What is an advantage of a factorial design over a one way design?

7.3.1.2.1 Full Factorial or Factorial Design (2k)

In a factorial design, the influence of all experimental factors and their interaction effects on the response(s) are investigated. If the combinations of k factors are investigated at two levels, a factorial design will consist of 2k experiments. In Table 7.1, the factorial designs for 2, 3, and 4 experimental parameters are shown. To continue the example with higher numbers, six parameters would give 26 = 64 experiments; seven parameters would render 27 = 128 experiments, etc. The levels of the factors are given by (−) minus for low level and (+) plus for high level. A zero level is also included, a center, in which all parameters are set at their midvalue. Three or four center experiments should always be included in factorial designs, for the following reasons:

Table 7.1. Factorial Designs

the risk of missing nonlinear relationships in the middle of the intervals is minimized, and

repetition allows for the determination of confidence intervals.

What − and + should correspond to for each parameter is defined from what is assumed to be a reasonable variation to investigate. In this way, the size of the experimental domain has been established. For two and three parameters, the experimental domain and design can be illustrated in a simple way. For two parameters, the experiments will describe the corners in a quadrate (Figure 7.8), while in a design with three parameters, they are the corners of a cube (Figure 7.9).

3.5.2.1 Factorial designs

Factorial designs are the most common designs used in QbD. These are used to identify important factors as well as any interactions that may exist between factors (see Chapter 5 in Ref. 13). These designs are used for method development as well as for showing the ruggedness or robustness of a method over a region. They consist of two or more factors with each factor set at two or more levels. The total number of combinations that could be tested is the product of all the levels. Each combination of factors and levels is called a treatment combination. So if there are two factors at two levels and one factor at four levels, there would be 2 × 2 × 4 = 16 treatment combinations. The design before randomization would look like the following (Table 3.2):

Table 3.2. Three-factor design (two at two levels, low, L, and high, H, and one at four levels, L1-4) prior to randomization

RunFactor A
(Two Levels: L and H)Factor B
(Two Levels: L and H)Factor C
(Four Levels: L1, L2, L3, L4)1LLL12LLL23LLL34LLL45LHL16LHL27LHL38LHL49HLL110HLL211HLL312HLL413HHL114HHL215HHL316HHL4

It is important to run the 16 experiments in a random order to eliminate any systematic errors. Performing a full factorial design allows estimation of the effects that each factor has on the response as well as the possible interactions between the factors. In the example above, there are three factors, so the analysis would include estimation of the main effects, 2-way and 3-way interactions. Main effects of a factor are computed by determining the difference between the average of one level of the factor averaged over all the other factors to the average of another level of the factor averaged over all the other factors. So the main effect of A is the average of the responses corresponding to runs 1–8 to the computed average response from runs 9–16. The 2-way interaction between factors A and B would compare the four combinations of the A and B levels as shown in Table 3.3A. An example of an AB interaction is shown in Table 3.3B and Fig. 3.2.

Table 3.3A. Two-way interactions are determined by comparing the average AB levels

Factor ALHFactor BLAverage runs (1–4)Average runs (9–12)HAverage runs (5–8)Average runs (13–16)

Table 3.3B. An example of the AB interaction

Factor ALHFactor BL78.391.1H80.881.3

In this example, the effect of factor A depends on the level of factor B. At the low level of factor B, increasing factor A from low to high increases the response by 13 but at the high level of B, increasing factor A from low to high increases the response by only 0.5. If a factor is involved in an interaction, then interpreting the factor's main effect can be very misleading. Notice that if an OFAT strategy was performed and the scientist held factor B at the high level first and performed a run at the low and high level of factor A, there would be no difference since they both would result in a measured value around 82. If the scientist followed up holding the factor A at the low level and performed a run at the low and high level of B, then the low level of B would indicate a lower response than the high level of B. The scientist would decide that the high level of B is optimum and factor A has little effect, completely missing the fact that the low level of factor B and high level of factor A would result in a response of over 90.

In running a factorial design, replication of points is highly recommended. In the above example, suppose the experimenter only obtained one result from each of the four combinations of factors A and B of 78.3, 91.1, 80.8, and 81.3 as shown in the table. It is possible that these four results could have been obtained by performing the same combination of factors A and B four times. The variability in the results may not be due to changing the factor levels but rather just the natural variation in the method upon repeating the same treatment combinations four times. If the factors are all quantitative, then replication can be accomplished in a factorial design by adding center points at the mean of each quantitative factor. If the design also contains qualitative factors, then there is no “true” center. For example, if the design consists of the factors time (quantitative) at 5 and 10 min and two solvents (qualitative), then the replicates would be performed at 7.5 min for each solvent. Replication is used to test the significance of factor effects on the response and to provide an estimate of the reproducibility of the treatment combination. Another benefit of center points is that if the factors are quantitative and the center is the average of the high and low levels, then it is possible to obtain an estimate of curvature over the experimental region. If there is a significant curvature, then predicting the response within the factor ranges cannot be done accurately because the design cannot determine which factor is causing the curvature. Additional design points are needed to determine what factor(s) are causing the curvature. Response surface designs (Section 3.5.2.4) are often used to estimate curvature.

Performing a full factorial design with several factors each at several levels becomes large very quickly. For example, seven factors each at two levels would require 128 separate experiments! In this case, fractional factorial designs, which are subsets of full factorial designs, are generally used since they require fewer treatment combinations (see Chapter 6 in Ref. 13). These subsets are chosen in a special way so that the maximum information can be gained from the experiment. Fractional factorials generally provide less information on higher order interactions. For example, a full factorial design with five factors each at two levels would require 25 = 32 treatment combinations. But a half fraction (expressed as 25−1) would require testing only 16 treatment combinations as shown in Table 3.4. The 2 represents the number of levels, the 5 represents the number of factors, and the −1 represents the fraction of the full factorial (a −2 would mean a quarter fraction).

Table 3.4. 25−1 Fractional factorial design

RunFactorsABCDE1LLLLH2LLLHL3LLHLL4LLHHH5LHLLL6LHLHH7LHHLH8LHHHL9HLLLL10HLLHH11HLHLH12HLHHL13HHLLH14HHLHL15HHHLL16HHHHH

There is some loss of information because the entire 32 run design is not performed. This is called confounding, meaning that certain terms are not separable from each other. In this design, the loss of information arises from the fact that the main effects are confounded with the four-way interactions and the two-way interactions are confounded with the three-way interactions. For example, the interaction between A and B is confounded with the three-way interaction of C and D and E. If the AB interaction is significant in the analysis, the experimenter will not know whether the AB interaction is causing significance or the CDE interaction because they are indistinguishable. If the experimenter believes that the only possible effects are the main effects and the two-way interactions, and that the three- and four-way interactions do not exist or are very small, then not much is lost by running the ½ fraction.

22.4 Conclusion

The factorial design of experiments for the batch experimental study of Pb2 + and Zn2 + ions using EC and LM were studied. Four adsorption parameters such as pH, adsorbent dosage, initial concentration, and time were optimized using CCD. The maximum percentage removal (93%) was obtained at optimum condition pH (5), adsorbent dosage (1 g/L), initial concentration (10 mg/L), time (48 h) for Zn2 + ions removal with respect to LM with desirability 1. Equilibrium and kinetic data were analyzed and the results reveal that Langmuir isotherm and pseudo second order be the best-fitted model. The experimental values were in good conformity with the simulated values from the response surface analysis and it confirmed that the RSM using the statistical design of experiments can be efficiently used to optimize the process parameters. The results from the present adsorption experimental studies reported that the Lemna Minor (LM) can be utilized as an effective low-cost adsorbent for the removal of Pb2 + and Zn2 + ions from battery effluent.

6.3.1.1 Factorial design for the formulation of polymeric nanocarriers

A 32 factorial design was performed to study about two factors using three levels as −1, 0, and 1, identified as low, medium, and high, respectively. Here, the polymer (X1) and surfactant concentration (X2) were used as independent variables and the particle size and entrapment efficiency were measured as responses Y1 and Y2, respectively [21]. Fig. 6.1A and B shows the 3D image depicting the effect of polymer and surfactant on particle size and entrapment efficiency, respectively. The values of the dependent and independent variables were shown in Table 6.2.

Table 6.2. 32 factorial design for the preparation of nanoformulation.

Formulation codeIndependentVariablesDependentVariables (response)X1 (PCL)X2 (PVA)Y1 (particle size) (nm)Y2 (Entrapment efficiency) (%)F1−1−1453.485.11F2−10403.285.115F3−11316.756.66F40−1332.887.3F500333.963.22F601549.769.79F71−1404.776.36F810148969.79F911797.756.66For X1For X2Low−1=5 mg−1=0.5%Medium0=15 mg0=1.5%High1=25 mg1=2.5%

5.2.5 Experimental Results and Discussion

A 33 factorial design was used to evaluate the ELID grinding of sapphire. The effect of kinematic parameters such as wheel speed, feed rate and depth of cut on the surface roughness was investigated. The depths of cut used were 1, 2 and 3 μm per pass. The rpms used were 600, 800 and 1000, which have the corresponding wheel speeds 9.6 m/s, 12.8 m/s and 16 m/s, respectively. The feed rates applied were 0.5 m/min, 1 m/min, and 2 m/min. ELID grinding was conducted under E0=60 V, Iinitial=9 A, τon=τoff=3 μs, gap: 0.1~0.2 mm, and pre-dressing time: 30 min. For each set of parameters, 5 repeated grinding tests were run, and 15 roughness measurements were taken in total.

The results of these grinding trials show that the achieved surface roughness can be as good as 50 nm when the super-abrasive wheel and ELID technique are used. The effects of wheel speed, feed rate, and depth of cut on surface finish in ELID grinding of sapphire are shown in Figures 5.17 to 5.21. They show that wheel speeds of 9.8 m/s to 16 m/s have a significant effect on the surface finish of sapphire. Feed rates from 0.5 m/min to 2 m/min also have an effect but this is not as significant as wheel speed.

The ANOVA analysis of the experimental results is shown in Table 5.2.

Table 5.2. ANOVA for 33 Factorial Design

SourceSum of SquaresDOFMean SquareF ValueProb>FModel0.05830.019318.35A: wheel speed0.04410.044726.01<0.0001B: feed rate1.318E-311.318E-321.60<0.0001C: depth of cut0.01310.013207.45<0.0001Residual0.0244016.104E-5Lack of Fit2.358E-3231.025E-41.750.0182Pure Error0.0223785.851E-5Cor Total0.083404

The Model F-value of 318.35 implies the model is significant. There is only a 0.01% chance that a ‘Model F-Value’ this large could occur due to noise. Values of “Prob>F” less than 0.05 indicate model terms are significant.

The regression equation was determined as follows:

(5.21)Ra=0.11344−4.00347E-003×Vs(m/s)+2.89312E-003×Vw(m/min)+6.84815E-003×DOC(μm)

The estimated response surface is shown in Figures 5.22 to 5.24.

From the experimental results and the regression model, it is clear that the depth of cut, wheel speed, and feed rate affect the surface finish in the ELID grinding of sapphire, which is consistent with the theoretical surface roughness model. Increasing wheel speed, reducing the feed rate, or reducing the depth of cut reduces the chip thickness, and thus improves the quality of surface finish.

4 Results and discussion

4.2 Response surfaces

The response surfaces for the products prices are obtained according to the amount of sugarcane processed to cogenerate energy, which has a direct relation with the hydrogen production. Then, scenarios must be compared: a scenario stopping in the hydrogen production and another one aggregated with the methanol plant. The profit is obtained according with the production of the alternative expressed in terms of the respective CO2 emission in the distillery (Table 2) multiplying the prices interval purposed in Table 1. The alternatives are compared through the factorial design and presented in Figure 3.

Figure 3. Response surface of the total profit obtained in the alternatives.

The results show that the problem of high-energy consumption for the production of hydrogen via electrolysis was bypassed using co-generated energy, being possible and viable to design this process in distilleries able to emit more than 350 kton.year- 1 of CO2 with cogeneration systems. In Figure 3 is possible to note that the methanol production combined with sales of oxygen is the most promising alternative. However, the electricity and hydrogen production also produce great profitability on the scale presented.

Formulation components

Box-Behnken factorial design can help in the optimization of nanocarrier formulations. For instance, formulation variables of NPs loaded with the anti-HIV microbicide Tenofovir, formulated following a gelation method, were investigated [28]. The influence of three formulation factors was considered: chitosan concentration, sodium triphosphate pentabasic (STP)/chitosan weight ratio, and Tenofovir/chitosan weight ratio. The optimal formulation based on a mathematical optimization process produced NPs with a size of ≈  210 nm with a relatively low encapsulation efficiency (EE%, ≈  5%). To improve the EE values, a 50% (v/v) ethanol/water mixture was used as a solvent for chitosan. While the use of ethanol resulted in EE of ≈  20%, size increased to ≈  600 nm. Increased mucoadhesive ability from 6% to 12% was observed when the particle size decreased from 900 to ≈  190 nm. Considering factors of EE and mucoadhesive ability, larger-sized NPs were proved the most efficient. Similarly, Tenofovir-loaded solid lipid nanoparticles (SLNs) were prepared by a modified phase inversion technique, and the effect of bovine serum albumin (BSA) concentration, pH of the aqueous phase, and lipid amount (Softisan 100) on the particle mean diameter [29] was evaluated. Box-Behnken factorial design predicted that an increase in both the concentration of BSA and pH has a significant decreasing effect on the particle size. In a more recent work, Box-Behnken factorial design enabled the optimization of PLGA NPs loaded with the anti-HIV microbicide dapivirine [30]. Factors such as the volume of ethyl acetate, the concentration of PVA solution, and the intensity of sonication were explored for their effects on the NP characteristics. Optimized NPs produced by an emulsion solvent evaporation method were monodisperse with a diameter of ≈  170 nm. Such statistical design can be used as a useful tool in the design of nanocarriers with defined properties.

1.11.2.9 Least-Squares Fitting of Response Models

Another way to evaluate factorial designs is to fit a linear response model to the observed data. As shown above, such a model can be seen as an approximation of the underlying, but unknown, response function, f, by a truncated Taylor expansion. For this, the experimental variable should be continuous and the Taylor polynomial defines a response surface. We can also fit polynomial models with discrete variables at two levels. For this, we use dummy variables, xi, to define the discrete settings. The value xi = − 1 is assigned to one of the discrete settings, and xi = 1 to the other. The model is fitted and from the coefficients of the discrete variables it is possible to evaluate how the discrete changes influence on the response. It is also possible to evaluate interactions between discrete variables and other variables from their cross-product coefficients. However, such models cannot be interpreted geometrically as response surfaces.

The models are usually fit using scaled and centered variables. How such scaling is done was shown in Section 1.11.1.8. The designs are defined by their design matrices, D, and these are identical to the corresponding sign tables in which (−) signs are replaced by the scaled setting (−1), and the (+) are replaced by (1). The design matrix for the 22 is shown in Table 12.

Table 12. Design matrix of a 22 factorial design

Experimentx1x21−1−121−13−11411

A Taylor model that accounts for the average responses, the linear effects, and the interaction effect is

y=β0+β1x1+β2x2+β12x1x2+e

The model matrix, X, contains a column corresponding to each term in the model. It is obtained from the design matrix D by appending a column of ones. I, corresponding to the constant term β0, and a column for the cross-product x1x2. The model matrix is shown in Table 13.

Table 13. Model matrix of a 22 factorial design

Ix1x2x1x21−1−1111−1−11−11−11111

Let y be the vector defined by the response in the experiments, y = [y1 y2 y3 y4]T, let β = [β0 β1 β2 β12]T be the vector of model parameters to be estimated, and let e = [e1 e2 e3 e4]T be the vector of errors. The results can be summarized as the matrix equation

(3)[y1y2y3y4]=[1−1−1111−1−11−11−11111][β0β1β2β3]+[e1e2e3e4]

which can be simplified to

(4)y=Xβ+e

The ‘true’ parameter vector β cannot be known with certainty; it contains the true coefficients of the Taylor polynomial, but an estimate can be obtained from the observed responses and the design. A least-squares estimate, b = [b0 b1 b2 b12],T of β is obtained by the corresponding relation

(5)b=(XTX)−1XTy

For this, we have to compute the ‘information matrix’, XTX, and its inverse (XTX)−1, which is called the ‘dispersion matrix’. The information matrix is computed as

(6)XTX=[1111−11−11−1−1111−111][1−1−1111−1−11−11−11111]=[4000040000400004]

It is seen that XTX is a diagonal matrix in which the diagonal elements equals the number of experiments, N = 22 = 4. The columns of X are mutually orthogonal. An experimental design for which the model matrix is a diagonal matrix is called an ‘orthogonal design’. The dispersion matrix (XTX)−1 is also a diagonal matrix in which the diagonal elements are 1/N = 1/4, that is the dispersion matrix is 14I. The matrix I is the identity matrix, a diagonal matrix in which all diagonal elements aii = 1, and the off-diagonal elements aij = 0, for i ≠ j.

(7)(XTX)−1=[14000014000014000014]=14⋅[1000010000100001]

We will now apply these principles to the data from the enamine reduction shown in Table 3. The design matrix and the yields, y, of amine are shown in Table 14:

b=(XTX)−1XTy

can be simplified to

b=14⋅IXTy

since I XT = XT we obtain

b=14⋅XTy

The computations are

[b0b1b2b3]=14[1111−11−11−1−1111−1−11][80.472.494.490.6]

[b0b1b2b3]=14[80.4+72.4+94.4+90.6−80.4+72.4−94.4+90.6−80.4−72.4+94.4+90.680.4−72.4−94.4+90.6]=14⋅[337.8−11.632.24.2]

[b0b1b2b3]=[84.45−2.908.051.05]

The model will thus be

y=84.45−2.9x1+8.05x2+1.05x1x2+e

The model can be interpreted geometrically as a response surface and the features of the response surface is shown in Figure 7. Four model parameters have been computed from four experimental results. There are no degrees of freedom left for an analysis of the validity of the model. Figure 7 is shown for the sole purpose of illustrating a response surface.

Table 14. Design matrix and yields obtained in the enamine reduction

Experimentx1x2y1−1−180.421−172.43−1194.441190.6

The computations shown above are the same as for the computations of effects using sign tables with one exception; in the least-squares fitting, the divisor of the sums is 1/N = 2−k, whereas the divisor for computing the effect is 21−k. Hence, to obtain the coefficients of the Taylor polynomial, the corresponding effect is divided by 2, and to obtain the effects, the corresponding polynomial coefficient should be multiplied by 2. The methods used to compute the effects are therefore equivalent. To evaluate experiments in chemistry, it is suggested that least-squares fitting to Taylor models is the preferred method. The reasons for this are discussed below.

3 Process optimization with designed experiments

Some special types of factorial designs are very useful in process development and improvement. One of such kinds are factorials of the type 2k with k factors, each at two levels usually referred as low level (-1) and high level (+1) of the factor. As the number of factors in a factorial experiment grows the number of effects to estimate also grows. In the present case we have a total of 8 factors and so we would need a total of 256 simulations (experiments) with no replication in order to perform all the combinations. In order to reduce the number of simulations and assuming the sparsity of effects principle a fractional factorial design can be used to obtain information on the main effects and low order interactions. The fractional factorial chosen was 2IV8‒4. In this design no main effect is aliased with any other main effect or two factor interaction being a good design to use in a screening experiment due to its high resolution. This design only requires 16 experiments reducing considerably the number of runs required for a full factorial experiment. For the fractional factorial design four generators were used, E=BCD, F=ADC, G=ABC and H=ABD. In order to interpret the results of fractional factorial designs it is necessary to take into account the alias relationships (Montgomery, 2009).

For the design of experiments simulations, a variation of ±1 stage was used for the structural factors and a variation of ± 5 kgmol/h for the operational factors in relation to the starting values. The response variable selected was the total cost obtained with the Aspen Economic Evaluator using the default definition. With this tool, it is possible to obtain a rapid estimation of the capital and operational cost of each run. After performing the 16 experiments, the effects were estimated and a normal probability plot of the effects was built in order to graphically judge the relevance of the factors and interactions. The estimates that behave like a random sample drawn from a normal distribution have zero mean and the plotted effects will lie approximately along a straight line. Those effects that do not plot on the line are probably the significant effects as we can observe in Figure 2a).The analysis of variance (ANOVA) was performed in order to test which factors and interactions are significant. The mean square of each of the factors and interactions that did not plot on the line (i.e. a total of seven) were calculated and divided by the mean square error. Each of these ratios follows an F distribution, with the numerator degrees of freedom equal to the number of levels minus one (i.e 1 degree of freedom) an the denominator degrees of freedom equal to 8 (i.e 15-7). The computed F should be compared with the tabular value (i.e. F5%,1,8 = 5.32) and the null hypothesis is rejected if the computed F exceed the tabular value for the significance level of 5 %. After ANOVA computation we were able to conclude that all the seven, factors and interactions, affect significantly the total cost (response variable), a result already observed graphically (Figure 2a) and tested with ANOVA. We concluded that factors A, B, G and H are significant as well as the interactions AB, BG and GH. Figure 2(b) represents the plots of the AB, BG and GH interactions and A, B, G and H main effects. The plot of the interaction GH shows that the interaction is very strong, and the effect of changing H from the lower level to the higher level is dependent of the level in which factor G is settled (the interaction hide the main effects). Looking at Figure 2(b) it was easy to conclude that it is better to work with factors A and B in their higher levels in order to minimize the cost. In relation to factors G and H it is better to work with factor H in the lower level an also factor G in the lower level due to the effect of the interaction GH that is stronger than the effect of the individual factors.

After performing the fractional factorial design for process characterization and once the appropriate set of structural and operational factors as well as their levels is identified, the next step was the process optimization in order to find the set of conditions that result in the lowest total cost. In order to optimize we used the method of steepest descent, which is a procedure for moving sequentially along the path of steepest descent that is in the direction of the minimization of the response. A second cycle of simulations were performed varying the factors considered significant. The experiments were conducted along the path of steepest descent with a full factorial design24 with the factors A, B, G, and H varying in the direction of the better level in which the total cost reduces. The results of ANOVA for a level of significance of 5 % allowed to the conclusion that the factors A, B, G and H still affect the response. After that, a third designed experiment was performed in order to further move along the path of steepest descent. After the whole process of optimization, the better conditions of the eight factors (structural and operational variables) in order to minimize the total cost are: A –NF=13; B –NTP=32; C –NTS1=15; D –NTS2=51; E -NSD=42; F –NTS3=8; G – V3=265kgmol/h; H – L1 = 225 kgmol/h. The use of these conditions in the Petlyuk column allows a reduction of the total cost of 9.6 % relative to the starting values.

What are the advantages of factorial experimental designs over one way experimental designs?

What is an advantage of a factorial design?

What are the advantages of factorial Anova over one way Anova?

What is a benefit of a factorial design over a parallel design?