Friedman FR TEST FOR RANDOMIZED BLOCK DESIGN.pptx

Discussant
Melody J. Oreta
Friedman Fr Test for Randomized Block
Design
MA.DIVINA B. REYES
Discussat
Professor
dr. Michelle BALUIS
Discussant
STATISTICS IN EDUCATION
Topic # 14

• The Friedman Fr Test is the
nonparametric equivalent of the
randomized block design with k
treatments and b blocks.
• All k measurements within a block are
ranked from 1 to b.
• We use the sums of the ranks of the k
treatment observations to compare the k
treatment distributions.
The Friedman Fr Test

• What is a dependent sample (repeated
measure)? In a dependent sample, the
measured values are connected.

• Why are ranks used? The big advantage is that
if you don't look at the mean difference, but at
the rank sum the data doesn't have to be
normally distributed

• Hypotheses in the Friedman test
• Null hypothesis: there is no significant
difference between the dependent groups.
• Alternative hypothesis: there is a significant
difference between the dependent groups.

Rank the k measurements within each block from
from 1 to k. Tied observations are assigned average of
the ranks they would have gotten if not tied.
Calculate
Ti = rank sum for the ith treatment i = 1, 2,…,k
and the test statistic
)
1
(
3
)
1
(
12 2




 k
b
T
k
bk
F i
r

H0: the k treatments are identical versus
Ha: at least one distribution is different
Test statistic: Friedman Fr
When H0 is true, the test statistic Fr has an
approximate chi-square distribution with df
= k-1.
Use a right-tailed rejection region or p-
value based on the Chi-square distribution.

Example 1
A student is subjected to a stimulus and
we measure the time until the student reacts
by pressing a button. Four students are used in
the experiment, each is subjected to three
stimuli, and their reaction times are measured.
Do the distributions of reaction times differ for
the three stimuli? Stimuli
Subject 1 2 3
1 .6 .9 .8
2 .7 1.1 .7
3 .9 1.3 1.0
4 .5 .7 .8

Reaction Times
Stimuli
Subject 1 2 3
1 .6 .9 .8
2 .7 1.1 .7
3 .9 1.3 1.0
4 .5 .7 .8
(1) (3) (2)
(1.5) (3) (1.5)
(1) (3) (2)
(1) (2) (3)
Ti 4.5 11 8.5
Rank the 3 measurements for each
subject from 1 to 3, and calculate the
three rank sums.

Reaction Times
H0: the distributions of reaction times are the same
Ha: the distributions differ in location
375
.
5
)
4
)(
4
(
3
)
5
.
8
11
5
.
4
(
)
4
(
12
12
)
1
(
3
)
1
(
12
2
2
2
2









 k
b
T
k
bk
F i
r
:
statistic
Test
Rejection region: Use Table 5.
For a right-tailed chi-square test
with a = .05 and df = 3-1 =2,
reject H0 if H  5.99.
Do not reject H0. There is
insufficient evidence to
indicate that there is a
difference in reaction times for
the three stimuli.

Summary
•The Friedman Fr test is the rank equivalent
of the randomized block design two-way
analysis of variance F test.

Key Concepts
Nonparametric Methods
These methods can be used when
• the data cannot be measured on a quantitative scale,
or when
2. the numerical scale of measurement is arbitrarily set
by the
researcher, or when
3. the parametric assumptions such as normality or
constant
variance are seriously violated.

Key Concepts
The Friedman Fr Test: Randomized Block Design
1. Rank the responses within each block from 1 to k.
Calculate the rank sums T1, T2, ¼, Tk, and the test
statistic
2. If the null hypothesis of equality of treatment
distributions is false, Fr will be unusually large,
resulting in a one-tailed test.
3. For block sizes of five or greater, the rejection region
for Fr is based on the chi-square distribution with (k -
1) degrees of freedom.
)
1
(
3
)
1
(
12 2




 k
b
T
k
bk
F i
r

--END OF REPORT--
• --THE END—

Friedman FR TEST FOR RANDOMIZED BLOCK DESIGN.pptx

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