Chapter 13
LURN… To Perform Analyses of Variance (ANOVA)

This chapter explains the basic use of the Analysis of variance (ANOVA) model for analyzing appropriate data. This is commonly (but not always) obtained through experimentation. To truly learn about the appropriate analysis of experimental data you will need to consult a suitable textbook on experimental design.

As there are insufficient data sets in the datasets package, the examples in this chapter draw on data supplied with the MASS package. This package supports a very famous resource by Venables and Riply, often abbreviated to V&R. The full reference is:

Venables, W.N. and Ripley, B.D. (1999) Modern Applied Statistics with S-PLUS. Third Edition. Springer.

This book is very useful for many topics in statistics and given the similarity of the syntax between S-PLUS and R, remains a good resource for users of R today.

13.1 Data and suitable exploratory graphs

The data used to fit models discussed in this chapter require a single response variable which is assumed continuous, and one or more explanatory variables that are discrete valued. More specifically, these explanatory variables are called factors; in an experiment, each observation takes a preplanned level for each factor under consideration.

13.2 One-way ANOVA

13.3 Model validation using diagnostic plots

Readers should note that the same code was used to generate the residual analysis plots as would be done for a linear regression model. Note that the plot() command is sensitive to the type of object being used, and that the set of residual graphs is therefore different to those created for an lm object.

13.4 Two-way ANOVA

There are two different situations where a two-way analysis of variance is called for. First, a simple randomized complete block design and second, a factorial experiment with two factors.

13.4.1 Randomized complete block designs

We use the sleep data used to show the analysis of paired data in Section11.2.1, which has two measurements on ten individuals. As the two observations on each person are often thought to be related, we use the known structure of the experiment’s design to incorporate this knowledge. The individual people are therefore thought of as a blocking variable. Rather than using the attach() command, we employ the data argument inside the aov() command. As before, we use the summary() method to present the analysis:

> summary(aov(extra~ID+group, data=sleep))

            Df Sum Sq Mean Sq F value Pr(>F)
ID           9   58.1    6.45    8.53 0.0019 **
group        1   12.5   12.48   16.50 0.0028 **
Residuals    9    6.8    0.76
Signif. codes:
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The group variable is the treatment variable for this data set. Note that (depending on the level of significance required) leaving the ID variable out of the analysis could lead to a different conclusion for this experiment. The above analysis shows that there were differences among the ten individuals as well as a significant difference between the two treatments.

13.4.2 Factorial design with two factors

13.4.3 Graphing the results for an experiment with two factors

13.4.4 What is an interaction?

13.4.5 Two-way ANOVA with no interaction

13.4.6 Two-way ANOVA with an interaction