Evaluation and optimization of variables using response surface methodology

Evaluation and
Optimization ofVariables
using Response surface
Methodology
Dr. Mohammed Abdullah Issa
Department of Chemical and Environmental Engineering
University of Putra Malaysia (UPM)
mohbaghdadi1@yahoo.com

Response Surface Methodology
Tutorial
 This tutorial shows the use of Design-Experts software for response surface
methodology (RSM). This class of designs is aimed toprocess optimization.
 The case study in this tutorial includes synthesis optimization of nanoparticles
(carbon dots).
 Three process factors (input variables):
 A - Temperature
 B – Time
 C – Solvent ratio
 The response of interest (output variable) is:
 Y – Quantum yield

1) Start the program by clicking the Design-Expert software icon then press New Design to start
new design and build new model
2) Click on the Response Surface folder tab to show the designs available for RSM.

 Factors and Levels?
 You will study the optimization process with a standard RSM
designcalled a central composite design (CCD). It’s well
suited for fitting a quadratic surface, which usually works well
for processoptimization.
Factors Levels

3) Select the Central Composite design.
Then click on the entry field for Numeric Factors and enter 3.
Factor Selection
4) Replace the default entries for factor Name (A, B, C), and levels for
Low (-1) and High (1) and entering the details given in this case study

Leave the Type at its default value of Full.
Use Face centered of “Alpha,” set at 1 in coded units
• in case you want to work within the applied ranges.
Use Rotatable Alpha, set at 1.68179 in coded units
• if you want to work slightly out of selected ranges.
You can change alpha to other values via Options.

Click on the Continue button to reach the second page of the “wizard” for building a response
surface design. Select 1 from the pull down list for Responses.Then enter the response Name
and Units for response as shown below.
Press the Continue button to get the design layout.

The three columns on the left of
the design layout identify the
experimental runs.
Design-Expert will randomize the
order, so your runs will probably
not match the layout shown above.

You will now start analyzing the responses numerically. Click on the node labeled
QY
Design-Expert provides a full array of response transformations via the
Transform option. For now, accept the default transformation selection
of None.
1 2 3 4 5 6
Follow orders starting from 1 to 6, then return back to 1 to continue analyze the data

Click on the Fit Summary button next. At this point Design-Expert fits linear, two-
factor interaction (2FI), quadratic and cubic polynomials to the response.
The “Sequential Model Sum of Squares” summary table shows how terms of
increasing complexity contribute to the total model. The model hierarchy is
described below:
 • “Linear”: the significance of adding the linear terms to the mean.
 “2FI”: the significance of adding the two factor interaction terms to the mean
and linear terms already in the model.
 • “Quadratic”: the significance of adding the quadratic (squared) terms to the
mean, linear and two factor interaction terms already in the model.
 • “Cubic”: the significance of the cubic terms beyond all other terms.

For each source of terms (linear, etc.), examine the probability (“PROB > F”) to
see if it falls below 0.05 (or whatever statistical significance level you choose). So
far, the quadratic model looks best – these terms are significant, but adding the
cubic order terms will not significantly improve the fit.

• The “Lack of Fit Tests” table compares the residual error to the “Pure Error” from
replicated design points. If there is significant lack of fit, as shown by a low
probability value (“Prob>F”), then be careful about using the model as a response
predictor. The quadratic model, identified earlier as the likely model, does not show
significant lack of fit. Remember that the cubic model is aliased, so it should not be
chosen.

The “Model Summary Statistics” table lists other statistics useful in comparing
models. The quadratic model comes out best: It exhibits low standard
deviation (“Std. Dev.”), high “R-Squared” values and a low “PRESS.”
R-squared should be equal or higher than 0.8

Click on the Model button at the top of the screen next to see the terms in the model.
For this case study, we’ll leave
the selection at Quadratic.
If we got significant model with
significant LOF, then some
terms needed to be eliminated
for obtaining insignificant LOF
and thus fitting the observed
data.

Once the model was selected, in order to fit the quadratic model to the experimental data for
identifying the relevant model terms, regression method is employed.
Click on the ANOVA button to produce the ANOVA table for the selected model.
the P-values of all three
variables (A, B, C) and their
interactions (AC), except for
the interactions between time
with both temperature (AB) and
solvent ratio (BC) and the
quadratic effect of time (X2),
were found to be significant
("Prob> F" less than 0.05) and
hence they were included in the
final regression equation.

Again scroll down to bring the next
section to your screen.
16
Final Equation for QY Response:
Coded &Actual

You can see the experimental (actual) result, predicted values, residuals and diagnostic
information on the individual experiments in the report.

We can also check the fitness of the suggested model through Fig. below.
The predicted values, which were obtained from model fitting techniques, showed
a strong correlation with the actual values.

Click on the Model Graphs button. The 3D plot of factors A versus B comes up by default.
19
In this case you see a 3d plot of QY as a function
of time and temperature at a mid-level slice of
solvent ratio.
ert® Software
ng: Actual
perature
e
or
ght = 16.7
2
3
4
5
6
230
240
250
260
270
0
5
10
15
20
25
QY(%)
A: Temperature (°C)B: Time (hr)

Click on the Numerical node to start the validation process.
Now you get to the crucial phase of numerical
optimization: assignment of “Optimization Parameters.”
The program uses five possibilities for a “Goal” to
construct desirability indices: maximum, minimum, is
target, is in range, is equal to (factors only)
For this tutorial case study, assume that you
need the QY to be as high as possible. Click
on QY and set its Goal at is maximum.
Process Validation

The ramp display combines the individual graphs for easier interpretation. The dot
on each ramp reflects the factor setting or response prediction for that solution.
Start the optimization by clicking on the Solutions icon

Model represents the terms estimating factor effects.
Residual is the unexplained variation seen as the difference between the observed response and the
value predicted by the model for a particular design point. It is used to estimate experimental error.
Lack of fit It is used to estimate experimental error.
Pure error is the normal variation in the response which appears when an experiment is repeated.
Corrected total is the total sum of squares corrected for the mean.
Sum of squares (SS) is the sum of squared distances from the mean due to an effect.
Model SS is the sum of squares for the terms in the model.
Residual SS is the sum of squares for all the terms not included in the model.
Residual SS = Corrected total SS ˗ Model SS
Lack of fit SS is the residual SS after removing pure error SS
Lack of fit SS = Corrected total SS ˗ Pure error SS ˗ Model SS
Pure error SS is the pure error SS for replicated points.
Corrected total is the sum of squared deviations of each point from the mean.
Degree of freedom (DF) is the number of independent comparisons available to estimate a
parameter.
DF of Residual is the estimation of variance around the model.
DF Residual = Corrected total DF ˗ DF Model
ANOVA (Summary Statistics)

Lack of fit DF is the amount of information available from replicated points.
Lack of fit DF = DF Residual ˗ DF Pure error
DF pure error is the amount of information available from replicated points.
DF Corrected total is the total degrees of freedom for the experiment, minus one for the mean.
Mean Square (MS) is sum of squares divided by number of degrees of freedom and it is used to
estimate model variance.
Mean Square of Model is the estimate of the model variance, calculated by the model sum of squares
divided by model degrees of freedom.
Residual MS is the estimate of process variance.
Residual MS = Residual SS / DF Residual
Lack of fit MS is the estimate of lack of fit
Lack of fit MS = Lack of fit SS / Lack of fit DF
F Value is the test for comparing model variance with residual (error) variance. variances are close to
the same, the ratio will be close to one and it is less likely any of the factors have a significant effect on
the response.
F Value of model compares model variance with residual variances.
F value of Model = Model MS / Residual MS
F Value of lack of fit compares lack of fit variance with pure error variance.
F value of lack of fit = lack of fit MS / Pure error MS
P-Value is a function of the observed sample results (a statistic) that is used for testing a statistical
hypothesis. The level of marginal significance within a statistical hypothesis test, representing the
probability of the occurrence of a given event. The smaller the p-value, the stronger the evidence is in
favor of the alternative hypothesis.

Prob > F is the probability of seeing the observed F value if the null hypothesis is true (there is no
factor effect). Small probability values call for rejection of the null hypothesis. The probability equals
the proportion of the area under the curve of the F-distribution that lies beyond the observed F value. If
the Prob>F value is very small (less than 0.05) then the terms in the model have a significant effect on
the response.
PRESS is predicted Residual Error Sum of Squares. A measure of how the model fits each point in the
design. The PRESS is computed by first predicting where each point should be from a model that
contains all other points except the one in question. The squared residuals (difference between actual
and predicted values) are then summed.
R-Squared: A measure of the amount of variation around the mean explained by the model.
R2 = 1 ˗ [Residual SS / (Model SS + Residual SS)]
Adj R-Squared = Measure of the amount of variation about the mean explained by themodel
adjusted for the number of parameters in themodel.
Pred R-Squared = A measure of the predictive capability of themodel.

Evaluation and optimization of variables using response surface methodology

More Related Content

Similar to Evaluation and optimization of variables using response surface methodology (20)

Recently uploaded (20)

Evaluation and optimization of variables using response surface methodology