Module Directory

This module is concerned with the application of linear models to the analysis of data. The underlying assumptions are discussed and general results are obtained using matrices.

The standard approach to the analysis of normally distributed data using ANOVA is introduced. Methods for the design and analysis of efficient experiments are introduced.

The aim of this module is:

• To provide the essential foundations of linear models by studying important topics of statistical modelling achieved by an in-depth study of the main methods to analyse experimental data.

By the end of this module, students will be expected to be able to:

1. Calculate confidence intervals for parameters and prediction intervals for future observations.


2. Understand how to represent a linear model in matrix form.


3. Check model assumptions and identify influential observations.


4. Identify simple designed experiments.


5. Construct factorial experiments in blocks.


6. Work efficiently in small groups to analyse data.


7. Analyse linear models using R.

Indicative syllabus

1. Data Analysis Fundamentals
   
   
   
   • Understand the aims of Data Analysis.
   
   
   • Be able to identify and apply the different stages of model building.
   
   
   • Able to identify and manipulate different data sources and files (CSV, text, tab).
   
   
   • Foundations of reproducible research.
   
   
   
   


2. Understand Variables in Regression Analysis:
   
   
   
   • Define and differentiate between response and explanatory variables in the context of linear regression.
   
   
   
   


3. Simple and Multiple Linear Regression Models:

Model Structure:

• Simple Regression: Understand a simple linear model, the representation, the role of the slope and the intercept.


• Multiple Regression: Extend the simple linear model to a domain where there are more than one explanatory variables.


• ANOVA: One way ANOVA, scope, computation and implementation in R

Calculation and Application Using R Software:

• Least Squares Estimates: Estimate the parameters using the least squares method and maximum likelihood.


• Software Application: Employ R statistical software to fit both types of models to data. This process involves estimating parameters and interpreting the output.

Interpretation and Evaluation:

• Statistical Inference: Perform statistical inference on parameters, particularly the slope in simple regression, to understand their significance.


• Goodness of Fit: Evaluate the model's fit using statistical measures (like R-squared) to determine how well the model explains the variability of the response variable.


• Prediction: In simple and multiple regression, predict responses with confidence intervals using the fitted linear relationship.

Model Assessment:

• Model Selection in Multiple Regression: Use measures of model fit to choose appropriate explanatory variables, ensuring the model is neither overfitted nor underfitted.


• Prediction: Be able to perform and assess prediction based on the fitted model(s)


• Residual Analysis: For both models, assess suitability and validity through residual analysis, checking for patterns that might indicate violations of regression assumptions.


• Variable Selection: Use of Cp, AIC, BIC and shrinkage methods (Ridge regression and Lasso regression).

Dimensionality Reduction:

• Introduction to the curse of dimensionality and to Principal Component Analysis

4. Generalized Linear Model
   
   
   
   • Introduction to the Exponential Family of distributions in the canonical form.
   
   
   • Fit of generalized linear models using R in case of Normal, Binomial, Poisson and Gamma distribution and interpret the results.
   
   
   • Define the Pearson and deviance residuals and howe they may be used.
   
   
   • Apply measures of model assessment: Pearson’s chi square and likelihood ratio test.
   
   
   
   

Teaching in the School will be delivered using a range of face to face lectures, classes and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.