Latin Square Designs

Dynamite Example

Program Listing

OPTIONS LS=76 NONUMBER NODATE NOCENTER;

DATA dynamite;
 DO batch = 1 TO 5;   /* row block factor, 5 batches of raw materials */
  DO operator = 1 TO 5;   /* column block factor, 5 operators */
   INPUT formula $ response @@;   /* Latin letters, treatment level and response */
   OUTPUT;
  END;
 END;
 DATALINES;
 A 24   B 20   C 19   D 24   E 24
 B 17   C 24   D 30   E 27   A 36
 C 18   D 38   E 26   A 27   B 21
 D 26   E 31   A 26   B 23   C 22
 E 22   A 30   B 20   C 29   D 31
;

TITLE1 'Latin Square Designs';
TITLE2 'Dynamite Example';
PROC GLM DATA=dynamite;
 CLASS formula batch operator;
 MODEL response = formula batch operator;
 RUN;
 TITLE3 'Treatment Means and Block Means';
 MEANS formula batch operator;
 RUN;
* Upon inspection of the ANOVA table, the treatment factor
  formula is judged significant.  Hence, it is desirable
  to perform stage 2 analysis. ;
 TITLE3 'Protected LSD for Treatment Means';
 MEANS formula/LSD;
 RUN;
 TITLE3 'Least Square Means for Treatments';
 LSMEANS formula/TDIFF;
QUIT;

Output Listing

Latin Square Designs
Dynamite Example

The GLM Procedure

     Class Level Information
 
Class         Levels    Values
formula            5    A B C D E 
batch              5    1 2 3 4 5 
operator           5    1 2 3 4 5 


Number of Observations Read          25
Number of Observations Used          25
 

Dependent Variable: response   
                            Sum of
Source            DF       Squares   Mean Square  F Value  Pr > F
Model             12   548.0000000    45.6666667     4.28  0.0089
Error             12   128.0000000    10.6666667                 
Corrected Total   24   676.0000000                               


R-Square     Coeff Var      Root MSE    response Mean
0.810651      12.85821      3.265986         25.40000


Source            DF     Type I SS   Mean Square  F Value  Pr > F
formula            4   330.0000000    82.5000000     7.73  0.0025
batch              4    68.0000000    17.0000000     1.59  0.2391
operator           4   150.0000000    37.5000000     3.52  0.0404


Source            DF   Type III SS   Mean Square  F Value  Pr > F
formula            4   330.0000000    82.5000000     7.73  0.0025
batch              4    68.0000000    17.0000000     1.59  0.2391
operator           4   150.0000000    37.5000000     3.52  0.0404



Latin Square Designs
Dynamite Example
Treatment Means and Block Means

The GLM Procedure

Level of           -----------response----------
formula      N             Mean          Std Dev

A            5       28.6000000       4.66904701
B            5       20.2000000       2.16794834
C            5       22.4000000       4.39317653
D            5       29.8000000       5.40370243
E            5       26.0000000       3.39116499


Level of           -----------response----------
batch        N             Mean          Std Dev

1            5       22.2000000       2.48997992
2            5       26.8000000       7.04982269
3            5       26.0000000       7.64852927
4            5       25.6000000       3.50713558
5            5       26.4000000       5.02991054


Level of           -----------response----------
operator     N             Mean          Std Dev

1            5       21.4000000       3.84707681
2            5       28.6000000       6.91375441
3            5       24.2000000       4.60434577
4            5       26.0000000       2.44948974
5            5       26.8000000       6.45755372



Latin Square Designs
Dynamite Example
Protected LSD for Treatment Means

The GLM Procedure
 
t Tests (LSD) for response

NOTE: This test controls the Type I comparisonwise error rate, not the 
experimentwise error rate.


Alpha                            0.05
Error Degrees of Freedom           12
Error Mean Square            10.66667
Critical Value of t           2.17881
Least Significant Difference   4.5005


Means with the same letter are not significantly different.
 
  t
Groupi
  ng            Mean      N    formula
     A        29.800      5    D      
     A                                
     A        28.600      5    A      
     A                                
B    A        26.000      5    E      
B                                     
B    C        22.400      5    C      
     C                                
     C        20.200      5    B      



Latin Square Designs
Dynamite Example
Least Square Means for Treatments

The GLM Procedure
Least Squares Means

               response      LSMEAN
formula          LSMEAN      Number
A            28.6000000           1
B            20.2000000           2
C            22.4000000           3
D            29.8000000           4
E            26.0000000           5


                  Least Squares Means for Effect formula
                 t for H0: LSMean(i)=LSMean(j) / Pr > |t|
                                     
                       Dependent Variable: response
 
i/j              1             2             3             4             5
   1                    4.066633      3.001562      -0.58095       1.25872
                          0.0016        0.0110        0.5720        0.2321
   2      -4.06663                    -1.06507      -4.64758      -2.80791
            0.0016                      0.3078        0.0006        0.0158
   3      -3.00156       1.06507                    -3.58251      -1.74284
            0.0110        0.3078                      0.0038        0.1069
   4      0.580948       4.64758       3.58251                    1.839667
            0.5720        0.0006        0.0038                      0.0907
   5      -1.25872      2.807913      1.742843      -1.83967              
            0.2321        0.0158        0.1069        0.0907              

NOTE: To ensure overall protection level, only probabilities associated 
      with pre-planned comparisons should be used.