Statnote 12: the split-plot analysis of variance

In some experimental situations, the factors may not be equivalent to each other and replicates cannot be assigned at random to all treatment combinations. A common case, called a ‘split-plot design’, arises when one factor can be considered to be a major factor and the other a minor factor. Investigators need to be able to distinguish a split-plot design from a fully randomized design as it is a common mistake for researchers to analyse a split-plot design as if it were a fully randomised factorial experiment.

Statnote 31:analysis of covariance

Analysis of covariance (ANCOVA) is a useful method of ‘error control’, i.e., it can reduce the size of the error variance in an experimental or observational study. An initial measure obtained before the experiment, which is closely related to the final measurement, is used to adjust the final measurements, thus reducing the error variance. When this method is used to reduce the error term, the X variable must not itself be affected by the experimental treatments, because part of the treatment effect would then also be removed. Hence, the method can only be safely used when X is measured before an experiment. A further limitation of the analysis is that only the linear effect of Y on X is being removed and it is possible that Y could be a curvilinear function of X. A question often raised is whether ANCOVA should be used routinely in experiments rather than a randomized blocks or split-plot design, which may also reduce the error variance. The answer to this question depends on the relative precision of the difference methods with reference to each scenario. Considerable judgment is often required to select the best experimental design and statistical help should be sought at an early stage of an investigation.

Data methods in optometry Part 9: experimental design and analysis of variance

The key to the correct application of ANOVA is careful experimental design and matching the correct analysis to that design. The following points should therefore, be considered before designing any experiment: 1. In a single factor design, ensure that the factor is identified as a 'fixed' or 'random effect' factor. 2. In more complex designs, with more than one factor, there may be a mixture of fixed and random effect factors present, so ensure that each factor is clearly identified. 3. Where replicates can be grouped or blocked, the advantages of a randomised blocks design should be considered. There should be evidence, however, that blocking can sufficiently reduce the error variation to counter the loss of DF compared with a randomised design. 4. Where different treatments are applied sequentially to a patient, the advantages of a three-way design in which the different orders of the treatments are included as an 'effect' should be considered. 5. Combining different factors to make a more efficient experiment and to measure possible factor interactions should always be considered. 6. The effect of 'internal replication' should be taken into account in a factorial design in deciding the number of replications to be used. Where possible, each error term of the ANOVA should have at least 15 DF. 7. Consider carefully whether a particular factorial design can be considered to be a split-plot or a repeated measures design. If such a design is appropriate, consider how to continue the analysis bearing in mind the problem of using post hoc tests in this situation.

The application of analysis of variance (ANOVA) to different experimental designs in optometry

Analysis of variance (ANOVA) is the most efficient method available for the analysis of experimental data. Analysis of variance is a method of considerable complexity and subtlety, with many different variations, each of which applies in a particular experimental context. Hence, it is possible to apply the wrong type of ANOVA to data and, therefore, to draw an erroneous conclusion from an experiment. This article reviews the types of ANOVA most likely to arise in clinical experiments in optometry including the one-way ANOVA ('fixed' and 'random effect' models), two-way ANOVA in randomised blocks, three-way ANOVA, and factorial experimental designs (including the varieties known as 'split-plot' and 'repeated measures'). For each ANOVA, the appropriate experimental design is described, a statistical model is formulated, and the advantages and limitations of each type of design discussed. In addition, the problems of non-conformity to the statistical model and determination of the number of replications are considered. © 2002 The College of Optometrists.

Statnote 13: a more complex analysis of variance incorporating a 'repeated measures' factor

Experiments combining different groups or factors and which use ANOVA are a powerful method of investigation in applied microbiology. ANOVA enables not only the effect of individual factors to be estimated but also their interactions; information which cannot be obtained readily when factors are investigated separately. In addition, combining different treatments or factors in a single experiment is more efficient and often reduces the sample size required to estimate treatment effects adequately. Because of the treatment combinations used in a factorial experiment, the degrees of freedom (DF) of the error term in the ANOVA is a more important indicator of the ‘power’ of the experiment than the number of replicates. A good method is to ensure, where possible, that sufficient replication is present to achieve 15 DF for the error term of the ANOVA testing effects of particular interest. Finally, it is important to always consider the design of the experiment because this determines the appropriate ANOVA to use. Hence, it is necessary to be able to identify the different forms of ANOVA appropriate to different experimental designs and to recognise when a design is a split-plot or incorporates a repeated measure. If there is any doubt about which ANOVA to use in a specific circumstance, the researcher should seek advice from a statistician with experience of research in applied microbiology.

Design and analysis of a 2-DoF split-stator induction motor

A two degrees of freedom (2-DOF) actuator capable of producing linear translation, rotary motion, or helical motion would be a desirable asset to the fields of machine tools, robotics, and various apparatuses. In this paper, a novel 2-DOF split-stator induction motor was proposed and electromagnetic structure pa- rameters of the motor were designed and optimized. The feature of the direct-drive 2-DOF induction motor lies in its solid mover ar- rangement. In order to study the complex distribution of the eddy current field on the ferromagnetic cylinder mover and the motor’s operating characteristics, the mathematical model of the proposed motor was established, and characteristics of the motor were ana- lyzed by adopting the permeation depth method (PDM) and finite element method (FEM). The analytical and numerical results from motor simulation clearly show a correlation between the PDM and FEM models. This may be considered as a fair justification for the proposed machine and design tools.

Field experiments on competition between three foliose lichens using the de Wit experimental design

Competition between three foliose, saxicolous lichens common on slate rock in South Gwynedd, Wales, U.K. was studied experimentally using the de Wit design. Fragments of the three species were cut from the edges of large thalli, glued to 5 x 5 cm plots marked out on pieces of slate which were then placed on boards in the field. For each combination of pairs of species, the two species were grown either in monoculture at a density of 24 fragments per plot or together in three mixtures in differing proportions, i.e. species A:B with 16:8, 12:12 and 8:16 fragments per plot; the density remaining constant throughout. Area of the species in the plots after 3 years was used as an estimate of growth. Physcia orbicularis and Parmelia glabratula ssp. fuliginosa grew similarly in monoculture. In mixtures of the two, growth of each species was linearly related to its proportion in a mixture, suggesting little competition had occurred during three years. By contrast, the growth of Parmelia conspersa in monoculture was significantly greater than that of P. orbicularis or P. glabratula. In addition, the growth of both species was substantially reduced in mixtures with P. conspersa; P. glabratula being eliminated in the mixture in which it was the minority species. These results suggest that P. conspersa should predominate in communities with either of the other two species and, in the absence of P. conspersa, communities dominated by P.orbicularis and P. glabratula should be more stable.

Integrated manufacturing systems design in a multi-product company

Using a hydraulic equipment manufacturing plant as the case study, this work explores the problems of systems integration in manufacturing systems design, stressing the behavioural aspects of motivation and participation, and the constraints involved in the proper consideration of the human sub-system. The need for a simple manageable modular organisation structure is illustrated, where it is shown, by reference to systems theory, how a business can be split into semi-autonomous operating units. The theme is the development of a manufacturing system based on an analysis of the business, its market, product, technology and constraints, coupled with a critical survey of modern management literature to develop an integrated systems design to suit a specific company in the current social environment. Society currently moves through a socio-technical revolution with man seeking higher levels of motivation. The transitory environment from an autocratic/paternalistic to a participative operating mode demands systems parameters only found to a limited extent in manufacturing systems today. It is claimed, that modern manufacturing systems design needs to be based on group working, job enrichment, delegation of decision making and reduced job monotony. The analysis shows how negative aspects of cellular manufacture such as lack of flexibility and poor fixed asset utilisation are relatively irrelevant and misleading in the broader context of the need to come to terms with the social stresses imposed on a company operating in the industrial environment of the present and the immediate future.

Reduced postprandial proinsulinaemia and 32-33 split proinsulinaemia after a mixed meal in type 2 diabetic patients following sensitization to insulin with pioglitazone

Objective: Reduced insulin sensitivity associated with fasting hyperproinsulinaemia is common in type 2 diabetes. Proinsulinaemia is an established independent cardiovascular risk factor. The objective was to investigate fasting and postprandial release of insulin, proinsulin (PI) and 32-33 split proinsulin (SPI) before and after sensitization to insulin with pioglitazone compared to a group treated with glibenclamide. Design and patients: A randomized double-blind placebo-controlled trial. Twenty-two type 2 diabetic patients were recruited along with 10 normal subjects. After 4 weeks washout, patients received a mixed meal and were assigned to receive pioglitazone or glibenclamide for 20 weeks, after which patients received another identical test meal. The treatment regimes were designed to maintain glycaemic control (HbA1c) at pretreatment levels so that ß-cells received an equivalent glycaemic stimulus for both test meals. Measurements: Plasma insulin, PI, SPI and glucose concentrations were measured over an 8-h postprandial period. The output of PI and SPI was measured as the integrated postprandial response (area under the curve, AUC). Results: Pioglitazone treatment resulted in a significant reduction in fasting levels of PI and SPI compared to those of the controls. Postprandially, pioglitazone treatment had no effect on the insulin AUC response to the meal but significantly reduced the PI and SPI AUCs. Glibenclamide increased fasting insulin and the postprandial insulin AUC but had no effect on the PI and SPI AUCs. Conclusions: Sensitization to insulin with pioglitazone reduces the amount of insulin precursor species present in fasting and postprandially and may reduce cardiovascular risk. © 2007 The Authors.