School of Mathematics and Statistics (SoMaS)

MAS223 Statistical Inference and Modelling

This unit develops methods for analysing data, and provides a foundation for further study of probability and statistics at Level 3. It introduces some standard distributions beyond those met in MAS113, and proceeds with study of continuous multivariate distributions, with particular emphasis on the multivariate normal distribution. Transformations of univariate and multivariate continuous distributions are studied, with the derivation of sampling distributions of important summary statistics as applications. The concepts of likelihood and maximum likelihood estimation are developed. Data analysis is studied within the framework of linear models. There will be substantial use of the software package R.

Prerequisites: MAS113 (Introduction to Probability and Statistics)
Corequisites: MAS211 (Advanced Calculus and Linear Algebra)

The following modules have this module as a prerequisite:

Outline syllabus

• Univariate distribution theory
• Multivariate distribution theory
• Likelihood
• Likelihood case studies
• Linear models

Aims

• extend students' familiarity with standard probability distributions;
• give practice in handling discrete and continuous distributions, especially continuous multivariate ones;
• instil an understanding of the rationale and techniques of likelihood exploration and maximisation;
• consider linear regression models in detail;
• extend the comparison of means from two to several groups through ANOVA models;
• give students experience in the use of R for fitting linear models;

Learning outcomes

• handle a wide range of standard distributions, including the multivariate normal;
• calculate joint, marginal and conditional continuous distributions;
• manipulate multivariate means, variances and covariances;
• transform univariate and multivariate continuous random variables;
• derive, manipulate and interpret likelihood functions, and find maximum likelihood estimators;
• understand regression and ANOVA models as examples of linear models;
• estimate parameters in a linear model;
• make inferences about model parameters through appropriate model comparisons;
• develop a 'best-fitting' model in a systematic and pragmatic way;
• undertake model checking procedures through the use of residuals;
• use R to implement methods covered in the course;
• prepare a structured word processed report of the statistical analysis of an open-ended problem.

Assessment

80% exam in Spring

Full syllabus

Univariate distribution theory
Revision of sample spaces, events and random variables; distribution functions, probability functions, probability density functions; moments; random variables without a mean (Cauchy as example); discrete standard distributions: hypergeometric, negative binomial; revision of normal; gamma and beta functions; gamma (χ² as special case) and beta distributions; visualising distributions in R; transformations of univariate random variables including monotonic case and non-monotonic examples.

• Matrix representation of a linear model: linear regression, polynomial regression, multiple regression and ANOVA models as examples of linear models.
• Least squares estimation: least squares estimators in matrix notation; distributional properties of least squares estimators and the residual sum of squares.
• Hypothesis testing: the F-test for comparing nested linear models; t-tests.
• Prediction: confidence intervals and prediction intervals
• Model checking: diagnostics using standardized residuals; transformations; R².
• Factor independent variables: ANCOVA and one-way and two-way ANOVA.
• Fitting and analysing linear models using R.

Reading list