Cylindrically Rotatable Designs of Type 3: Further Considerations

The cylindrically rotatable designs of type 3 are generalised to consider the factors split up into any number of groups. The conditions which must be satisfied by such a design are derived from the moment generating function of the design. A method of construction of second-order cylindrically rotatable designs of type 3 with three or more groups of factors is provided.     

Group Divisible Second Order Rotatable Designs

The Group Divisible Rotatable (GDR) designs are the designs in which the factors get divided into groups such that for the factors within group, the designs are rotatable. In the present paper we have obtained a series of Group Divisible Second Order Rotatable designs, by decomposing the v-dimensional space corresponding to v-factors under consideration into three mutually orthogonal spaces. We have given the least squares estimates of the parameters, the analysis and construction of such designs.     

On Construction of Some Augmented Block Designs

Augmented block designs are frequently used in practice. In this paper procedures of construction of some types of augmented block designs are discussed using the concept of efficiency-balance and partially efficiency-balance.     

Designs for restricted exploration in mixture experiments

Extreme Vertices designs were developed by MCLEAN and ANDERSON (1966) for situations where components of a mixture are restricted by lower and upper bounds, SNEE and MARQUARDT (1974) and SNEE (1975) gave algorithms to construct optimum designs in these situations. SAXENA and NIGAM (1975) evolved a transformation which provides designs for restricted exploration using Symmetric Simplex designs. In this paper a procedure has been given which provides alternative designs with uniform exploration in constrained mixture experiments. The procedure is illustrated by an example.     

Some Bounds on Parameters of Efficiency-Balanced Block Designs

Bounds on the eigenvalue of the C -matrix with respect to R and the number of blocks for efficiency-balanced block designs are given. AMS 1970 Subject Classification: 62K10     

Experiment with Split-Plot Generated by PBIB Designs

The paper refers to the construction of two-factor experiments carried out in incomplete split-plot design in which levels of both factors occur as treatments in PBIB designs. It presents also the analysis of variance of these experiments based on partial analyses in the so called strata. These strata are connected with the variability between subplots inside wholeplots, between the wholeplots inside blocks and between blocks inside the total experiment.     

On G-Efficiency of some Second Order Rotatable Designs

The effect of adding central points to a given design on the G-efficiency has been examined for certain classes of second order rotatable designs.     

Construction of Resolvable Designs with Nested Treatment Structure

The construction of resolvable incomplete block and row-column designs is investigated when the treatments have a nested structure. Some theoretical results are derived for lattice designs. Efficient designs for unequal-sized treatment groups are obtained by defining a multiple objective function and carrying out a computer search using an interchange algorithm.     

A New Class of Matrices with Application in Experimental Designs

A very general class of balanced matrices is defined in this paper. These matrices are generalisations of the balanced orthogonal designs of Rao (1970) and generalised balanced matrices of Dey and Midha (1976). Some methods of construction of these balanced matrices are discussed. An application of these matrices in experimental designs is also given.     

Incomplete Split-Block Designs

The paper deals with modelling of experiments carried out in incomplete split-block designs. Moreover, the characterization of the design with respect to a general balance property is given. Also, some statistical properties of the design are examined. The problems considered are illustrated with examples.     

Enzymatic synthesis of betulinic acid ester as an anticancer agent: Optimization study

Abstract

Immobilized Candida antarctica lipase, Novozym 435, was used to catalyze the esterification reaction between betulinic acid and phthalic anhydride to synthesize 3-O-phthalyl betulinic acid in n-hexane/chloroform. Response surface methodology based on a five-level, four-variable central composite rotatable design was employed to evaluate the effects of synthesis parameters such as reaction time, reaction temperature, enzyme amount and substrate molar ratio on the yield of ester. Based on the response surface model, the optimal enzymatic synthesis conditions were predicted to be: reaction time 20.3 h, reaction temperature 53.9°C, enzyme amount 145.6 mg, betulinic acid to phthalic anhydride molar ratio 1:1.11. The predicted yield was 65.8% and the actual yield was 64.7%.     

Estimation of Several Missing Values in F-Square Designs

The present study deals with the estimation of several missing values in F-square designs. The estimating equations for the non-iterative least squares estimation of Missing Values and explicit expressions for the estimators of the particular patterns of Missing Values are presented. This procedure is illustrated with the help of a numerical example.     

Rank Tests for Complete Block Designs

In this article a general univariate K-sample rank test for complete block designs with proportional cell frequencies is derived. It is shown that the test statistic has under H₀ and for arbitrary scores asymptotically a X²-distribution with K — 1 degrees of freedom. Special cases of this test are the Kruskal-Wallis test and the Friedman test. The test is compared with the Benard-van-Elteren test, the Mack-Skillings test and a test proposed by Downton. Finally the application of the test is illustrated by two examples.     

Simultaneous and comparative assessment of parent and mutant strain of Rhizobium meliloti for nutrient limitation and enhanced polyhydroxyalkanoate (PHA) production using optimization studies

Nutrient limitation conditions, optimization and comparison of polyhydroxyalkanoate (PHA) yields and biomass production by parent and mutant strains of Rhizobium meliloti were investigated. Complex interactions among concentrations of sucrose (5–55 g/l), urea (0.05–0.65 g/l) inoculum (10–250 ml/l) and K₂HPO₄ (0.5–2 g/l), were studied using central composite rotatable design (CCRD) experiments. Phosphate-limiting medium (0.33 g K₂HPO₄/l) in the presence of excess carbon (sucrose 42.5 g/l) results in more production of PHA (2.2 g/l) in the parent strain. In comparison, the mutant strain required moderate levels of sucrose (30 g/l), along with excess of phosphate (1 g/l) for high PHA content of cell biomass (80%) and PHA yield (3.3 g/l). Optimised PHA production (biomass 4.8 g/l and PHA 3.09 g/l) by the parent strain occurred at: sucrose 51.58 g/l, urea 0.65 g/l, K₂HPO₄ 0.48 g/l and inoculum 10 ml/l. In the mutant strain, higher yields of biomass (9.05 g/l) and PHA (5.66 g/l) were obtained in Optimised medium containing: sucrose 55 g/l, urea 0.65 g/l, K₂HPO₄ 1.0 g/l and inoculum 150.58 ml/l.     

Optimization of polyhydroxybutyrate production by <Emphasis Type="Italic">Bacillus</Emphasis> sp. CFR 256 with corn steep liquor as a nitrogen source

Polyhydroxyalkanotes (PHAs), the eco-friendly biopolymers produced by many bacteria, are gaining importance in curtailing the environmental pollution by replacing the non-biodegradable plastics derived from petroleum. The present study was carried out to economize the polyhydroxybutyrate (PHB) production by optimizing the fermentation medium using corn steep liquor (CSL), a by-product of starch processing industry, as a cheap nitrogen source, by Bacillus sp. CFR 256. Response surface methodology (RSM) was used to optimize the fermentation medium using the variables such as corn steep liquor (5–25 g l⁻¹), Na₂HPO₄ 2H₂O (2.2–6.2 g l⁻¹), KH₂PO₄ (0.5–2.5 g l⁻¹), sucrose (5–55 g l⁻¹) and inoculum concentration (1–25 ml l⁻¹). Central composite rotatable design (CCRD) experiments were carried out to study the complex interactions of the variables. The optimum conditions for maximum PHB production were (g l⁻¹): CSL-25, Na₂HPO₄ 2H₂O-2.2, KH₂PO₄ − 0.5, sucrose − 55 and inoculum − 10 (ml l⁻¹). After 72 h of fermentation, the amount of PHA produced was 8.20 g l⁻¹ (51.20% of dry cell biomass). It is the first report on optimization of fermentation medium using CSL as a nitrogen source, for PHB production by Bacillus sp.     

A Simple Procedure for Constructing Experiment Designs with Incomplete Blocks of Sizes 2 and 3

In order to maximize control of heterogeneity within complete blocks, an experimenter could use incomplete blocks of size k = 2 or 3. In certain situations, incomplete blocks of this nature would eliminate the need for such spatial types of analyses as nearest neighbor. The intrablock efficiency factors for such designs are relatively low. However, with recovery of interblock information, FEDERER and SPEED (1987) have presented measures of design efficiency factors which demonstrate that efficiency factors approach unity for certain ratios of the intrablock and interblock variance components. Hence with recovery of interblock information, even incomplete block designs with k = 2 or 3 have relatively high efficiency factors. The reduction in the intrablock error variance over the complete block error variance in many situations will provide designs with high efficiency. A simple procedure for constructing incomplete blocks of sizes 2 and 3 is presented. It is shown how to obtain additional zero-one association confounding arrangements when v = 4 t, t an integer, and for v = pk, k ≤ p. It is indicated how to do the statistical analysis for these designs.     

Efficiency Indices for Second-Order Designs

Design efficiencies are studied for inferences regarding the coefficients of second-order models. Subspaces of parameters are identified wherein each of two designs dominates the other locally, or the two designs are equally efficient. Both Fisher efficiency in estimation and Pitman efficiency in hypothesis testing are considered. Eight small second-order designs in k=3 regressor variables are screened using a four-part efficiency index. Of these, the central composite, Box-Behnken, small composite, and hybrid H310 and H311 B designs are found to be generally superior and roughly comparable. Detailed comparisons among these five designs are then undertaken. This work provides further tools for use in design evaluation, requiring only basic matrix operations independently of experimental data.