When either a matched-subjects or repeated-measure design is used and the hypothesis of a difference among three
or more treatments is to be tested, the Friedman 2-way ANOVA by ranks provides an acceptable test. Formally,
Friedman's test is a nonparametric test for treatment differences in a randomized complete block design. Each
block (row) of the design is a subject. Treatments are randomly assigned to subjects within each block. The subjects
are repeatedly measured once under each treatment. The order of treatments is randomized for each subject.
As an example, suppose a teacher, having read that the time of day can affect the performances of children on
cognitive tasks, decides to carry out an action research study to determine when best to test children-early morning
(EM) late morning (LM) early afternoon (EA) or late afternoon (LA).
She has a class of 19 students.
She begins with a list of 100, age/grade-appropriate spelling words and then constructs four unique spelling tests
(A, B,C, and D) of 25 words each by randomly selecting 25 words from the list of 100 for the first test, then
randomly selecting 25 words from the remaining 75 for the second test, then randomly selecting 25 words from the
remaining 50 for the third test, and using the remaining 25 words for the fourth test. After spending two weeks
teaching the complete list of 100 words, she waits a week and then performs the "experiment" by administering the
four tests over four successive days under the following conditions.
On Monday she administers a spelling test at 8:30 am; on Tuesday she administers a test at 11: am; on Wednesday she administers a test at 12:30 pm; and on Thursday she administers a test at 2:30 pm. The tests are randomly assigned over children. For instance, the order of tests administered to the first child (S1) might be CADB; for the second child (S2) it might be BCDA, and so on.
The data (spelling test scores) collected over the four days are shown in the table to right.
| S
EM LM
EA LA s1 17 23 24 21 s2 16 17 24 19 s3 16 17 21 13 s4 13 15 19 18 s5 19 18 22 24 s6 18 15 18 24 s7 16 17 20 13 s8 12 15 15 14 s9 16 16 22 16 s10 17 18 18 17 s11 16 17 24 19 s12 9 14 17 15 s13 18 20 25 22 s14 17 21 24 20 s15 20 20 25 24 s16 15 15 19 13 s17 18 17 20 20 s18 12 17 17 15 s19 17 20 21 21 |
The Friedman test statistic (Qk) is:
To compute Qk work through the following steps.
1. Within each row rank-order the k = 4 scores from low to high. The new scores (ranks) are the R's given in the equation above.
2. Sum (over the N=19 cases) the ranked scores for each column,
.
3. Square each of the column totals,
; then plug the values into the equation and solve for QR in the equation given above.
For the data in this example, the ranks (within rows), the column rank sums, and the squares of the column rank sums are given in the table below.
| RANKS (within rows): S REM RLM REA RLA s1 1.0 3.0 4.0 2.0 s2 1.0 2.0 4.0 3.0 s3 2.0 3.0 4.0 1.0 s4 1.0 2.0 4.0 3.0 s5 2.0 1.0 3.0 4.0 s6 2.5 1.0 2.5 4.0 s7 2.0 3.0 4.0 1.0 s8 1.0 3.5 3.5 2.0 s9 2.0 2.0 4.0 2.0 s10 1.5 3.5 3.5 1.5 s11 1.0 2.0 4.0 3.0 s12 1.0 2.0 4.0 3.0 s13 1.0 2.0 4.0 3.0 s14 1.0 3.0 4.0 2.0 s15 1.5 1.5 4.0 3.0 s16 2.5 2.5 4.0 1.0 s17 2.0 1.0 3.5 3.5 s18 1.0 3.5 3.5 2.0 s19 1.0 2.0 3.5 3.5
Sum
28 43.5
67.5 43.5 |
N = 19,
k = 4,
R12 = 784,
R22 = 1892.25,
R32 = 5041.00, and
R42 = 2256.25
and we can compute,
QR = [12 / Nk(k+1)] × [R12 + R22 + R32 + R42] - 3N(k + 1)
= [12 / (19)(4)(5)] × [784 + 1892.25 + 5041 + 2256.25] - 3(19)(5)
= (12 / 180) × 9973.5 - 285
= .06667 × 9973.5 - 285
= 314.65 - 285
= 29.95,
which can be compared to a c2 with k-1 = 3 degrees of freedom.
A computer program is available for computing this statistic. Click here: Friedman's ANOVA by ranks.