Statistics for experimenters: design, innovation, and discovery

This book on experimental design is somewhat similar in flavor to the classic book by G.E

In many mixture-process experiments, restricted randomization occurs and split-plot designs are commonly employed to handle these situations. The objective of this study was to obtain an optimal split-plot design for performing a mixture-process experiment. A split-plot design composed of a combination of a simplex centroid design of three mixture components and a 2² factorial design for the process factors was assumed. Two alternative arrangements of design points in a split-plot design were compared. Design-Expert® version 10 software was used to construct I-and D-optimal split-plot designs. This study employed A-, D-, and E- optimality criteria to compare the efficiency of the constructed designs and fraction of design space plots were used to evaluate the prediction properties of the two designs. The arrangement, where there were more subplots than whole-plots was found to be more efficient and to give more precise parameter estimates in terms of A-, D- and E-optimality criteria. The I-optimal split-plot design was preferred since it had the capacity for better prediction properties and precision in the measurement of the coefficients. We thus recommend the employment of split-plot designs in experiments involving mixture formulations to measure the interaction effects of both the mixture components and the processing conditions. In cases where precision of the results is more desirable on the mixtures as well as where the mixture blends are more than the sets of process conditions, we recommend that the mixture experiment be set up at each of the points of a factorial design. In situations where the interest is on prediction aspects of the system, we recommend the I-optimal split-plot design to be employed since it has low prediction variance in much of the design space and also gives reasonably precise parameter estimates.

The objective of this study was to evaluate the effect of dosages NPK and harvest day on seeds viability. The study was conducted at the experiment station of Polinela, Bandarlampung and Seed Laboratory, Unila from Januari to June 2008.Treatment consicted of two factors arranged in factorial design (5x7). A Split plot design with three replications was used. The main plot factor was NPK dosages, i.e., 0, 25, 50,75, and 100 kg.ha-1. The Subplot factor was harvest day, i.e., R7, R8, R8+3, R8+6, R8+9, R8+12, R8+15. Examination of hypothesis done with regression analysis at significance level 5%. Interaction between fertilizer doses NPK supplement and harvest day influences initial vigour of variety soy seed Anjasmoro. Germanation uniform and germination speed can be improved with addition of NPK supplement to 100 kg/ha though harvested at R8+15 day while at harvesting since R8 until R8+12 day, initial vigour not reality; phase R8 or day 81stafter planting. Harvest day 88-89th after planting or R8+15 day to yield maximum initial vigour based on variable germination speed ( 30,19 %/day) if dose NPK supplement is improved to 100 kg/ha. Harvest R7, germination seed are still increase at various doses NPK supplement while harvesting since R8 until R8+15 day, the influence of germinates power is not significant. Keywords: fertilization, seed, soybean, harvest

ABSTRACTIn chemical engineering applications such as optimizing plant operations and product quality, factorial experiments are often conducted to obtain empirical models for systems. While the mathematical underpinning for two‐level factorial designs is well understood, a methodology for analyzing higher‐level experiments is not readily available. Often an orthogonal or orthonormal basis is selected for a factorial design matrix. In factorial design, an orthonormal basis is defined as an orthogonal matrix where the Euclidean two‐norms of the column vectors are equal. This investigation examines, for full factorial design, the selection of parameters using normal probability plots and the effect that the design basis has on parameter determination. When using normal probability plots to determine parameter significance, the traditional orthogonal basis for higher‐level experiments may result in erroneous conclusions. A Monte‐Carlo experiment was developed to simulate 3‐level and mixed‐level factorial experiments with different types of measurement error. The basis chosen is shown to affect the shape of probability plots, and measurement errors from a given normal distribution are shown to result in a constant standard deviation for all parameter estimates only when using an orthonormal basis.

The performance of age estimation methods may vary due to a combination of method- and sample-related factors. Method development and evaluation necessitates understanding what influences these factors have on age estimation outcomes. In the specific context of juvenile dental age estimation, we used a single dataset and complete factorial design to systematically test four potential sources of difference: age distributions of reference and target sample (uniform, unimodal, U-shaped), Bayesian (multivariate Bayesian cumulative probit) vs. classical regression modeling (multivariate adaptive regression splines i.e. MARS), and model selection bias. The dataset consisted of 850 sets of left mandibular molar scores from London children 5-18years old. True age and estimated age intervals in target samples were compared for bias, root-mean-squared error, precision, and accuracy using locally weighted smoothing of performance measures across the age range and means of performance metrics between factor-level combinations. We found interactions of model type, reference distribution, and target distribution. MARS models showed consistent evidence of age mimicry. Central tendency of the reference sample corresponded with increased bias while central tendency of the target sample corresponded with reduced RMSE and reduced precision for both model types. We found evidence of model selection bias, mitigated through averaging model metrics. We conclude that reference and target sample distribution influences and model selection bias are sufficient to cause difference in model performance within a single population. We suggest using Bayesian modeling, drawing uniform reference and target samples, and calculating test error on a hold-out sample to mitigate these challenges in method development.

In this article, we present a study carried out to compare the effectiveness of the normal probability plot (NPP) and a simple dot plot in assessing the significance of the effects in experimental designs with factors at two levels (2 k−p designs). Several groups of students who had just completed a course that covered factorial designs were asked to identify the significant effects in a total of 32 situations, 16 of which were represented using NPPs and the other 16 using dot plots. Although the 32 scenarios were said to be different, there were really only 16 different situations, each of which was represented using the two methods to be compared. A simple graphical analysis shows no evidence that there is a difference between the two procedures. However, in designs with 16 runs there are some cases where NPP seems to give slightly better results.

An n by n magic square with entries 1 to n 2 has all row and column sums equal. If n has k factors, then a factorial design with n 2 runs and up to 2 k factors can be obtained from this for which the main effects and at least some of the two‐factor interactions are linear‐trend‐free (their estimates are unaffected if there is a straight line trend over time). An unconnected experimental design called a magic Latin square is a Latin square with additional blocking obtained from structured rows and columns.

Split-plot designs and normal probability graphs for the optimization of chemical systems

The majority of plant tissue culture experiments are set up as factorial experiments in completely randomized designs (CRD), randomized complete block designs (RCBD) or split-plot designs (SPD) that require a great number of stabilized cultures. For various reasons, an insufficient number of explants may prevent the employment of CRD, RCBD or SPD with factorial treatments and hinder the optimization of factors affecting culture response. To prioritize the optimization of “the most important” factors affecting culture response, we explored the applicability of Plackett–Burman Design (PBD) and Fractional Factorial Design (FFD) with a limited number of explants. To test the effects of 8 factors (genotype, 6-benzyladenine (BA), CuSO4 5H2O, Fe source, agar, pH, myo-inositol and cold treatment following inoculation of explants) at 2 levels on response of single-node grape vine cultures, 12 treatment combinations were generated according to the PBD and 16 treatment combinations were generated according to the FFD. These designs require many fewer explants since a typical experiment with eight factors, each at two levels of will require 256 (28) treatment combinations. However, the costs of the PBD and FFD are that there is limited, if any information, on interactions and if available may be more difficult to interpret. Of the factors tested, cultivar, BA and agar concentrations were found to be the most important factors affecting culture response. The types of culture response were in agreement with previous reports indicating that PBD and FFD can effectively be employed with limited number of explants to test effects of several factors and prioritize the “most important” factors.

While drought negatively affects plant production, vermicompost and grafted seedlings offer a promising strategy to reduce these effects. This study, conducted in Ankara (greenhouse) and Antalya (field) provinces in 2020–2021, investigated the effects of grafting and vermicompost on morphophysiological traits, yield, and water efficiency of eggplant under drought stress. The greenhouse experiment was conducted in a factorial design (n = 3), and the field experiment was conducted using a split-split plot design (n = 3). In the greenhouse, grafted and non-grafted plants were subjected to three irrigation levels—100%, 70%: moderate stress (MS), 30%: severe stress (SS)—and vermicompost doses (0, 1, 2, 3%). Under MS conditions, grafted plants supplemented with 2% vermicompost showed significant improvements: shoot fresh weight increased by 48.81%, dry weight by 42.92%, shoot length by 49.52%, and stomatal conductance by 88.89%, while yield losses were reduced by 96% and irrigation water productivity improved by 62.79%. Field experiments applied the same irrigation levels (SS = 0.30 Epan, MS = 0.70 Epan, control = 1.00 Epan), and the optimal vermicompost dose (2%) identified in the greenhouse trials. Under MS, shoot fresh and dry weights increased by 121.20 and 162.18%, respectively, with total productivity rising by 101.13%. The innovative contribution of this study lies in its synergistic integration of grafting and vermicompost, two sustainable practices previously untested in eggplant cultivation to mitigate drought effects across both fields. The results demonstrated that this strategy reduces yield losses by 96% under severe stress and doubles productivity under moderate drought, offering a sustainable and environmentally friendly solution for water-limited agriculture.

This study was conducted to determine the effect of pollination methods (PMs) and integrated fertilizer (IF) application on the pollination rate, yield, and fruit quality of rockmelon (Cucumis melo L. var. Reticulatus cv. Glamour). This factorial experiment was arranged in a split plot design with four replications. Two rates of IF between chemical (CF) and organic (OF) fertilizer were investigated, namely, T1 (100% CF), which served as a control, and T2 (75% CF + 25% OF). The three PMs used in this study were natural (NP), bee (BP), and human (HP) pollination. Data collection included the pollination rate, yield, postharvest quality (soluble solids content, color, firmness, sugar, and organic acid), and macronutrient content in leaf tissue. The results of this study showed that there was an interaction of the pollination rate between PM and IF; however, the number of bees that survived decreased over time. The yield quality showed an interaction between PM and IF for both the fruit diameter and weight, while no interaction between PM and IF for the organic acid and sugar contents was observed. In conclusion, T2 application combined with BP reduced CF utilization, which benefits the development of a sustainable rockmelon production system.

An experiment was conducted in the farmer’s field located at latitude is N 23º1¢19.9¢¢ and longitude is E 88º34¢31.8¢¢ in an arsenic affected village Goentra of west Bengal, India during winter season 2009-10, 2010-11 and 2011-12. In arsenic contaminated village of Goentra, the wide variation of arsenic in shallow tube well water and pond water was observed. The level of variation is almost three times more in shallow tube well water as compare to the pond water. Rain water harvested pond water (0.04 mg L-1) is safer than the underground source of water (0.14 mg L-1). The experiment was conducted in factorial experiment in split-split plot design. Two fodder crops were laid as main plot, i.e. Oats and lucerne, two sources of irrigation i.e. pond water and shallow tube well water was in sub plot and frequency of irrigation was three in sub-sub plot. The crops was, therefore, tested against two source of irrigations viz.: (i) Pond water and (ii) shallow tube well water; and frequency of irrigation water viz.: (i) Irrigation at 20 DAS, (ii) Irrigation at 20 and 35 DAS and (iii) Irrigation at 20, 35 and 50 DAS. The Lucerne fodder crop received comparatively less arsenic uptake than oats fodder. Pond water irrigated fodder crops shows significantly less arsenic concentration in plant parts comparatively to shallow tube well water irrigated fodder crops. Shoots arsenic content reduction with once irrigation is 11.93 % whereas in two irrigations shoots arsenic content is only 4.0 % at harvest stage of crops. The frequency of irrigation considerably shows the arsenic concentration variation in fodder crops. The fodder crops irrigated with single irrigation shows less arsenic (2.97 mg kg-1) in plant than the two or three irrigation.

Separation of the few control factors with significant effect from the remaining ones at the early phase of experimental studies is a recommended practice for systems development and improvement. The normal probability plot is a widely accepted graphical technique for screening, although it does not provide a unique interpretation of results in some cases. Thus, more formal procedures have been proposed to supplement it. The performance of popular methods along with lesser known and used ones is assessed using four evaluation standards based on a simulation study. The results show that Al-Shiha and Yang’s method stands out as the best performer when compared with other computational methods so it is the best choice to supplement the often used graphical technique. A strategy for analysing unreplicated fractional factorial designs is also outlined.

In each of three experiments, in vitro-matured and -fertilised zygotes were cultured to Day 7 post insemination in synthetic oviductal fluid (SOF). In Experiment 1, zygotes were cultured in groups in either SOF plus albumin (SOFA) or serum (SOFS) and then blastocysts were cultured individually for a further 24 h without a change of media. In Experiment 2, zygotes were cultured in groups using a 2 x 2 factorial design in SOFA or SOFS, with or without recombinant ovine granulocyte-macrophage colony stimulating factor (GM-CSF; 5 ng mL(-1)). Blastocysts were then cultured individually using a split-plot design in SOFA or SOFS with or without GM-CSF. In Experiment 3, zygotes were cultured in SOFA in which GM-CSF was absent (A) or present (P) during Days 1-3, Days 3-5 or Days 5-7 of IVC in six combinations as follows: AAA, AAP, APP, PPP, PPA and PAA. Serum or GM-CSF increased secretion of interferon (IFN)-tau in Experiments 1 and 2 both between Days 5 and 7 of group culture and during individual culture. Secretion of IFN-tau during individual culture was determined by the medium in which embryos were group cultured and the effects of GM-CSF and serum were not additive. In Experiment 3, the presence of GM-CSF between Days 1 and 3 of culture was responsible for stimulation of secretion of IFN-tau between Days 5 and 7; IFN-tau secretion was detected as early as Day 3 post insemination.

Aiming to study the effect of planting dates and spraying with the compound DSPER Size on the growth of (Zea mays L.), a factorial experiment was conducted using a randomized complete block design (RCBD) with split-plot design and three replications. The experiment was carried out in one of the agricultural fields designated for the Field Crops Department, College of Agriculture, University of Kerbala, at the Ibn Al-Bitar Vocational School in the Al-Husseiniya region during the summer season of 2024. The experiment included two factors Main plots planting dates (July 10, July 20, July 30, and August 9) and Subplots concentrations of the DSPER Size compound (0, 1.0, 2.0, 3.0, and 4.0) g l-1. The results showed that the third planting date (July 30) outperformed in some of the studied traits, recording the highest mean plant height (223.71 cm) and stem diameter (29.781 mm). Meanwhile, the fourth planting date excelled in leaf area, with a mean of (6853.6 cm²), and chlorophyll content, of (60.581 SPAD). The treatment with the compound concentration of 3 g L⁻¹ significantly outperformed in giving the highest means for all traits plant height (213.25 cm), stem diameter (29.810 mm), number of leaves (13.240 leaf plant-1), leaf area (6697.4 cm²), and chlorophyll content (61.230 SPAD).The results of the study also showed the superiority of the interaction treatment between the third planting date (July 30) and the concentration of 3 g L⁻¹, which resulted in the highest plant height of (225.23 cm) and the number of leaves reaching (14.630 leaf plant-1).