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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