252anovaex2
11/15/06
MINITAB EXAMPLE
Two-way ANOVA
Explanation: The data set has already been prepared and stored as 2f2.mtb. The first column represents costs. The
second column tells to what fuel the cost refers and the third column tells to what insulation the cost refers. For
example, one of the values for fuel 2 and insulation 1 is 120. The data set is retrieved and three ‘table’ commands are
issued. The first table tells us how many measurements there are in each cell, each row and each column. The second
table tells the actual measurements in each row and column. The third table gives cell, row, column and overall means.
Finally the ‘twoway’ command is issued and an ANOVA table is produced. There are three null hypotheses: (i) equal
row means, (ii) equal column means and (iii) no interaction.
Minitab Output:
————— 11/15/2006 4:44:20 PM ————————————————————
Welcome to Minitab, press F1 for help.
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\2F2.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\2F2.MTW'
Worksheet was saved on Wed Nov 15 2006
Results for: 2F2.MTW
MTB > print c1-c3
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
COST
89
101
87
87
120
110
98
104
100
98
86
96
FUEL
1
1
1
1
2
2
2
2
3
3
3
3
INS
1
1
2
2
1
1
2
2
1
1
2
2
MTB > table fuel ins
Tabulated statistics: FUEL, INS
Rows: FUEL
1
2
3
All
Columns: INS
1
2
All
2
2
2
6
2
2
2
6
4
4
4
12
Cell Contents:
Count
1
252anovaex2
11/15/06
MTB > table fuel ins;
SUBC> data cost.
Tabulated statistics: FUEL, INS
Rows: FUEL
Columns: INS
1
2
1
89
101
87
87
2
120
110
98
104
3
100
98
86
96
Cell Contents:
COST
:
DATA
MTB > table fuel ins ;
SUBC> mean cost.
Tabulated statistics: FUEL, INS
Rows: FUEL
1
2
3
All
Columns: INS
1
2
All
95
115
99
103
87
101
91
93
91
108
95
98
Cell Contents:
COST
:
Mean
MTB > Twoway c1 c2 c3;
SUBC>
GBoxplot;
SUBC>
GNormalplot;
SUBC>
NoDGraphs.
Two-way ANOVA: COST versus FUEL, INS
Source
DF
SS
MS
FUEL
2
632 316
INS
1
300 300
Interaction
2
24
12
Error
6
192
32
Total
11 1148
S = 5.657
R-Sq = 83.28%
F
9.88
9.38
0.38
P
0.013
0.022
0.702
R-Sq(adj) = 69.34%
2,6  and F 1,6  that could be used for significance tests. Minitab
Comment: You should be able to find the values of F.05
.05


takes a p-value approach, which means that it computes pvalue  P F 2,6   9.88  0.13 and
pvalue P F 1,6  9.38  0.022 . Since the first two are below our significance level of 5%, we reject the null


hypothesis of equal fuel (row) means and the null hypothesis of equal insulation (column) means. However, the p-value
for interaction is well above 5%, so we cannot reject the null hypothesis of no interaction.
The two diagrams that were printed out follow on the next page. The first diagram shows the range of costs for
each combination of fuel and insulation and how the change in insulation affects the costs for each fuel type. The
second diagram is a Normal plot of the residuals. To do this the computer calculates an expected value for each number
based on the row, column and cell means. The residual is the difference between the expected and observed value. The
fact that the points on the plot show no curvature seems to indicate that the underlying data is Normally distributed.
2
252anovaex2
11/15/06
Boxplot of COST by FUEL, INS
Normplot of Residuals for COST
MTB >
3