Explore the vast range of titles available.

It 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.

Textbooks on reponse surface methodology emphasize the importance of lack-of-fit tests when fitting response surface models and stress that, to be able to test for lack of fit, designed experiments should have replication and allow for pure-error estimation. In this paper, we show how to obtain pure-error estimates and how to carry out a lack-of-fit test when the experiment is not completely randomized, but a blocked experiment, a split-plot experiment, or any other multi-stratum experiment. Our approach to calculating pure-error estimates is based on residual maximum likelihood (REML) estimation of the variance components in a full treatment model (sometimes also referred to as a cell means model). It generalizes the approach suggested by Vining et al. (2005) in the sense that it works for a broader set of designs and for replicates other than center-point replicates. Our lack-of-fit test also generalizes the test proposed by Khuri (1992) for data from blocked experiments because it exploits replicates other than center-point replicates and works for split-plot and other multi-stratum designs as well. We provide analytical expressions for the test statistic and the corresponding degrees of freedom and demonstrate how to perform the lack-of-fit test in the SAS procedure MIXED. We re-analyze several published data sets and discover a few instances in which the usual response surface model exhibits significant lack of fit.

In industrial experiments, there are often restrictions in randomization caused by equipment and resource constraints, as well as budget and time constraints. Next to the split-plot and the split-split-plot design, the staggered-level design is an interesting design option for experiments involving two hard-to-change factors. The staggered-level design allows both hard-to-change factors to be reset at different points in time, resulting in a typical staggering pattern of factor-level resettings. It has been shown that, for two-level designs, this staggering pattern leads to statistical benefits in comparison to the split-plot and the split-split-plot design. In this paper, we investigate whether the benefits of the staggered-level design carry over to situations where the objective is to optimize a response and where a second-order response surface model is in place. To this end, we study several examples of D- and I-optimal staggered-level response surface designs.

This paper discusses two types of split-plot designs, split-plot designs with few whole-plot factors and blocked split-plot designs. We provide a list of efficient designs and compare them with the designs available in the literature with respect to a surrogate for the information capacity criterion derived in Cheng and Tsai (2011). In many cases, either better designs are found or additional designs are identified as possible alternatives. In the remaining cases, the optimality of the designs tabulated in design literature is confirmed.

Ecological field experiments initiate successional and evolutionary changes among resident species, yet effect sizes are often reported as if they were constants. Few ecological studies have addressed their questions through long-term, experimental approaches, and many questions remain unanswered regarding temporal patterns in ecological effect sizes. We document temporal variation in effect sizes in response to pulse and press manipulations in a long-term factorial field experiment at Nash's Field, England. The experiment comprises seven treatments applied in a split-plot design to test the single and interactive effects of herbivory by insects, molluscs, and rabbits, liming, nutrient limitation (applied as press experiments), competition (exclusion of grasses or herbs with specific herbicides), and seed limitation (pulse experiments) on plant community dynamics. The response of all vascular plant species was followed for two decades. High species richness was positively related to the minus-grass herbicide in the first decade and negatively related to both nitrogen addition and the abundance of dominant species in both decades. Many significant effects appeared quickly, but some large effects were not detected until year 15. Press experiments produced some long-lasting effects, but effect sizes changed due to both idiosyncratic “year effects” and secular trends. For pulse experiments, most effects, including positive and negative responses to herbicide application and the invasion of most of the sown species, disappeared quickly. However, some endured or grew monotonically, such as the invasion of two sown species that benefited from particular combinations of the press treatments. The fastest effects to appear were the responses from established species. Many of these responses were negative, likely resulting from reduced niche dimensionality and competitive exclusion by new dominant species. Contrarily, one of the largest community-level effects took well over a decade to appear: the natural invasion by one species, which responded to a four-way interaction between experimental treatments. The insights gained from individual effects increased with the duration of the lag before their first appearance, drawing attention to the importance of long-term, manipulative field experiments. This experiment also reinforces the point that factorial experiments are the most insightful way to explore ecological interactions.

Additive manufacturing of continuous carbon fibre-reinforced polymer (CCFRP) parts enables the production of high-strength parts for aerospace, engineering and other industries. Continuous fibres allow for parts to be reinforced along the load path, multiplying their mechanical properties. However, current additive manufacturing processes for producing CCFRP parts do not optimally meet the requirements of the matrix. With resin- and extrusion-based processes, the time-consuming and costly post-processing required to remove support structures severely limits design freedom, and producing small batches requires increased effort. In contrast, laser sintering has proven to be a promising alternative in an industrial environment, allowing the production of robust parts without support structures in a time-efficient and economical manner for single and small-batch production. Based on a novel laser-sintering machine with the automated integration of continuous fibres, a combination of the advantages of the laser-sintering process and the advantages of continuous fibres is to be achieved. This paper describes an experimental analysis and optimisation of this laser-sintering machine using design of experiments. The processing time for fibre integration could be reduced by a factor of three compared to the initial state.

In industrial experimentation, there is growing interest in studies that span more than one processing step. Convenience often dictates restrictions in randomization in passing from one processing step to another. When the study encompasses three processing steps, this leads to split-split-plot designs. We provide an algorithm for computing D-optimal split-split-plot designs and several illustrative examples.

In many industrial experiments, some of the factors are not independently set for each run. This is due to time and/or cost constraints and to the hard-to-change nature of the levels of these factors. Most of the literature restricts attention to split-plot designs in which all the hard-to-change factors are independently reset at the same points in time. This constraint is to some extent relaxed in split-split-plot designs because these allow the less hard-to-change factors to be reset more often than the most hard-to-change factors. A key feature of the split-split-plot designs, however, is that the less hard-to-change factors are reset whenever the most hard-to-change factors are reset. In this article, we relax this constraint and present a new type of design, which allows the hard-to-change factor levels to be reset at entirely different points in time. We show that the new designs are cost-efficient and that they outperform split-plot and split-split-plot designs in terms of the D- and A-optimality criteria. Because of the fact that the hard-to-change factors are independently reset alternatingly, we name the new designs staggered-level designs. Supplementary materials for this article are available online.

Strip-plot designs are commonly used in situations where the production process consists of two process stages involving hard-to-change factors and where it is possible to apply the second stage to semifinished products from the first stage. In this paper, we focus on three-stage processes. As opposed to the three-stage strip-plot designs in the literature, the third stage does not involve hard-to-change factors but easy-to-change factors that are reset independently for each run. For this scenario, the split-split-plot design is a well-known alternative design option. However, we prefer the more statistically efficient strip-plot designs and, therefore, we construct D-optimal strip-plot designs for three-stage processes with no randomization restriction in the third stage. The coordinate-exchange algorithm we use to construct our designs can handle any type of factor and any number of factor levels, runs, rows, and columns.

Renewable energy from photovoltaic power plants has increased in amount globally as an alternative energy to combat global climate change by reducing fossil fuel burning and carbon dioxide (CO2) emissions. The agro-photovoltaic (APV) approach can be a solution to produce solar energy and crop production at the same time by installing solar panels on the same farmland to increase land use efficiency. This study aimed to compare the yield and yield components of rice (Oryza sativa L.) between a vertical APV system and a control field across two years. The solar panels were installed around the rice field in four directions of rice cultivation. Based on the analysis of variance, the primary factor influencing measured yield and yield components was the year effect, whereas the direction effect did not show significance, except for amylose content and ripened grains. Especially for rice production, the rice yield in 2023 was 6.8 t/ha, which was significantly higher by 0.8 t/ha than in 2022. Compared with the control condition, however, there was no significant negative impact on the year-to-year rice yield of the vertical APV system across two years. As rice yield was mainly affected by year, rice yield trials will be required for multiple years to understand the genetic and environmental factors influencing rice production under the vertical APV system.