Roy E Welsch.

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LIBRARY

OF THE

MASSACHUSETTS INSTITUTE
OF TECHNOLOGY

A MODIFICATION OF THE NEWMAN-KEULS PROCEDURE

FOR MULTIPLE COMPARISONS*

by
Roy E. Welsch

Working Paper bl4 B. , i=3,4,...,10.

1 — 1-1 ' ' '

In this case we used p^ = (i/10)«(.05) except that p _ was set equal

to .05. Table I lists the values of p for each r. Using these values,

the only cases where B > B occurred was with i = r-1 and in these

cases B , was set equal to B „.
r-1 ^ r-2

We used the method of inverse interpolation described in Harter
(1959) to obtain the critical values from tables of the studentized
range. The tables in this paper were computed to an accuracy of one
unit in the fourth significant digit and then rounded to three significant
digits. Linear harmonic v-wise interpolation is recommended.

4. Examples

The tables that form a part of this paper can be used for the
Tukey test, the N-K test, and the test described above. Consider the
following data:

sample means:

8.29 8.96

12.57

15.41

16.50

18.92

"l "'2

"^3

"4

m5

"•6

standard error of m. : 1.0

1

degrees of freedom: 30

An underscore is used to indicate that the means cannot be
declared significantly different.

Tukey

Test

k

Test Value

Comparisons

6

4.30

(1,6)

5

4.30

(1,5) (2,6)

4

4.30

(1,4) (2,5) (3,6)

3

4.30

(L

.3) (2,4) (3,5) (4,6)

2

4.30

The test value is just the single entry for k = 6 , r = 6 in the table
with u = 30.

10

N-K Test

Test Value

Comparisons

4.30

(1,6)

4.10

(1,5) (2,6)

3.85

(1,4) (2,5) (3,6)

3.49

(1

,3) (2,4) (3,5) (4,6)

2

2,89 .

(L

l2)

(2,3) (3,4) (4,5) (5,6)

In this case the test values are the diagonal entries (k=6 , r=6) ,
(k=5, r=5) ,. . . ,(k=2, r=2) in the same table.

Test Value

New

Te;

3t

k

Comparisons

6

4.30

(1,6)

5

4.10

(1,5) (2,6)

4

4.10

(1,4) (2,5) (3,6)

3

3.92

(1:

,3) (2,4) (3,5) (4,6)

2

3.59

(1

^)

(2,3) (3,4)

The test values are the entries in the r = 6 column of the same table.

11

p

(k)

Table I
Probabilities Used to Compute Critical Values

10

Total Number of Means (r)
8 7 6 5

.0500

.0500* .0500

.0400 .0500* .0500

.0350 .0389 .0500* .0500

.0300 .0333 .0375 .0500* .0500

.0250 .0278 .0312 .0357 .0500* .0500

.0200 .0222 .0250 .0286 .0333 .0500* .0500

,0150 .0167 .0187 .0214 .0250 .0300 .0500 .0500

,0100 .0111 .0125 .0143 .0167 .0200 .0250 .0500 .0500

Actual table entries in some cases are largei* than this
probability requires in order to insure that B _„ < B ,

12
Table II

0F=

Total Number of Means (r)
10 98765^32

10 6.99

S 9 6.97 6.80
i

z 8 6.97 6.75 6.58
e

7 6,93 6.75 6.50 6.33
o

f 6 6.87 6.69 6.50 6.19 6.03

G 5 6.78 6.60 6.A1 6.19 5.81 5.67
r

° A 6.61 6.44 6.26 6.05 5.81 5.30 5.22
u

P 3 6.32 6.16 5.98 5.78 5.56 5.30 4.60 4.60

^^^ 2 5.70 5.55 5.39 5.21 5.00 4.76 4.47 3.o4 3.64

DF= 6

Total Number of Means (r)
10 98765432

10 6.49

^ 9 6.44 6.32

1

^ 8 6.44 6.25 6.12

e

7 6.40 6.25 6.02 5.90

o

^ 6 6.33 6.18 6.02 5.74 5.63

^ 5 6,23 6.08 5.92 5.74 5.40 5.31

r

° 4 6.07 5.93 5.77 5.60 5.40 4.94 4.90

^ 3 5.80 5.66 5.51 5.35 5.16 4.94 4.34 4.34

fk')

2 5.24 5,12 4,98 4.83 4.65 4.44 4.20 3.46 3.46

0F= 7

Total Number of Means (r)
10 98 7 6 5 A 3 2

10 6.16

S 9 6.09 6.00

i

z 8 6.09 5.91 5.82

e

7 6.04 5.91 5.70 5.61

o

f 6 5.97 5.84 5.70 5.45 5.36

G 5 5.87 5.74 5.60 5.45 5,13 5.06

r

° 4 5.71 5.59 5.46 5.30 5.13 4.70 4.68

u

P 3 5.A6 5.34 5.21 5.06 4.90 4.70 4.17 ^.17

^^) 2 4.95 4.84 4.72 4.58 4.42 4.24 4.02 3.34 3,34

DF =

Total Number of Means (r)
10 9 8 7 6 5

10 5.92

S

i 9 5.35 5.77

z

e 8 5.85 5.68 5,60

o 7 5.79 5.68 5.48 5.40
f

6 5.72 5.60 5.^8 5.24 5.17
G

r 5 5.62 5.50 5.38 5.24 4.94 4.89
o

u 4 5.46 5.36 5.23 5.10 4,94 4.54 ^,53
P

3 5.22 5,12 5.00 4.87 4.71 ^.54 4.04 4.04
(k)

2 4.75 4.65 4,53 4.41 4.27 4.10 3.89 3.26 3.26

DF= 9

Total Number of Means (r)
10 98765^32

10 5.74

S

i 9 5.66 5.60

z
e 8 5.66 5.50 5. A3

o 7 5.61 5.50 5.31 5.24

f

6 5.53 5.43 5.31 5.08 5.02

G

r 5 5.43 5.33 5.21 5.08 4.80 4.76

o

u 4 5.28 5.18 5.07 4.94 4.80 4. '♦I 4.41

P

3 5.05 4.95 4.84 4.72 4,58 4.41 3.95 3.95

(k)

2 4.60 4.50 4.40 4.28 4.15 3.99 3.80 3.20 3.20

0F= 10

Total Number of Means (r)
10 9 8 7 6 5 4

10 5.60

S

i 9 5.52 5.46

e 8 5.52 5.36 5.30

o 7 5,46 5.36 5.18 5.12
f

6 5.39 5.29 5.18 4.96 4.91
G

r 5 5.29 5.1-9 5.08 4.96 4.69 4.65
o
u A 5.14 5.05 4.94 4.82 4.69 4.33 4.33

P

3 4.92 4,82 4.72 4.61 4.47 4.32 3.88 3.88
(k)

2 4.48 4.39 4.30 4.19 4.06 3.91 3.73 3.15 3.15

0F= 11

Total Number of Means (r)
10 9 a 7 6 5 4

10 5.49

S

i 9 5.40 5.35

e 8 5.40 5.25 5.20

o 7 5.35 5.25 5.08 5.03
f

6 5.27 5.18 5.08 4.86 4.82
G

r 5 5.17 5.08 4.98 4.86 4.60 4.57
o

u 4 5.03 4.94 4.84 4.73 4.60 4.26 4.26
P

3 4.81 4.72 4.63 4.52 4.39 4.24 3.82 3.82
(k)

2 4.39 4.31 4.22 4.11 3.99 3.84 3.67 3.11 3.11

DF= 12

Total Number of Means (r)
10 98765432

10 5.40

S
i 9 5.31 5.27

e 8 5.31 5.16 5.12

o 7 5.25 5.16 4.99 4.95

f

6 5.18 5.09 4.99 4.79 4.75

G

r 5 5.08 4,99 4.90 4.79 4,53 4.51

o

u 4 4.94 4.86 4.76 4,65 4.53 4.20 4.20

P

3 4.73 4.64 4.55 4.45 4.32 4.18 3.77 3.77

(k)

2 4.32 4.24 4.15 4.05 3.93 3.79 3.62 3.08 3.08

DF= 13

Total Number of Means (r)
10 9 8 7 6 5

10 5.32
S
i 9 5,23 5.19

z

e 8 5.23 5.09 5,05

o 7 5.18 5.09 4.92 4.88
f

6 5.10 5.02 4.92 4.72 4.69
G
r 5 5.00 4.92 4.83 4.72 4.47 4,45

u 4 4.87 4.79 4.69 4.59 4.47 4.15 -!< . 1 5
P

4,66 4.53 4.A9 4.39 4.27 4.13 3.73 3.73

3

(k)

2

4.26 4,18 4.10 4,00 3.88 3.75 3.58 3.06 3.06

DF= 14

Total Number of Means (r)
10 9 8 7 6 5

10 5.25
S

i 9 5.17 5.13
z
e 8 5.17 5.03 4,99

o 7 5.11 5,03 4,86 4.83
f

6 5.04 4.96 4.86 4.67 4,64
G

r 5 4.94 4.86 4.77 4.67 4.42 4.41
o

u 4 4.80 4.73 4.64 4,54 4.42 4.11 ^,lI
P

3 4.60 4.52 4.^3 4.34 4.22 A. 09 3.70 3.70

(k)

2

4.21 4.14 4.05 3,95 3,84 3,71 3,55 3.03 3.03

DF= 15

Total Number of Means (r)
10 9 8 7 6 5

10 5.20

S

i 9 5.11 5.08

z
e 8 5.11 A, 97 4.94

o 7 5.05 4.97 4.81 4.78

f

6 4.98 4.90 4.81 4.62 4.59

G

r 5 4.89 4.81 4.72 4.62 4.38 4.37

o
u 4 4.75 4.68 4.59 4.49 4.38 4.08 4.08

P

3 4.55 4.47 4,39 4.29 4.18 4,05 3.67 3.67

(k)

2 4.17 4.09 4.01 3.92 3.81 3.68 3.52 3.01 3.01

DF= 16

Total Number of Means (r)
10 98765432

10 5.15

S

i 9 5.06 5.03

I 8 5.06 4.93' 4.90

o "7 5.01 4.93 4.77 4.74

f

6 4.93 4.86 4.77 4.58 4.56

G

r 5 4.84 4.76 4.68 4,56 4. 3A 4.33

u ^ ^•^l 4.63 4.55 4.45 4.34 4.05 4.05

p

3 4.51 4.43 4.35 4.26 4.15 ^.02 3.65 3.65

(k)

2 4.13 4.06 3.98 3.69 3,78 3.65 3.50 3.00 3.00

DF= 17

Total Number of Means (r)
10 9 8 7 6 5 A

10 5.11

I 9 5.02 4.99

I 8 5.02 4.89 4.86

Q 7 4.96 4.89 4.73 4.71

f

6 4.89 ^.82 4.73 4.55 4.52

G

^ 5 4.80 4.72 4.64 4.55 4.31 4.30

u 4 4.67 4.59 4.51 4.42 4.31 4.02 4.02

3 4.47 4.40 4.32 4.22 4.12 3.99 3.63 3.63
(k)

2 4.10 4.03 3.95 3.86 3.76 3.63 3.48 2.98 2.98

DF= 18

Total Number of Means (r)
10 9 8 7 65 4 3 2

10 5.07

S

i 9 4,98 4.96

e 8 4.98 4.85 4.82

o 7 4.93 4.85 4.70 4.67

f

6 4.85 4.78 4.70 4.51 4.49

G

r 5 4.76 4.69 4.61 4.51 4.28 4.28

o
u 4 4.63 4.56 4.48 4.39 4.28 4.00 4.00

P

3 4.^,4 4.37 4.29 4.20 4.09 3.97 3.61 3.61

(k)

2 4.07 4.00 3.92 3.84 3.73 3.61 3.46 2.97 2.97

" DF= 19

Total Number of Means (r)
10 9 8 7 6 5 A

10 5.04

^ 9 4.95 4,92

I 8 4.95 4.82 4.79

p 7 4.89 4.82 4.67 4.65

6 4.82 4.75 4.67 4.49 4,47

r 5 4.73 4.66 4.58 4.49 4.26 4.25

° 4 4.60 4.53 4.45 4.36 4.26 3.98 3.98

^ 3 4.41 '!i.34 4.26 4.17 4.07 3.95 3.59 3.59

(k)

2 4.C5 3.98 3.90 3.81 3.71 3.59 3.44 2.96 2.96

DF= 20

Total Number of Means (r)
10 98765432

10 5.01

S

i 9 4.92 4.90

e 8 4.92 ^.79 4.77

o 7 4.86 4.79 4.64 4,62

f

6 4,79 4.72 4.64 4.46 4.45

G

r 5 4.70 4,63 4.55 4.46 4.23 4.23

o

u ^ 4.57 4.50 4. '{3 4.34 4.23 3.96 3.96

P

3 4.38 4.31 4.24 4.15 4.05 3,93 3.58 3.58

(k)

2 4.02 3.96 3.88 3,80 3.70 3,58 3.43 2.95 2.95

DF= 2^

Total Number of Means (r)
10 9 8 7 6 5 4

10 4.92

S

9 4.83 4.81

I 8 4.83 4.70 4.68

o 7 4.77 4.70 4.56 4,54

6 4.70 4.63 4.56 4.38 4.37

r 5 4.61 4.54 4.47 4.38 4.17 ^.17

° 4 4.^9 4.42 4.35 4.26 4.17 3,90 3.90

^ 3 4.30 4.24 4.16 4.08 3.98 3.87 3.53 3.53

(k)

2 3.96 3.89 3.82 3.74 3.64 3.52 3.38 2.92 2.92

0F= 30

Total Number of Means (r)
10 9 8 7 6 5 4

10 4.82

S

i 9 4.74 4.72

e 8 4.74 4.62 4.60

o 7 4.68 4.62 4.A8 4.46

f

6 4.61 4.55 4.A8 4.31 4.30

G

r 5 4.52 4.46 4.39 4.31 4.10 A. 10

u 4 4.40 4.34 4.27 4.19 4.10 3.85 3.85

3 4.22 4.16 4.09 4,01 3.92 3.81 3.'=f9 3.49

(k)

2 3.89 3.83 3.76 3.68 3,59 3.48 3.3A 2.89 2.89

21

DF= AO

Total Number of Means (r)
10 9 8 7 6 5

10 4.73

i 9 4.65 4.63

I 8 4.65 4.53 4.52

^ 7 4.59 4.53 4.^0 4.39

6 4.53 4.47 4.^0 4.24 4.23

G

J. 5 4.44 4.38 4.31 4.24 4.04 4.04

o

u ^ 4.32 4.26 4.20 4.12 4.03 3.79 3.79

p

3 4.15 4.09 4.02 3.95 3.86 3.75 3.44 3.44
(k)

2 3.82 3.77 3.70 3.62 3.53 3.43 3.29 2.86 2.36

DF= 60

Total Number of Means (r)
10 9 8 7 6 5 4

10 4.65
S

i 9 4.56 4.55
z
e 8 4.56 4.45 4.44

o 7 4.51 4.45 4.32 4.31
f

6 4.44 4.38 4.32 4.17 4.16
G

r 5 4.36 4.30 4.24 4.17 3.98 3.98
o

u ^ 4.24 4.19 4.12 4.05 3.97 3.74 3.74
P

3 4.07 4.02 3.95 3,88 3.80 3.70 3.40 3.40
(k)

2 3.76 3.71 3.64 3.57 3.48 3.38 3.25 2.83 2.83

DF= 120

Total Number of Means (r)
10 9 8 7 6 5 4

10 4.56

S

i 9 4.48 4.47

z
e 8 4.48 A. 37 4.36

o 7 4.42 4.37 4.25 4.24

f

6 4.36 A. 31 4.25 4.10 4.10

G

r 5 4.28 A. 22 4.16 4.10 3.92 3.92

o
u A 4.17 A. 11 4.05 3.98 3.90 3.68 3.68

P

3 4.00 3.95 3.89 3.82 3.74 3.64 3.36 .3.36

(k)

2 3.70 3.65 3.59 3.52 3.43 3.33 3.21 2.80 2.80

DF =

Total Number of Means (r)
10 98765432

10 4.47

S

i 9 4.39 4.39

e 8 4.39 4.29 4.29

o 7 4.34 4.29 4.17 4.17

f

6 4.28 4.23 4.17 4.03 4.03

G

r 5 4.20 A. 15 4.09 4.03 3.86 3.86

o
u 4 4.09 4.04 3.98 3.92 3.84 3.63 3.c3

P

3 3.93 3.88 3.83 3.76 3.66 3« 59 3.31 3.31

(k)

2 3.64 3.59 3.53 3.46 3.39 3.29 3.17 2.77 2.77

23

References

Duncan, D. B. (1955). Multiple range and multiple F tests. Biometrics ,
11, 1-42.

Feder, P. I. (1972). Studentized range graph paper — a new tool for

the graphical comparison of treatment means. TIS Report No. 72-CRD-193.

Harter, H. L. , Clemm, D. S., and Guthrie, E. H, (1959). The probability
integrals of the range and of the studentized range-probability integral
and percentage points of jthe studentized range; critical values for
Duncan's new multiple range test. Wright Air Development Center Technical
Report 58-484, Vol. II. (ASTIA Document No. AD231733) .

Kurtz, T. E., Link, R. F. , Tukey, J. W. and Wallace, D. L. (1965).
Short-cut multiple comparisons for balanced single and double classifi-
cations: Part 1, Results. Technometrics , 1,, 95-165.

Miller, R. G. (1966). Simultaneous Statistical Inference . New York:
McGraw-Hill.

O'Neill, R. and Wetherill , G. B. (1971). The present state of multiple
comparison methods. JRSS B, 33, 218-250.

Tukey, J. W. (1953). The problem of multiple comparisons. Unpublished
Dittoed Notes, Princeton University.

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Online LibraryRoy E WelschA modification of the Newman-Keuls procedure for multiple comparisons → online text (page 1 of 1)