Response central composite design

The experimental designs discussed in Chapters 24-26 for optimization can be used also for finding the product composition or processing condition that is optimal in terms of sensory properties. In particular, central composite designs and mixture designs are much used. The analysis of the sensory response is usually in the form of a fully quadratic function of the experimental factors. The sensory response itself may be the mean score of a panel of trained panellists. One may consider such a trained panel as a sensitive instrument to measure the perceived intensity useful in describing the sensory characteristics of a food product. 

This model is capable of estimating both linear and non-linear effects observed experimentally. Hence, it can also be used for optimization of the desired response with respect to the variables of the system. Two popular response surface designs are central composite designs and Box-Behnken designs. Box-Behnken designs were not employed in the experimental research described here and will therefore not be discussed further, but more information on Box-Behnken designs can be obtained from reference [15]. 

Replication is often included in central composite designs. If the response surface is thought to be reasonably homoscedastic, only one of the factor combinations (commonly the center point) need be replicated, usually three or four times to provide sufficient degrees of freedom for s. If the response surface is thought to be heteroscedastic, the replicates can be spread over the response surface to obtain an average purely experimental uncertainty. 

Based on the obtained response surface, a second roimd of optimization follows, using the steepest ascent method where the direction of the steepest slope indicates the position of the optimum. Alternatively, a quadratic model can be fitted around a region known to contain the optimum somewhere in the middle. This so-called central composite design contains an imbedded factorial design with centre... 

If the F-test is significant then there is evidence of a quadratic effect due to at least one of the variables. With the present design, however, the investigator will not be able to determine which of the variables has a quadratic effect on the response. Additional experimentation, perhaps by augmenting the current design with some star points to construct a central composite design (see section on central composite designs below), will need to be conducted to fully explore the nature of the quadratic response surface. 

A class of augmented designs, first proposed by Box and Wilson [6] and frequently applied in response surface work, is the central composite design. Composite designs consist of ... 

Now if each of the design points in the central composite design is replicated five times, so that the complete design has 75 runs, then at each design point we can calculate the average response and the standard deviation of the response. The analysis techniques associated with response surface methodology can then be applied to fit separate models to... 

If a method of analysis is fast or can be fully automated and requires the testing of few factors (three or less) then the larger designs can be considered. Good choices are central composite designs, or if a linear factor response is expected a full factorial design at two levels. 

Figure 3.8. Possible configurations of nine sets of factor values (experiments) that could be used to discover information about the response of a system. Dotted ellipses indicate the (unknown) response surface, (a) A two-factor, three-level factorial design (b) a two-factor central composite design.

Five factors for hydride generation were studied to develop a method to determine As in gasolines. A central composite design was used to develop the response surface and, thus, optimise the extraction procedure... 

A screening design detected significant instrumental and chemical variables to volatilise and measure Sb. They were optimised using response surfaces derived from central composite designs. Findings were confirmed using artificial neural networks... 

FAAS, ETAAS The variables implied on an ultrasound-assisted acid leaching procedure were evaluated by experimental designs. The relevant variables were subsequently optimised by a central composite design and response surface... 

A combination of reversed-phase chromatography and ETAAS was proposed. A central composite design studied the chromatographic conditions. Response curves were deployed for each analyte in order to select the best experimental conditions and evaluate robustness... 

The conditions to determine the three metals were assessed by experimental design and, Anally, optimised by response surfaces developed after applying central composite designs... 

A central composite design for three factors was used to generate 20 combinations. The effects of independent variables—acid/glycerol molar ratio (R), temperature (T), and enzyme concentration (E)—on the response (i.e., the monolaurin molar fraction at 4 h) were investigated. The upper and lower limits of each variable were chosen based on published data and preliminary studies (12,13). Actual independent variables or factors and their corresponding coded levels are presented in Table 1. 

The full factorial central composite design includes factorial points, star points, and center points. The corresponding model is the complete quadratic surface between the response and the factors, as given by Eq. 1 ... 

Central Composite Design of Factors in Coded Levels with Nisin and Lactic Acid Concentration as Response... 

Once the variables having the greatest influence on the responses were identified, a 20-run central composite design was used to optimize the levels of these variables (18). A design matrix was developed (Table 2) and the true values for the variables were determined (Table 3). 

In the second optimization step, the exact values of the three variables that were identified to have significant effects on nisin and/or lactic acid production were determined using a central composite design (Table 2). The coded and actual values of each variable are given in Table 3. The fermentation media (pH 6.5) were composed of 50 g/L of whey, 5 g/L of polypeptone, 1 g/L of Tween-80, and 30 g/L of CaC03, and the predetermined amount of the three variables was assigned by the central composite design. The content of nisin and lactic acid after 24 h of fermentation at 30°C was measured and are presented as responses in Table 2. 

The design matrix is a key concept. A design may consist of a series of experiments performed under different conditions, e.g. a reaction at differing pHs, temperatures, and concentrations. Table 2.6 illustrates a typical experimental set-up, together with an experimental response, e.g. the rate constant of a reaction. Note the replicates in the final five experiments in Section 2.4 we will discuss such an experimental design commonly called a central composite design. 

Many designs for use in chemistry for modelling are based on die central composite design (sometimes called a response surface design), die main principles of which will be illustrated via a three factor example, in Figure 2.29 and Table 2.31. The first step,... 

Three factors, namely (1) irradiation power as a percentage, (2) irradiation time in seconds and (3) number of cycles, are used to study the focused microwave assisted Soxhlet extraction of olive oil seeds, the response measuring the percentage recovery, which is to be optimised. A central composite design is set up to perform the experiments. The results are as follows, using coded values of the variables ... 

Cocaine has been extracted from coca leaves and the optimization procedure was investigated by means of a central composite design [17]. Pressure, temperature, nature, and percentage of polar modifier were studied. A rate of 2 mL/min CO2 modified by the addition of 29 % water in methanol at 20 M Pa for 10 min allowed the quantitative extraction of cocaine. The robustness of the method was evaluated by drawing response surfaces. The same compound has also been extracted by SEE from hair samples [18-20]. 

The central composite design was often selected because of the limited number of experiments needed to sample the response surfaces. In the separation of As and Se species in tap water, the analysis of isoresponse curves allowed the determination of optimum chromatographic conditions and the robustness of the method [77]. The same design was also used to study the influence of an organic modifier and IPR concentration on retention of biogenic amines in wines. To obtain a compromise between resolution and chromatographic time, optimization through a multi-criteria approach was followed [78]. 

Probably, the very long time used in most cases could be significantly reduced by using a multivariate optimization approach focusing on interrelated variables. Figure 5.4 shows a surface response from a central composite design for [HCI]-ultrasound exposure for the determination of tin in coal acidified slurries [6]. 

Figure 5.4. Response surface estimated from the central composite design obtained for the pair [HCIJ-US exposure time In the determination of tin in coal acidified slurries. (Reproduced with permission of Elsevier, Ref [6].)...

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