Two-level Fractional Factorial Designs Research Articles

Bayesian-inspired minimum contamination designs under a double-pair conditional effect model

In two-level fractional factorial designs, conditional main effects can provide insights by which to analyze factorial effects and facilitate the de-aliasing of fully aliased two-factor interactions. Conditional main effects are of particular interest in situations where some factors are nested within others. Most of the relevant literature has focused on the development of data analysis tools that use conditional main effects, while the issue of optimal factorial design for a given linear model involving conditional main effects has been largely overlooked. Mukerjee, Wu and Chang [Statist. Sinica 27 (2017) 997–1016] established a framework by which to optimize designs under a conditional effect model. Although theoretically sound, their results were limited to a single pair of conditional and conditioning factors. In this paper, we extend the applicability of their framework to double pairs of conditional and conditioning factors by providing the corresponding parameterization and effect hierarchy. We propose a minimum contamination-based criterion by which to evaluate designs and develop a complementary set theory to facilitate the search of minimum contamination designs. The catalogues of 16- and 32-run minimum contamination designs are provided. For five to twelve factors, we show that all 16-run minimum contamination designs under the conditional effect model are also minimum aberration according to Fries and Hunter [Technometrics 22 (1980) 601–608].

A Note on Early Termination in the 2^(k-p) Designs

A complete factorial experiment can resolve problems that occur in experiments that to determine suitable tolerances for the factors in a manufacturing process. Two-level fractional factorial designs have been widely used to investigate the effect of factors in which several factors are involved. In many circumstances, the experiment may be stopped before all the points have been run. This article argues that one should not arrange the points of the experiment in random order. Instead, one should consider adjusting the run order to protect against the risk of early termination, that is, a systematic run order should be carefully applied to the case. In this article, we will use semi-folding design as examples to illustrate the idea of how to take advantage of the sequential run order.

Construction of Optimal Split-Plot Designs for Various Design Scenarios

When performing fractional factorial experiments in a completely random order is impractical, fractional factorial split-plot designs are suitable options as an alternative. It is well recognized that the more there are lower order effects of interest at lower order confounding, the better the designs. From this viewpoint, this paper considers the construction of optimal regular two-level fractional factorial split-plot designs. The optimality criteria for two different design scenarios are proposed. Under the newly proposed optimality criteria, the theoretical construction methods of optimal regular two-level fractional factorial split-plot designs are then proposed. In addition, we also explore the theoretical construction methods of some optimal regular two-level fractional factorial split-plot designs under the widely adopted general minimum lower order confounding criterion.

Structure of Nonregular Two-Level Designs

Two-level fractional factorial designs are often used in screening scenarios to identify active factors. This article investigates the block diagonal structure of the information matrix of nonregular two-level designs. This structure is appealing since estimates of parameters belonging to different diagonal submatrices are uncorrelated. As such, the covariance matrix of the least squares estimates is simplified and the number of linear dependencies is reduced. We connect the block diagonal information matrix to the parallel flats design (PFD) literature and gain insights into the structure of what is estimable and/or aliased using the concept of minimal dependent sets. We show how to determine the number of parallel flats for any given design, and how to construct a design with a specified number of parallel flats. The usefulness of our construction method is illustrated by producing designs for estimation of the two-factor interaction model with three or more parallel flats. We also provide a fuller understanding of recently proposed group orthogonal supersaturated designs. Benefits of PFDs for analysis, including bias containment, are also discussed.

Microwave-assisted extraction of antioxidant compounds from Russian olive leaves and flowers: Optimization, HPLC characterization and comparison with other methods

Russian olive is a medicinal plant containing phytochemicals with antioxidant activities, which are of great interest due to their beneficial health effects. Our preliminary studies produced extracts from Russian olive leaves and flowers and determined their potential antioxidant activity and their most abundant phenolic compounds were detected. However, the maximum retention of these antioxidant compounds during microwave-assisted extraction (MAE) requires optimized process conditions. Therefore, two-level fractional factorial designs in preliminary tests were used for assessing the magnitude of microwave extraction factors' effects on the extraction of bioactive compounds from Russian olive samples and the experimental domains were determined. Subsequently, Response surface methodology (RSM) was conducted to determine the optimized MAE process variables including citric acid’s molarity (X1: 1–5 M), extraction temperature (X2: 60–110 °C), ethanol concentration (X3: 0–100%) and solid to solvent ratio (X4: 5–15) for maximum recovery of total phenolic content, total flavonoid content and total antioxidant compounds in the extracts. Based on the results, the optimal extraction conditions were as follows: solid to solvent ratio of 7.5 (w/v), citric acid molarity of 2 M, ethanol concentration of 59.8% and 66.4%, and temperature of 97.4 °C and 97.5 °C for Russian olive leaves and flowers, respectively. In addition, the influence of extraction parameters on the responses was modeled using a second-order regression equation and a good agreement between the predicted values and the experimentally obtained values was indicated. Finally, among phenolic compounds which were previously identified through Quadrupole Time of Flight Mass Spectrometer (QTOF-MS) analysis, three abundant flavonoids of the extracts with therapeutic properties including isorhamnetin, luteolin and rutin contained in the optimized extract from MAE were compared with that of ultrasonic extraction and conventional extraction using high-performance liquid chromatography (HPLC) analysis. Results confirmed MAE as an efficient method for extraction of bioactive phytochemicals from Russian olive.

Pilot-Scale Protease Production by Bacillus sp. C4 for Silk Degumming Processes

ABSTRACT The serine protease of Bacillus sp. C-4 has been reported as candidate proteases in the degumming process of the silk industry. The optimization of protease production by Bacillus sp. C4 was carried out using two-level fractional factorial and central composite designs. The investigation was followed by laboratory and pilot-scale protease production using molasses in batch and fed-batch bioreactors to reduce the production cost. The highest protease activity observed (1546.5 U/mL, equivalent to 34.4 U/mL/h) in a 5-L bioreactor was obtained under fed-batch fermentation compared with batch process. The production of protease was subsequently scaled up to a pilot-scale 300-L fed-batch bioreactor, and a maximum protease activity of 1260 U/mL (28 U/mL/h) was observed. The degumming efficiencies of the protease produced by Bacillus sp. C4 in the CCD and bioreactors were 88.3% and 81.6%, respectively, relative to the alkaline degumming method. Furthermore, the stability test of the protease showed that the addition of CaCl2 prolonged the stability of the protease activity from 4 to 8 weeks at room temperature. Furthermore, the protease can be conveniently preserved at room temperature for 1 month and at 4°C for more than 2 months without the addition of any chemicals/cryoprotectants.

Integer programming approaches to find row–column arrangements of two-level orthogonal experimental designs

Design of experiments is an effective, generic methodology for problem solving as well as for improving or optimizing product design and manufacturing processes. The most commonly used experimental designs are two-level fractional factorial designs. In recent years, nonregular fractional factorial two-level experimental designs have gained much popularity compared to the traditional regular fractional factorial designs, because they offer more flexibility in terms of run size as well as the possibility to estimate partially aliased effects. For this reason, there is much interest in finding good nonregular designs, and in orthogonal blocking arrangements of these designs. In this contribution, we address the problem of finding orthogonal blocking arrangements of high-quality nonregular two-level designs in scenarios with two crossed blocking factors. We call these blocking arrangements orthogonal row-column arrangements. We propose two strategies to find row-column arrangements of given two-level orthogonal treatment designs such that the treatment factors’ main effects are orthogonal to both blocking factors. The first strategy involves a sequential approach which is especially useful when one blocking factor is more important than the other. The second strategy involves a simultaneous approach for situations where both blocking factors are equally important. For the latter approach, we propose three different optimization models, so that, in total, we consider four different methods to obtain row-column arrangements. We compare the performance of the four methods by looking for good row-column arrangements of the best two-level 24-run orthogonal designs in terms of the G-aberration criterion, and apply the best of these methods to 64- and 72-run orthogonal designs.

An Evaluation of Estimation Capacity Under the Conditional Main Effect Parameterization

The conditional main effect (CME) parameterization system can enable more informative analyses of regular two-level fractional factorial designs compared to the traditional orthogonal components system. However, formal evaluations of estimation capacities for designs under CME models have yet to be performed. We establish a necessary and sufficient condition for a model consisting of all of the main effects and a selection of CMEs to be estimable in a regular two-level design of resolution at least III. Our condition illuminates the implications of the maximum estimation capacity criterion for analyses of such traditional and conditional effects. A novel aspect of our evaluations is the direct derivation of D-efficiencies for regular designs of resolution at least III with respect to CME models in which the selected CMEs are not siblings or family members, and their corresponding two-factor interactions are not completely aliased with a main effect.

Robust two-level regular fractional factorial designs

In this paper, we introduce the concept of model quality for two-level regular fractional factorial designs. Under the effect hierarchy principle, this paper raises the definition of model quality and introduces robust model-number pattern (RP) to choose the optimal robust design. Some theoretical results on this optimality and comparisons with GMC and MEC criterion are given.

Some new results on Triple designs

The method of doubling has been used to construct two-level fractional factorial designs. Zhang (2016) extended the method of doubling to the method of tripling, which has been used to construct three-level fractional factorial designs. Various popular screening criteria, such as E(fNOD) criterion, generalized minimum aberration (GMA), minimum moment aberration (MMA), B criterion and uniformity criterion, have been proposed from different viewpoints to compare and construct designs. In this paper, the analytic connections between Triple designs constructed by tripling and initial designs under the above screening criteria are considered. These connections are suitable to general original three-level fractional factorial designs, whether regular or nonregular. Furthermore, the connections provide theoretical basis for constructing a kind of optimal three-level designs with large sizes by tripling successively.

A note on extending the alias length pattern

The alias length pattern (ALP) was proposed to characterize the aliasing among two-factor interactions for regular two-level fractional factorial designs of resolution IV. Experimenters often pay more attention to joint estimation of main effects and some two-factor interactions. Hence, resolution III and IV designs are commonly used in practice. In this article, we first extend the ALP for regular two-level fractional factorial designs of resolution III and IV by modeling justification. Then we study the properties of the ALP and provide the relationships between the ALP and other popular criteria.

A fresh look at effect aliasing and interactions: some new wine in old bottles

Interactions and effect aliasing are among the fundamental concepts in experimental design. In this paper, some new insights and approaches are provided on these subjects. In the literature, the “de-aliasing” of aliased effects is deemed to be impossible. We argue that this “impossibility” can indeed be resolved by employing a new approach which consists of reparametrization of effects and exploitation of effect non-orthogonality. This approach is successfully applied to three classes of designs: regular and nonregular two-level fractional factorial designs, and three-level fractional factorial designs. For reparametrization, the notion of conditional main effects (cme’s) is employed for two-level regular designs, while the linear-quadratic system is used for three-level designs. For nonregular two-level designs, reparametrization is not needed because the partial aliasing of their effects already induces non-orthogonality. The approach can be extended to general observational data by using a new bi-level variable selection technique based on the cme’s. A historical recollection is given on how these ideas were discovered.

A note on constructing clear compromise plans

The regular two-level fractional factorial designs of n factors and N runs, having resolution IV and allowing experimenters to clearly estimate all main effects and a set of required two-factor interactions (2fi’s), are called clear compromise plans. Four classes of clear compromise plans have been discussed in the literature. The general minimum lower order confounding (GMC) is an elaborate criterion, which was proposed to select optimal fractional factorial designs. This paper gives a theory on constructing a set of class three clear compromise plans with 5N∕32+3≤n≤N∕4+1, called partially general minimum lower order confounding (P-GMC) designs. We first prove that each P-GMC design is constructed by a GMC design and two specified columns. Then we study the properties of these designs. For N=32,64 and 128, we illustrate that the P-GMC designs are admissible designs. Furthermore, they all have GMC, except for the P-GMC 213−7 and 223−16 designs.

Generalized Subset Designs in Analytical Chemistry.

Design of experiments (DOE) is an established methodology in research, development, manufacturing, and production for screening, optimization, and robustness testing. Two-level fractional factorial designs remain the preferred approach due to high information content while keeping the number of experiments low. These types of designs, however, have never been extended to a generalized multilevel reduced design type that would be capable to include both qualitative and quantitative factors. In this Article we describe a novel generalized fractional factorial design. In addition, it also provides complementary and balanced subdesigns analogous to a fold-over in two-level reduced factorial designs. We demonstrate how this design type can be applied with good results in three different applications in analytical chemistry including (a) multivariate calibration using microwave resonance spectroscopy for the determination of water in tablets, (b) stability study in drug product development, and (c) representative sample selection in clinical studies. This demonstrates the potential of generalized fractional factorial designs to be applied in many other areas of analytical chemistry where representative, balanced, and complementary subsets are required, especially when a combination of quantitative and qualitative factors at multiple levels exists.

Compromise designs under baseline parameterization

We consider estimation of main effects using two-level fractional factorial designs under the baseline parameterization. Previous work in the area indicates that orthogonal arrays are more efficient than one-factor-at-a-time designs whereas the latter are better than the former in terms of minimizing the bias due to non-negligible interactions. Using efficiency criteria, this paper examines a class of compromise designs obtained by adding runs to one-factor-at-a-time designs. A theoretical result is established for the case of adding one run. For adding two or more runs, we develop a complete search algorithm to find optimal compromise designs.

Some properties of double designs in terms of lee discrepancy

Doubling is a simple but powerful method of constructing two-level fractional factorial designs with high resolution. This article studies uniformity in terms of Lee discrepancy of double designs. We give some linkages between the uniformity of double design and the aberration case of the original one under different criteria. Furthermore, some analytic linkages between the generalized wordlength pattern of double design and that of the original one are firstly provided here, which extend the existing findings. The lower bound of Lee discrepancy for double designs is also given.

Copper nanoparticles applied to the preconcentration and electrochemical determination of β-adrenergic agonist: An efficient tool for the control of meat production

A novel method for preconcentration and electrochemical detection of zinterol in bovine urine samples was developed. In order to improve the limit of detection, the surface of a screen-printed carbon electrode was modified with electrodeposited metal copper nanoparticles. The experimental electrodeposition optimization was performed using a central composite design (CCD), involving the variables: precursor concentration, potential and time applied. Copper nanoparticles were characterized by transmission electron microscopy, scanning electron microscopy, cyclic voltammetry, and energy dispersive X-ray spectroscopy. Mesoporous shuttle-like copper oxide nanoparticles were used for the preconcentration step to avoid interferences with many compounds present in the sample matrix. The optimal working conditions for the preconcentration approach were found by means of both two-level fractional factorial and CCD designs. The obtained enhancement factor for a sample volume of 30mL was 35 fold. The calibration curve showed linearity between 0.5 and 45ngmL−1 and the limit of detection was 0.16ngmL−1. The intra and inter assay coefficients of variability were below 4% and 5%; respectively.

Uniformity in double designs

In constructing two-level fractional factorial designs, the so-called doubling method has been employed. In this paper, we study the problem of uniformity in double designs. The centered L 2-discrepancy is employed as a measure of uniformity. We derive results connecting the centered L 2-discrepancy value of D(X) and generalized wordlength pattern of X, which show the uniformity relationship between D(X) and X. In addition, we also obtain lower bounds of centered L 2-discrepancy value of D(X), which can be used to assess uniformity of D(X).

Assessing the bias in blocked two level fractional factorial designs with respect to confounding pattern

Blocked two-level fractional factorial designs are very useful in screening experiment and other areas of scientific research. In some experiments, apart from the main effects, some two-factor interactions may be important and should be estimated. Thus, the postulated model should include all main effects, some important two-factor interactions and blocking effects. The remaining two factor interaction and some other higher order interactions not included in the postulated model may confound and bias the estimate of the effects in the model which in turn lower the precision of the parameter estimate. It is therefore necessary to select an optimal design from the design space that will minimize this bias (contamination). In this article, two designs were selected with respect to confounding pattern and the bias accounted in the estimation of the regression coefficient of the postulated models were evaluated and compared. The confounding pattern of design I and II is (0, 6, 1) and (4, 2, 2) respectively; it was observed that bias accounted in the estimation of regression coefficient is higher in design II.

Robustness Testing in HPLC Analysis of Clarithromycin, Norfloxacin, Doxycycline, Tinidazole and Omeprazole in Pharmaceutical Dosage forms Using Experimental Design

Robustness tests were performed on a gradient RP-HPLC method that had been developed for rapid simultaneous separation and determination of clarithromycin, norfloxacin, doxycycline, tinidazole and omeprazole in their pharmaceutical dosage forms. Separation was carried out on a 250 x 4.6 mm (i.d.), 5µm ODS column (Inertsil, Tokyo, Japan) and multiwavelength overlay chromatograms at max of 210 and 310 nm were used for quantitative analysis. The standard curve was linear in the concentration range of 100 – 900, 5 – 150, 20 – 220, 10 - 150 and 0.5 - 110 µg ml-1 for clarithromycin, norfloxacin, doxycycline, tinidazole and omeprazole, respectively. Robustness tests were performed using twolevel fractional factorial designs with resolution III, so only main effects can be estimated. Factors chosen for this study were the column temperature (˚C), pH, ion pairing agent concentration (g/l), phosphate buffer concentration (g/l), the fraction of mobile phase B (B% start) at 4 min of the gradient and fraction of mobile B (B% end) at 7 min of the gradient run. The significance of the factor effects was determined statistically, using algorithm of Dong in the calculation of critical effects, and graphically, by means of half-normal plots. Non-significant intervals for significant factors and system suitability limits were calculated based on robustness tests' results. The method was validated according to ICH guidelines. The proposed method was found to be accurate, reproducible, and consistent which is useful for the routine determination of clarithromycin, norfloxacin, doxycycline, tinidazole and omeprazole in combined mixtures and pharmaceutical dosage forms.