The factors (parameters) involved in an experiment can   be

either quantitative or qualitative. When the initial design and analysis are considered, both types of factors are treated identically. The experimenter tries to determine the differences between the levels of factors. The experimenter is usually interested in creating an interpolation equation for the response variable in the experiment. This equation is an empirical model of the process that has been evaluated. In general, the procedure used for fitting empirical models is called regression analysis (Ref 19).

3.1 Regression Modeling Approach

The aim of multiple regression modeling is to determine the  quantitative  relations  between  independent  variables ðx1; x2; ... ; xk Þ and dependent variable (y). The   relationship

1 1

3.2 Analysis of Variance

The objective of the analysis of variance (ANOVA) is to evaluate the effects of the process parameters on the response and to measure the adequacy of the statistics obtained from the multiple regression equations using the experimental data. In other words, ANOVA checks whether the effect of process parameters (factors) on the desired response is important or not. In addition, the ANOVA method is associated with the regression modeling approach. Therefore, it is essential to perform the general regression significance test by integrating the ANOVA method and the regression modeling approach. This situation can be expressed more clearly by the following equations:

between  these  variables  is  characterized  by  a mathematical n 2

model which is called a regression model. The regression model is fit to set of sample data (Ref 19). Commonly used the mathematical models are represented as follows:

y ¼ f ðx1; x2; ... ; xk Þ ðEq 1Þ

A linear regression equation can be written as  follows:

y ¼ b0  þ b1x1  þ b2x2  þ e ðEq 2Þ

This  equation  is  a  multiple  linear  regression  model     with

two factors. The linear term is used because the, b0, b1, b2, unknown parameters in Eq 2 and, e, experimental error are a linear function. In general, the response (y) is associated   with

k regressor variables. In this case, the multiple linear regres- sion models can be written as  follows:

y ¼ b0 þ b1x1 þ b2x2 þ ··· þ bkxk þ e ðEq 3Þ

These models are more complex  than Eq 3  can  be analyzed by  the  multiple  linear  regressions  modeling  approach.  The

first-order  and  the  second-order  models  can  be  written   as

where  n  is  the  number  of  experiments  yi   is  the   observed

x3  ¼ x1x2; b3  ¼ b12 ð Þ

(measured) response, ^yi  is the fitted (desired) response, and ¯^yi

Y1  ¼ b0  þ b1x1  þ b2x2  þ b12x1x2  þ b3x3  þ e ðEq 5Þ

is the mean value of yi. Also, SSE  is the error sum of   squares,

SST is the total sum of squares, F is the test tool to control whether  the  regression  model  is  statistically  appropriate or