1,720,964 research outputs found
Robustness of diallel cross designs to the loss of one or more observations.
No full textThe effects of missing observations on complete and partial diallel cross designs are examined. A-efficiencies, based on average variances of the elementary contrasts of the line-effects, suggest that these designs are fairly robust. Simple g-inverses may be found for the information matrices of the line effects which allow evaluation of expressions for the variances of the line-effect differences with and without the missing observations. It is shown that, for small designs or when the number of lines is large, the reduction in efficiency for individual line comparisons can be quite large. When these designs are employed, care should be taken to ensure that individual observations are not lost
Robustness of a class of partial diallel cross designs to the unavailability of a complete block of observations.
No full textComplete and partial diallel cross designs are examined as to their construction and robustness against the loss of a block of observations. A simple generalized inverse is found for the information matrix of the line effects, which allows evaluation of expressions for the variances of the line-effect differences with and without the missing block. A-efficiencies, based on average variances of the elementary contrasts of the line-effects, suggest that these designs are fairly robust. The loss of efficiency is generally less than 10%, but it is shown that specific comparisons might suffer a loss of efficiency of as much as 40%
The Effect of Missing Values on Designed Experiments
No full textIn this thesis, the effect of different configurations of missing values on block designs, row-column designs, and diallel cross designs is investigated. The average variance of all pairwise treatment comparisons has been used as a measure of robustness by the majority of researchers. The maximum variance of comparisons is computed numerically, or developed theoretically, in this thesis for most patterns of missing data. The reduced normal equations can be solved with a suitable choice of generalised inverse, and formulae for the individual variances of pairwise treatment differences can also be derived.The effect of missing values on block designs, in particular randomised and balanced incomplete block designs is studied. It is shown that designs with a small number of treatments and a small number of blocks are severely affected by the loss of one, two or three observations. Larger designs are not as seriously affected when the average variance is considered, but there are a small number of pairwise treatment comparisons that suffer a large loss of efficiency.Row-column designs have also been investigated for similar patterns of missing data. The lack of orthogonality introduced by the loss of data in many situations complicates the analysis and derivation of general expressions of the variances. The loss of efficiency for small Latin square designs is substantial after the removal of only one or two units. Constructing a design with multiple squares is shown to reduce the impact of the missing data. Youden square designs also suffer a similar loss of information after the loss of a few observations, and it is also shown that the structure of the design affects the distributions of efficiencies for a given number of missing values.</p
Missing observations in Youden square designs
No full textThe reduction in efficiency in estimating treatment differences in Youden square designs from which individual observations have been lost is considered. A simple generalised inverse of the associated information matrix is used to develop expressions for the variances of the pairwise treatment comparisons. Results on the robustness of Youden squares to the loss of one or two observations are given, and it is shown that for two observations missing there are eight possible cases of resulting design which need to be considered. The frequencies of these cases depend on the form of the initial design, as well as on the design parameters. Examples of similar designs are used to illustrate these different frequencies of resulting designs
Robustness of balanced incomplete block designs to randomly missing observations
No full textPractical experimenters must always be aware of the possibility that some of their observations could become unavailable for analysis. In an experiment involving treatments and blocks, it could be desirable to select a design that is resistant to the loss of a complete block or treatment, or a small number of observations distributed at random throughout the initial design. In this paper, we examine the robustness of binary, variance-balanced, incomplete block designs using the eigenvalues of the associated information matrix when specific observations are missing. Results are presented for up to three missing observations and the procedure is illustrated using an example involving eight treatments arranged in 14 blocks of four treatments per block. On the basis of these considerations, it is recommended that, to guard against a substantial loss of efficiency due to a small number of randomly missing observations, it is preferable to use designs with as few treatments common to pairs of blocks as possible
Missing values in replicated Latin squares.
No full textDesigns based on any number of replicated Latin squares are examined for their robustness against the loss of up to three observations randomly scattered throughout the design. The information matrix for the treatment effects is used to evaluate the average variances of the treatment differences for each design in terms of the number of missing values and the size of the design. The resulting average variances are used to assess the overall robustness of the designs. In general, there are 16 different situations for the case of three missing values when there are at least three Latin square replicates in the design. Algebraic expressions may be determined for all possible configurations, but here the best and worst cases are given in detail. Numerical illustrations are provided for the average variances, relative efficiencies, minimum and maximum variances and the frequency counts, showing the effects of the missing values for a range of design sizes and levels of replication
Designing experiments for an application in laser and surface Chemistry
No full textWe consider the design used to collect data for a Second Harmonic Generation (SHG) experiment, where the behaviour of interfaces between two phases, for example the surface of a liquid, is investigated. These studies have implications in surfactants, catalysis, membranes and electrochemistry. Ongoing work will be described in designing experiments to investigate nonlinear models used to represent the data, relating the intensity of the SHG signal to the polarisation angles of the polarised light beam. The choice of design points and their effect on parameter estimates is investigated. Various designs and the current practice of using equal-spaced levels are investigated, and their relative merits compared on the basis of the overall aim of the chemical study
Statistical analysis of second harmonic generation experiments: a phenomenological model
No full textWe discuss issues arising in fitting theoretically derived nonlinear models with complex coefficients to data from surface Second Harmonic Generation (SHG) experiments conducted at the air/liquid interface. We explore different parametrisations for the complex parameters and show that the Enter (magnitude and phase angle) parametrisation is preferable to the real and imaginary parts parametrisation, both theoretically and empirically. We emphasise the importance and value of diagnostic plots for evaluating the quality of model fit. We derive approximate standard errors for the parameter estimates and discuss issues of making inference about ratios of parameters. We consider approximate confidence intervals (using the approximate standard errors), profile likelihood intervals, Fieller's method and bootstrap intervals. Fieller's method (and the bootstrap intervals) provide useful information on the value of the simpler approximate confidence intervals. We also propose and implement a likelihood ratio test to assess whether a common model can be fitted to several independent data sets. Finally, the methods are applied to data sets obtained from SHG experiments on L-phenylalanine at the air/water interface and toluene
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