Incomplete Factorial Experiment Research Articles

Efficient block designs for incomplete factorial experiments for two factors with unequal block sizes

In this article, the problem of obtaining efficient block designs with unequal block sizes for incomplete factorial treatment structure with two factors excluding one treatment combination for estimation of dual versus single treatment contrasts is considered. A method of construction for obtaining efficient block designs with unequal block sizes for incomplete factorial treatment structure is proposed.

DETERMINATION OF THE OPTIMAL EXTRACTION CONDITIONS OF DANDELION HERB AND LARGE BURDOCK LEAF SPECIE BY METHODS OF MATHEMATICAL PLANNING OF A MULTIFACTORIAL EXPERIMENT

Introduction. Use of medicinal species at home entails some inconvenience. Therefore, it is topical to obtain phytopreparations, in particular dry extracts, from medicinal plant raw materials. This ensures the maximum yield of a complex of biologically active substances, and simplifies the way of its application, dosing and storage. The department of pharmacy of the Altai State Medical University has developed a specie of the following composition: dandelion medicinal herb and large burdock leaf (1:1). On the basis of this species, a technology for obtaining the dry extract is being developed. Variety factors influence on the output of a complex of biologically active substances from plant raw materials: the technological properties of raw materials, the methods and conditions for carrying out the extraction process, and the equipment used. Aim. To determine the optimal conditions for extracting a specie of dandelion herb and a large burdock leaf. Materials and methods. To solve this problem, methods of mathematical planning of a multifactorial experiment have been used: simplex method of mathematical planning, incomplete factorial experiment of type 25-2 for compiling the equation of the response surface, the method of steep ascent (the Box – Wilson method) along the response surface. Based on the theoretical basis of extraction, parameters and factors of process optimization were chosen, intervals of their variation are indicated. Variable factors – shredding, the ratio of «raw materials : extractant», time of maceration, temperature and the number of maceration. The extraction was carried out by remaceration, aqua purificata serves as extractant. The parameter of optimization was the exhaustion of the specie of extractive substances. Results and discussion. As a result of the simplex method, a number of studies were carried out until the area of the supposed optimum was reached. To describe the experiment in the region of the optimum and to compose the equation of the response surface, an incomplete factor experiment of the type 25-2 were put. A linear equation of the extraction process were obtained and the adequacy of the work of the obtained mathematical model is proved. To refine the extraction conditions and establish the optimal extraction regime, the method of steep ascent on the response surface was used. Conclusion. As a result of the experiment, optimal extraction conditions were established: crushing – 2 mm, ratio «raw material : extractant» – 1:10, time of maceration – 30 minutes, number of maceration – 3, temperature – 100 °С. These results will be used in the development of technology and the technological scheme for obtaining the dry extract of the specie.

Analysis of hydrogen production by anaerobic fermentation from urban organic waste

The production of hydrogen by anaerobic fermentation of urban organic waste from fruits and vegetables was studied. Ten tests were carried out under an incomplete factorial experiment design with four factors and three levels. The factors were: acidification pH, acidification time, operational pH, and organic load. The response variables were: maximum hydrogen content (%) and maximum hydrogen production (L/day). The results were fitted to a quadratic polynomial model. In the case of the maximum hydrogen content, an R2 and an R2adjusted of 0.9987 and 0.988 were obtained, respectively. For the maximum production, an R2 and an R2adjusted of 0.9815 and 0.833 were obtained, respectively. Three mathematical techniques were used to obtain the maximum values in the response variables. Fuzzy logic showed the best fit between the experimental values and estimated values. In addition, it predicts a maximum hydrogen content of 58.5% and a maximum production of 63.4 L/day.

Influence of competition and rainfall manipulation on the growth responses of savanna trees and grasses

In this study, we explored how rainfall manipulation influenced competitive interactions between grasses and juvenile trees (small nonreproductive trees capable of resprouting) in savanna. To do this, we manipulated rainfall amount in the field using an incomplete factorial experiment that determined the effects of rainfall reduction, no manipulation, rainfall addition, and competition between grasses and trees on grass and tree growth. As response variables, we focused on several measures of tree growth and Disc Pasture Meter settling height as an estimate of grass aboveground biomass. We conducted the study over four years, at two sites in the Kruger National Park, South Africa. Our results show that rainfall manipulation did not have substantial effects on any of the measures of tree growth we considered. However, trees at plots where grasses had been removed grew on average 15 cm more in height and 1.3-1.7 times more in basal area per year than those in plots with grasses. Grass biomass was not influenced by the presence of trees but was significantly and positively influenced by rainfall addition. These findings were not fundamentally influenced by soil type or by prevailing precipitation, suggesting applicability of our results to a wide range of savannas. Our results suggest that, in savannas, increasing rainfall serves to increase the competitive pressure exerted by grasses on trees. The implication is that recruitment into the adult tree stage from the juvenile stage is most likely in drought years when there is little competition from grass for resources and grass fuel loads are low.

Effective data analysis methods for incomplete two-level factorial experiments

Designing and analysing two-level factorial or fractional factorial experiments require all experimental runs should be completed. However, in many actual industrial experiments, a portion of experimental runs cannot be completed due to either infeasibility or budget limitation, resulting in incomplete experiments. In this paper, two effective methods to estimate the factorial effects in incomplete 2 level factorial or fractional factorial experiments are developed. In the first method, each traditional factorial main effect or interaction effect is decomposed as the average of many individual elementary effects. Under this model, missing runs in factorial experiments only affect some individual elementary effects; the main effects or interactions can still be estimated by the partial average of the remaining individual elementary effects. The second method is based on estimation of missing runs by using response surface model on incomplete experimental data set. These two methods are compared with other two existing methods for incomplete factorial experiments on eight testing problems and the results show our two methods perform well.

Level of cottonseed meal but not frequency of feeding regulates whole-body protein synthesis and growth of sheep fed a roughage diet

An incomplete factorial experiment was conducted to determine the effect of level and frequency of feeding of a protein-rich supplement on the growth and whole-body protein metabolism of young sheep fed a medium quality roughage diet. Cottonseed meal (CSM) was used as the protein supplement and provided at 0, 0.2 or 0.4% liveweight per day at a frequency of 1 or 3 times each week and chopped oaten (0.95) and lucerne (0.05) hay was the roughage. Growth rate more than doubled (P < 0.01) following provision of CSM but there was no advantage of feeding CSM at the highest level. Frequency of feeding CSM did not alter growth rate. Intake of hay was little affected by CSM and as a consequence the food conversion ratio declined (P < 0.01) favourably from 22 : 1 (nil CSM) to 9 : 1 as a result of supplementation. The rate of whole-body protein synthesis increased (P < 0.01) in response to the highest level of CSM with no apparent change in protein degradation, underpinning an increase (P < 0.01) in protein retention. These results highlight the role of protein supplements for promoting growth of young sheep on roughage diets and indicate that these supplements need to be provided only once a week.

Incomplete factorial experiments in completely randomized and randomized complete block designs

The estimation of main and interaction effects is considered, when some treatment combinations are excluded from the experiment and one or more factorial contrasts can be assumed to be negligible. For equal treatment replication, it is shown that the increase in variance due to the exclusion of some treatment combinations depends on the degree of involvement of the effect of the excluded treatment combinations in the negligible factorial contrasts. Further, linear combinations of the set of negligible contrasts are identified which maximizes the precision of the estimators of the contrasts of interest.

Incomplete Factorial and Response Surface Methods in Experimental Design: Yield Optimization of tRNATrp from In vitro T7 RNA Polymerase Transcription

We have studied the yield of Escherichia coli tRNA(Trp) obtained from in vitro T7 RNA polymerase transcription using incomplete factorial and response surface methods. Incomplete factorial experiments were first used to estimate the relative impact of six variables on the yield of tRNA(Trp). Fifteen trials were performed according to a balanced and randomized design. The correlation between observed yield and all experimental variables was identified by stepwise multiple linear regression analysis. The concentrations of T7 RNA polymerase, DNA template, NTP and MgCl2 proved to be significantly correlated with the yield of tRNA(Trp). We then optimized the yield with respect to each of these four variables simultaneously with a designed, response surface experiment based on the Hardin-Sloane minimum prediction variance algorithm. Twenty experiments were performed, in duplicate, to sample the quadratic surface relating the yield to the four significant variables. Coefficients of the quadratic function with all two-factor interactions were evaluated by stepwise regression using least squares, and significant coefficients were retained. Partial differentiation of the resulting quadratic model showed it to possess an optimum. Transcription performed at the corresponding conditions yielded 6-fold more tRNA(Trp) than the initial conditions, confirming the predictive value of the experimentally determined response surface.

Entropy, likelihood and phase determination.

Entropy, likelihood and phase determination.

Rapid access to an updated biological macromolecule crystallization database through artificial intelligence

We have written a computer program (CRISTAL) by using PASCAL and PROLOG II to interpretate the crystallization database originally compiled by G. Gilliland and recently updated by us. A large repertoire of questions can be asked using natural language (English). Very rapid retrieval of crystallization data has been made possible thanks to the use of direct-access files and efficient coding of the key-words corresponding to the contents of the database. A brief statistical analysis is reported concerning the use of different precipitating agents and the publication trends between 1940 and May 1989. Finally, we mention the possible use of CRISTAL in the design of incomplete factorial experiments for macromolecule crystallization.

Influence of Gonadotropin-Releasing Hormone and Timing of Insemination Relative to Estrus on Pregnancy Rates of Dairy Cattle at First Service

The objective was to determine the influence of gonadotropin-releasing hormone on pregnancy rates of dairy cattle at first services, when both the timing of hormone injection and insemination were altered relative to the onset of estrus. Cows (n=325) were assigned randomly to six groups making up a 2×2×2 incomplete factorial experiment; dose of GnRH (100μg versus saline), timing [1h (early) or 12 to 16h (late) after first detected estrus] of AI, and timing of hormone injection (early versus late) were the three main effects. Cows were observed for estrus 4 times daily. Treatments and resulting pregnancy rates were: 1) hormone injection early plus AI early (35%), 2) hormone injection late plus AI early (34%), 3) saline injection early plus AI early (30%), 4) hormone injection late plus AI late (30%), 5) hormone injection early plus AI late (46%), and 6) saline injection late plus AI late (43%). Pregnancy rate in the first four groups (32%) was less than that in the latter two groups (44%). Concentrations of LH in serum were greater for cows given hormone or saline injections in early estrus than for cows injected with either hormone of saline during late estrus. Concentrations of LH in serum 2h after GnRH were elevated above those of controls, whether GnRH was injected during early or late estrus. Neither concentrations of LH during estrus nor concentrations of progesterone 8 to 14 d after estrus explained the possible antifertility effect of GnRH given during late estrus. It was concluded that pregnancy rates at first services accompanied by treatment with GnRH were not improved by altering either timing of AI or timing of hormone injection relative to the onset of estrus.

Protein crystallization using incomplete factorial experiments.

We have studied the crystallization of Bacillus ste- arothermophilus tryptophan tRNA synthetase. Exper- imental conditions used in 35 different crystallization trials were chosen by systematic assignment of varia- bles in accordance with the two principles of randomi- zation and balance, which are characteristic of incom- plete factorial experiments. Polymorphism in the re- sulting precipitated, microcrystalline, or crystalline protein samples was identified and rated numerically on a rough but quantitative scale. Correlations between these estimates of crystal quality and all experimental variables were identified by stepwise multiple regres- sion analysis. Several variables proved to be signifi- cantly correlated with crystal quality. Having established a favorable precipitation proce- dure and ionic composition within broad limits, we next carried out a complete factorial experiment with four factors using two pH values and two temperatures in the presence or absence of two low molecular weight substrates, tryptophan and ATP. These experiments clearly established a favorable pH and produced large (0.5 mm) single crystals with each of the four possible substrate permutations. At least one crystal form, en- zyme plus tryptophan, is suitable for structural studies by x-ray diffraction. The factorial approach appears to be an advantageous alternative to classical controlled single factor and full factorial designs for establishing cause and effect relationships in protein crystallization. Rational exploration of cause and effect relationships gov- erning the crystallization of proteins has traditionally been eschewed, owing to the potentially very large number of controlled experiments required for a systematic study. The essential difficulty is that a number of different experimental conditions, pH, temperature, various mono- and divalent ions, and different organic solutes, may interact in apparently ca- pricious ways to provide a favorable environment in which crystals will eventually grow (an excellent review is Mc- Pherson, 1976a). Despite this seemingly overwhelming difficulty, small quan- tities of protein can be used quite efficiently by using the factorial approach to experimental design (Fisher, 1942). This approach is counter-intuitive: no attempt is made to identify the effects of varying individual factors by controlled experi- ments in which other factors remain constant. Rather, effi- ciency in a factorial experiment arises from two rather differ- * This work was supported by Research Grant GM21991 from the National Institutes of Health and by a fellowship in academic medi- cine from the Jefferson Pilot Corporation (C!. W. C. Jr.). The costs of publication of this article were defrayed in part by the payment of

Corrigenda: Small Incomplete Factorial Experiment Designs for Two- and Three-Level Factors

Corrigenda: Small Incomplete Factorial Experiment Designs for Two- and Three-Level Factors

Small Incomplete Factorial Experiment Designs for Two- and Three-Level Factors

This catalogue of small incomplete factorial designs was compiled to provide designs for two- and three-level factors with 20 or fewer runs. It is assumed that the three-level factors are quantitative. The designs are suitable for estimating a selection of interactions, including the linear-by-linear components of interactions between the three-level factors. Ad hoc methods were used to construct the designs, so that there is no particular uniformity in the designs and no general property of optimality which they satisfy. Designs were selected on the basis of a measure of efficiency which is presented. No claim is made that the given designs are the best possible, and others should be encouraged to try to develop more efficient designs. A detailed example based on a real experiment is presented.

Small Incomplete Factorial Experiment Designs for Two- and Three-Level Factors

Small Incomplete Factorial Experiment Designs for Two- and Three-Level Factors