1,820 research outputs found
Testing for lack of fit in blocked and split-plot response surface designs
No full textTextbooks on response 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. It generalizes the one 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
The performance of subset response surface designs for estimating third order terms
No full textResponse surface designs are widely used in industries like chemicals, foods, pharmaceuticals, bioprocessing, agrochemicals, biology, biomedicine, agriculture and medicine. One of the major objectives of these designs is to study the functional relationship between one or more responses and a number of quantitative input factors. However, biological materials have more run to run variation than in many other experiments, leading to the conclusion that smaller response surface designs are inappropriate. Thus designs to be used in these research areas should have greater replication. Gilmour (2006) introduced a wide class of designs called "subset designs" which are useful in situations in which run to run variation is high. These designs allow the experimenter to fit the second order response surface model. However, there are situations in which the second order model representation proves to be inadequate and unrealistic due to the presence of lack of fit caused by third or higher order terms in the true response surface model. In such situations it becomes necessary for the experimenter to estimate these higher order terms. In this study, the properties of subset designs, in the context of the third order response surface model, are explored.</p
Analysis of data from non-orthogonal multistratum designs in industrial experiments
No full textSplit-plot and other multistratum structures are widely used in factorial and response surface experiments. Residual maximum likelihood (REML) and generalized least squares (GLS) estimation is seen as the state of the art method of data analysis for non-orthogonal designs. We analyse data from an experiment that was run to study the effects of five process factors on the drying rate for freeze-dried coffee and find that the main plot variance component is estimated to be 0. We show that this is a typical property of REML–GLS estimation in non-orthogonal split-plot designs with few main plots which is highly undesirable and can give misleading conclusions. Instead, we recommend a Bayesian analysis, using an informative prior distribution for the main plot variance component and implement this by using Markov chain Monte Carlo sampling. Paradoxically, the Bayesian analysis is less dependent on prior assumptions than the REML–GLS analysis. Bayesian analyses of the coffee freeze-drying data give more realistic conclusions than REML–GLS analysis, providing support for our recommendation.<br/
Robustness of subset response surface designs to missing observations
No full textExperiments designed to investigate the effect of several factors on a process have wide application in modern industrial and scientific research. Response surface designs allow the researcher to model the effects of the input variables on the response of the process. Missing observations can make the results of a response surface experiment quite misleading, especially in the case of one-off experiments or high cost experiments. Designs robust to missing observations can attract the user since they are comparatively more reliable. Subset designs are studied for their robustness to missing observations in different experimental regions. The robustness of subset designs is also improved for multiple levels by using the minimax loss criterion.<br/
A general strategy for analyzing data from split-plot and multistratum experimental designs
No full textIncreasingly, industrial experiments use multistratum designs, such as split-plot and strip-plot designs. Often, these experiments span more than one processing stage. The challenge is to identify an appropriate multistratum design, along with an appropriate statistical model. In this article, we introduce Hasse diagrams in the response surface context as a tool to visualize the unit structure of the experimental design, the randomization and sampling approaches used, the stratum in which each experimental factor is applied, and the degrees of freedom available in each stratum to estimate main effects, interactions, and variance components. We illustrate their use on several responses measured in a large study of the adhesion properties of coatings to polypropylene. We discuss quantitative, binary, and ordered categorical responses, for designs ranging from a simple split-plot to a strip-plot that involves repeated measurements of the response. The datasets discussed in this article are available online as supplementary materials, along with sample SAS programs
A general criterion for factorial designs under model uncertainty
No full textMotivated by two industrial experiments in which rather extreme prior knowledge was used to choose the design, we show that the QB criterion, which aims to improve the estimation in as many models as possible by incorporating experimenters’ prior knowledge along with an approximation to the As criterion, is more general and has a better statistical interpretation than many standard criteria. The generalization and application of the criterion to different types of designs are presented. The relationships between QB and other criteria for different situations are explored. It is shown that the E(s2) criterion is a special case of QB and several aberration-type criteria are limiting cases of our criterion, so that QB provides a bridge between alphabetic optimality and aberration. The two case studies illustrate the potential benefits of the QB criterion. R programs for calculating QB are available online as supplemental materials. <br/
New families of QB-optimal saturated two-level main effects screening designs
Full text linkIn this paper, we study saturated two-level main effects designs which are commonly used for screening experiments. The QB criterion, which incorporates experimenters' prior beliefs about the probability of factors being active is used to compare designs. We show that under priors with more weight on models of small size, p-efficient designs should be recommended; when models with more parameters are of interest, A-optimal designs would be better. We identify new classes of saturated main effects designs between these two designs under different priors. The way in which the choice of designs depends on experimenters' prior beliefs will be demonstrated for the cases when the number of runs N = 2 mod 4. A novel method of construction of QB-optimal designs using conference matrices is introduced. Complete families of optimal designs are given for N = 6; 10; 14; 18; 26; 30
Multilevel augmented pairs second-order response surface designs and their robustness to missing data
No full textMissing observations can occur even in a well-planned experiment. The effect of missing observations can be much more serious when the design is saturated or near saturated. The levels of factor settings that make a design more robust to missing observations are of great importance in the sense that the loss for missing observations becomes minimum. In this study, new augmented pairs minimax loss designs are constructed, which are more robust to one missing design point than the augmented pairs designs presented by Morris (2000). New designs are compared with augmented pairs designs, central composite designs, and small composite designs under generalized scaled standard deviations. The model used is also studied for the regression coefficient estimates
Designs for first-order interactions in paired comparison experiments with two-level factors
No full textFor paired comparison experiments involving options described by a common set of two-level factors a new method for generating exact designs is presented. These designs allow the efficient estimation of main effects and first order interactions and perform better than alternative designs available in the literature
Fractional polynomial models for constrained mixture experiments
No full textA new - parsimonious but flexible - class of non-linear models, based on Fractional Polynomials, is proposed for fitting the data from constrained mixture experiments. These Mixture Fractional Polynomial (MFP) Models are easily fitted by nonlinear least squares using a partially linear algorithm. They are compared with the recently proposed class of nonlinear General Blending Models, and with several commonly used linear models from the literature. It is shown that the new class of MFP Models outperforms those competing models in several practical applications
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