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A Bayesian Experimental Design Approach for Assessing New Product Performance: An Application to Disinfectant Formulation Navid Omidbakhsh, Thomas A. Duever,* Ali Elkamel and Park M. Reilly Department of Chemical Engineering, University of Waterloo, Waterloo, Canada

This paper presents a Bayesian methodology for computer-aided experimental design for hydrogen peroxide formulations. Hydrogen peroxide is one of the oldest known active antimicrobial chemicals and is used in many cleaning/disinfecting formulations. It is favourable as an active antimicrobial in that it degrades only to water and oxygen, and does not contaminate the environment. However hydrogen peroxide is difficult to stabilise, and disinfecting products based on it soon lose their antimicrobial activity. Moreover, regulatory agencies such as U.S. Environmental Protection Agency (EPA) and Health Canada require that disinfecting products do not lose more than 5–10% of their active concentration throughout their shelf life. Therefore, it is very important while formulating hydrogen peroxide-based products to test for their stability. An effective way to improve hydrogen peroxide stability in a solution is to use stabilisers. It is desired to use these chemicals in as low concentrations as possible for environmental and economic considerations. On the other hand, due to tight market competition, the new products need to be formulated as quickly as possible, and therefore there is limited time to ensure product stability. In this paper, prior information has been used in the form of a model, based on historical experiments. A Bayesian D-optimality criterion is used to design a few additional experiments so that the resulting model can have an acceptable prediction power. It is shown that a design which uses the Bayesian D-optimality criterion taking advantage of prior information can be more efficient than even a resolution IV fractional factorial design in the sense that using fewer trials gives a model with equivalent prediction capability. This can be critical where experiments are expensive to perform. Ce mémoire présente une méthodologie bayésienne pour la conception assistée par ordinateur de formulations de peroxyde d’hydrogène. Le peroxyde d’hydrogène est un agent antimicrobien e´ cologique mais difficile a` stabiliser et les produits désinfectants ayant celui-ci comme base perdent rapidement leur activité antimicrobienne. De plus, les organismes de réglementation dont la U.S. Environmental Protection Agency (EPA) et Santé Canada exigent que les produits désinfectants ne contiennent pas plus de 5 a` -10% de leur concentration active tout au long de leur durée de conservation. Il est donc très important lors de la formulation de produits a` base de peroxyde d’hydrogène de mettre leur stabilité a` l’essai. Une façon efficace d’augmenter la stabilité du peroxyde d’hydrogène en solution est de faire appel a` des stabilisateurs. Il est désirable d’utiliser ces produits chimiques a` des concentrations aussi faibles que possible pour des raisons e´ conomiques et environnementales. Par contre, compte tenu de la concurrence féroce sur le marché, les nouveaux produits doivent eˆ tre formulés aussi rapidement que possible. Il y a donc un temps limité pour assurer la stabilité du produit. Dans ce mémoire, des données antérieures ont e´ té utilisées sous forme de modèle, fondé sur des expériences historiques. Un critère d’optimalité-D bayésien est utilisé pour concevoir quelques expériences supplémentaires de sorte que le modèle en résultant ait une capacité prédictive acceptable. Il est démontré qu’un concept faisant appel au critère d’optimalité-D bayésien bénéficiant des données antérieures peut eˆ tre plus efficace qu’un concept de résolution IV fractionnelle factorielle en ce sens que bien que le recours a` un nombre d’essais soit moindre, le modèle a une capacité prédictive e´ quivalente. Cela peut eˆ tre critique là ou` l’exécution d’expériences peut eˆ tre coˆ uteuse. Keywords: biochemical engineering, industrial chemicals, mathematical modelling, multivariate analysis, statistical theory

INTRODUCTION

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xperimentation plays a very important role in the scientific method. There are different methods of design used for experimentation. One approach is to use the traditional “one-factor-at-a-time” (OFAT) method. This involves varying one factor while keeping the others at constant levels. Although

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∗ Author to whom correspondence may be addressed. E-mail address: [email protected] Can. J. Chem. Eng. 88:88–94, 2010 © 2010 Canadian Society for Chemical Engineering DOI 10.1002/cjce.20254

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simple, this often requires a considerable amount of experimental work and may be costly. Factorial designs of experiments (DOE) are an alternative to the OFAT methodology to design experiments. They have been proven to be more efficient and have many advantages over OFAT including: • They require less resource (experiments, time, material, etc.) for a given amount of information obtained. This is very important since experiments can be very expensive and time consuming. • The estimates of the effects of each factor are more precise, since more observations are used to estimate a single effect (Czitrom, 1999; Montgomery, 2004). • The interaction between factors can be estimated systematically, while they cannot be estimated in OFAT (Czitrom, 1999). • There is experimental information in a larger region of the factor space in DOE (Czitrom, 1999; Montgomery, 2004). Factorial designs are widely used in experiments involving several factors where it is necessary to study the joint effect of the factors of a response. The most widely used design is the 2k factorial, which studies k factors, each at only two levels. These levels maybe quantitative, such as two values of temperature, pressure, or time, or they maybe qualitative, such as two machines, two operators, or perhaps the presence and absence of a factor. These designs are extensively explained in the literature (Montgomery, 2004; Montgomery and Runger, 2006). The response is assumed linear over the range of the chosen factor levels. This is often adequate, particularly in the early stages of a study, though caution is required. As the number of factors in a 2k factorial design increases, the number of runs required for a complete replicate of the design rapidly outgrows the resources of most experiments. The full factorial experiment provides an experimenter with enough information to evaluate the whole set of main effects as well as all interaction effects. The main effects and lower-order interactions are usually the most significant terms. In fact, one is usually capable of determining the main effects and the lower-order interactions by performing a fraction of the complete factorial design with little loss of information. Such a design is called a “fractional factorial” design (FFD). A 2k−p fractional factorial design containing 2k−p runs is called a (1/2p ) fraction of the 2k complete design, or more simply a 2k−p fractional factorial design (Montgomery, 2004). It occurs quite often however that the standard theory of factorial design does not provide an experimental program which is optimal under the existing circumstances. For example, for a large class of situations the engineer has more prior knowledge than can be accommodated by using the standard theory. As a result he is unable to apply fully his judgement, which is an essential part of the art of making engineering designs. Prior knowledge may be present in several ways. It may simply take the form that there is some advance knowledge, however vague, of the parameters in the linear model implied by the factorial experiment. In other situations the engineer or another person with expert knowledge of the system may be able to mentally estimate values, again possibly vaguely, of the response variable for known values of the factors (manipulated variables). The technique is then to analyse the estimates as if they were measured data, taking into account their estimated accuracy. This provides prior parameter estimates and their associated variance–covariance structure. Another situation where a factorial experiment is to be designed under prior information is where a factorial, or possibly another

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type of experiment, has been partially completed and by some misadventure the original design cannot be completed or has been wrongly applied. If further trials are contemplated they should be redesigned taking into account the results of the completed work. All of these examples represent types of experiments which can be designed under conditions where there is a set of possible trials, a known number of which are to be chosen in an optimal manner; given some prior knowledge. While there are different criteria which may be used to define optimality, minimising the determinant of the posterior covariance matrix, often called D-optimality is used here.

Bayesian D-Optimality In many practical cases, for example in the case study of formulating disinfectant products, some prior data are available. If conventional fractional factorial design or D-optimal techniques are applied, one might not fully take advantage of the information hidden in the prior data to “optimally” design further tests. To address this issue, a Bayesian design can be contemplated, which can fully take advantage of the prior knowledge and lead to the optimal design. We use the standard linear regression model to further illustrate the concept of Bayesian D-optimality: y = X ∗ + ε

(1)

In this model y is an n × 1 vector of responses and X is an n × p regression matrix which contains the perfectly known values of the independent variables. The vector is a p × 1 vector of parameters, and * indicates true values. The error term, ε is independently normally distributed with mean zero and variance 2 . Note that every term in the model corresponds to an element of and a column of X. A row of X corresponds to an experimental trial. Prior information about parameter values is expressed as : N(˛,U) in which the p × p covariance matrix U is positive definite. After the data y are observed, Bayes’ theorem (Bayes, 1764) may be applied to give the posterior distribution of the parameter values: ∗ :N y

U

−1

1 + 2 XX

−1 U

−1

1 1 ˛ + 2 X y ; U −1 + 2 X X

−1

(2) It follows that the D-optimal experiment is the one which minimises det{[U−1 + (1/ 2 )X X]−1 }, the determinant of the posterior covariance matrix. This is equivalent to maximising G = det{U−1 + (1/ 2 )X X} which may be written in the form G = M × det(U−1 ), where M = det{I + (1/ 2 )XUX } (Mardia et al., 1988), and I is an n × n identity matrix. The quantity det(U−1 ) is a constant and the determinant in the definition of G is p × p while that in M is n × n. In nontrivial cases n < p. Since U is positive definite, the [I + (1/ 2 )XUX ] matrix never becomes singular, even in the case of trying to design fewer number of trials than the total number of model parameters. Consequently it is advantageous to achieve D-optimality by maximizing M and therefore the design process will consist of choosing the n rows of X which yield the largest M. These rows are selected from a candidate set of trials, which are simply the full 2k factorial experiment. In this study a Bayesian D-optimality criterion will be utilised to augment a set of historical data on a disinfectant formulation. We will then compare Bayesian D-optimality to fractional factorial design for the model prediction and offer appropriate conclusions.

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Case Study Hydrogen peroxide is one of the oldest antimicrobial active ingredients and is generated naturally in many settings (Omidbakhsh, 2006). Its major drawback is that it is not highly stable and easily degrades to oxygen and water over time. Regulatory agencies such as US Environmental Protection Agency (EPA) or Health Canada require disinfecting products to retain at least 95% of their active ingredient content throughout their shelf life. In order to stabilise hydrogen peroxide solutions, stabilisers can be used. However many of them such as phosphonates, EDTA, and NTA have environmental problems or like ethylenediamine-N,N -disuccinic acid (EDDS) are expensive. Other than stabilisers, there are several other ingredients in the formulation such as pH buffers, surfactants, solvents, etc. The objective is to carry out the least possible number of trials to obtain a robust model for peroxide prediction loss so that the peroxide loss of future formulations can be accurately predicted. This prediction capability will enable us to optimize the concentrations of these ingredients to reduce the environmental impact as much as possible.

Disinfectant Formulations Disinfectant formulations contain at least one antimicrobially active ingredient. These actives can be one or a mixture of: • Oxidizing agents such as chlorine based compounds, peroxides such as hydrogen peroxide, peracids including peracetic acid. • Quaternary ammonium compounds such as benzalkonium chloride. • Alcohols such as ethanol, isopropyl alcohol. • Aldehydes such as formaldehyde, glutaraldehyde. • Phenols such as orthophenylphenol. Arguably hydrogen peroxide is one of the most popular actives due to its low toxicity and biodegradability. Hydrogen peroxide can be combined with other ingredients in a formulation to make a disinfectant solution. Surfactants are usually added to the solutions since they improve the cleaning performance. The formulation should also contain pH buffers to maintain pH in a specified range since pH changes can result in more peroxide decomposition and product ineffectiveness. Solvents might also be used in the formulation to improve solubility of different ingredients, as well as enhancing disinfecting and cleaning power. Peroxide stabilisers are the most important part of the formulation since they delay the decomposition of hydrogen peroxide. Peroxide stabilisers chelate heavy metals, which are mostly present as impurities of other ingredients or water and are known as peroxide decomposition catalysts. They also delay the peroxide decomposition process.

For product development purposes, it is not feasible to age the solution at room temperature since this approach requires a few months for each test. Instead, an accelerated stability method is implemented so that the stability test is carried out at a higher temperature for a shorter period of time. The decomposition rate for hydrogen peroxide can be written by an expression of the form: −d(H2 O2 ) = Kg(CH2 O2 , CH2 O , CO2 , . . .) dt

(3)

where K is the kinetic constant, given by the Arrhenius expression (Levenspiel, 1999), g is the function of the concentration of

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A [T1 − T2 ] = log10

t 2

(4)

t1

For our temperature range: A = 0.0342; T1 , test temperature (◦ C); T2 , 20◦ C; t1 , test duration (days); t2 , shelf-life at 20◦ C. In our stability tests, we age the samples at 70◦ C for a 1 week period in a water bath. A volume of 230 mL of each sample is placed in a glass precleaned Erlenmeyer and is placed in the water bath. For precleaning, Erlenmeyers are washed with detergent and then filled with a 5% Etidronic acid solution for 5 min to eliminate any possible metal contamination, rinsed with deionised water and dried in the oven at 50◦ C for an hour. During the stability test, Erlenmeyers are closed with caps to avoid evaporation of the test solution. The hydrogen peroxide in the solution is measured before and after the stability test using an iodometric titration method as described by Blanco (Blanco et al., 2006), and the peroxide loss is calculated as: Peroxide loss =

Initial Peroxide − Final Peroxide Initial Peroxide

(5)

Historical Data Analysis Formulation chemists have usually some knowledge about the different ingredients that they intend to use. Often, they have test results from previous projects which also relate to the ingredients of the current project. These models can be used to save time and cost in product development as will be illustrated here. In this study, a few stability test results were available from the past (Table 1). Also the following model is available based on similar previous studies: log loss = ˇ0 + ˇ1 X1 + ˇ2 X2 + ˇ3 X3 + ˇ4 X4 + ˇ5 X1 X4 + ˇ6 X3 X4 (6) where y is the peroxide loss, X1 is a chelating agent, X2 is an anionic surfactant, X3 is a carboxylic acid, and X4 is the pH of the solution. X1 to X4 represent the coded factors throughout this paper. Other ingredients in the formulations including a pH buffer, and a non-ionic surfactant are known to not affect peroxide stability, and are kept constant in all formulations.

Table 1. Historical stability data

Peroxide Stability Tests

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the relevant species affecting the decomposition rate, and t is the reaction time. For a function g which is commonly used, integrating Equation (3) for time t1 and t2 results in (Schumb et al., 1955):

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No.

X1

X2

X3

X4

y

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0.086

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0.142

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4.84

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−1

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Table 3. Actual experiment for Bayesian D-optimality along with the test results for four trials X1

X2

X3

−1

−1

1

0.886

−1

1

1

−0.943

0.1

1

1

−1

−0.943

0.02

1

1

1

1

1.95

X4

y 7.94

Figure 1. log10 (det(1 + XUX )) versus the number of optimal trials.

The traditional approach is to use OFAT testing such that in each experiment only one factor is changed. OFAT is very inefficient and in particular, fails to find the interactions. A better alternative is to use experimental design methods such as a full or fractional 2k factorial design. A full 2k factorial design for 4 factors would require 16 trials, but there is a resolution IV fractional factorial design with only 8 trials. However, since we already have some existing knowledge, it is more efficient if we design our further experiments in the light of this knowledge. One such method is Bayesian D-optimality. In Figure 1, the determinant of the (I + (1/ 2 )XUX ) matrix is shown for different numbers of trials based on the Bayesian D-optimality criterion explained above. In each case, the X matrix is that of the Bayesian D-optimal design for the number of trials shown. In the calculations below, the prior covariance matrix is calculated as: U = [A−1 + Xp Xp ]−1 where A = 400I and I is a p × p identity matrix. A is also known as the vague initial covariance matrix where the knowledge of each parameter is independently judged as having variance 400, which is a reasonable assumption in our case study. Xp is an n × p matrix of the prior available trials (Table 1). Furthermore, 2 of 0.05 was assumed based on similar analysis. It can be seen from Figure 1 that the det(I + (1/ 2 )XUX ) increases significantly for one to six optimal trials and the rate of the increase decreases afterwards. In the following, we demonstrate different scenarios to design respectively 4, 5, and 6 trials based on the Bayesian D-optimality approach as well as a resolution IV fractional factorial design which contains eight trials. Case 1: augmenting the prior data with four trials using Bayesian D-optimality. Table 2 shows the optimal four-trial experiment designed to augment the previous data. In practice, it was not possible to set X4 (pH) as shown in Table 2 since it is difficult to adjust it precisely. The experiment actually performed is shown in Table 3. The observations of Table 3 were added to the prior data (Table 1) and the whole dataset was analysed and the

Table 2. Bayesian D-optimality design to augment the prior data with four trials

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X1

X2

X3

X4

−1

−1

1

1

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1

1

−1

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1

−1

−1

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Figure 2. Measured peroxide loss versus calculated peroxide loss for a model obtained by augmenting historical data with four experiments.

following model was obtained: log10 y = − 0.174 − 0.306X1 − 0.272X2 − 0.192X3 + 0.975X4 + 0.255X1 X4 + 0.0126X3 X4

(7)

Even though R2 and adjusted R2 are high (respectively 98.3% and 95%), the predicted R2 which is an indication of the model predictive capability was very low (close to zero) and therefore this model is not desirable for prediction purposes. The predicted R2 is defined as 1 − PRESS/SST, where PRESS represents the prediction error sum of squares and SST is the total sum of squares (Allen, 1971; Quan, 1988) PRESS =

n i=1

[yi − yˆ (i) ] = 2

n i=1

ei 1 − hii

2 (8)

where e(i) = yi − yˆ (i) , yˆ (i) is the predicted value of the ith observed response based on a model fit to the remaining n − 1 sample points and hii is the ith diagonal element of the hat matrix H. PRESS is generally regarded as a measure of how well a regression model will perform in predicting new data. A model with a small value of PRESS is desired. The PRESS statistic can be used to compute an R2 -like statistic for prediction say. One of the most important features of PRESS statistic is in comparing regression models. Generally, a model with a small value of PRESS is preferable to one where PRESS is large. Figure 2 shows the measure of peroxide loss versus the predicted values. As seen, the prediction is very poor. Case 2: augmenting the prior data with five trials using Bayesian D-optimality. Table 4 shows the resulting optimal five-trial design, however as mentioned above, it is not possible to set X4 exactly to −1 and 1 and Table 5 shows the actual experiment that was carried out.

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Table 4. Bayesian D-optimality design to augment the prior data with five trials

Table 6. Bayesian D-optimality design to augment the prior data with six trials

X1

X2

X3

X4

X1

X2

X3

X4

−1

−1

−1

−1

−1

−1

−1

1

1

1

−1

−1

−1

1

1

−1

−1

−1

1

1

−1

1

1

1

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1

1

−1

1

−1

1

−1

1

1

1

1

1

−1

1

1

1

1

−1

−1

Table 5. Actual experiment for Bayesian D-optimality along with the test results for five trials Table 7. Actual experiment for Bayesian D-optimality along with the test results for six trials

X1

X2

X3

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−1

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−1

1

X1

X2

X3

1

1

−1

−0.943

0.02

−1

−1

−1

0.828

−1

−1

1

0.886

7.94

−1

1

1

−0.943

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1

1

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−1

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1

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0.01

1

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X4

y

1

1.95

X4

y 13.33

Figure 3. Measured peroxide loss versus calculated peroxide loss for a model obtained by augmenting historical data with five experiments.

The results of Table 5 were added to the prior dataset (Table 1) and the whole dataset was analysed and the following model was obtained:

analysed and the following model was obtained:

log10 y = − 0.17 − 0.313X1 − 0.279X2 − 0.204X3 + 0.973X4 + 0.269X1 X4 + 0.0214X3 X4

Figure 4. Measured peroxide loss versus calculated peroxide loss for a model obtained by augmenting historical data with six experiments.

(9)

log10 y = − 0.181 − 0.425X1 − 0.111X2 − 0.187X3 + 1.04X4 + 0.196X1 X4 + 0.131X3 X4

It is observed here again that even though the number of optimal designs has been increased from 4 to 5, the predictive capability of the model is not increased (still close to zero) although R2 and the adjusted R2 were high (98.2% and 95.5% respectively, Figure 3). Case 3: augmenting the prior data with six trials using Bayesian D-optimality. Table 6 shows the resulting six-trial optimal design and Table 7 shows the actual experiment that was carried out. The results of Table 7 were added to the prior data and the whole dataset were

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(10)

A significant increase in the model predictive capability is observed now by adding 6 optimal trials to the prior data. Figure 4 illustrates the improvements in the model predictive capability. (Predicted R2 = 75.7%). Case 4: augmenting the prior data using a resolution IV fractional factorial design. Table 8 shows eight trials designed based on the resolution IV fractional factorial design, while Table 9 shows the actual experiment conducted. The results of Table 9 were added to the

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Table 8. Prior data augmentation using eight trials based on resolution IV X1

X2

X3

X4

−1

1

−1

1

1

1

1

1

1

−1

−1

1

−1

−1

−1

−1

1

−1

1

−1

1

1

−1

−1

−1

−1

1

1

−1

1

1

−1

Table 9. Actual design based on the resolution IV fractional factorial design X1

X2

X3

X4

y

−1

1

−1

1.171

15.6

1

1

1

1

1.95

1

−1

−1

1

3.9

−1

−1

−1

−1

1

1

−1

1

−0.943

0.01

1

1

−1

−0.943

0.02

−1

−1

1

0.886

7.94

−1

1

1

−0.943

0.1

prior data of Table 1 and the whole dataset was analysed and the following model was obtained: log10 y = − 0.198 − 0.431X1 − 0.122X2 − 0.215X3 + 0.986X4 + 0.201X1 X4 + 0.125X3 X4

(11)

Even though the predicted R2 for eight trials based on fractional factorial design is greater than that of the 6 Bayesian D-optimal design (87.88% vs. 75.7%), comparing Figures 4 and 5 shows that the Bayesian D-optimality results in better predictive capability. The higher value of the predicted R2 for the fractional factorial design can be due to the positioning of the errors, and in general figures are more accurate tools for such comparison purposes than numeric indicators such as predicted R2 .

DISCUSSION AND CONCLUDING REMARKS The Bayesian approach provides a coherent framework where prior information regarding unknown quantities can be combined to find an experimental design that optimizes the goals of the experiment. It is demonstrated in this study that using a Bayesian Doptimality method, one can develop a model that shows a comparable prediction capability to a model based on a fractional

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Figure 5. Measured peroxide loss versus calculated peroxide loss for a model obtained by augmenting historical data with eight experiments based on resolution IV.

factorial which requires more trials. In this way, it is possible to reduce the number of trials without sacrificing the precision of the results. This method can be very useful especially where the experiments are expensive to perform. It should be pointed out that in all this work the model is assumed known. However the concept here is to take advantage of the prior information and in practice there are many situations where this can be done. It should also be noted that this approach can be used to modify the model or develop a new model. It is of interest to note, that the four-trial experiment is contained within the five-trial design, however the five-trial design is not contained within the six-trial design. One approach, which one might envision taking when expanding from a four-trial to a five-trial design, would be to add one additional trial to the previously calculated four-trial optimal design. However this would not necessarily lead to the optimal five-trial experiment. Therefore, the fact that the four-trial experiment is contained within the five-trial experiment is in this case coincidence. It is therefore not unusual that the five-trial design is not included in the six-trial design. In summary the presented methodology is superior to factorial designs in that the number of experiments is flexible and can be any arbitrary number. Furthermore, it is advantageous over conventional D-optimality in that it easily deals with the singularity problem, and more importantly fully takes advantage of the prior information. In other words, if sufficient prior information is available, then one can estimate the parameters no matter how few additional trials are performed.

REFERENCES Allen, D. M., “The prediction sum of squares as a criterion for selecting predictor variables,” Technical report 23, University of Kentucky, Department of statistics (1971). Bayes, T., “An Essay Toward Solving a Problem in the Doctrine of Chances,” Phil. Trans. R. Soc. London 53, 370–418 (1764). Blanco, M., J. Coello and M. J. Sanchez, “Experimental Design for Optimization of Peroxide Formulation Stability and Cost,” J. Surfact. Detergents 9(4), 341–347 (2006). Czitrom, V., “One-Factor-at-a-Time Versus Designed Experiments,” Am. Statist. 53(2), 126–131 (1999). Levenspiel, O., “Chemical Reaction Engineering,” 3rd ed., John Wiley & Sons Inc., New York, USA (1999). Mardia, K. V., J. T. Kent and J. M. Bibby, “Multivariate Analysis,” Academic Press (1988).

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Montgomery, D. C., “Design and Analysis of Experiments,” 6th ed., Wiley, NJ, USA (2004). Montgomery, D. C. and G. C. Runger, “Applied Statistics and Probability for Engineers,” 4th ed., Wiley, New York (2006). Omidbakhsh, N., “A New Peroxide-Based Flexible Endoscope-Compatible High-Level Disinfectant,” Am. J. Infect. Control 34(9), 571–577 (2006). Quan, N. T., “The Prediction Sum of Squares as a General Measure for Regression Diagnostics,” J. Business Econ. Statist. 6(4), 501–504 (1988). Schumb, W. C., C. N. Satterfield and R. L. Wentworth, “Hydrogen Peroxide,” Reinhold Publishing Corp., New York (1955).

Manuscript received April 10, 2009; revised manuscript received August 29, 2009; accepted for publication August 31, 2009.

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