Two-Factor Analysis of Variance
Using MINITAB
EXAMPLE An experiment was performed to determine the effects of four different pesticides on the yield of fruit from three
different varieties of a citrus tree. Eight different trees were chosen from each variety and the four different pesticides were applied
randomly such that two trees of each variety were exposed to the same pesticide. The data is given as follows:
Variety
1
2
3
1
49, 39
(44)
55, 41
(48)
66, 68
(67)
Pesticide
2
3
50, 55
43, 38
(52.5)
(40.5)
67, 58
53, 42
(62.5)
(47.5)
85, 92
69, 62
(88.5)
(65.5)
4
53, 48
(50.5)
85, 73
(79)
85, 99
(92)
INPUTTING DATA:
VARIETY
1
1
1
PESTICIDE
1
1
2
YIELD
49
39
50
.
.
.
.
.
.
3
3
4
4
85
99
Assessing the reasonableness of the normality assumption
COMMANDS IN MINITAB: STAT > ANOVA > TWO-WAY > GRAPHS > NORMAL PLOT OF
RESIDUALS
Normal Probability Plot
(response is yield)
99
95
90
Percent
80
70
60
50
40
30
20
10
5
1
-10
-5
0
Residual
5
10
Assessing the reasonableness of the equal variance assumption
COMMANDS IN MINITAB: STAT > ANOVA > TWO-WAY > GRAPHS > RESIDUALS vs fits
Versus Fits
(response is yield)
8
6
Residual
4
2
0
-2
-4
-6
-8
40
50
60
70
Fitted Value
80
90
Perform Two-Factor Analysis of Variance
COMMANDS IN MINITAB: STAT > ANOVA > Two-Way > Row factor VARIETY
Column factor PESTICIDE
Response YIELD
OUTPUT:
Two-way Analysis of Variance
Analysis of Variance for YIELD
Source
DF
SS
MS
VARIETY
2
3996.1
1998.0
PESTICID
3
2227.5
742.5
Interaction
6
456.9
76.2
Error
12
507.5
42.3
Total
23
7188.0
F
47.24
17.56
1.80
P
0.000 (H0: 1=2=3)
0.000 (H0: 1=2=3=4)
0.182 (H0: The two factors do not interact)
Construct Profile Plot
COMMANDS IN MINITAB: STAT > ANOVA > INTERACTIONS PLOT > RESPONSES YIELD
FACTORS VARIETY PESTICIDE
Interaction Plot - Data Means for YIELD
VARIETY
1
2
3
90
Mean
80
70
60
50
40
1
2
3
PESTCIDE
4
Perform Multiple Comparisons
COMMANDS IN MINITAB: STAT > ANOVA > General Linear Model > RESPONSES> YIELD
MODEL > VARIETY PESTICIDE VARIETY*PESTICIDE
COMPARISONS > Pairwise comparisons > Terms > VARIETY PESTICIDE
General Linear Model
Source
VARIETY
PESTICID
VARIETY*PESTICID
Error
Total
DF
2
3
6
12
23
Seq SS
3996.08
2227.46
456.92
507.50
7187.96
Adj SS
3996.08
2227.46
456.92
507.50
Adj MS
1998.04
742.49
76.15
42.29
F
47.24
17.56
1.80
P
0.000
0.000
0.182
Tukey 95.0% Simultaneous Confidence Intervals
Response Variable YIELD
All Pairwise Comparisons among Levels of VARIETY
VARIETY = 1 subtracted from:
VARIETY
2
3
Lower
3.707
22.707
Center
12.37
31.37
Upper
21.04
40.04
-------+---------+---------+--------(-------*--------)
(-------*--------)
-------+---------+---------+--------10
20
30
Upper
27.67
-------+---------+---------+--------(--------*--------)
-------+---------+---------+--------10
20
30
VARIETY = 2 subtracted from:
VARIETY
3
Lower
10.33
Center
19.00
Tukey 95.0% Simultaneous Confidence Intervals
Response Variable YIELD
All Pairwise Comparisons among Levels of PESTICID
PESTICID = 1 subtracted from:
PESTICID
2
3
4
Lower
3.68
-12.98
9.68
Center
14.833
-1.833
20.833
Upper
25.984
9.317
31.984
----+---------+---------+---------+-(----*-----)
(----*-----)
(----*-----)
----+---------+---------+---------+--20
0
20
40
Upper
-5.516
17.151
----+---------+---------+---------+-(-----*----)
(-----*-----)
----+---------+---------+---------+--20
0
20
40
Upper
33.82
----+---------+---------+---------+-(----*-----)
----+---------+---------+---------+--20
0
20
40
PESTICID = 2 subtracted from:
PESTICID
3
4
Lower
-27.82
-5.15
Center
-16.67
6.00
PESTICID = 3 subtracted from:
PESTICID
4
Lower
11.52
Center
22.67