Recall that the purpose of doing an ANOVA is to see if some difference between treatment means is sufficiently large is to be unlikely to occur by chance (less than 5% chance)

When we test that … we get an estimate of the difference we are interested in, and divide it by an estimate of variation due to chance

Specifically:

Notice that this F value will increase if the difference between the means is large OR if the measurement of error is small

As you will see, repeated-measures designs allow us to reduce the error term, thereby resulting in larger Fs (more power)

This experiment will show the importance of the "articulatory loop" for retaining information in short-term memory

Between-Subjects Version

Within-Subject Version

An assumption of the between-subjects ANOVA is that the observations in one level of the treatment are independent of those in the other level(s)

Hopefully you will notice that this assumption does not hold in our within-subjects version of the experiment 

The use of the same subject in more than one level of the treatment almost always builds in a dependency because subjects who do well in one level tend to also do well in the other(s)

Can we remove this dependency? In fact we can, and when we do, there is a bonus! (the kind of thing that makes statistics geeks real happy J )

The "dependency in observations" is due to some subjects doing better than others

What we are going to do to deal with this is to literally remove the variation due to subjects from the error term

For demonstration purposes only .. you can think of this as subtracting each subjects mean from all the scores they contribute

Using the data from out class:

Remember, F values are increased if the difference of interest in larger OR if the measure of variance (MS_error) gets smaller

While removing the sum of squares due to subjects does make the observations independent across levels of the treatment variable, it OFTEN reduces the MS_error, thereby resulting in increased power (larger F values)

This only occurs though if the reduction in MS_error is more than compensates for the loss in df_error .. so it is not always true

Note that you cannot remove the variance (sum of squares) due to subjects when using a between subjects design because you only have one observation per subject … thus the variance due to subjects must remain as part of the error term

Moral: Usually, it is better to use within-subject (repeated measures) designs … not only do they let you use less subjects, but they are also more powerful, statistically speaking

Remember when we did between-subject ANOVAs, one of the assumptions was that the variance in our various treatment groups were homogenous (i.e., roughly equivelant)

A similar but slightly more complex assumption underlies repeated measures designs

Specifically, we need to satisfy the "compound symmetry" assumption which is that in addition to the variances being equal, the covariances between pairs of variables are also equal

For this to make sense, I think we may have to do a B07 time travel to re-introduce the notion of covariance ….

Imagine any two variables such as …

The covariance formula should look familiar to you. If all the Ys were exchanged for Xs, the covariance formula would be the variance formula

Note what this formula is doing, however, it is capturing the degree to which pairs of points systematically vary around their respective means 

If paired X and Y values tend to both be above or below their means at the same time, this will lead to a high positive covariance

However, if the paired X and Y values tend to be on opposite sides of their respective means, this will lead to a high negative covariance

If there is no systematic tendencies of the sort mentioned above, the covariance will tend towards zero

Given its similarity to the variance formula, it shouldn’t surprise you that there is also a computationally more workable version of the covariance formula:

For our height versus weight example then:

OK, now let’s assume we ran a repeated measures study in which we were looking at practice effects on some task over 3 days

These variances and covariances are often presented in a matrix such as the following:

So, the assumption of compound symmetry is simply that the variances must all be approximately equal and the covariances must all be approximately equal

The variances need not (and often do not) equal the variances though

So far in this chapter, we have been dealing with only one variable that has been manipulated in a within-subject manner

However, as we saw in Chapter 13, studies usually manipulate more than one variable which raises several possibilities

2 variables

3 variables

Computationally, we will only focus on the 2 new "2 variable" situations

However, as was the case with 3 between subject variables, I will expect you to be able to interpret 3 variable results … we will spend time doing this as well

Imagine the following study (raw data is presented in the text, pp. 459)

Similar to Siegel’s morphine tolerance study, King (1986) was interested in conditioned tolerance to another drug … midazolam

Just like when we had two between-subject variables, there are three effects of interest in the current experiment:

However, recall that we can (an do) use a different error term when testing within-subject effects than when testing between subject effects

So, the first thing we must do is to decide which effects are purely between-subjects, and which have a within-subject component

For this study, Group was manipulated between-subjects, but both Interval and the Group x Interval interaction have a between subjects component (i.e., Interval)

OK, now we separately deal with our between and within-subject effects

We treat between subject effects like we always have. We calculate SS_treat as the sum of squares of the treatment means times the relevant n, and we calculate SS_error as the sum of the variance of subjects within the group

OK, for starters, the sums of squares for the Interval and interaction effects are calculated like we did in the 2 between-subject case

Remember than when we are dealing with within subject effects, we use a different error term (one that does not include the variability due to subject by subject variation)

Given the computations we have done so far, we can get the rest by subtraction …

²obtained by subtracting SS_between from SS_total

³obtained by subtracting SS_interval and SS_{grp * int} from SS_within

⁴obtained by subtracting df_group, df_ss/group, df_interval and df_{grp * int} from df_total

Critical F’s:

Main Effect of Group 

We can reject the null hypothesis that there was no effect of group. The F-obtained for the main effect of group was greater than the critical F suggesting the there are differences among the three group means. From looking at the means it appears that this is mostly due to the mean for the "Same" group being much higher than the other two means.

Main Effect of Interval

We can also reject the null hypothesis that there was no effect of interval. The F-obtained for the main effect of interval was greater than the critical F suggesting that there are differences among the six interval means. From the means, it appears as though activity was very high in the first interval, then dropped of and stayed relatively constant.

Interaction of Group * Interval

Finally, we can also reject the null hypothesis that the effect of interval was the same for the three groups. The F-obtained for the interaction was greater than the critical F suggesting that the effect of interval is different for the three groups. From the means, it appears as though the "Same" group stayed active longer (across more of the early intervals) than the other groups.

Simple Effects for the effect of time at each level of group.

So, we could describe the interaction by saying that fear increased over time for phobics, but fear did not change at all over time for the controls

As was the case when we had two between subject variables, we will often want to do simple-effects analyses to gain a better understanding of the interaction 

Recall that there are two ways we could approach these analyses, we could ask

Here it makes sense to look at the interaction and consider the experimental predictions to determine which of these approaches is likely to yield the information you want

Since the predictions are focused primarily on potential differences between groups (or lack of differences), the first approach is the one we would want to take in this case 

Nonetheless, we will briefly consider both situations

We had decided that in our situations we were not interested in looking at the effect of interval separately for each group

But, if we had been, then we would have been examining the effect of a within-subject variable (interval)

For reasons that are not important, whenever you are doing simple-effects that are focused on the effect of a within-subject variable, you cannot use some general error term (like, for example SS_{s/grp * int})

Instead, what you do is a separate one-way, repeated measures analysis of variance for each simple effect

So, for example, if you were interested in the effect of interval for the control group, you would run a complete repeated measures ANOVA examining the interval variable but using only the data from the control group

Since there are three groups at each interval, there are 2 degrees of freedom for each contrast

MS = SS/df, so …

OK, here is where we differ from the Chapter 13 way of doing things

The appropriate error term SS is the SS_Ss/Cell

We could calculate that by hand but it would take a lot of work

In the "trust me" category, I give you the following:

So, for our example …