video 11.3. within-subjects ANOVA

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There is one more kind of ANOVA that I want to talk about. Sometimes it is called a "within-subjects ANOVA" or "repeated-measures ANOVA". It is a lot like a paired-samples t-test in the way that the data is structured.

What we are going to do is work with another example where people have drank caffeinated or decaffeinated coffee. Then we measure their alertness. Here we are going to set up our data like this. This is our participant, and this is when the participant drank decaf. Here is when they drank caffeinated coffee. This is the drink variable. We will just have two participants to keep the math simple here. We will say when this person, person A, drank decaffeinated coffee, their alertness measure was 4. It was 6 when they drank caffeine. For the other person, it was also 4 when they drank decaffeinated coffee, and 10 when they drank the caffeinated coffee.

Here, just like with a paired-samples t-test, each person has got two different measures. Or this could have been, for example, a couple, where this is the husband and this is the wife. It would have paired that way, but the way it is set up you could consider this more of a repeated measure setup. Remember, with a paired-samples t-test, we calculated the difference scores, and then we did a one-sample t-test on those different scores. With the null hypothesis being that there is no difference. Well, with within-subjects ANOVAs, we are not going to do that. In fact, we are going to do an ANOVA like we are familiar with. Except we are going to treat the participants' data a little bit differently.

What we are going to do is we are going to treat participants as a factor (or as a variable). But, we are going to remove their sums of squares and degrees of freedom from the mean-square within. Our F statistic has this on the bottom. Our bottom is going to be extra small, because we are going to get rid of any of the variability that is consistent between our participants. By treating it as if ... We will calculate its sums of squares and degrees of freedom, just the same way we will with the drink. We will go through that in a second. That is why I wanted to do a really simple data set, because I want to show you how the math works. This simple demonstration within-subjects ANOVA. The easiest way to do it is with the sums of squares approach.

The first thing we are going to need to do is calculate our marginal means. 4+4=8, divided by 2. So the marginal mean here is 4. The marginal mean there, 16, divided by 2 is 8. The marginal mean over here is going to be 5. And the marginal mean here, 14, is 7. And then the overall mean is going to be 6. You could calculate as the average of these two, or the average of these two.

The first thing we will do is calculate the sums-of-squares total. Remember, this ignores any kind of grouping. For this we are going to take these four scores, calculate their sums of squares, with respect to the grand mean. Here we have 4 minus 6, squared, plus 4 minus 6, squared, plus 6 minus 6, squared, plus 10 minus 6, squared. Negative 2 squared, plus negative 2 squared, plus 0 squared, plus 4 squared 4 plus 4, plus 0, plus 16. That is going to be 24. Our sums of squares total is going to be 24. Remember, just like with the other ANOVAS we talked about, all of the sums of squares that exist across all the different ways of breaking down the data, those are all going to add to the sums-of-squares total. This is the total sums of squares available to work with.

Next, we will calculate the sums of squares for the drink variable. This is exactly like was for between-groups ANOVA. We will calculate the marginal means, and compare them to the grand mean. For each marginal mean, you do it one time for each data point in there. So 4 minus 6 (squared), plus 4 minus 6 (squared). And that is because there is two here. We do 8 minus 6, squared, two times. Once for each of these two. This is -2 squared, plus -2 squared, plus 2 squared, plus 2 squared. 4x4, that is going to be 16. So our sums of squares for the drink independent variable is going to be 16.
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