16. F this!

Motivating scenario: You can compare two groups a two-sample t-test. You have also been told that a two-sample t-test is “just a specific linear model.” So you are now ready to build from this t-test understanding to the wider world of linear models.

Learning goals: By the end of this chapter, you should be able to

  1. Explain how total variability can be partitioned into model (explained) and residual (unexplained) components, and interpret visual representations of this partitioning.
  2. Calculate and interpret key values in an ANOVA analysis including:
    • Mean squares for the model and error terms.
    • The F statistic as the ratio of model to error mean squares.
    • \(R^2\) as a measure of effect size in one-way models.
  3. Recognize the connection between the two sample t test and the F test when comparing two groups.
  4. Recognize the assumptions of ANOVA and interpret correlations with care.
A picture of one white and one pink parviflora flower.
Figure 1: Our white and pink parviflora flowers!
The \(F\) distribution is often introduced after the t-distribution to clear a path to comparing means of more than two groups. This makes sense, as the \(F\) distribution lends itself to this application. But, in my view at least, that introduction to the \(F\) does it a disservice. More than simply offering a way to compare more than two groups, the \(F\) provides a new way to think about variability in data. Specifically, the \(F\) allows us to “partition variance” into components associated with predictors in our model and “residual” variance that is not in our model.


In this chapter, we focus on the simplest case, comparing two groups, to better understand what the F statistic represents (or What the \(F\) is going on!) We’ll connect this to concepts you already know (t-tests, sampling distributions), work through the math of partitioning variance, learn how to run ANOVA in R, and end by showing how these methods are all part of a unified linear model framework.

The utility of \(F\)

In the coming chapters you will see that this view of “partitioning variance” has broad utility. In addition to its utility in partitioning variability within and among two groups, the approach in this chapter can be applied to

  • More than two groups.
  • A numeric explanatory variable.
  • Multiple explanatory variables, and even
  • Explanatory variables whose interaction influences the response variable.

We will work through these applications in coming chapters!

Data and hypotheses for this section

In the previous section, we saw that at Sawmill Road, pink parviflora RILs attracted more pollinators than did white parviflora RILs. We cared about this because we want to know if petal color influences hybridization – a question we will address later.

But there is a bigger motivating question than what happens in an experiment. That is, we want to know what has actually happened (and what is happening) in nature. In this chapter we get at the question of “does petal color influence hybridization?” in a somewhat backwards way. Instead of asking if pink flowers set more hybrid seed than white flowers, we analyze sequence data from pink and white flowered parviflora found in natural hybrid zones and ask if one petal color is associated with more introgression of xantiana ancestry.

Introgression is the incorporation of ancestry from one lineage into the genome of another. This happens through hybridization and subsequent backcrossing.

What’s ahead

In this section

  • We go through the mathemagical foundation of partitioning variance and how this relates to our understanding of populations and samples as draws from the sampling distribution.

  • We then show how to calculate the sum of squares error, sum of squares model, and related measures.

  • Next, we introduce \(R^2\) as the “effect size” in such a model, and provide guidance on how to interpret this common summary.

  • We then show how to use this partitioning of variance to test the null hypothesis that all groups represent samples from the same population.

  • Before the chapter summary, we reflect a bit on correlation, causation and artifacts that may influence the patterns that we observe.