Biometrical Applications  

1. Bias in Research

     
2. RCBD vs CRD      
Questions    
Stability Analysis (excel)    
Before combining locations, do you test for homogeneity of error terms?

data one;
input loc rep nrate yield MP;
cards;
1 1 20 20 24
1 1 40 23 23
1 1 60 24 22
1 1 80 26 25
1 1 100 35 29
1 2 20 22 20
1 2 40 24 22
1 2 60 30 33
1 2 80 31 34
1 2 100 34 36
1 3 20 21 17
1 3 40 25 20
1 3 60 26 26
1 3 80 28 32
1 3 100 30 34
2 1 20 33 35
2 1 40 34 32
2 1 60 40 41
2 1 80 45 46
2 1 100 50 47
2 2 20 33 32
2 2 40 34 33
2 2 60 39 40
2 2 80 44 44
2 2 100 45 47
2 3 20 30 22
2 3 40 31 23
2 3 60 40 40
2 3 80 45 46
2 3 100 46 44
3 1 20 50 41
3 1 40 40 35
3 1 60 50 44
3 1 80 44 44
3 1 100 44 41
3 2 20 51 48
3 2 40 48 40
3 2 60 47 44
3 2 80 50 49
3 2 100 49 44
3 3 20 49 43
3 3 40 44 42
3 3 60 43 41
3 3 80 44 40
3 3 100 45 47
4 1 20 30 23
4 1 40 35 34
4 1 60 36 36
4 1 80 37 32
4 1 100 37 33
4 2 20 29 22
4 2 40 32 34
4 2 60 33 33
4 2 80 35 32
4 2 100 34 30
4 3 20 33 33
4 3 40 35 36
4 3 60 38 40
4 3 80 40 39
4 3 100 38 36
proc glm;
classes loc rep nrate;
model mp yield = loc rep(loc) nrate loc*nrate;
means loc nrate loc*nrate;
run;
proc glm;
classes loc rep nrate;
model yield = loc rep(loc) nrate loc*nrate mp;
lsmeans loc nrate loc*nrate;
run;
proc glm;
model yield=mp;
run;
proc plot;
plot yield*mp='x';
run;


LOCATION*TRT INTERACTIONS:
Dependent Variable: yield
                                               Sum of
       Source                      DF         Squares     Mean Square    F Value    Pr > F
       Model                       27     4182.666667      154.913580      49.11    <.0001
       Error                       32      100.933333        3.154167
       Corrected Total             59     4283.600000
                        R-Square     Coeff Var      Root MSE    yield Mean
                        0.976437      4.826080      1.775997      36.80000
       Source                      DF       Type I SS     Mean Square    F Value    Pr > F
       loc                          3     3132.933333     1044.311111     331.09    <.0001
       rep(loc)                     8      121.066667       15.133333       4.80    0.0006
       nrate                        4      484.933333      121.233333      38.44    <.0001
       loc*nrate                   12      443.733333       36.977778      11.72    <.0001
       Source                      DF     Type III SS     Mean Square    F Value    Pr > F
       loc                          3     3132.933333     1044.311111     331.09    <.0001
       rep(loc)                     8      121.066667       15.133333       4.80    0.0006
       nrate                        4      484.933333      121.233333      38.44    <.0001
       loc*nrate                   12      443.733333       36.977778      11.72    <.0001
       Level of            --------------MP-------------     ------------yield------------
       loc           N             Mean          Std Dev             Mean          Std Dev
       1            15       26.4666667       6.08119662       26.6000000       4.59502837
       2            15       38.1333333       8.30547725       39.2666667       6.38599658
       3            15       42.8666667       3.54293395       46.5333333       3.31375201
       4            15       32.8666667       4.96943036       34.8000000       3.05193147
       Level of            --------------MP-------------     ------------yield------------
       nrate         N             Mean          Std Dev             Mean          Std Dev
       20           12       30.0000000       10.1533693       33.4166667       11.0161727
       40           12       31.1666667        7.3587713       33.7500000        7.6767062
       60           12       36.6666667        6.9718047       37.1666667        7.8836232
       80           12       38.5833333        7.4767316       39.0833333        7.6450618
       100          12       39.0000000        6.7823300       40.5833333        6.6668561
 Level of     Level of           --------------MP-------------     ------------yield------------
 loc          nrate        N             Mean          Std Dev             Mean          Std Dev
 1            20           3       20.3333333       3.51188458       21.0000000       1.00000000
 1            40           3       21.6666667       1.52752523       24.0000000       1.00000000
 1            60           3       27.0000000       5.56776436       26.6666667       3.05505046
 1            80           3       30.3333333       4.72581563       28.3333333       2.51661148
 1            100          3       33.0000000       3.60555128       33.0000000       2.64575131
 2            20           3       29.6666667       6.80685929       32.0000000       1.73205081
 2            40           3       29.3333333       5.50757055       33.0000000       1.73205081
 2            60           3       40.3333333       0.57735027       39.6666667       0.57735027
 2            80           3       45.3333333       1.15470054       44.6666667       0.57735027
 2            100          3       46.0000000       1.73205081       47.0000000       2.64575131
 3            20           3       44.0000000       3.60555128       50.0000000       1.00000000
 3            40           3       39.0000000       3.60555128       44.0000000       4.00000000
 3            60           3       43.0000000       1.73205081       46.6666667       3.51188458
 3            80           3       44.3333333       4.50924975       46.0000000       3.46410162
 3            100          3       44.0000000       3.00000000       46.0000000       2.64575131
 4            20           3       26.0000000       6.08276253       30.6666667       2.08166600
 4            40           3       34.6666667       1.15470054       34.0000000       1.73205081
 4            60           3       36.3333333       3.51188458       35.6666667       2.51661148
 4            80           3       34.3333333       4.04145188       37.3333333       2.51661148
 4            100          3       33.0000000       3.00000000       36.3333333       2.08166600
1. Test Treatment Significance of Potential Covariate
If TRT effect is not significant using the potential covariate (MP) as a dependent variable
it can be used as a covariate in AOV
Dependent Variable: MP
                                               Sum of
       Source                      DF         Squares     Mean Square    F Value    Pr > F
       Model                       27     3936.583333      145.799383      15.35    <.0001
       Error                       32      304.000000        9.500000
       Corrected Total             59     4240.583333
                        R-Square     Coeff Var      Root MSE       MP Mean
                        0.928312      8.785388      3.082207      35.08333
       Source                      DF       Type I SS     Mean Square    F Value    Pr > F
       loc                          3     2235.650000      745.216667      78.44    <.0001
       rep(loc)                     8      289.333333       36.166667       3.81    0.0031
       nrate                        4      855.333333      213.833333      22.51    <.0001
       loc*nrate                   12      556.266667       46.355556       4.88    0.0002
       Source                      DF     Type III SS     Mean Square    F Value    Pr > F
       loc                          3     2235.650000      745.216667      78.44    <.0001
       rep(loc)                     8      289.333333       36.166667       3.81    0.0031
       nrate                        4      855.333333      213.833333      22.51    <.0001
       loc*nrate                   12      556.266667       46.355556       4.88    0.0002
2.  Enter Covariate as independent effect in the model, but do not include
it in the classes statements (not an effect we are controlling)
Use LSMEANS (adjusted means)

Dependent Variable: yield
                                               Sum of
       Source                      DF         Squares     Mean Square    F Value    Pr > F
       Model                       28     4198.509956      149.946784      54.63    <.0001
       Error                       31       85.090044        2.744840
       Corrected Total             59     4283.600000
                        R-Square     Coeff Var      Root MSE    yield Mean
                        0.980136      4.502054      1.656756      36.80000
       Source                      DF       Type I SS     Mean Square    F Value    Pr > F
       loc                          3     3132.933333     1044.311111     380.46    <.0001
       rep(loc)                     8      121.066667       15.133333       5.51    0.0002
       nrate                        4      484.933333      121.233333      44.17    <.0001
       loc*nrate                   12      443.733333       36.977778      13.47    <.0001
       MP                           1       15.843289       15.843289       5.77    0.0225
       Source                      DF     Type III SS     Mean Square    F Value    Pr > F
       loc                          3     270.9265184      90.3088395      32.90    <.0001
       rep(loc)                     8      48.2687576       6.0335947       2.20    0.0555
       nrate                        4      78.3412862      19.5853215       7.14    0.0003
       loc*nrate                   12     150.5722362      12.5476864       4.57    0.0003
       MP                           1      15.8432895      15.8432895       5.77    0.0225
 
                                        The GLM Procedure
                                       Least Squares Means
                                       loc    yield LSMEAN
                                       1        28.5670943
                                       2        38.5703838
                                       3        44.7564803
                                       4        35.3060417
                                      nrate    yield LSMEAN
                                      20         34.5771382
                                      40         34.6441338
                                      60         36.8052083
                                      80         38.2843202
                                      100        39.6891996
                                   loc    nrate    yield LSMEAN
                                   1      20         24.3672697
                                   1      40         27.0628838
                                   1      60         28.5120066
                                   1      80         29.4177083
                                   1      100        33.4756031
                                   2      20         33.2365680
                                   2      40         34.3126645
                                   2      60         38.4681469
                                   2      80         42.3266996
                                   2      100        44.5078399
                                   3      20         47.9644189
                                   3      40         43.1058662
                                   3      60         44.8593750
                                   3      80         43.8883224
                                   3      100        43.9644189
                                   4      20         32.7402961
                                   4      40         34.0951206
                                   4      60         35.3813048
                                   4      80         37.5045504
                                   4      100        36.8089364
3.  Plot covariate versus dependent variable (simple linear model)

Dependent Variable: yield
                                               Sum of
       Source                      DF         Squares     Mean Square    F Value    Pr > F
       Model                        1     3660.722778     3660.722778     340.87    <.0001
       Error                       58      622.877222       10.739262
       Corrected Total             59     4283.600000
                        R-Square     Coeff Var      Root MSE    yield Mean
                        0.854590      8.905112      3.277081      36.80000
       Source                      DF       Type I SS     Mean Square    F Value    Pr > F
       MP                           1     3660.722778     3660.722778     340.87    <.0001
       Source                      DF     Type III SS     Mean Square    F Value    Pr > F
       MP                           1     3660.722778     3660.722778     340.87    <.0001
                                                  Standard
                Parameter         Estimate           Error    t Value    Pr > |t|
                Intercept      4.203462574      1.81551271       2.32      0.0242
                MP             0.929117456      0.05032392      18.46      <.0001
                              Plot of yield*MP.  Symbol used is 'x'.
 yield 
       
    51                                                                                  x
    50                                                                 x       x      x    x
    49                                                                      x  x
    48                                                               x
    47                                                                         x
    46                                                                         x
    45                                                                             x  x
    44                                                               x x  x    x
    43                                                                 x
    42 
    41 
    40                                                   x         x x x
    39                                                               x
    38                                                     x         x
    37                                            x x
    36                                                     x
    35                                     x      x    x   x
    34                                       x    x x      x
    33                                            x x    x
    32                                                 x
    31                      x                          x
    30                    x x                       x  x
    29                    x
    28                                            x
    27 
    26                           x x
    25               x
    24                    x
    23                      x
    22               x
    21        x
    20                         x
       
       
         15          20          25          30          35          40          45          50
                                                   MP
NOTE: 5 obs hidden.