the analysis of the uncertainty inherent in statistical information, with the goal of providing a correct evaluation and communication of risk. Basic notions of elementary probability theory and of Bayesian probability are introduced and discussed, and their application is illustrated in problems connected with the medical and psychological practice, also within the framework of recent Italian legislation on informed consent which imposes to all health care professionals a correct risk assessment and the adequate communication of it to patients. The course discusses topics including uncertainty in statistical information; problems related to the evaluation of risk and communication of risk; real-world examples; Bayesian inferences through the use of probabilities and by means of natural frequencies; suitability of the natural frequencies for a more intuitive and direct insight in both risk estimation and in a transparent representation of risk; examples focusing on the correct judgement of the probabilistic predictive value of medical diagnostic tests, and aiming at avoiding misleading risk information; cases related to the ongoing Covid-19 public-health emergency; evaluation of the effect of interventions, including relative risk and absolute risk, and relative and absolute risk reduction (or increase); and number needed to treat or to harm.

This course examines the generalized linear model and extensions to fit data arising from a range of sources including multiple regression models, logistic regression models, and log-linear models.