Level Fractional Factorial Designs Research Articles

Mixed 2- and [formula omitted]-level fractional factorial split-plot designs with clear effects

Fractional factorial split-plot (FFSP) designs are often used when the levels of some factors are difficult to change or control. If not all experimental factors have the same number of levels, mixed-level designs are natural choices. This paper provides the necessary and sufficient conditions for mixed 2- and 2r-level FFSP designs of resolution III or IV to contain clear main effects or two-factor interaction components. Particularly, the sufficient conditions are proved through constructing the corresponding designs. The new results here are more general and include Zhao and Chen (2012a,b)’s results as special cases for r=2.

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.

Construction and Nonparametric Testing of Orthogonal Arrays through Algebraic Strata and Inequivalent Permutation Matrices

In this article we use a new methodology, based on algebraic strata, to generate the class of all the orthogonal arrays of given size and strength. From this class we extract all the non isomorphic orthogonal arrays. Then, using all these non isomorphic orthogonal arrays, we suggest a method based on the inequivalent matrices permutations testing procedures Basso et al. (2004) in order to obtain separate permutation tests for the effects in unreplicated mixed level fractional factorial designs. In order to validate the proposed method we perform a Monte Carlo simulation study and find out that the permutation tests appear to be a valid solution for testing effects, in particular when the usual normality assumptions cannot be justified.

Optimization of microwave assisted extraction for determination of iron and manganese in cereal flour using FAAS: Doehlert design

The present work describes the development by response surface methodology of a method for manganese and iron determination by flame atomic absorption spectrometry (FAAS) in cereal flours, which is based on microwave assisted extraction. The optimization process was performed using two level fractional factorial designs (24-1) and Doehlert design matrix. Four variables such as irradiation power, time, temperature and concentration of nitric acid were regarded as factors in the optimization. Results of two level fractional factorial designs with 10 runs for manganese and iron extraction, based on the Pareto chart, demonstrated that the factors irradiation power, time and temperature are statistically significant, and concentration of nitric acid is not significant. Then, a Doehlert design was carried out in order to determine the optimum conditions for the procedure through response surface study. The working conditions were established as a compromise between optimum values found for each analyte. These values were 87.0°C, 573.0 W and 40.0 min, for temperature, irradiation power and time, respectively. The accuracy of the optimized procedure was evaluated by analysis of certified reference material. The procedure proposed was applied for the determination of iron and manganese in cereal flours.

A Class of Partially Replicated Two Level Fractional Factorial Designs

Resolution III fractional factorial designs permit the uncorrelated estimation of all main effects under the assumption of no interactions among the factors and are often used in the initial phases of an experimental program to screen factors. Even in these experimental situations, it may be desirable to include partial replication of the design, thereby allowing for the unbiased estimation of the experimental error while maintaining the orthogonality of the main effects. It is often then possible to assess the no-interactions assumption by comparing the sample of confounded interactions with the pure error estimate. In this paper, a new class of partially replicated two-level fractional factorial designs based on Hadamard matrices is proposed. A method of constructing these designs is given and the relative performance of the designs is evaluated by comparing them with competing fractional factorial designs.

Optimal foldover plans for regular [formula omitted]-level fractional factorial designs

This article introduces a general decomposition structure of the foldover plan. While all the previous work is limited to two-level designs, our results here are good for general regular s -level fractional factorial designs, where s is any prime or prime power. The relationships between an initial design and its combined designs are studied. This is done for both with and without consideration of the blocking factor. For illustration of the usage of our theorems, a complete collection of foldover plans for regular three-level designs with 27 runs is given that is optimal for aberration and clear effect numbers.

Fractional factorial designs applied to enzymatic sugar beet pulps pressing improvement

The effects of several plant cell wall polysaccharides degrading enzymes on sugar beet pulps pressing were studied. Study was carried out using three two level fractional factorial experiment designs. With only 36 experiments, the effects of the presence of pectin methylesterase, pectin lyase, polygalacturonase, cellulase, arabinase, xylanase and two rhamnogalacturonases on pressing were examined. Pectin lyase, pectin methylesterase and cellulase had a negative effect and caused the decrease of sugar beet pulp pressability. On the contrary, the presence of polygalacturonase, arabinase and xylanase increased pressing efficiency. When increasing enzymes concentrations, these effects varied and positive interactions between xylanase and polygalacturonase appeared. The presence of each of the two rhamnogalacturonases improved pressability despite their antagonistic effects. These enzymes had a complex effect and strongly interacted with polygalacturonase, arabinase and xylanase.

Application of factorial designs for optimization of cyclodextrin glycosyltransferase production fromKlebsiella pneumoniae pneumoniae AS-22

Production of cyclodextrin glycosyltransferase (CGTase) from Klebsiella pneumoniae pneumoniae AS-22 was optimized in shake flasks using a statistical experimental design approach. Effect of various components in the basal medium, like carbon, nitrogen, phosphorus, and mineral sources as well as initial pH and temperature, were tested on enzyme production. The optimum concentrations of the selected media components were determined using statistical experimental designs. Two level fractional factorial designs in five variables, namely, dextrin, peptone, yeast extract, ammonium dihydrogen orthophosphate, and magnesium sulphate concentrations were constructed. The optimum medium composition thus found consisted of 49.3 g/L dextrin, 20.6 g/L peptone, 18.3 g/L yeast extract, 6.7 g/L ammonium dihydrogen orthophosphate, and 0.5 g/L magnesium sulphate. The maximum CGTase activity obtained was 21.4 U/mL in 28 h of incubation. The cell growth and CGTase production profiles were studied with the optimized medium in shake flasks and in 1-L fermenters. It was observed that the enzyme production was growth associated both in shake flask and in fermenter, although it was slower in shake flask. The maximum CGTase activity obtained in the fermenter was 32.5 U/mL in 16 h. The optimized medium resulted in about 9-fold increase in the enzyme activity as compared to that obtained in the basal medium in shake flask as well as in fermenter.

Four Factor Balanced Optimal Designs of Resolution V in Terms of a Modified Trace Criterion

AbstractBalanced optimal two level fractional factorial designs of resolution V have been reported in series of papers by SRIVASTAVA and CHOPRA (1971a, 1971b, 1974) and CHOPRA and SRIVASTAVA (1973a, 1973b). The optimality criterion considered by them is the trace optimality which implies minimisation of the average variance of the estimated effects which include the general mean μ, main effects, Ai, and first order interactions, Aij. In the present work some new optimal designs have been obtained on the basis of a modified criterion of trace optimality. The criterion requires minimisation of the average variance of each of the main effects, two factor interactions, etc. separately instead of minimising average variance of all the estimated effects.

Partially Duplicated Fractional Factorial Designs

This paper describes two level fractional factorial designs in which some of the treatment combinations are duplicated. As is well known, the duplicated runs provide an unbiased estimate of error variance and more precise estimates of the effects. An example is given with the analysis procedure. Block designs for the corresponding fractional factorial designs are also given.

Reduced Designs of Resolution Five

The two level fractional factorial designs of resolution five enable the experimenter to estimate independently all main effects and two-factor interactions under the assumptions that higher order interaction effects are negligible. By relaxing, very slightly, the requirement that all two-factor interactions be estimable, or that all estimated effects be orthogonal, the number of runs required for many resolution five designs can be greatly reduced.

Reduced Designs of Resolution Five

The two level fractional factorial designs of resolution five enable the experimenter to estimate independently all main effects and two-factor interactions under the assumptions that higher order interaction effects are negligible. By relaxing, very slightly, the requirement that all two-factor interactions be estimable, or that all estimated effects be orthogonal, the number of runs required for many resolution five designs can be greatly reduced.

Partial Duplication of Factorial Experiments

Two level factorial and fractional factorial designs are described in which a subset of the treatment combinations are duplicated. The duplicate runs provide an unbiased estimate of the experimental error variance and more reliable estimates of the effects. Several analysis procedures are described and a numerical example is given. Designs for partially duplicated two level factorial and fractional factorial designs are proposed for as many as eleven factors.