6.3: Summary And Software Solution
 Page ID
 2906
Summary
Twoway analysis of variance allows you to examine the effect of two factors simultaneously on the average response. The interaction of these two factors is always the starting point for twoway ANOVA. If the interaction term is significant, then you will ignore the main effects and focus solely on the unique treatments (combinations of the different levels of the two factors). If the interaction term is not significant, then it is appropriate to investigate the presence of the main effect of the response variable separately.
Software Solutions
Minitab
General Linear Model: yield vs. fert, irrigation
Factor 
Type 
Levels 
Values 

fert 
fixed 
4 
100, 
150, 
200, 
C 
irrigation 
fixed 
4 
A, 
B, 
C, 
D 
Analysis of Variance for Yield, using Adjusted SS for Tests 

Source 
DF 
Seq SS 
Adj SS 
Adj MS 
F 
P 
fert 
3 
1128272 
1128272 
376091 
12.76 
0.000 
irrigation 
3 
161776127 
161776127 
53925376 
1830.16 
0.000 
fert*irrigation 
9 
2088667 
2088667 
232074 
7.88 
0.000 
Error 
64 
1885746 
1885746 
29465 

Total 
79 
166878812 

S = 171.653 RSq = 98.87% RSq(adj) = 98.61% 
Unusual Observations for yield 

Obs 
yield 
Fit 
SE 
Fit 
Residual 
St 
Resid 
4 
2390.00 
2700.20 
76.77 
310.20 
2.02 
R 

28 
2250.00 
2646.00 
76.77 
396.00 
2.58 
R 

35 
4250.00 
3327.60 
76.77 
922.40 
6.01 
R 

R denotes an observation with a large standardized residual. 

Grouping Information Using Tukey Method and 95.0% Confidence 

irrigation 
N 
Mean 
Grouping 

A 
20 
3120.60 
A 

B 
20 
3040.05 
A 

C 
20 
352.85 
B 

D 
20 
129.55 
C 

Means that do not share a letter are significantly different. 

Grouping Information Using Tukey Method and 95.0% Confidence 

fert 
N 
Mean 
Grouping 

150 
20 
1797.90 
A 

200 
20 
1749.95 
A 

100 
20 
1592.55 
B 

C 
20 
1502.65 
B 

Means that do not share a letter are significantly different. 

Grouping Information Using Tukey Method and 95.0% Confidence 

fert 
irrigation 
N 
Mean 
Grouping 

200 
A 
5 
3381.00 
A 

150 
B 
5 
3327.60 
A 

100 
A 
5 
3232.20 
A 

150 
A 
5 
3169.00 
A 

200 
B 
5 
3097.00 
A 

C 
B 
5 
3089.60 
A 

C 
A 
5 
2700.20 
B 

100 
B 
5 
2646.00 
B 

150 
C 
5 
623.80 
C 

100 
C 
5 
340.60 
C 
D 

200 
C 
5 
338.00 
C 
D 

200 
D 
5 
183.80 
D 

100 
D 
5 
151.40 
D 

C 
D 
5 
111.80 
D 

C 
C 
5 
109.00 
D 

150 
D 
5 
71.20 
D 

Means that do not share a letter are significantly different. 
Excel
Anova: TwoFactor With Replication 

SUMMARY 
Bcontrol 
B100 
B150 
B200 
Total 

AA 

Count 
5 
5 
5 
5 
20 

Sum 
13501 
16161 
15845 
16905 
62412 

Average 
2700.2 
3232.2 
3169 
3381 
3120.6 

Variance 
35700.2 
4679.2 
11167.5 
40930 
87716.57 

AB 

Count 
5 
5 
5 
5 
20 

Sum 
15448 
13230 
16638 
15485 
60801 

Average 
3089.6 
2646 
3327.6 
3097 
3040.05 

Variance 
5839.8 
76917.5 
269901.3 
7432.5 
139929.4 

AC 

Count 
5 
5 
5 
5 
20 

Sum 
545 
1703 
3119 
1690 
7057 

Average 
109 
340.6 
623.8 
338 
352.85 

Variance 
351.5 
2525.8 
1079.7 
6782.5 
37326.03 

AD 

Count 
5 
5 
5 
5 
20 

Sum 
559 
757 
356 
919 
2591 

Average 
111.8 
151.4 
71.2 
183.8 
129.55 

Variance 
1485.2 
4135.3 
997.7 
1510.7 
3590.366 

Total 

Count 
20 
20 
20 
20 

Sum 
30053 
31851 
35958 
34999 

Average 
1502.65 
1592.55 
1797.9 
1749.95 

Variance 
2069464 
1977134 
2317478 
2359637 

ANOVA 

Source of Variation 
SS 
df 
MS 
F 
pvalue 
F crit 
Sample 
1.62E+08 
3 
53925376 
1830.164 
5.98E62 
2.748191 
Columns 
1128272 
3 
376090.7 
12.76408 
1.23E06 
2.748191 
Interaction 
2088667 
9 
232074.2 
7.876325 
1.02E07 
2.029792 
Within 
1885746 
64 
29464.78 

Total 
1.67E+08 
79 