Simple ANOVA with Multiple Mean Comparisons 1
Response variable: Strength
Treatment : Cotton
Levels of treatment are 15, 20, 25, 30 and 35
Obs COTTON STRENGTH
1 15 7
2 15 7
3 15 15
4 15 11
5 15 9
6 20 12
7 20 17
8 20 12
9 20 18
10 20 18
11 25 14
12 25 18
13 25 18
14 25 19
15 25 19
16 30 19
17 30 25
18 30 22
19 30 19
20 30 23
21 35 7
22 35 10
23 35 11
24 35 15
25 35 11
GLM APPROACH TO THE SAME PROBLEM 2
The ANOVA Procedure
Class Level Information
Class Levels Values
COTTON 5 15 20 25 30 35
Number of observations 25
GLM APPROACH TO THE SAME PROBLEM 3
The ANOVA Procedure
Dependent Variable: STRENGTH
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 4 475.7600000 118.9400000 14.76 <.0001
Error 20 161.2000000 8.0600000
Corrected Total 24 636.9600000
R-Square Coeff Var Root MSE STRENGTH Mean
0.746923 18.87642 2.839014 15.04000
Source DF Anova SS Mean Square F Value Pr > F
COTTON 4 475.7600000 118.9400000 14.76 <.0001
GLM APPROACH TO THE SAME PROBLEM 4
The ANOVA Procedure
Duncan's Multiple Range Test for STRENGTH
NOTE: This test controls the Type I comparisonwise error rate, not the
experimentwise error rate.
Alpha 0.05
Error Degrees of Freedom 20
Error Mean Square 8.06
Number of Means 2 3 4 5
Critical Range 3.745 3.931 4.050 4.132
Means with the same letter are not significantly different.
Duncan Grouping Mean N COTTON
A 21.600 5 30
B 17.600 5 25
B
B 15.400 5 20
C 10.800 5 35
C
C 9.800 5 15
GLM APPROACH TO THE SAME PROBLEM 5
The ANOVA Procedure
Student-Newman-Keuls Test for STRENGTH
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 20
Error Mean Square 8.06
Number of Means 2 3 4 5
Critical Range 3.7454541 4.5427098 5.0256318 5.3729606
Means with the same letter are not significantly different.
SNK Grouping Mean N COTTON
A 21.600 5 30
B 17.600 5 25
B
B 15.400 5 20
C 10.800 5 35
C
C 9.800 5 15
GLM APPROACH TO THE SAME PROBLEM 6
The ANOVA Procedure
t Tests (LSD) for STRENGTH
NOTE: This test controls the Type I comparisonwise error rate, not the
experimentwise error rate.
Alpha 0.05
Error Degrees of Freedom 20
Error Mean Square 8.06
Critical Value of t 2.08596
Least Significant Difference 3.7455
Comparisons significant at the 0.05 level are indicated by ***.
Difference
COTTON Between 95% Confidence
Comparison Means Limits
30 - 25 4.000 0.255 7.745 ***
30 - 20 6.200 2.455 9.945 ***
30 - 35 10.800 7.055 14.545 ***
30 - 15 11.800 8.055 15.545 ***
25 - 30 -4.000 -7.745 -0.255 ***
25 - 20 2.200 -1.545 5.945
25 - 35 6.800 3.055 10.545 ***
25 - 15 7.800 4.055 11.545 ***
20 - 30 -6.200 -9.945 -2.455 ***
20 - 25 -2.200 -5.945 1.545
20 - 35 4.600 0.855 8.345 ***
20 - 15 5.600 1.855 9.345 ***
35 - 30 -10.800 -14.545 -7.055 ***
35 - 25 -6.800 -10.545 -3.055 ***
35 - 20 -4.600 -8.345 -0.855 ***
35 - 15 1.000 -2.745 4.745
15 - 30 -11.800 -15.545 -8.055 ***
15 - 25 -7.800 -11.545 -4.055 ***
15 - 20 -5.600 -9.345 -1.855 ***
15 - 35 -1.000 -4.745 2.745
GLM APPROACH TO THE SAME PROBLEM 7
The ANOVA Procedure
Tukey's Studentized Range (HSD) Test for STRENGTH
NOTE: This test controls the Type I experimentwise error rate.
Alpha 0.05
Error Degrees of Freedom 20
Error Mean Square 8.06
Critical Value of Studentized Range 4.23186
Minimum Significant Difference 5.373
Comparisons significant at the 0.05 level are indicated by ***.
Difference
COTTON Between Simultaneous 95%
Comparison Means Confidence Limits
30 - 25 4.000 -1.373 9.373
30 - 20 6.200 0.827 11.573 ***
30 - 35 10.800 5.427 16.173 ***
30 - 15 11.800 6.427 17.173 ***
25 - 30 -4.000 -9.373 1.373
25 - 20 2.200 -3.173 7.573
25 - 35 6.800 1.427 12.173 ***
25 - 15 7.800 2.427 13.173 ***
20 - 30 -6.200 -11.573 -0.827 ***
20 - 25 -2.200 -7.573 3.173
20 - 35 4.600 -0.773 9.973
20 - 15 5.600 0.227 10.973 ***
35 - 30 -10.800 -16.173 -5.427 ***
35 - 25 -6.800 -12.173 -1.427 ***
35 - 20 -4.600 -9.973 0.773
35 - 15 1.000 -4.373 6.373
15 - 30 -11.800 -17.173 -6.427 ***
15 - 25 -7.800 -13.173 -2.427 ***
15 - 20 -5.600 -10.973 -0.227 ***
15 - 35 -1.000 -6.373 4.373
GLM APPROACH TO THE SAME PROBLEM 8
The GLM Procedure
Class Level Information
Class Levels Values
COTTON 5 15 20 25 30 35
Number of observations 25
GLM APPROACH TO THE SAME PROBLEM 9
The GLM Procedure
Dependent Variable: STRENGTH
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 4 475.7600000 118.9400000 14.76 <.0001
Error 20 161.2000000 8.0600000
Corrected Total 24 636.9600000
R-Square Coeff Var Root MSE STRENGTH Mean
0.746923 18.87642 2.839014 15.04000
Source DF Type I SS Mean Square F Value Pr > F
COTTON 4 475.7600000 118.9400000 14.76 <.0001
Source DF Type III SS Mean Square F Value Pr > F
COTTON 4 475.7600000 118.9400000 14.76 <.0001
GLM APPROACH TO THE SAME PROBLEM 10
The GLM Procedure
t Tests (LSD) for STRENGTH
NOTE: This test controls the Type I comparisonwise error rate, not the
experimentwise error rate.
Alpha 0.05
Error Degrees of Freedom 20
Error Mean Square 8.06
Critical Value of t 2.08596
Least Significant Difference 3.7455
Means with the same letter are not significantly different.
t Grouping Mean N COTTON
A 21.600 5 30
B 17.600 5 25
B
B 15.400 5 20
C 10.800 5 35
C
C 9.800 5 15
GLM APPROACH TO THE SAME PROBLEM 11
The GLM Procedure
Duncan's Multiple Range Test for STRENGTH
NOTE: This test controls the Type I comparisonwise error rate, not the
experimentwise error rate.
Alpha 0.05
Error Degrees of Freedom 20
Error Mean Square 8.06
Number of Means 2 3 4 5
Critical Range 3.745 3.931 4.050 4.132
Means with the same letter are not significantly different.
Duncan Grouping Mean N COTTON
A 21.600 5 30
B 17.600 5 25
B
B 15.400 5 20
C 10.800 5 35
C
C 9.800 5 15
GLM APPROACH TO THE SAME PROBLEM 12
The GLM Procedure
Student-Newman-Keuls Test for STRENGTH
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 20
Error Mean Square 8.06
Number of Means 2 3 4 5
Critical Range 3.7454541 4.5427098 5.0256318 5.3729606
Means with the same letter are not significantly different.
SNK Grouping Mean N COTTON
A 21.600 5 30
B 17.600 5 25
B
B 15.400 5 20
C 10.800 5 35
C
C 9.800 5 15
GLM APPROACH TO THE SAME PROBLEM 13
The GLM Procedure
Tukey's Studentized Range (HSD) Test for STRENGTH
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 20
Error Mean Square 8.06
Critical Value of Studentized Range 4.23186
Minimum Significant Difference 5.373
Means with the same letter are not significantly different.
Tukey Grouping Mean N COTTON
A 21.600 5 30
A
B A 17.600 5 25
B
B C 15.400 5 20
C
D C 10.800 5 35
D
D 9.800 5 15
GLM APPROACH TO THE SAME PROBLEM 14
The GLM Procedure
Dependent Variable: STRENGTH
Contrast DF Contrast SS Mean Square F Value
C30 vs. C35 1 291.6000000 291.6000000 36.18
C15 and C25 vs. C30 and C35 1 31.2500000 31.2500000 3.88
C15 vs. C25 1 152.1000000 152.1000000 18.87
C15, C25, C30 and C35 vs. C20 1 0.8100000 0.8100000 0.10
Contrast Pr > F
C30 vs. C35 <.0001
C15 and C25 vs. C30 and C35 0.0630
C15 vs. C25 0.0003
C15, C25, C30 and C35 vs. C20 0.7545