TOPIC 7.1 : Friedman Two

Way Analysis Of Variance By

Ranks

LEARNING OUTCOME

At the end of this lesson, students

should be able to determine if we

may conclude from the sample

that there is difference among

treatment effects by using

Friedman Test.

FRIEDMAN TEST

The test presented in this section

is a nonparametric analogue of

the parametric one-way ANOVA

repeated measurement.

The sample population are

normally distributed.

ASSUMPTIONS

1. The data consist of b mutually

independent samples(blocks) of size

k. The typical observation Xij is the

jth observation in the ith

sample(block). Rows represent the

blocks and the columns are called

treatments.

2. The variable of interest is

continuous.

3. There is no interaction between

blocks and treatments.

4. The observations within each block

may be ranked in order of

magnitude.

Continue…

Table 7.1 Data display for the Friedman two- way analysis of variance by ranks

bkbibbb

ikijiii

kj

kj

kj

XXXXXb

XXXXXi

XXXXX

XXXXX

XXXXX

kj

321

321

33333231

22232221

11131211

3

2

1

321Treatment

Block

HYPOTHESES

Ho : M1 = M2 = … = Mk

H1 : At least one equality is violated

TEST STATISTIC

First step:

Convert the original observations to

ranks

In Friedman test the observation

within each block are ranked

separately from smallest to largest

Second step: Obtain the sums of the ranks Rj in

each column.

The Friedman test statistic is defined

as:

2 2

1

123 ( 1)

( 1)

k

r jj

R b kbk k

2 2

1

123 ( 1)

( 1)

k

r jj

R b kbk k

Equation 7.1

Equation 7.2

22

1

12 ( 1)

( 1) 2

k

r jj

b kR

bk k

Show that Equation 7.1 = Equation 7.2

22

1

12 ( 1)

( 1) 2

k

r jj

b kR

bk k

1 1

( 1) and 12

k k

jj j

kGiven R k b k

k

j

k

jj

k

jj

kbkbRR

kbk 1

22

11

2

4

)1(

2

)1(2

)1(

12

2

1

126 ( 1) 3 ( 1)

( 1)

k

jj

R b k b kbk k

2 22 2

1

12 ( 1)[ ( 1)]

( 1) 2 4

k

jj

k b kR b k k

bk k

2 22

1

12 ( 1)( 1) [ ( 1)]

( 1) 2 4

k

jj

k b kR k b b k k

bk k

2

1

123 ( 1)

( 1)

k

jj

R b kbk k

(Equation

7.2)

DECISION

Reject Ho if

2 2(1 , 1)r k

Example 1

Hall et al. * compared three methods

of determining serum amylase values

in patients with pancreatitis. The

result are shown in table 7.2. We

wish to know whether these data

indicate a difference among the three

methods. Given

05.0

*Hall, F.F., T. W. Culp,T. Hayakawa, C. R. Ratliff, and N. C. Hightower,"An Improved Amylase Assay Using a New Starch Derivative,” Amer. J. Clin. Pathol.,53 (1970),627-634

Serum amylase values (enzyme units per 100 ml of serum) in patients with pancreatitis

Table 7.2Specimen

Methods of determination

A B C

1 4000 3210 6120

2 1600 1040 2410

3 1600 647 2210

4 1200 570 2060

5 840 445 1400

6 352 156 249

7 224 155 224

8 200 99 208

9 184 70 227

HYPOTHESES

Ho : MA = MB = MC

H1 : At least one equality is violated

(claim)

TEST STATISTIC

b = 9, k = 3

After convert the original observations to ranks, we have

Specimen

Methods of determination

A B C

1 2 1 3

2 2 1 3

3 2 1 3

4 2 1 3

5 2 1 3

6 3 1 2

7 2.5 1 2.5

8 2 1 3

9 2 1 3

RA = 19.5 RB = 9 RC = 25.5

By equation, we have

5.15

1085.123

2 2

1

3, 9

123 ( 1)

( 1)

k

r jj

k b

R b kbk k

2 2 212(19.5 9 25.5 ) (3)(9)(4)

(9)(3)(4)

From table A.11, ,

Since then we reject

Enough evidence to support the claim

that the three methods do not all yield

identical results.

0H

2(0.95,2) 5.991

2(1 , 1)k 0.05, 3k

15.5 5.991

DECISION

CONCLUSION

EXERCISE

1.A study of effects of three drugs on reaction time of human subjects resulted in the data in table below. Do these data provide sufficient evidence to indicate that the three drugs differ in their effects? Let α = 0.05

Change in response time (milliseconds) of 10 subjects after

receiving one of three drugs

Answer: 8.45 > 5.991, reject Ho

Drug Subject

1 2 3 4 5 6 7 8 9 10

A 10 10 11 8 7 15 14 10 9 10

B 10 15 15 12 12 10 12 14 9 14

C 15 20 12 10 9 15 18 17 12 16

2.Perry et al.* determined plasma epinephrine concentrations during isoflurane, halothane, and cyclopropane anesthesia in 10 dogs. The results are shown in table below. Do these data suggest a difference in treatment effects? Let α = 0.05

Concentrations, nanogram per milliliter, of free catecholamines in arterial plasma response to isoflurane, halothane, and cyclopropane.

Drug 1 2 3 4 5 6 7 8 9 10

Isoflurane 0.28 0.51 1.00 0.39 0.29 0.36 0.32 0.69 0.17 0.33

Halothane 0.30 0.39 0.63 0.38 0.21 0.88 0.39 0.51 0.32 0.42

Cyclopropane

1.07 1.35 0.69 0.28 1.24 1.53 0.49 0.56 1.02 0.30

*Perry, Lawrence B., Russell A. Van Dyke, and Richard A. Theye, "Sympathoadrenal and Hemodynamic Effects of Isoflurane, Halothane, and Cyclopropane in Dogs,”Anesthesiology, 40 (1974), 465-470.

Answer : 2.6 < 5.991 , do not reject Ho

3.Syme and Pollard* conducted an

experiment to investigate the effect of

different motivation levels on measures of

food-getting dominance in the laboratory

rat. The data shown in table below are the

amounts of food in grams eaten by eight

male hooded rats following 0, 24 and 72

hours of food deprivation. Do these data

provide sufficient evidence to indicate a

difference in the effects of the three

levels of food deprivation? Let α = 0.05

Subject Hours of food deprivation

0 24 72

1 3.5 5.9 13.9

2 3.7 8.1 12.6

3 1.6 8.1 8.1

4 2.5 8.6 6.8

5 2.8 8.1 14.3

6 2.0 5.9 4.2

7 5.9 9.5 14.5

8 2.5 7.9 7.9

Amount of food, grams, eaten by eight rats under three level of food deprivation

 * Syme, G. J., and J. S. Pollard, "The Relation between Differences in Level of Food Deprivation and Dominance

in Food Getting in the Rat," Psychon, Sci.,29 (1972),297-298.

Answer : , thus we reject 991.525.12 0H

For the solution, you can refer to

this link :

http://appliednonparametricstatistic.blogspot.com/

Table A.11

Example 1

Example 2

This is the link to watch video on Youtube :

http://www.youtube.com/watch?v=T9klHQk8s2A&feature=youtu.be