Lab Exercise 10 (Repeated Measures)
The goal of this exercise is to contrast the randomized block design with the repeated
measures design. We will introduce the Friedman Rank Test for repeated measures
along with multiple comparisons.
In a randomized block design (two-way layout with one observation per cell) we
randomly assign independent subjects to the treatments. The assignment is made
within blocks in order to compare similar subjects. You worked with this type of data in
Lab Exercise 9 where we developed an aligned rank test (Kruskal-Wallis computed on
the residuals after ½ cycle (1 iteration) of median polish). The experiment entailed
randomly assigning broccoli to packaging types within facilities as the blocks.
In repeated measures we randomly order the treatments that are repeatedly given to a
subject. The subject constitutes a block. The data collected are dependent and
correlated within a subject.
To decide whether you have a randomized block design or a repeated measures design
think about whether subjects are assigned to treatments or whether treatments are
assigned to the subject over time.
The aligned rank test is not appropriate for analyzing data in a repeated measures
design. However, the Friedman test introduced next can be used in a randomized block
design but it is not as powerful as the aligned rank test. Now we consider an example
of a repeated measures design and introduce statistical analysis. See section 4.5 of the
text for more discussion along with the formula for the Friedman statistic.
Example: Twenty two professional baseball players participated in a study to
determine if there is a best way to run from home plate to second base. In particular
they wanted to minimize the time. They tried three methods of rounding first base:
round out method (RO), narrow angle (NA), wide angle (WA). Each player (subject) ran
six times, two times each for each method. The methods (treatments) were assigned in
random order and the data is the average of the two times for each of the same
methods. Hence, the data is a set of three repeated measures on each of the 22
players. Get the data called baseball.txt from the 464 web site.
1. State the research hypothesis.
2. To prepare the data for analysis you will need to stack the three columns of data and
then have two columns of subs: one column for treatment (method of running the
bases) and one column for blocks (players). Use the help search Friedman in Minitab
to help you do this. You may want to simply put the subs in by hand. This is part of the
exercise.
3. Once you have the data in proper form use the menu Stat>Nonparametrics>Friedman
to apply the Friedman test. The main difference between the Kruskal-Wallis statistic
and the Friedman statistic is that with the Friedman statistic you only rank within
blocks but you don’t remove the block effect. With the Kruskal-Wallis statistic you first
remove the block effect and then rank all the data.
4. You will need to compute the multiple comparisons by hand. And you will need the
fact that
Ri  R j
k ( k  1)
6b
can be referred to a standard normal table in order to check the p-value for the
comparison between treatment i and treatment j. Here R i is the rank average for the ith
treatment, k is the number of treatments, and b is the number of blocks. The
expression under the square root is the variance of the difference.
5. Write a report on the baseball data. You should use Minitab to apply the Friedman
test. Then you should carry out multiple comparisons based on the formula in 4 above.
Finally, you should do ½ cycle (1 iteration) of median polish on the data and get the CIBoxplots for the residuals. Make a case for which method you would recommend for
the coaches to teach for running the bases based on your analysis.