All the ANOVA stuff we have done so far has had different subjects in the various cells of the experimental design
That kind of experiment is called a between-subjects design
Sometimes, however, we run the same subjects in some or all cells of the design
Such a within-subjects (or repeated measures) design has two advantages:
- you get more data per subject
- you actually get more power because you can factor between-subject differences out of the error term
Memories of the ANOVA logic
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
As you will see, repeated-measures designs allow us to reduce the error term, thereby resulting in larger Fs (more power)
An example: Within versus Between
Between-Subjects Version
1
2
3
4
5
| X X2
5 25
4 16
6 36
4 16
7 49
|
1
2
3
4
5
| X X2
7 49
6 36
5 25
6 36
6 36
|
1
2
3
4
5
| X X2
5 25
3 9
7 49
2 4
3 9
| X X2
7 49
4 16
7 49
4 16
5 25
|
Computations for Between-Subject
How is the within-subjects version different from the between-subjects version?
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 )
Getting Rid of the Variability Due to Subjects
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:
1
2
3
4
5
| X X¢
5 -1
3 -.5
7 0
2 -1
3 -1
| X X¢
7 +1
4 +.5
7 0
4 +1
5 +1
|
Within-Subjects Computations
- Another way of doing what is essentially the same thing is to remove the sum of squares due to subjects from the error term.
Source Tables
- Between-Subjects
| ||||
| ||||
|
| |||
|
- Within-Subjects
| ||||
| ||||
| ||||
|
|
|
| |
|
The Advantage of Within-Subject Designs
While removing the sum of squares due to subjects does make the observations independent across levels of the treatment variable, it OFTEN reduces the MSerror, thereby resulting in increased power (larger F values)
This only occurs though if the reduction in MSerror is more than compensates for the loss in dferror .. 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
Assumption of Compound Symmetry
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 of these variables is computed as:
But what does it mean?
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
The Computational Formula for Cov
Back to Compound Symmetry
| |||
- S X = 700 + 520 + 600 + 650 + 750 = 3220
- S Y = 650 + 450 + 540 + 630 + 700 = 2970
- S XY = (700 * 650) + �. (750 * 700) = 1947500
The Covariance (Variance/Covariance) Matrix
The variances need not (and often do not) equal the variances though
Complicating it all
However, as we saw in Chapter 13, studies usually manipulate more than one variable which raises several possibilities
2 variables
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
One Between - One Within
Similar to Siegel�s morphine tolerance study, King (1986) was interested in conditioned tolerance to another drug � midazolam
- initially midazolam decreases motor activity
- however, tolerance develops quickly
- 3 groups � 2 got 2 injections of midazolam prior to test .. the other (the control group) got saline injections
- at test, all groups got midazolam, but one of the experimental groups received it in the same context as the had before (same group) whereas the other received it in a different context (the different group)
- motor activity measured in 6 five-minute intervals producing the following data
The Data, Steve Style
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
|
|
| ||||||||
The Dreaded Computations
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
Between-Subject Effects
Within-Subject Effects
The Within-Subject Error Term
Given the computations we have done so far, we can get the rest by subtraction �
| ||||
| ||||
| ||||
|
| |||
|
|
|
|
|
| ||||
| ||||
| ||||
|
| |||
|
|
|
3obtained by subtracting SSinterval and SSgrp * int from SSwithin
4obtained by subtracting dfgroup, dfss/group, dfinterval and dfgrp * int from dftotal
Conclusions from the Anova
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.
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.
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
Recall that there are two ways we could approach these analyses, we could ask
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
Simple Effects for Within-Subject Variables
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 SSs/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
Simple Effects for Between-Subject Variables
Step 1: Computing sums of squares for the effect of group at each interval
Step 2: Mean Squareds for the group effects at each interval
MS = SS/df, so �
The appropriate error term SS is the SSSs/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:
dfSs/Cell = dfSs/Group + dfSs/Grp X Int
Step 4: Source table depicting results
| ||||
| ||||
| ||||
| ||||
| ||||
| ||||
| ||||
|
|
|
|
|
|
| |||
|
0 comments:
Post a Comment