Experimental design application and interpretation in pharmaceutical technology - Computer-Aided Applications in Pharmaceutical Technology - page 45

[3.3]

regression coeffi cients β i can be determined by solving a system of
equations. It is therefore necessary that the number of experiments
performed is equal to or exceeds the number of factors being investigated.
The regression model can also be written in the form of vector components:

y = X β + ε [3.4]

where response vector y is an N × 1 matrix, model X is an N × t matrix
( t is the number of terms included in the model), β is the t × 1 vector of
regression coeffi cients, and ε is the N × 1 error vector. Regression
coeffi cient b is usually calculated using least squares regression:

b = ( X T X ) −1 X T y [3.5]

where X T is the transposed matrix of X .
Also, the effect of each factor x on each response y is estimated as:

[3.6]

where Σ y (+1) and Σ y (−1) represent the sums of the responses, where
factor x is at the (+1) and (−1) levels, respectively, and N is the number of
design experiments.
Because effects estimate the change in response when changing the
factor levels from −1 to +1, and coeffi cients between levels 0 and +1, both
are related as follows:

￿

￿ ￿

E

x = 2 b x

[3.7]

In order to determine the signifi cance of the calculated factor effect,
graphical methods and statistical interpretations are used. Graphically,
normal probability or half-normal probability plots are drawn
(Montgomery, 1997). On these plots, the unimportant effects are found on
a straight line through zero, while the important effects deviate from this
line (Dejaegher and Heyden, 2011). Statistical interpretations are usually
based on t -test statistics, where the obtained t value or the effect E x value
is compared to critical limit values t critical and Ex critical . All effects greater
than these critical values (in absolute terms) are then considered signifi cant:

[3.8]