in a balanced factorial experiment
with two factors, 
and 
, is given as follows:
This section explains the reason behind the use of regression in DOE++ in all calculations
related to the sum of squares. A number of textbooks present the method
of direct summation to calculate the sum of squares. But this method is
only applicable for balanced designs and may give incorrect results for
unbalanced designs. For example, the sum of squares for factor in a balanced factorial experiment
with two factors, and , is given as follows:
where 
represents the levels of factor 
, 
represents the levels of factor 
, and 
represents the number of samples for
each combination of 
and 
. The term 
is the mean value for the 
th level of factor 
, 
is the sum of all observations at
the 
th level of factor 
and 
is the sum of all observations.
The analogous term to calculate 
in the case of an unbalanced design
is given as:
where 
is the number of observations at the
th level of factor 
and 
is the total number of observations.
Similarly, to calculate the sum of squares for factor 
and interaction 
, the formulas are given as:
Applying these relations to the unbalanced data of Table 6.6, the sum
of squares for the interaction 
is:
|
Table 6.6: Example of an unbalanced design. |
which is obviously incorrect since the sum of squares cannot be negative. For a detailed discussion on this refer to [23].
The correct sum of squares can be calculated as shown next. The 
and 
matrices for the design of Table 6.6
can be written as:
Then the sum of squares for the interaction 
can be calculated as:
where 
is the hat matrix and 
is the matrix of ones. The matrix
can be calculated using 
where 
is the design matrix, 
, excluding the last column that represents
the interaction effect 
. Thus, the sum of squares for the
interaction 
is:
This is the value that is calculated by DOE++ (see Figure 6.15 for the experiment design and Figure 6.16 for the analysis).
|
Figure 6.15: Unbalanced experimental design for the data in Table 6.6. |
|
Figure 6.16: Analysis for the unbalanced data in Table 6.6. |
See Also:
