Environmental Engineering Reference
In-Depth Information
residual error to the actual error from replication, a lack-of-fit test was
performed using ANOVA. When residual error significantly exceeded actual
error, the model showed significant lack of fit in which another model found
more appropriate.
It was confirmed that the desired result in a lack-of -fit test was achieved,
when the model selected in step 1 di not showed significant lack of fit (where
F test was insignificant). This showed the probability was greater (Prob>F)
and F value was less than the desired significance levels at 99.5% confidence
interval (0.05). To verify the model adequacy, statistical analysis was done and
the statistics include root mean square error (RMSE), adjusted r
2
, predicted r
2
,
and prediction error sum of squares (PRESS). The RMSE was the standard
deviation associated with experimental error. The adjusted r
2
was a measure of
the variation about the mean explained by the model, adjusted for the number
of parameters in the model. The predicted r
2
measured the amount of variation
in new data explained by the model. PRESS measured how well the model fits
each point in the design. To calculate PRESS, the model was used to estimate
each point using all of the design points except the one being estimated.
PRESS was the sum of the squared differences between the estimated values
and the actual values over all points. A good model show a low RMSE, a large
predicted r
2
, and a low PRESS.
A simple linear regression technique (least squares) was used to fit the
model to the data having rough linear relationship. ANOVA was performed
and an overall F-test and lack-of-fit test confirmed the applicability of the
model. Also, summary statistics (r
2
, adjusted r
2
, PRESS, etc.) and the standard
error for each model coefficient were calculated. After the model fitting was
performed, validated the assumptions used in the ANOVA residual analysis.
This analysis includes calculating case statistics to identify outliers and
examining diagnostic plots such as normal probability plots and residual plots.
When these analyses were satisfactory, the model was considered adequate,
and response surface (contour) plots were generated. Contour plots were used
for interpretation and optimization.
‗Design Expert' computer software was used to design and analyze the
experimental data. The program selected 14 points from a list of candidate
points that is known to include the best points for fitting a quadratic
polynomial. The goal was to optimize the matrix composition to produce
desirable quality of composite bricks in compliance with the specifications
For each of the four responses, a model was fit to data using ANOVA and
the least-squares methods, validated by examining the residuals for trends and
outliers and interpreted graphically using trace plots, contour plot, and 3D
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