Latin Square Design 

  A manufacturer conducted a small pilot study of the effect of the
price of one of its products on sales of this product in hardware
stores. Since it might be confusing to customers if prices were switched
repeatedly within a store, only 1 price was used for any one store
during the 6-month study period. Sixteen stores were employed in the
study. To reduce experimental error variability, they were chosen so
that there would be 1 store for each sales volume-geographic location
class. The 4 prices levels (A: $1.79, B: $1.69, C: $1.59, D: $1.49) were
assigned to the stores according to the latin square design below. Data
on sales during the 6-month period (in $000's) are shown:

                                   Geographic Location Class

  Sales Volume Class               
                               NE      NW       SE        SW

         1 (smallest)        1.2(B)   1.5(C)   1.0(A)   1.7(D)    
	 2                   1.4(A)   1.9(D)   1.6(B)   1.5(C)
	 3                   2.8(C)   2.1(B)   2.7(D)   2.0(A)
	 4                   3.4(D)   2.5(A)   2.9(C)   2.7(B)


 Does the price level affect mean sales?

 The SAS code to analyze this latin square design follows. We see we can
use PROC ANOVA. Our data file has information on the rows, columns and
treatments in our latin square, along with the response variable.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

title 'Latin Square Design: Hardware Sales';

options linesize=79;

data hardware;
   infile 'sales.dat';

input Volume Location Price Sales;

PROC PRINT;

PROC ANOVA;
    CLASS Volume Location Price;
    MODEL Sales = Volume Location Price;

RUN;

                      Latin Square Design: Hardware Sales                     

                  Obs    Volume    Location    Price    Sales

                    1       1          1         2       1.2 
                    2       1          2         3       1.5 
                    3       1          3         1       1.0 
                    4       1          4         4       1.7 
                    5       2          1         1       1.4 
                    6       2          2         4       1.9 
                    7       2          3         2       1.6 
                    8       2          4         3       1.5 
                    9       3          1         3       2.8 
                   10       3          2         2       2.1 
                   11       3          3         4       2.7 
                   12       3          4         1       2.0 
                   13       4          1         4       3.4 
                   14       4          2         1       2.5 
                   15       4          3         3       2.9 
                   16       4          4         2       2.7 

                      Latin Square Design: Hardware Sales                     

                              The ANOVA Procedure

                           Class Level Information
 
                       Class         Levels    Values

                       Volume             4    1 2 3 4 
                       Location           4    1 2 3 4 
                       Price              4    1 2 3 4 

                         Number of observations    16
                      Latin Square Design: Hardware Sales                     

                              The ANOVA Procedure
 
Dependent Variable: Sales   

                                      Sum of
Source                     DF        Squares    Mean Square   F Value   Pr > F

Model                       9     7.24062500     0.80451389     40.65   0.0001

Error                       6     0.11875000     0.01979167                   

Corrected Total            15     7.35937500                                  

              R-Square     Coeff Var      Root MSE    Sales Mean

              0.983864      6.841720      0.140683      2.056250

Source                     DF       Anova SS    Mean Square   F Value   Pr > F

Volume                      3     5.98187500     1.99395833    100.75   <.0001
Location                    3     0.12187500     0.04062500      2.05   0.2081
Price                       3     1.13687500     0.37895833     19.15   0.0018