## Analysis of Covariance (ANCOVA)

This section of the website is being **actively developed**. Please send any comments/suggestions to graham.currell@uwe.ac.uk

**Study Text**: "Essential Mathematics and Statistics for Science", 2nd ed, G Currell and A A Dowman (Wiley-Blackwell)

### Quick Link

**Video answer to Case Study: **Use of Minitab to perform Analysis of Covariance (ANCOVA)

### Introduction

A co-variate is an additional numerical factor that may affect the experimental value being measured.

In some cases it may not be possible to control the value of this co-variate in an experimental design, but it is still necssary to take it into account when analysing the results of the analysis.

See:

Introduction to the Analysis of Covariance (ANCOVA) Excel file: ANCOVA.xls

**Case Study**

The aim of a student forensic project is to test whether there is a significant difference between the numbers of fibres transferred between different pairs, A and B, of fabrics of different types.

In separate measurements, the materials are brought together with different pressures created by a beaker filled with different amounts of water.

However, the experiment is badly designed in that the different pressures are not set at specific pre-determined values for both pairs of fabrics, but with ranges of random values.

The experimental results are as follows:

Pair A | 'Pressure' (kg of water) | 0.9 | 1.0 | 1.5 | 1.6 | 1.9 | 2.1 |

Number of fibres | 46 | 38 | 60 | 72 | 73 | 89 | |

Pair B | 'Pressure' (kg of water) | 0.8 | 1.1 | 1.4 | 1.5 | 1.8 | 2.2 |

Number of fibres | 33 | 40 | 45 | 53 | 74 | 77 |

A two sample, 2-tailed, *t*-test for the numbers of fibres (ignoring the effect of pressure) gives a *p*-value = 0.405, which does NOT give a significant difference between A and B.

However, pressure appears to have an effect on the numbers transferred, and obviously needs to be taken into account. Can we use a 2-way ANOVA with fabric type as one factor and pressure as a second factor?

Enter the data into Minitab to find out why a 2-way ANOVA will not work.

Pressure is a co-variate in this problem. The number of fibres transferred is related to the value of pressure.

Use an ANCOVA, with pressure as the covariate, to test whether there is a significant difference in the numbers of fibres transferred between the different pairs of fabrics.

**Video answer: **Use of Minitab to perform Analysis of Covariance (ANCOVA)

**NOTE**: In good experimental design, it would be important to **choose** the values of the relevant factors (e.g. pressure) to produce a balanced design and use an ANOVA, rather than use the ANCOVA as in the above example.

The ANCOVA should only be used when it is not possible to control the values of the co-variate.

This on-line Study Guide has been developed by Graham Currell in association with:

University of the West of England,

"Essential Mathematics and Statistic for Science", 2nd Edition,

Graham Currell and Antony Dowman, Wiley-Blackwell, 2009