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The ANOM Procedure

Example 4.7 ANOM for Cell Means in Presence of Interaction

[See ANMXEX6 in the SAS/QC Sample Library]This example illustrates the use of analysis of means in an experiment with two factors where an interaction effect is present. The following data set cleaninginteract is a modified version of the data set Cleaning, which includes an interaction effect for position and depth.

Consider the following data set cleaninginteract:

data cleaninginteract;  
   do position = 1 to 5;
      do depth = 1 to 3;
         do rep = 1 to 2;
            input concentration @@;
            output;
         end; 
      end;
   end;
   datalines;
   15 16 15 14 19 5
   15 16 14 14 0  1
   19 15 16 16 11 8
   18 16 24 23 8  14
   15 12 23 24 8  11
   ;  
run;

The following statements use PROC GLM to test for an interaction:

proc glm data=cleaninginteract;
   class position depth;
   model concentration = position depth position*depth;
run;

The analysis of variance results in Output 4.7.1 indicate a significant interaction between position and depth.

Output 4.7.1 GLM Results
ANOM for Effect of Position

The GLM Procedure
 
Dependent Variable: concentration

Source DF Sum of Squares Mean Square F Value Pr > F
Model 14 885.666667 63.261905 6.66 0.0004
Error 15 142.500000 9.500000    
Corrected Total 29 1028.166667      

R-Square Coeff Var Root MSE concentration Mean
0.861404 21.75676 3.082207 14.16667

Source DF Type I SS Mean Square F Value Pr > F
position 4 169.0000000 42.2500000 4.45 0.0144
depth 2 515.4666667 257.7333333 27.13 <.0001
position*depth 8 201.2000000 25.1500000 2.65 0.0496

Since an interaction effect is present, an appropriate way to analyze the data is to create an ANOM chart for the cell means.

In order to create the chart you first need to compute the cell means and a new group variable which designates the cells. The following statements use PROC MEANS for this purpose.

proc means data=cleaninginteract n mean std;
   class position depth; 
   var concentration;
   types position*depth;
   output out=cellmeans mean=concentrationX std=concentrationS;
run;
data cellmeans; set cellmeans;
   rename _FREQ_ = concentrationN;
   pos = put(position, z1.);
   dep = put(depth, z1.);
   cell  = cat('P',pos, 'D', dep);
   drop _TYPE_ pos dep;
run;

The cell means are stored in the data set cellmeans shown in Output 4.7.2:

Output 4.7.2 Data Set cellmeans
ANOM for Effect of Position

The MEANS Procedure

Analysis Variable : concentration
position depth N Obs N Mean Std Dev
1 1 2 2 15.5000000 0.7071068
  2 2 2 14.5000000 0.7071068
  3 2 2 12.0000000 9.8994949
2 1 2 2 15.5000000 0.7071068
  2 2 2 14.0000000 0
  3 2 2 0.5000000 0.7071068
3 1 2 2 17.0000000 2.8284271
  2 2 2 16.0000000 0
  3 2 2 9.5000000 2.1213203
4 1 2 2 17.0000000 1.4142136
  2 2 2 23.5000000 0.7071068
  3 2 2 11.0000000 4.2426407
5 1 2 2 13.5000000 2.1213203
  2 2 2 23.5000000 0.7071068
  3 2 2 9.5000000 2.1213203



ANOM for Effect of Position

position depth concentrationN concentrationX concentrationS cell
1 1 2 15.5 0.70711 P1D1
1 2 2 14.5 0.70711 P1D2
1 3 2 12.0 9.89949 P1D3
2 1 2 15.5 0.70711 P2D1
2 2 2 14.0 0.00000 P2D2
2 3 2 0.5 0.70711 P2D3
3 1 2 17.0 2.82843 P3D1
3 2 2 16.0 0.00000 P3D2
3 3 2 9.5 2.12132 P3D3
4 1 2 17.0 1.41421 P4D1
4 2 2 23.5 0.70711 P4D2
4 3 2 11.0 4.24264 P4D3
5 1 2 13.5 2.12132 P5D1
5 2 2 23.5 0.70711 P5D2
5 3 2 9.5 2.12132 P5D3

The data set cellmeans has the structure of a SUMMARY= input data set for the ANOM procedure. For details concerning a SUMMARY= data set, see the section Creating ANOM Charts for Means from Group Summary Data.

The following statements use cellmeans to create the ANOM chart for the cell means using SUMMARY= option:

title "ANOM for Cell Means of Position and Depth";
proc ANOM summary = cellmeans;
   xchart concentration * cell / turnhlabels;      
   label concentrationX = 'Mean of Concentration';
   label cell           = 'Cell';
run;

The chart is shown in Output 4.7.3:

Output 4.7.3 ANOM for Cell Means of Position and Depth
ANOM for Cell Means of Position and Depth

The chart shows that the cell means for P2D3, P4D2, and P5D2 are significantly different from the average concentration level.

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