ST 524
Multiple Comparisons Procedures
NCSU - Fall 2008
Cherry Spraying Example
Nine sprays were tested for their ability to hold fruit on the tree, the fruit crop being cherries. The
experimental design was a randomized complete block design with 9 blocks. Plots were single
trees.
The number of trees in dour one-pound random samples of the crop from each of the 81 trees
was obtained. An average count was obtained for the four samples from each tree. The data are
reported below.
TRT
T1
T2
T3
T4
T5
T6
T7
T8
T9
REP1
REP2
REP3
REP4
REP5
REP6
REP7
REP8
REP9
126.50
117.75
145.00
109.50
124.25
128.50
117.00
113.75
123.50
111.00
116.00
179.50
119.50
120.75
121.00
128.75
112.75
126.75
113.00
104.25
159.50
121.25
118.50
131.50
123.75
111.25
126.50
113.25
110.75
125.75
109.25
125.00
134.75
119.00
114.25
117.25
117.00
114.75
149.00
104.25
123.25
129.00
115.50
109.00
117.50
106.75
107.00
139.75
114.25
132.75
124.00
110.50
119.75
107.50
115.00
117.00
145.75
120.50
127.25
106.75
117.50
117.00
115.50
98.75
126.50
142.75
103.50
114.25
113.00
118.75
104.50
121.50
113.75
113.50
179.50
107.25
128.75
127.75
101.50
106.25
121.00
Mean and variance of Number of fruit (count/lb) per tree
Cherry Spraying
var_
range_
Obs
trt
mn_count
count
count
1
2
3
4
5
6
7
8
9
10
T1
T2
T3
T4
T5
T6
T7
T8
T9
120.827
112.778
114.167
151.833
112.139
123.861
124.028
116.917
112.056
119.667
212.037
56.069
42.313
323.781
48.455
30.877
82.335
59.453
24.075
36.578
80.75
27.75
22.25
53.75
17.75
18.50
28.00
27.25
15.25
19.25
Decision was taken to drop treatment 3 from analysis. This spray was found to destroy the flesh
of the cherry, leaving only the stone.
Analysis of variance and Multiple Comparison Procedures to analyze treatment effects.
ThursdayOctober 16, 2008 Multiple Comparisons Procedures
1
ST 524
Multiple Comparisons Procedures
NCSU - Fall 2008
Number of fruit (count/lb) cherries per tree
Cherry Spraying - Multiple Comparison Procedures
27
The GLM Procedure
Class Level Information
Class
Levels
rep
trt
Values
9
8
1 2 3 4 5 6 7 8 9
T1 T2 T4 T5 T6 T7 T8 T9
Number of Observations Read
Number of Observations Used
72
72
Dependent Variable: count
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
15
1973.204861
131.546991
2.76
0.0030
Error
56
2665.500000
47.598214
Corrected Total
71
4638.704861
Ho: All spray treatments have same
effect
R-Square
Coeff Var
Root MSE
count Mean
0.425378
5.899157
6.899146
116.9514
Source
rep
trt
Source
rep
trt
Ho:  i
0
DF
Type I SS
Mean Square
F Value
Pr > F
8
7
375.736111
1597.468750
46.967014
228.209821
0.99
4.79
0.4562
0.0003
DF
Type III SS
Mean Square
F Value
Pr > F
8
7
375.736111
1597.468750
46.967014
228.209821
0.99
4.79
0.4562
0.0003
, i  1,
t Tests (LSD) for count
NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha
0.05
Error Degrees of Freedom
56
Error Mean Square
47.59821
Critical Value of t
2.00324
Least Significant Difference
6.5151
t(0.975, 56)
lsd0.05,56df  2.00324 47.59821  2 9 
Means with the same letter are not significantly different.
t Grouping
Mean
N
trt
A
A
A
A
A
124.028
9
T6
123.861
9
T5
119.667
9
T9
C
C
C
C
C
C
C
C
C
116.917
9
T7
114.167
9
T2
112.778
9
T1
112.139
9
T4
112.056
9
T8
B
B
B
B
B
ThursdayOctober 16, 2008 Multiple Comparisons Procedures
2
9
ST 524
Multiple Comparisons Procedures
NCSU - Fall 2008
Waller-Duncan test
The GLM Procedure
Waller-Duncan K-ratio t Test for count
Note: This test minimizes the Bayes risk under additive loss and certain other assumptions.
Kratio
100
Error Degrees of Freedom
56
Error Mean Square
47.59821
F Value
4.79
Critical Value of t
2.01945
Minimum Significant Difference
6.5678
Means with the same letter
are not significantly different.
Waller Grouping
Mean
N
trt
124.028
9
T6
123.861
9
T5
A
119.667
9
T9
B
C
116.917
9
T7
B
C
B
C
114.167
9
T2
112.778
9
T1
112.139
9
T4
112.056
9
T8
A
A
A
A
B
B
C
C
C
C
C
C
ThursdayOctober 16, 2008 Multiple Comparisons Procedures
3
ST 524
Multiple Comparisons Procedures
NCSU - Fall 2008
Duncan's Multiple Range Test for count
NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha
0.05
Error Degrees of Freedom
56
Error Mean Square
47.59821
Number of Means
Critical Range
2
6.515
3
6.853
4
7.076
5
7.238
6
7.363
7
7.462
8
7.544
Means with the same letter are not significantly different.
Duncan Grouping
Mean
N
trt
A
A
A
A
A
124.028
9
T6
123.861
9
T5
119.667
9
T9
C
C
C
C
C
C
C
C
C
116.917
9
T7
114.167
9
T2
112.778
9
T1
112.139
9
T4
112.056
9
T8
B
B
B
B
B
B
B
> 124.028 - 123.861
0.167 compared to
> 124.028 - 116.917
7.111 compared to
> 119.667 - 112.139
7.528 compared to
= 0.167
6.515
= 7.111
7.076
= 7.528
7.238
Tukey's Studentized Range (HSD) Test for count
NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher
Type II error rate than REGWQ.
Alpha
0.05
Error Degrees of Freedom
56
Error Mean Square
47.59821
Critical Value of Studentized Range 4.45236
Minimum Significant Difference
10.239
> qtukey, 0.05, 8 trt,
w  q  p, f e  s y
56 df
Means with the same letter are not significantly different.
Tukey Grouping
Mean
N
trt
A
A
A
A
A
A
A
A
A
124.028
9
T6
123.861
9
T5
119.667
9
T9
116.917
9
T7
114.167
9
T2
112.778
9
T1
112.139
9
T4
112.056
9
T8
B
B
B
B
B
B
B
B
B
B
B
ThursdayOctober 16, 2008 Multiple Comparisons Procedures
4
ST 524
Multiple Comparisons Procedures
NCSU - Fall 2008
Student-Newman-Keuls Test for count
NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis
but not under partial null hypotheses.
Alpha
0.05
Error Degrees of Freedom
56
Error Mean Square
47.59821
Number of Means
Critical Range
2
6.5152872
3
7.8301065
4
8.6118701
5
9.1670319
6
9.5964097
7
9.9455437
8
10.239151
Means with the same letter are not significantly different.
SNK Grouping
Mean
N
trt
A
A
A
A
A
A
A
124.028
9
T6
123.861
9
T5
119.667
9
T9
116.917
9
T7
114.167
9
T2
112.778
9
T1
112.139
9
T4
112.056
9
T8
B
B
B
B
B
B
B
B
B
B
B
Number of fruit (count/lb) cherries per tree
Cherry Spraying - Multiple Comparison Procedures
34
Bonferroni (Dunn) t Tests for count
NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher
Type II error rate than REGWQ.
Alpha
0.05
Error Degrees of Freedom
56
Error Mean Square
47.59821
Critical Value of t
3.28057
Minimum Significant Difference
10.669
  2 k  k  1  0.001785
t10.001785/ 2,56 df  3.28057
Means with the same letter are not significantly different.
Bon Grouping
Mean
N
trt
A
A
A
A
A
A
A
A
A
124.028
9
T6
123.861
9
T5
119.667
9
T9
116.917
9
T7
114.167
9
T2
112.778
9
T1
112.139
9
T4
112.056
9
T8
Sidak:
  1  1   
2 k  k 1
 0.001830
t10.001830/ 2,56 df  3.27227
B
B
B
B
B
B
B
B
B
B
B
ThursdayOctober 16, 2008 Multiple Comparisons Procedures
5
ST 524
Multiple Comparisons Procedures
NCSU - Fall 2008
Scheffe's Test for count
NOTE: This test controls the Type I experimentwise error rate.
Alpha
0.05
Error Degrees of Freedom
56
Error Mean Square
47.59821
Critical Value of F
2.17816
Minimum Significant Difference
12.699
Means with the same letter are not significantly different.
Scheffe Grouping
Mean
N
trt
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
124.028
9
T6
123.861
9
T5
119.667
9
T9
116.917
9
T7
114.167
9
T2
112.778
9
T1
112.139
9
T4
112.056
9
T8
Ryan-Einot-Gabriel-Welsch Multiple Range Test for count
NOTE: This test controls the Type I experimentwise error rate.
Alpha
0.05
Error Degrees of Freedom
56
Error Mean Square
47.59821
Number of Means
Critical Range
2
8.3699169
3
9.0960312
4
9.4906375
5
9.7573197
6
9.9553297
7
9.9553297
8
10.239151
Means with the same letter are not significantly different.
REGWQ Grouping
Mean
N
trt
A
A
A
A
A
A
A
124.028
9
T6
123.861
9
T5
119.667
9
T9
116.917
9
T7
114.167
9
T2
112.778
9
T1
112.139
9
T4
112.056
9
T8
B
B
B
B
B
B
B
B
B
B
B
ThursdayOctober 16, 2008 Multiple Comparisons Procedures
6
ST 524
Multiple Comparisons Procedures
ThursdayOctober 16, 2008 Multiple Comparisons Procedures
NCSU - Fall 2008
7