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Statistics LibreTexts

Section 3: Summary And Software Solution

[ "article:topic", "authorname:dkiernan" ]
  • Page ID
    2906
  • 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