• 16. F the ratio of variance

Motivating scenario: You want to understand what goes into calculating F and where this logic comes from.

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

1. Explain how variance among sample means relates to variance in the population.
2. Define mean squares for the model and for error.
3. Interpret the \(F\) ratio as a comparison of variance among vs. within groups.
4. Recognize that if all groups represent samples from the same population, the expected value of \(F\) is one (allowing for sampling error).

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Predicting \(\sigma_\bar{x}\) from \(\sigma\)

Remember our “statistical” view of where data come from. We imagine that the data we observe are a sample of size \(n\) from a population with a true mean \(\mu\) and standard deviation \(\sigma_x\). Although we can’t know these true parameters, we estimate them using the sample mean \(\bar{x}\) and standard deviation \(s\).

We envision the distribution of sample means we would get by repeatedly sampling from the same population as the sampling distribution. If the population is normally distributed with standard deviation \(\sigma\), then the standard deviation of the sample means (aka the standard error) is:

\[\sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}}\]

Figure 1: The F distribution is magic.

Squaring both sides gives the variance of the sampling distribution of the mean as the population variance divided by the sample size:

\[\sigma_{\overline{x}}^2 = \frac{\sigma_x^2}{n}\] Multiplying both sides by \(n\) shows that the variance among sample means (for the same population), multiplied by the sample size, equals the population variance:

\[\sigma_{\overline{x}}^2 \times n = \sigma_x^2\]

Components of the \(F\) statistic

So we can turn these ideas into parameters to estimate.

Source	Parameter	Estimate	Notation
Model	\(n \times \sigma_\bar{x}^2\)	Mean square model	\(\text{MS}_\text{Model}\)
Error	\(\sigma^2_x\)	Mean square error	\(\text{MS}_\text{Error}\)
Total	\(n \times \sigma_\bar{x}^2 + \sigma^2_x\)	Mean square total	\(\text{MS}_\text{Total}\)

We can use these values to calculate the ratio of variance among groups to variance within groups, \(F\). Notice that when our two samples come from the same population, we expect \(F\) to equal one (save for some sampling error).

\[F = \frac{\text{MS}_\text{Model}}{ \text{MS}_\text{Error} }\]

In one-way ANOVA, what I call “Mean square model” (\(\text{MS}_\text{Model}\)) is often called “Mean squares group” (\(\text{MS}_\text{Groups}\)). I use mean square model to highlight that this extends beyond ANOVA to regression and other linear models.

Implications for NHST:

The derivation above assumes that samples are all drawn from the same population. So this assumption is how we generate our expectation from the null model:

• If all groups are drawn from the same population (the null hypothesis), then this equality holds: the variance among group means is exactly what we’d expect from sampling error.

• If groups come from different populations (the alternative), then the variance among group means will be larger than \(\sigma^2\).

In the next section we will see how we can use this framework to test the null that all samples come from the same statistical population.