Split-plot Structure Research Articles

Nutrient Contents and Productivity of Triticum aestivum Plants Grown in Clay Loam Soil Depending on Humic Substances and Varieties and Their Interactions

Due to an extreme increase in population growth, Egypt suffers from a widening gap in the quantity of imported wheat compared with production and local consumption. Two field trials were conducted during the 2018/2019 and 2019/2020 seasons with three levels of humic substances (HSs) as a foliar spray (1.0, 2.0 and 4.0 g L−1; HS1, HS2 and HS3) and three levels (5.04, 7.56 and 10.08 kg ha−1; HS4, HS5 and HS6) as a soil application. These were applied three times (30, 45 and 60 days after sowing) in comparison with the control (HS0) to evaluate the performance of three wheat varieties (Seds1 (V1), Misr2 (V2) and Giza168 (V3)) grown in clay loam soil. The experiment was set up according to the split-plot structure in a randomized complete block design; however, the varieties were set as the main plot and treatments were a sub-main plot. Generally, the data indicated that the soil application treatments recorded maximum values for most growth and yield attributes, except for spike length and grain weight per spike, SPAD reading and total grain yield in the first season, and leaf area and biological yield in the second season. HS1, HS2, HS5, and HS6 were the superior treatments for most of the nutrient contents studied. Regarding the influence of variety, the results showed that V3 recorded maximum values for LA, SpL, TGW, TGY and leaf Zn and Cu contents in both seasons; PH, GWS and leaf N content in the first season; and SPAD reading, BY and leaf K, Fe and Mn contents in the second season. V1 was the superior variety for GWS, BY, leaf K and Mn contents in the 2018/2019 season and PH, GNS in the second season, followed by V2, which had the greatest values for leaf P contents in both seasons, and SPAD reading, GNS and leaf Fe content in the 2018/2019 season and GWS and leaf N content in the second season.

Simultaneous optimization of quality and censored reliability characteristics with constrained randomization experiment

ABSTRACT Product quality and reliability characteristics are important considerations for all manufacturers in the product and process design. Industrial experiments may include both quality and reliability characteristics with the goal to obtain a compromise optimization of the two responses. In many cases, such experiments do not use a completely randomized design. Instead, they involve a more complicated experimental protocol, for example, subsampling, blocking, and split-plot structure. This paper presents a framework for the simultaneous optimization of quality and reliability characteristics with random effects. The paper provides a linear mixed model for quality characteristic and a nonlinear mixed model for Type I censored lifetime to incorporate random effects in the analysis. Subsequently, the desirability function approach is used to obtain a trade-off between the quality and reliability characteristics. The mixed models in this paper can incorporate information from all censored test stands and random effects. The proposed framework provides engineers with an appropriate approach to simultaneously optimize the quality and reliability characteristics with random effects. The paper used a case study to illustrate the proposed framework. A simulation study is also considered to present the necessary of incorporating random effects in the modelling stage.

Split-plot Central Composite Experimental Design Method for Optimization of Cake Height to Achieve desired Texture

In many industrial experimental situations, the levels of certain factors under investigation are much harder to change than others due to time and/or cost constraints. An appropriate approach to such situations is to restrict the randomization of the hard-to-change (HTC) factors, which leads to a split-plot structure. This work designs and conducts a split-plot central composite experiment for optimizing cake height using oven temperature(Factor A) as the HTC factor, amount of flour (B), baking powder (C), and amount of milk (D) as the easy-to-change (ETC) factors. A second-order split-plot central composite design (CCD) model was fit to the generated data and analyzed using generalized least squares (GLS). A stationary point, which gives optimum cake height, was then determined. The results show that main effects of oven temperature, flour, baking powder, and milk were highly significant on the cake height . Their quadratic effects were also significant except that of the flour. The flower/baking powder interaction effect was significant. The fitted model accounted for about 95% of the total variability in the cake height data. The observed optimum cake height was ̂ at a stationary point: A . This study has established the potentials of response surface experiments in optimizing products in food industries.
 Keywords: Experiment, split-plot CCD, Cake height, Design, Stationary point.

Optimization of Split-Plot Design in the Context of Mixture Process Variable Settings

In the presence of process variables, a mixture design has become well-known in statistical modeling due to its utility in modeling the blending surface, which empirically predicts any mixture's response and serves as the foundation for optimizing the expected response blends of different components. In the most common practical situation involving a mixture-process variable, restricted randomization occurs frequently. This problem is solved when the split-plot layout arrangement is used within the constraints. This study's primary goal was to find the best split-plot design (SPD) for the settings mixture-process variables. The SPD was made up of a simplex centroid design (SCD) of four mixture blends and a factorial design with a central composite design (CCD) of the process variable and compared six different context split-plot structure arrangement. We used JMP software version 15 to create D-optimal split-plot designs. The study compared the constructed designs' relative efficiency using A-, D-, I-, and G- optimality criteria. Furthermore, a graphical technique (fraction of design space plot) was used to display, explain, and evaluate experimental designs' performance in terms of precision of the six designs' variance prediction properties. We discovered that arranging subplots with more SCD points than pure mixture design points within SPD with two high process variables is more helpful and provides more precise parameter estimates. We recommend using SPDs in experiments involving mixture process settings developments to measure the mixture components' interaction effects and the processing conditions. Also, the investigation should be set up at each of the points of a factorial design.

Missing Observations in Split-Plot Central Composite Designs: The Loss in Relative A-, G-, and V- Efficiency

The trace (A), maximum average prediction variance (G), and integrated average prediction variance (V) criteria are experimental design evaluation criteria, which are based on precision of estimates of parameters and responses. Central Composite Designs(CCD) conducted within a split-plot structure (split-plot CCDs) consists of factorial (𝑓), whole-plot axial (𝛼), subplot axial (𝛽), and center (𝑐) points, each of which play different role in model estimation. This work studies relative A-, G- and V-efficiency losses due to missing pairs of observations in split-plot CCDs under different ratios (d) of whole-plot and sub-plot error variances. Three candidate designs of different sizes were considered and for each of the criteria, relative efficiency functions were formulated and used to investigate the efficiency of each of the designs when some observations were missing relative to the full one. Maximum A-efficiency losses of 19.1, 10.6, and 15.7% were observed at 𝑑 = 0.5, due to missing pairs 𝑓𝑓, 𝛽𝛽, and 𝑓𝛽, respectively, indicating a negative effect on the precision of estimates of model parameters of these designs. However, missing observations of the pairs- 𝑐𝑐, 𝛼𝛼, 𝛼𝑐, 𝑓𝑐, and 𝑓𝛼 did not exhibit any negative effect on these designs' relative A-efficiency. Maximum G- and Vefficiency losses of 10.1,16.1,0.1% and 0.1, 1.1, 0.2%, were observed, respectively, at 𝑑 = 0.5, when the pairs- 𝑓𝑓, 𝛽𝛽, 𝑐𝑐, were missing, indicating a significant increase in the designs' maximum and average variances of prediction. In all, the efficiency losses become insignificant as d increases. Thus, the study has identified the positive impact of correlated observations on efficiency of experimental designs.
 Keywords: Missing Observations, Efficiency Loss, Prediction variance

Physicochemical characteristics of unripe and ripe banana ‘FHIA 18’ submitted to phosphorus fertilizer over three production cycles

The banana fruits of ‘FHIA 18’ have sensory properties like the ‘Prata’ ones. However, there is no information regarding to phosphorus (P) requirement of this hybrid and its effects on postharvest quality. This study aims to evaluate the physicochemical properties of ‘FHIA 18’ banana fruits submitted to different phosphorus levels under the edaphoclimatic conditions of the municipality of São Manuel, state of São Paulo, during three successive production cycles. Therefore, different levels of Phosphorus (0 [0%], 20 [50%)], 30 [75%], 40 [100%], 50 [125%] and 60 kg ha-1 year-1 P2O5 [150%]) were applied during three production cycles. A completely randomized design and split plot structure was used, the plots consisted of six phosphorus fertilizers doses and three production cycles (2014, 2015 and 2016) in sub plots. For unripe fruits, we evaluated the contents of dry matter, starch, total sugars, reducing sugars and phosphorus. For ripe fruits, we analyzed firmness, peel colour descriptors, total titratable acidity, total soluble solids, maturity index, total sugars and reducing sugars. Results showed that P fertilizer provided an increase in P content in unripe fruits until the application of 100% of the recommended dose; besides an increase in the reducing sugar contents until the application of 61% over the first production cycle. Results also indicated that applying lower phosphorus doses ( ? 100% of the recommended dose) promoted lower total titratable acidity and higher maturity index (52-53% of recommendation) in ripe fruits. Regardless of P2O5 dose, the contents of dry matter, starch and phosphorus increased in unripe fruits from first to third crop cycle, as well as the firmness, total soluble solids contents, maturity index and sugar content in ripe fruits.

Stochastic search variable selection for split-plot and blocked screening designs

Split-plot definitive screening and blocked definitive screening designs have been developed for detecting active main effects and second-order effects in screening experiments when split-plot and block structures exist. In the literature, multistage regression and forward stepwise regression methods were proposed for data analysis on the two types of designs. However, classical regression approaches present limitations and potential problems. First, the degrees of freedom may not be large enough to estimate all active effects. Second, the restricted maximum-likelihood estimates for variances of whole-plot and block errors can be zero. To solve these problems and enhance detection capability, we propose a stochastic search variable selection (SSVS) method based on Bayesian theory. Different from the existing Bayesian approaches for split-plot and blocked designs, the proposed SSVS method can perform variable selections and choose models that follow the effect heredity principle. Markov chain Monte Carlo and Gibbs sampling are applied and a general WinBUGS code, which can be used for any split-plot and blocked screening design, is provided. Simulation studies are conducted and their results show that the proposed SSVS method effectively controls the false discovery rate and has higher detection capability than the two regression methods.

Optimal Blocked and Split-Plot Designs Ensuring Precise Pure-Error Estimation of the Variance Components

Textbooks on response surface methodology generally stress the importance of lack-of-fit tests and estimation of pure error. For lack-of-fit tests to be possible and other inference to be unbiased, experiments should allow for pure-error estimation. Therefore, they should involve replicated treatments. While most textbooks focus on lack-of-fit testing in the context of completely randomized designs, many response surface experiments are not completely randomized and require block or split-plot structures. The analysis of data from blocked or split-plot experiments is generally based on a mixed regression model with two variance components instead of one. In this article, we present a novel approach to designing blocked and split-plot experiments which ensures that the two variance components can be efficiently estimated from pure error and guarantees a precise estimation of the response surface model. Our novel approach involves a new Bayesian compound D-optimal design criterion which pays attention to both the variance components and the fixed treatment effects. One part of the compound criterion (the part concerned with the treatment effects) is based on the response surface model of interest, while the other part (which is concerned with pure-error estimates of the variance components) is based on the full treatment model. We demonstrate that our new criterion yields split-plot designs that outperform existing designs from the literature both in terms of the precision of the pure-error estimates and the precision of the estimates of the factor effects.

Split Plot Mixture Process Variable Experiment on Steel Slag Concrete

In many such mixture-process variable experiments, constraints such as cost or time prohibit the selection of treatments completely at random. This leads to a split-plot structure of the data and affects both the experimental design and the statistical modelling. The constraints of the mixture component affect the experimental region. The experimental region could be a regular or irregular shape. For the irregular shape, the design points are difficult to identify by hand. XVERT algorithm is one algorithm in mixture experiments to define the design points of mixture experiment. An algorithm used for selecting a subset of extreme vertices when the number of candidate vertices is large. This research a new split-plot designs of mixture experiments with process variables was developed. The study case was on cementitious mixture components for steelslag concrete. The five mixture components were cement, aggregate fine, aggregate coarse, percentage steel slag replaced the aggregate fine and water. The process variable was size of steel slag. The steel slag concrete experiment was run using a split plot mixture process variable design with 23 whole plots of three observations. The final design consisted of {8 7 8} and experiment design with 69 observations.

Impact of censoring types on the two-stage method for analyzing reliability experiments with random effects

Many reliability experiments are not completely randomized. Instead they involve subsamples, blocks, split-plot structures, etc. A common analysis often uses random effects to account for the impact of the experimental protocol. The two-stage method is an easy way for practitioners to incorporate random effects in the analysis. This article compares performance of the two-stage method under Type I, Type II censored, and uncensored data from a Weibull distribution. We evaluate the effects of censoring type, censoring rate, sample size, and shape parameter on the two-stage method. Then, we apply the two-stage method to a real experiment. Finally, we give practitioners some recommendations for designing and analyzing reliability experiments.

Split-Plot and Multi-Stratum Designs for Statistical Inference

ABSTRACTIt is increasingly recognized that many industrial and engineering experiments use split-plot or other multi-stratum structures. Much recent work has concentrated on finding optimum, or near-optimum, designs for estimating the fixed effects parameters in multi-stratum designs. However, often inference, such as hypothesis testing or interval estimation, will also be required and for inference to be unbiased in the presence of model uncertainty requires pure error estimates of the variance components. Most optimal designs provide few, if any, pure error degrees of freedom. Gilmour and Trinca (2012) introduced design optimality criteria for inference in the context of completely randomized and block designs. Here these criteria are used stratum-by-stratum to obtain multi-stratum designs. It is shown that these designs have better properties for performing inference than standard optimum designs. Compound criteria, which combine the inference criteria with traditional point estimation criteria, are also used and the designs obtained are shown to compromise between point estimation and inference. Designs are obtained for two real split-plot experiments and an illustrative split–split-plot structure. Supplementary materials for this article are available online.

Physical properties of an Orthic Black Chernozem after 5 years of liquid and solid pig manure application to annual and perennial crops

Pig (Sus scrofa) manure is added to the soil to supply nutrients and improve soil properties. To our knowledge, no direct comparison has been made on the effect of liquid pig manure (LPM) and solid pig manure (SPM) on the physical properties of a prairie soil. This study was established in 2009 at the University of Manitoba’s Ian Morrison Research Station in Carman, Manitoba. The treatment design was a split-plot structure with cropping system as the main plot and manure treatments as subplots. Five years after the study was initiated, soil samples were collected from the 0–10 cm and 10–20 cm depth intervals for determination of bulk density, saturated hydraulic conductivity (Ksat), and water retention at field capacity and permanent wilting point (PWP). For wet aggregate stability, samples were collected from the 0–5 cm layer. Land application of SPM significantly decreased bulk density by 14%, significantly increased Ksatby 110% in the 0–10 cm layer, and resulted in a 30% increase in wet aggregate stability (P < 0.05). In perennial plots, SPM increased water retention at field capacity, PWP, and available water in the 0–10 cm compared with annual plots. This was not the case for LPM-amended soils. We conclude that SPM has the potential as an organic amendment to improve the physical properties of the topsoil.

Response Surface Methodology Using Split-Plot Definitive Screening Designs

Definitive screening designs are a new class of three-level designs. We investigate the performance of definitive screening designs in split-plot structures for one-step response surface methodology. The result of the projection eligibility and the study of D-efficiency and I-efficiency show that split-plot definitive screening designs perform well when the number of important factors is small. To reduce the risk of being unable to fit second-order models for response surfaces, we provide the column indexes of projections. Experimenters can assign potentially important factors to those columns to avoid ineligible projections. An example is presented to demonstrate how to analyze data for response surface methodology using the split-plot definitive screening design.

A Prediction Interval Estimator for the Original Response When Using Box–Cox Transformations

Motivated by electron microscopy experiments, we develop an approximate prediction interval on the response variable Y, where it is assumed that a normal-theory linear model is fit using a transformed version of Y, and the transformation type is contained in the Box–Cox family. We derive a closed-form approximation to the kth moment of the original response variable Y, which is then used to estimate the mean and variance of Y, given parameter estimates obtained from fitting the model in the transformed domain. Chebychev's inequality is then used to construct a 100(1 – α)% prediction-interval estimator on Y. Using Monte Carlo simulation, we assess the width performance of our proposed Chebychev prediction interval, relative to that obtained by employing a more common interval-construction approach. General results suggest that, for a given level of expected coverage, the proposed interval estimator will achieve a smaller mean and variance of the interval width estimates, especially as the number of degrees of freedom beyond that required to estimate model terms is small. We apply our method to two experimental-data sets, one involving a standard 2k design and the other involving a 2k design with a split-plot error structure.

The use of equivalent estimation designs in tribological studies

Tribological experiments in which friction is examined as a function of multiple factors often involve some factors whose levels are harder to change than others. For example, changing the temperature in a test setup usually is more time consuming than changing the sliding speed or the contact pressure. In such cases, it is preferable to rearrange the runs in the experiment in a way that minimizes the number of changes of the factor levels that are hard to change. This restricts the randomisation of the experiment and leads to a split-plot structure. A new class of split-plot designs called equivalent estimation designs has properties which seem to be well suited for use in tribological studies. This paper compares the performance of these designs to two commonly used designs (a completely randomised three level factorial design and a one-factor-at-a-time experiment) in a tribological example study. The results show that using equivalent estimation designs in tribology can deliver results that are similar to those of conventional designs while requiring just a fraction of their effort.

The possibility of crop cultivation and utilization of edible gum from herb (Dorema ammoniacum D. Don) in dryland farming

Gum ammoniacum (Dorema ammoniacum D. Don) is one of the most important food, industrial and medicinal plants of Iran which is mainly distributed in semi-arid and desert areas. It is endangered due to superfluous and unsustainable harvesting methods. The objective of this study was to evaluate the possibility of crop cultivation in terms of dryland farming and the best operation method for collecting gum resin. For this purpose, the effects of different scarification methods (traditional, concave and staircase) and cut-off frequencies (5, 10 and 13 times) on gum yield of the five-year-old plants were studied at Toroq farm in Mashhad. The treatments were studied under a split plot structure in time experiment which was arranged in a completely randomized design with three replications. The rate of the root gum production and the rate of plant survival in the years after the withdrawal were measured, and finally the data were analyzed. Results showed no significant differences among cutting methods at p<0.05. The yield of gum was affected by times of harvesting and the highest yield was observed (p<0.01) in 13 cut-off times with 31.67 g/plant. The lowest gum yields were seen in 5 cut-off times with 5.84 g/plant. The traditional cutting method destroyed crown buds and caused the death of the plant (p<0.01). Reasonable gum yields and plant regeneration rates were obtained with the use of stairs method which is an easy method to be applied by the farmers or beneficiaries.

Caracterização do microclima dos diferentes layouts de caixas no transporte de ovos férteis

Quando as condições microclimáticas e de transporte são combinadas inadequadamente, os componentes dos ovos férteis podem ser danificados. Objetivou-se, com este trabalho, caracterizar e analisar a dependência espacial dos perfis microclimáticos no transporte de ovos férteis em diferentes layouts de caixas. A pesquisa foi conduzida na empresa integradora avícola, em Mogi Mirim, SP, por meio do acompanhamento de cinco carregamentos. Utilizou-se um caminhão climatizado do tipo baú, com capacidade para 592 caixas de ovos e se avaliaram dois layouts da carga (caixas superiores e inferiores) de 9 pilhas com seis caixas de ovos férteis. Um microprocessador foi instalado dentro das caixas de ovos, totalizando 18 microprocessadores. As variáveis meteorológicas analisadas foram: temperatura (ºC) umidade relativa (%) e entalpia específica (kJ kg-1 ar seco). O delineamento experimental adotado foi o inteiramente casualizado com parcelas subdividas e realizada a análise geoestatística (krigagem ordinária). No layout das caixas superiores foram registrados valores elevados de temperatura e baixos valores de umidade relativa e entalpia específica. As regiões de piores condições microclimáticas nos dois layouts se situaram nas partes central e traseira do caminhão, durante o transporte dos ovos férteis.

Supersaturated Split-Plot Designs

Methods to construct supersaturated split-plot designs (SSSPDs) are presented. We use an E(s2)-optimal supersaturated design (SSD) in the whole-plot and half fractions of Plackett–Burman designs (PBDs) in the split plot. It is shown that (1) the resulting split-plot design has orthogonal columns, (2) the split-plot columns are orthogonal to the whole-plot columns, and (3) the whole-plot columns are orthogonal to the whole-plot by split-plot interactions. As an example, an SSSPD is applied to an extrusion process to produce biodegradable packaging foams, and data are analyzed using stepwise selection procedures. It is concluded that SSSPD can be used productively as a screening design when the experiment has a split-plot structure and traditional designs are not feasible.

Evaluation of Common Bean (Phaseolus vulgaris L.) Genotypes for Adaptation to Low Phosphorus

Common bean production in Tanzania is constrained by soil phosphorus which is mainly due to inherently low phosphorus content, soil erosion, and fixation by oxides in acidic soils. A study was conducted to evaluate bean genotypes in a screen house pot experiment for their ability to thrive and produce on low phosphorus soil. Assessment of shoot biomass, root biomass, shoot P concentration, P uptake, and yield components was done using three phosphorus levels and seven bean genotypes. Phosphorus levels, namely, control (P0), medium P (40 mg P/kg), and high P (160 mg P/kg), were the main plot factor, while the genotypes were the subplot in split plot structure, arranged in a completely randomized design. Shoot and root biomass as well as P uptake increased significantly with increase in phosphorus levels. There was varying response of genotypes in performance in terms of shoot biomass P uptake, and yield in a treatment without P addition. Genotypes MILENIO, BAT477, and A785 were outstanding in terms of root and shoot biomass, P uptake and grain yield under low P treatment. Therefore, those genotypes can be recommended for use in low-phosphorus environment as well as breeding materials.

A Case Study Involving Mixture–Process Variable Experiments within a Split-Plot Structure

ABSTRACT When an experiment involves both mixture and process variables, the size of the experiment can increase dramatically and the assumption of complete randomization may be violated due to the cost or time to change some factor levels. In this situation, restrictions on the randomization of experimental runs are necessary, resulting in a split-plot structure. Furthermore, when some process variables are noise variables, it is important to consider the noise variables at the design stage of the process to find the robust parameter setting that makes the response “robust” to the variability transmitted from the noise factors. However, many mixture–process experiments are analyzed without considering this randomization issue. We provide a real example of a mixture–process experiment with noise variables within a split-plot structure. Our example demonstrates show to minimize the prediction error with noise variables in a situation where the standard analysis results in poor estimation for the prediction due to the restricted randomization. Without a proper analysis, the experiment leads to the wrong model and results in poor prediction. When the noise variables are ignored in the experiments, the model provides large random errors due to the effect of noise variables. We show that dual optimization using a mean model and a variance model can find the robust settings for the noise variables.