# Multilevel Arrangements and Combination Designs

The techniques for DoE designs using the orthogonal arrays for more than two or three levels are explored in this section. Multilevel arrangements can be made when columns representing factors are combined to form a new column with multiple levels. For example, combining factors 1 and 2 and their interaction column 3 in an L8 would allow for creating a substitute factor of four levels, as shown in Table 7.11. It is important to maintain the degrees of freedom (DoF) in this arrangement. DoF is the number of levels in the column minus

1. For an L8 with a four-level column, the column has DoF = 3. This is made up from three columns (1, 2, and 3) of two levels each (DoP = \ for each two-level column). In an L16, the combination of columns 1, 2, and 4 and their interactions 3(12), 5(14), 6(24), and 7(124), shown in Table 7.4, can be combined to form a new column with eight levels and seven degrees of freedom.

If it is desired to use less than the four or eight multilevel arrangement, then only the desired number of levels are used, and some levels are repeated until the end level is reached. For example, if five levels are desired, then an L16 can be used with a combined column of the first seven columns, which has eight levels. The levels used in the combined column could be 1, 2, 3, 4, 5, 1, 2, 3. If one factor level is deemed important, then it can be multiply assigned such as 1, 2,3,4, 5, 4, 4, 4.

Combination designs can also be used for the insertion of two-level factors into a three-level orthogonal array, resulting in the ability to analyze more factors than originally available in the array. For example, a column representing two two-level factors could substitute one of the columns in an L9. In this case, the two columns X and Y of two levels each could be changed to a combined column of three levels (X1Y1, X1Y2 and X2Y1). The analysis of the L9 would be performed, resulting in determining the three levels of the two factors X and Y. Individual factor effects could be further calculated as follows:

The main effect of X at constant Y level = X2YX -X1Y1

The main effect of Y at constant X level =X1Y2-X1YI