## Linear Models with Nonparametric Regression

 Unit code: MATH48011 Credit Rating: 15 Unit level: Level 4 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

#### Requisites

Prerequisite

MATH48011 pre-requisites

Students are not permitted to take more than one of MATH48011 or MATH68011 for credit in an undergraduate programme and then a postgraduate programme, as the contents of the courses overlap significantly.

#### Aims

• To familiarise students with the methodology and applications of standard techniques of Linear Models such as regression analysis, and analysis of variance and covariance, etc.
• To introduce students to advanced statistical methods in Nonparametric Regression.
• To explore some of the wide range of real-life situations occurring in the fields of biology, engineering, industrial experimentation, medical and social sciences that can be investigated using Linear Models and Nonparametric Regression.

#### Overview

In many areas of science, technology, social science and medicine one often wishes to explore the relationship between one observable random response and a number of 'factors' which may influence simultaneously the response. The techniques developed to study such relationships fall in three broad categories:

• Regression Analysis where the influence of the factors is quantitative;
• Analysis of Variance where each factor's influence is qualitative; and
• Analysis of Covariance where both qualitative and quantitative factors are present.

However, these three valuable techniques can be studied together as special cases of a unified theory of Linear Models. The course starts with a study of estimation and hypothesis testing in the general linear problem. Once the principles and techniques are established practical applications in the three types of analysis are examined in greater detail.

Nonparametric regression provides a very flexible approach to exploring the relationship between a response and an associated covariate but without having to specify a parametric model. The different techniques available are essentially based on forms of local averaging controlled by the value of a smoothing parameter. In this part of the module we will study a few different techniques, along with their statistical properties. We will also look briefly at how such estimators can be used in more inferential procedures.

#### Learning outcomes

On successful completion of this course unit students will be able to:

• estimate the regression relationship between covariates and response variables using both linear modelling and nonparametric techniques, making use of appropriate transformations where necessary,
• explain key ideas underpinning parametric and nonparametric approaches, such as the impact of colinearity in linear modelling and how to select an appropriate value of the smoothing parameter in nonparametric problems,
• make inferences about the fit of a linear model, the values of its parameters, and of simple functions of the parameters, using confidence intervals and hypothesis tests,
• derive key theoretical properties of both parametric and nonparametric estimators, such as the form of the estimators and their asymptotic mean squared error,
• use the statistical software R to analyse real data using both parametric and nonparametric approaches.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Coursework: weighting 20%
• End of semester examination: three hours, weighting 80%

#### Syllabus

Linear Models

• General Linear Models: Least squares estimators (l.s.e) and their properties. Residuals and residual sum of squares. Leverage. Distribution of l.s.e and of the residual sum of squares. [5]
• The general linear hypothesis. Extra sum of squares, sequential sum of squares, partial sum of squares. The test statistic of the general linear hypothesis and its distribution. Confidence intervals and prediction intervals. [5]
• Linear regression: Simple regression, multiple regression, dummy variables and analysis of covariance. [6]
• Analysis of Variance. One and two way analysis of variance. Use of comparisons. Interactions. [6]

Nonparametric Regression

• Least squares regression, local averaging. [2]
• Local polynomial kernel regression. [3]
• Choosing the value of the smoothing parameter. [1]
• Variability bands, checking the validity of a parametric regression model. [3]
• Introduction to spline regression. [2]

• Weisberg, S., Applied Linear Regression J. Wiley 2005
• Montgomery, D. C. and Peck, E. A., Introduction to Linear Regression Analysis, J. Wiley 2001.
• Rawlings, J. O., Applied Regression Analysis: A Research Tool, Wadsworth and Brooks/Cole 1998.
• Bowman, A. W. and Azzalini, A. Applied Smoothing Techniques for Data Analysis. Oxford University Press (1998)
• Wand, M.P. and Jones, M.C. Kernel Smoothing. Chapman and Hall (1995)
• Eubank, R.L. Spline Smoothing and Nonparametric Regression. Marcel Dekkar
• Hardle, W. Applied Nonparametric Regression. Cambridge University Press (1991)

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 33 hours
• Tutorials - 11 hours
• Independent study hours - 106 hours

#### Teaching staff

Timothy Waite - Unit coordinator