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StatsExamples-f-test-examples.pdf

TRANSCRIPT OF VIDEO:

Slide 1.

The F-max test is used to test when a set of more than two population variances may have any that differ from each other. Let's take a look at some examples of how to do this test.

Slide 2.

First let's review the reasons behind this test.
We want to know if population variances differ in a set of more than two populations.
We can't measure the populations so we take random samples from the populations and calculate sample variances.
We could compare each pair of variances using the variance ratio F-test, but if we do all the possible pairwise comparisons, the risk of type I error is too high. This is because the probability of a type I error for each test is alpha and they will add up.
We therefore use a test which compares the largest variance to the smallest to see if at least one pair of variances differ, that test is the F-max test.

Slide 3.

Here's the formal procedure.
First, we create our null and alternative hypotheses. For an F-max test, the null hypothesis is that the population variances are all equal to one another and the alternative hypothesis is that at least one is different from at least one other.
Then we calculate our F-max value, which is just the largeest of the set of variances divided by the smallest of the set of variances.
Then we compare the F-max value to F critical values from an F-max table.
Critical values depend on the number of groups and sample sizes for each group. These tables, like the one shown from the StatsExamples website, are often arranged with separate columns for each number of groups and each row corresponding to the degrees of freedom for each group.
Then, we determine the P-value, the probability of seeing an F calculated value as large as we do.
This is the smallest alpha value we can choose and reject the null hypothesis, in other words, the P-value is the probability that we would get an F-max value as large as we do if all the populations have the same variance.
Then we decide to "reject the null hypothesis" or "fail to reject the null hypothesis" based on the P-value.
The null hypothesis is consistent with p values that aren't small.
The alternative hypothesis being true, if the variances aren't all equal, would usually give us small p values.
At this point we would return to our data and ask ourselves what it means that the population variances appear the same or some appear different.
Some statistical tests, the ANOVA in particular, require equal variances and this test tells us if we can do an ANOVA of our data or have to transform that data in some way before doing an ANOVA.
Lastly, we should always keep in mind the risk of type one or type two error when doing any statistical test.

Slide 4.

For our first example, let's think about 4 samples, each of which have 6 values from their populations.
The questions we will ask are:
Are any of the population variances different from one another?
And.
With what degree of confidence do we make this conclusion?
► The first step will be to calculate the means and variances for each sample. If you need a reminder of how to do this, see our video about calculating summary statistics. We won't be testing for equality of the means, but it's nice to see these values to help think about our data.
Then we calculate the F-max ratio, putting the largest of the 4 values in the numerator and the smallest in the denominator.
► For our data this ends up being 14 divided by 4 which gives us an F-max value of 3.50.

Slide 5.

Now we need to compare our calculated F-max value to the appropriate critical values.
► We have 4 groups and each group has 6 values which gives us 6 - 1 equals 5 degrees of freedom for each group.
► Looking at our F-max critical values tables we look at the columns for 4 groups and the rows for 5 degrees of freedom. The critical values are 13.70 for an alpha value of 0.05 and 49.00 for an alpha value of 0.01.
Comparing our F-max calculated value of 3.50 to these values we see that it is smaller than the 13.70 which indicates that the P-value for this F test will be more than 0.05 and we will get a non-significant result.
► We can therefore say that "The variances of these populations are not significantly different from one another ( p > 0.05 )."
The data that we have is not enough to convince us that any of the variances are different so we would thereby assume that they can be treated as equal for the purposes of techniques like the ANOVA.

Slide 6.

For our second example, let's think about 3 samples, each of which have 5 values from their populations.
The questions we will ask are:
Are any of the population variances different from one another?
And.
With what degree of confidence do we make this conclusion?
► As before, the first step is to calculate the sample means and variances so we can get the F-max value.
Looking at the variances, the 102.50 is the largest and the 6.00 is the smallest.
► The calculated F-max value is therefore 102.50 divided by 6 equals 17.08.

Slide 7.

Now we compare our calculated F-max value of 17.08 to the appropriate critical values to answer our questions.
► We have 3 groups and each group has 5 values which gives us 5 - 1 equals 4 degrees of freedom for each group.
► Looking at our F-max critical values tables we look at the columns for 3 groups and the rows for 4 degrees of freedom. The critical values are 15.50 for an alpha value of 0.05 and 37 for an alpha value of 0.01.
Comparing our F-max calculated value of 17.08 to these values we see that it is larger than the 15.50 which indicates that the P-value for this F test will be less than 0.05, but it's not as large as 37 so the P-value will be larger than 0.01.
► We can therefore say that "At least one of these variances is significantly different from at least one of the others ( 0.01 < p < 0.05 ).."
The data that we have convinces us that the variances aren't all equal. This would mean that we can't use this data for a technique that requires equal variances like the ANOVA..

Slide 8.

Let's take another look at our data and see what we can do.
For these sets of data we can't use any homoscedastic techniques, the ones that require equal variances. Luckily, there is something we can do, we can transform the data values.
There are lots of transformations that are used to equalize variances, we'll try one of the most common.
► Let's use a square root transformation for the data. This will give us a set of new values we can then test for equality of variances.
► The first value in our new data sets would come from the first value in our original set. We take the square root of the 16, which gives us a 4 for the first value in our transformed samples.
► The second value is the square root of 19, which gives us a 4.3589 for the next value in our transformed data set.
► Then the square root of 22, which gives us 4.6904.
► Then the square root of 21, which gives us 4.5826.
► Then the square root of 22, which gives us 4.6904.
► Doing this for the rest of the values gives us these new transformed samples.
One thing to keep in mind that if we decide to do a transformation, we need to transform all the values, we can't pick and choose to transform some data sets and not others.
► Now we calculate the means and variances of these transformed sample for the F-max value.
Looking at the variances of our transformed data sets, the 1.0592 is the largest and the 0.0857 is the smallest.
► The calculated F-max value is therefore 1.0592 divided by 0.0857 equals 12.35.

Slide 9.

Now we compare this calculated F-max value of 12.35 to the appropriate critical values to answer our questions.
► As before, we have 3 groups and each group has 5 values which gives us 5 - 1 equals 4 degrees of freedom for each group.
► Looking at our F-max critical values tables, the critical values are the same as before, 15.50 for an alpha value of 0.05 and 37 for an alpha value of 0.01.
Now when we compare our F-max calculated value of 12.35 to the critical values its smaller than the 15.50 which gives P-value larger than 0.05 and a nonsignificant result.
► "The variances of these transformed populations are not significantly different from one another ( p > 0.05 )."
The transformed data would now satisfy the assumptions of statistical tests that require equal variances, we could do an ANOVA on these transformed values.
Strictly speaking, any conclusions we make using the transformed data only apply to the transformed values, but this can still be useful.
For example, if we just wanted to know if the average sizes in some areas were different, we may not care if the raw values were different or the square roots of the values were different - both results show some kind of difference.

Slide 10.

Finally, a couple of cautions about the F-max test.
A strong assumption of the F-max test is normal population distributions.
If the populations are not normally distributed, then the F-max test can easily give type one or type two errors.
For example, if we look at the distributions of the example data sets used in the video, they aren't normal.
They were fine for showing how this techniques works, but since they weren't normal, the second data set especially, they aren't really appropriate for the F-max test.
When this happens, we prefer a Levene's, Bartlett's, or Brown-Forsythe test, but the math is more complicated.
Also, we need to keep in mind that if we transform the data, our subsequent tests apply to the transformed values, not the originals.
Results from a one-factor ANOVA on square root transformed data sets would tell us about differences in the means of the square roots of the values, not the original values.
If we just want to know if there are differences in the average size in a general way that's probably fine, but if the question we care about is directly linked to the untransformed values then our results from the transformed values might not answer our question.

Zoom out.

I hope you found these examples of the F-max test useful.
As always, a PDF of this slide, organized links to other videos, and other useful resources are available on the StatsExamples website.

End screen.

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