Lecture
10
WithinSubjects Analysis of Variance
When to Use
WithinSubjects ANOVA
The withinsubjects ANOVA is
really just an extension of the paired ttest
discussed earlier. It is used whenever there are repeated measures or repeated
treatments of an individual, paired participants (e.g., triplets), or matched
participants (e.g., four participants are yoked according to the amount of
daily exercise they get). Daniel describes two forms of the withinsubjects
design, the design he calls the "randomized complete block design"
and repeated measures. The randomized complete block design is a study in
several participants are matched or yoked together (e.g., on age), then one
member of each grouping is randomly assigned to each of the treatment
conditions. If there are three treatment conditions (e.g., three different
drugs are given), each person in a group of three would be sent to a different
condition. But in order for it to be valid to use a withinsubjects analysis,
the groupings of participants has to be done on some basis. For instance,
triplets or three people matched because they have the same number of years
experience on the job. Repeated measures, on the other hand, indicates that
each person is measured several times. In general, the withinsubjects ANOVA
applies to matching, repeated measures, repeated treatments, or any time
participants are yoked together.
An example of a withinsubjects
design would be a hypothetical study on cholesterol which has four treatments:
high oat fiber diet, exercise, a low fat diet, and low calorie diet. In a
betweensubjects version of the study, different participants would be in each
of these four conditions. In a withinsubjects version of this study, each
participant might be exposed to each of these treatments for a period of time.
So, if I was in the withinsubjects cholesterol study (or, to be more specific,
we would call this a repeated measures design), I would spend a month eating oatmeal
for breakfast. Then, the next month, I would stop eating oatmeal, and begin an
exercise regimen. The following month, I would stop exercising, and start
reducing fat intake. And so on.
One can think of the scores on the
cholesterol measure as blocked together according to each individual. Each
person has four scores. At the end of each month (and, hence, each treatment) a
cholesterol count is taken. In a way, each participant has a block of four
scoresso you can see the similarity to the randomized block design.
Like the betweensubjects ANOVA and
the betweensubjects ttest, the withinsubjects ANOVA and the withinsubjects
ttest are related. The withinsubjects ANOVA can be used for two or more
groups, and when used with two groups the ttest and the Ftest will lead to
the same conclusion ( and ).
The same benefits of the
withinsubjects ttest over the betweensubjects ttest still apply. We need
fewer participants overall and we have more power, because each individual (or
block) acts as its own control.
The Analysis
The analysis approach is similar to that of the paired (or withinsubjects or
matched) ttest. We are intrested in the differences between scores for an
individual (remember when we calculated the difference score, d, for each
individual?). We will also use a similar ANOVA logic to the logic we used with
the betweensubjects ANOVA. We will calculate the variation of the scores for
an individual relative to the mean of scores for that individual. In other
words, we want to know how much each participant's scores change from treatment
to treatment. If one or more of our cholesterol treatments had an effect, we
would expect cholesterol counts to change fairly dramatically from
monthtomonth.
A Note on Notation
The notation for this analysis does not change dramatically, but we need to
extend it a little. Previously, we used a dot to indicate that we had combined
scores across individuals. For example referred to the sum of individual scores in
a group. This time we will do some adding the other way, so that we combine
scores across treatments rather than individuals. So, the sum of scores for one
participant in the study, summed across treatments is symbolized by The sum of
cholesterol counts for the second participants who has completed all four
treatment conditions would be symbolized as Thus, stands for the sum of individuals for the
second treatment, and stands for the some of treatments for participant 2. Similary,
represents
the mean cholesterol score for participant 2 across all treatments, and represents the
mean of all participants for treatment number 2.
Example
Participant # 
Oat Fiber 
Exercise 
Low Fat 
Low Calorie 


1 
180 
200 
160 
200 
740 
185 
2 
230 
250 
200 
220 
900 
225 
3 
280 
310 
260 
270 
1120 
280 
4 
180 
200 
160 
200 
740 
185 
5 
190 
210 
170 
210 
780 
195 
6 
140 
160 
120 
110 
530 
132.5 
7 
270 
300 
250 
260 
1080 
270 
8 
110 
130 
100 
100 
440 
110 
9 
190 
210 
170 
210 
780 
195 
10 
230 
250 
200 
220 
900 
225 

2000 
2220 
1790 
2000 



200 
222 
179 
200 


Now we have several sums of squares
to compute.
Formula 
Name 
Description 
Example 

Sum of Squares Treatment 
Represents variation due to treatment effect 


Sum of Squares Block 
Represents variation within an individual (within block) 


Sum of Squares Error 
Represents error variation 


Sum of Squares Total 
Represents total variation 

Now for the mean squares and the
computed Fvalue:
SSTr=924.75 
k  1=4  1=3 


SSBl=25705.63 
n  1=10  1=9 
not needed 

SSE=88067.12 
(n  1)(k  1) =(10  1)(4  1)=27 


SST=114.697.50 
kn  1 = 40  1 = 39 


Interpretation
To check whether the calculated
Fvalue of 31.75 is significant, we compare that value to the critical value of
obtained
from Table G, using the table labeled .95 (for a=.05) and 3 and 27 d.f.
Our test is significant, indicating that there were differences in cholesterol
treatments that are unlikely to be due to chance. What if we had set alpha to a
lower value, say .01, would the test still be significant? To see, look up the
same d.f. values in the .99 (a = .01) table. If our calculated value
exceeds that table value, we would have significance at p <.01. Often
authors report the lowest pvalue at which they would find significance (e.g.,
p < .05, p < .01, p < .001), given their calculated value. Computers
print out the exact pvalue, so all one really has to do is look to see if the
pvalue printed out is less than .05, .01, or .001.