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Chapter 2   Experiments with one factor

2.1   Introduction

2.1.1   Recap of notation

The terminology used for experiments is first reviewed -- experimental units, controlled variables and a response.

2.1.2   Experiments in practice

In practice, experiments can be quite complex. A few examples are described within the framework described in the previous page.

2.1.3   Simple experiments

Extremely simple experiments involving homogeneous experimental units and a single controlled variable will be used to introduce many concepts and methods that also apply to more complex experiments.

2.2   Design and estimates

2.2.1   Completely randomised experiment

In a completely randomised experiment for a single factor, the different levels of the factor are randomly allocated to the pool of experimental units.

2.2.2   Treatment means

The mean responses at the different factor levels give a summary of the response variation that has been explained by the factor.

2.2.3   Normal model for variation

To fully interpret differences between the treatment means, unexplained variation must also be taken into account. Both explained and unexplained variation can be modelled using a normal model with separate terms for these two types of variation.

2.2.4   Describing unexplained variation

The amount of unexplained variation is given by the standard deviation of the error term in a normal model. A pooled estimate can be found by combining the observed standard deviations within the separate factor levels.

2.2.5   Confidence intervals for treatment means

95% confidence intervals for the treatment means take unexplained variation into account and make it easier to compare the factor levels.

2.3   Analysis of variance

2.3.1   Variation between and within treatments

The evidence for a difference between the response mean for different levels of the factor depends on both variation between the treatment means and also variation within each treatment.

2.3.2   Sums of squares

Explained variation is summarised by a sum of squares involving only the treatment and overall means. Unexplained variation is summarised by the residual sum of squares and is found from differences between values and their treatment means. Their sum is the total sum of squares.

2.3.3   Coefficient of determination

The coefficient of determination is the ratio of the explained sum of squares to the total sum of squares. It is a proportion between 0 and 1 and summarises the strength of the relationship between the factor and the response.

2.3.4   Test for differences between levels

The sums of squares and their degrees of freedom can be arranged in an analysis of variance table from which a p-value can be calculated to test whether the factor affects the response.

2.3.5   Examples

This page shows analysis of variance tables for a few data sets and describes the conclusions that should be reached.