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Statistics LibreTexts

Section 3: Summary And Software Solution

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