6.3: Summary And Software Solution

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Summary

Two-way 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 two-way 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

113_1.tif

113_2.tif

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 R-Sq = 98.87% R-Sq(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

112_1.tif

112_2.tif

Anova: Two-Factor 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	p-value	F crit
Sample	1.62E+08	3	53925376	1830.164	5.98E-62	2.748191
Columns	1128272	3	376090.7	12.76408	1.23E-06	2.748191
Interaction	2088667	9	232074.2	7.876325	1.02E-07	2.029792
Within	1885746	64	29464.78

Total	1.67E+08	79