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Chapter 13 Summary of strategies to explore the experimental space
325
Chapter 13 Summary of strategies to explore the experimental space 1.
Benefits of a step-wise strategy
When the experimental conditions for a chemical reaction are studied with a view to developing a synthetic method, there are many problems which must be solved. The process may start at a moment when an idea is born by a sudden flash of inspiration, and may last until a series of confirmatory runs has been accomplished and found to give an optimum and reproducible result. In the preceding chapters it was discussed how experiments can be designed to provide answers to questions which may arise during this process. At the beginning, little is known and the questions are rather qualitative. A screening experiment will give a clearer picture and develop the questions for the next step. Towards the end, when precise information is needed, response surface modelling and canonical analysis of the model will give detailed information and also give clues to an understanding at the mechanistic level. This chapter shows how the various tools described in the preceding chapters can be integrated into a step-wise strategy. The benefits of such a step-wise approach to problem-solving are obvious: Knowledge is accumulated as the project is progressing and new experiments can be planned in the light of what has been gained, which gives an adaptive and flexible strategy. A well-designed screening experiment can accomodate many variables and cover a large number of different experimental conditions. If it should be found that the reaction under study does not give any promising results, it is unlikely that it can be developed into something useful. Poor ideas can therefore be abandoned at an early stage.
2.
Flow sheet to define a strategy
The different steps are numbered. At each step there are several points which should be considered. Breakpoints are marked with Q.
326 1: Dejine the problem.
(a): State the objective clearly. Define a criterion of success. (b): Analyze the problem and answer the questions: (i): What is already known? (ii): What is not known? What do I need to know? (iii): 2: Go through the experimental procedure.
(a): Determine the critical steps. (b): Pay attention to practical constraints. 3: Defne response(s) and variables. (a): Determine a suitable response to measure. If possible, use a method
by which the reaction mixture can be analyzed directly.
(b): Evaluate 1 and 2 and determine which experimental variables must be included. If there are discrete variables which are (i): difficult to change in a randomized manner, consider the possibility of a design divided into blocks. 4: Run a pilot experiment to check if it is reproducible. Q:
(a): If the answer is yes, then proceed to 5. (b): If the answer is no, something is out of control. Go back to 1 and reconsider the problem.
5: Design a screening experiment. (a): Assign a tentative experimental domain
(i):
With totally new procedures, this may be difficult. To ensure that a sufficient range of variation has been chosen, run a small pilot experiment with the settings of the variables at the extremes of the tentative domain.
327 (b): Suggest a response surface model for the screening experiment. Use a linear model if there are many variables (i): to consider Use directly a second-order interaction model, (ii): if it is suspected that there are several strong interaction effects. (c): Select a design to fit the model chosen: If possible, use a two-level factorial or (i): fractional factorial design. If there is a constraint on the number of (ii): possible runs, consider a Plackett-Burman design if a fractional factorial design cannot be used. (iii): If there are constraints on the possible settings of the experimental variables so that a twolevel design cannot be used, use a D-optimal design. If there are discrete variables which are (iv): difficult to vary in randomized order, run the experiments in blocks. (d): Run the experiments in random order. 6: Evaluate the screening experiment and determine the significant variables. (a): Evaluate the model. Check the model fit by ANOVA (i): (ii): Plot residuals (iii): Draw projections of the response surface to assist the interpretation. (6): Identify the significant variables. If the experimental error is known, evaluate (i): the significance of the estimated effects from their confidence limits. Use a Normal probability plot to identify (ii): significant variables if the experimental error is not known.
328 (c): If there are any ambiguities due to confounding of interaction effects
with main effects, analyze the alias pattern and consider a complementary run. Use a complementary fraction of a factorial (i): design. (ii): Use a fold-over design.
(d): Run a confirmatory experiment under the best conditions predicted from the model. Q (a): If this experiment gives a satisfactory result which fulfils the criterion of success, then repeat this experiment to make sure that it is reproducible. The goal has been reached. Stop and write a report. (b): If the experimental variables are not found to be significant and the experimental results are inferior to what is satisfactory, go back to I and reconsider the problem. If this does not improve the result, then stop and turn to a more promising project. (c)
If there are significant variables, but the result is below the level of satisfaction, then proceed to 7.
7: Determine a better experimental domain.
(a): Fix the settings of the discrete variables at their most favourable levels. (b): Adjust the settings of the continuous experimental variables so that the explored domain is left in a direction which gives an improvement. Use the method of the steepest ascent. To (i): determine the direction of the search path the linear coefficients from the screening experiment can be used. Use a simplex method with the significant variables. (ii): Q (a) If the best result obtained during the search is satisfactory and fulfils the criterion of success, then repeat this experiment to confirm the reproducibility. Stop and write a report.
(b): If the results are promising, but not satisfactory, and the best settings are far from the first explored domain, there is a risk that
329 the influence of the variables may have been changed. To avoid false conclusions, go to 5 and consider a new screening of the variables in the improved domain. (c):
If the result is promising but not satisfactory, and experimental conditions for the best result obtained during the search are close to the explored domain, then proceed to 8.
8: Determine a quadratic response surface model to map the optimum domain. (a): Analyze the model fit.
(i): (ii): (iii):
Use ANOVA to analyze the regression. Plot the residuals. If necessary, transform the response variable.
(b): Use the model to localize the optimum conditions. Visualize the shape of the response surface by (i): projections and make an intuitive interpretation from the plots. Make a canonical analysis to determine the (ii): nature of the stationary point, and to explore ridges. Use this analysis to achieve a feed-back to an understanding of underlying mechanisms. (iii): Consider a d i a r y responses, which may impose constraints on the solution. Map these responses by separate response surface models and evaluate the models simultaneously to cope with the constraints. (c): Predict the optimum conditions from the model. 9:
Confirm the conclusions by experiments.
References Suggestions to hwther reading
G.E.P. Box, W.G. Hunter and J.S. Hunter
Statislics for Experimenters
Wiley, New York 1978.
G.E.P. Box and N.R. Draper Emphicar Model-Building and Response Surfaces Wiley, New York 1987.
330 S.M. Deming and S.L.Morgan Experimental Design. A Chemomenic Approach Elsevier, Amsterdam 1987.
Chapter 13 Summary of strategies to explore the experimental space 1.
Benefits of a step-wise strategy
When the experimental conditions for a chemical reaction are studied with a view to developing a synthetic method, there are many problems which must be solved. The process may start at a moment when an idea is born by a sudden flash of inspiration, and may last until a series of confirmatory runs has been accomplished and found to give an optimum and reproducible result. In the preceding chapters it was discussed how experiments can be designed to provide answers to questions which may arise during this process. At the beginning, little is known and the questions are rather qualitative. A screening experiment will give a clearer picture and develop the questions for the next step. Towards the end, when precise information is needed, response surface modelling and canonical analysis of the model will give detailed information and also give clues to an understanding at the mechanistic level. This chapter shows how the various tools described in the preceding chapters can be integrated into a step-wise strategy. The benefits of such a step-wise approach to problem-solving are obvious: Knowledge is accumulated as the project is progressing and new experiments can be planned in the light of what has been gained, which gives an adaptive and flexible strategy. A well-designed screening experiment can accomodate many variables and cover a large number of different experimental conditions. If it should be found that the reaction under study does not give any promising results, it is unlikely that it can be developed into something useful. Poor ideas can therefore be abandoned at an early stage.
2.
Flow sheet to define a strategy
The different steps are numbered. At each step there are several points which should be considered. Breakpoints are marked with Q.
326 1: Dejine the problem.
(a): State the objective clearly. Define a criterion of success. (b): Analyze the problem and answer the questions: (i): What is already known? (ii): What is not known? What do I need to know? (iii): 2: Go through the experimental procedure.
(a): Determine the critical steps. (b): Pay attention to practical constraints. 3: Defne response(s) and variables. (a): Determine a suitable response to measure. If possible, use a method
by which the reaction mixture can be analyzed directly.
(b): Evaluate 1 and 2 and determine which experimental variables must be included. If there are discrete variables which are (i): difficult to change in a randomized manner, consider the possibility of a design divided into blocks. 4: Run a pilot experiment to check if it is reproducible. Q:
(a): If the answer is yes, then proceed to 5. (b): If the answer is no, something is out of control. Go back to 1 and reconsider the problem.
5: Design a screening experiment. (a): Assign a tentative experimental domain
(i):
With totally new procedures, this may be difficult. To ensure that a sufficient range of variation has been chosen, run a small pilot experiment with the settings of the variables at the extremes of the tentative domain.
327 (b): Suggest a response surface model for the screening experiment. Use a linear model if there are many variables (i): to consider Use directly a second-order interaction model, (ii): if it is suspected that there are several strong interaction effects. (c): Select a design to fit the model chosen: If possible, use a two-level factorial or (i): fractional factorial design. If there is a constraint on the number of (ii): possible runs, consider a Plackett-Burman design if a fractional factorial design cannot be used. (iii): If there are constraints on the possible settings of the experimental variables so that a twolevel design cannot be used, use a D-optimal design. If there are discrete variables which are (iv): difficult to vary in randomized order, run the experiments in blocks. (d): Run the experiments in random order. 6: Evaluate the screening experiment and determine the significant variables. (a): Evaluate the model. Check the model fit by ANOVA (i): (ii): Plot residuals (iii): Draw projections of the response surface to assist the interpretation. (6): Identify the significant variables. If the experimental error is known, evaluate (i): the significance of the estimated effects from their confidence limits. Use a Normal probability plot to identify (ii): significant variables if the experimental error is not known.
328 (c): If there are any ambiguities due to confounding of interaction effects
with main effects, analyze the alias pattern and consider a complementary run. Use a complementary fraction of a factorial (i): design. (ii): Use a fold-over design.
(d): Run a confirmatory experiment under the best conditions predicted from the model. Q (a): If this experiment gives a satisfactory result which fulfils the criterion of success, then repeat this experiment to make sure that it is reproducible. The goal has been reached. Stop and write a report. (b): If the experimental variables are not found to be significant and the experimental results are inferior to what is satisfactory, go back to I and reconsider the problem. If this does not improve the result, then stop and turn to a more promising project. (c)
If there are significant variables, but the result is below the level of satisfaction, then proceed to 7.
7: Determine a better experimental domain.
(a): Fix the settings of the discrete variables at their most favourable levels. (b): Adjust the settings of the continuous experimental variables so that the explored domain is left in a direction which gives an improvement. Use the method of the steepest ascent. To (i): determine the direction of the search path the linear coefficients from the screening experiment can be used. Use a simplex method with the significant variables. (ii): Q (a) If the best result obtained during the search is satisfactory and fulfils the criterion of success, then repeat this experiment to confirm the reproducibility. Stop and write a report.
(b): If the results are promising, but not satisfactory, and the best settings are far from the first explored domain, there is a risk that
329 the influence of the variables may have been changed. To avoid false conclusions, go to 5 and consider a new screening of the variables in the improved domain. (c):
If the result is promising but not satisfactory, and experimental conditions for the best result obtained during the search are close to the explored domain, then proceed to 8.
8: Determine a quadratic response surface model to map the optimum domain. (a): Analyze the model fit.
(i): (ii): (iii):
Use ANOVA to analyze the regression. Plot the residuals. If necessary, transform the response variable.
(b): Use the model to localize the optimum conditions. Visualize the shape of the response surface by (i): projections and make an intuitive interpretation from the plots. Make a canonical analysis to determine the (ii): nature of the stationary point, and to explore ridges. Use this analysis to achieve a feed-back to an understanding of underlying mechanisms. (iii): Consider a d i a r y responses, which may impose constraints on the solution. Map these responses by separate response surface models and evaluate the models simultaneously to cope with the constraints. (c): Predict the optimum conditions from the model. 9:
Confirm the conclusions by experiments.
References Suggestions to hwther reading
G.E.P. Box, W.G. Hunter and J.S. Hunter
Statislics for Experimenters
Wiley, New York 1978.
G.E.P. Box and N.R. Draper Emphicar Model-Building and Response Surfaces Wiley, New York 1987.
330 S.M. Deming and S.L.Morgan Experimental Design. A Chemomenic Approach Elsevier, Amsterdam 1987.