Nonorthogonal Designs Research Articles

A graphical method for evaluating the effect of blocking in response surface designs

When fitting a response surface model, the least squares estimates of the model's parameters will generally depend on how the response surface design is blocked. Then, the choice of a blocking arrangement for a response surface design can have a considerable effect on estimating the mean response, on the size of the prediction variance and on the variance of the least squares estimator of the model's parameters. These are all shown to be affected by the sizes of the blocks and the allocation of experimental runs to the blocks. In this paper, we propose a graphical method for evaluating the effect of blocking in response surface designs. This graphical method can be used to examine how blocking influences the variance of the least squares estimator of the model's parameters. and to compare the effect of blocking in the cases of the orthogonal and nonorthogonal block designs, respectively

Measures for evaluating the lotted of blocking in response surface designs

The conditions under which experimental trials are performed in a response surface design may not, in general, be homogeneous. In this case, blocking may be usually carried out for within-block homogeneity. When fitting a response surface model, the estimates of the mean response and its prediction variance generally depend on the way a given response surface design is blocked. That is, the choice of a blocking arrangement for a response surface design can have a considerable effect on estimating the mean response and on the size of the prediction variance. Therefore, care should be exercised in the selection of blocks. In this paper, we propose measures for evaluating the effect of blocking in response surface designs in terms of prediction variance when block effects are fixed. These measures can be used as useful tools to examine how blocking influences the prediction variance, and to compare the effect of blocking on the prediction variance in the cases of the orthogonal and non-orthogonal block designs under the given experimental region, respectively.

Effect of ondansetron, a 5-HT3 receptor antagonist, on the dynamic association between bulimic behaviors and pain thresholds

Thresholds for detection of both pressure and thermal pain are elevated in patients with bulimia nervosa. The present study was aimed at determining (1) if pressure pain detection thresholds (PDT) varied dynamically with the primary disease symptoms of binge eating and vomiting and (2) if the elevation in PDT was effected by treatment with ondansetron (ONDAN), a 5-HT3 receptor antagonist. PDT was defined as the mean of the minimal amount of pressure (measured in g) perceived as painful when exerted by a 1 mm 2 blunted point onto the center of the ventral surface of the ungual phalanx of digits 2-5 of the non-dominant hand. Fourteen female patients with severe bulimia nervosa (currently >seven binge/vomit episodes per week;>2 years illness duration) served as participants. PDT were evaluated at weekly intervals during the course of ongoing treatment studies (double-blind and `open' label) investigating the therapeutic effects of ONDAN. Data were analyzed by random regression analyses, allowing for the repeated-measures and non-orthogonal design. Data collected from 14 patients under the no-drug condition indicated that PDT increased over the interval between binge/vomit episodes, with significant elevations occurring at times when patients had naturally exceeded their average inter-binge interval. Eleven of these 14 patients underwent 4 weeks of ONDAN treatment. Under this drug condition, the time since the last binge/vomit episode was no longer a significant predictor of PDT. These patients also experienced a significant reduction in the frequency of bulimic behaviors, a finding reported in detail elsewhere. The above finding from untreated patients support the involvement of a common underlying mechanism driving both the increase in pain detection thresholds and the occurrence of the next bulimic episode. This possibility is further supported by the findings that ONDAN treatment is associated with a significant moderation of both variables. The effect of ONDAN may be mediated by blockade of afferent vagal neurotransmission, although other mechanisms must be considered.

Robust nonorthogonal analyses revisited: An update based on trimmed means

Three approaches to the analysis of main and interaction effect hypotheses in nonorthogonal designs were compared in a 2×2 design for data that was neither normal in form nor equal in variance. The approaches involved either least squares or robust estimators of central tendency and variability and/or a test statistic that either pools or does not pool sources of variance. Specifically, we compared the ANOVA F test which used trimmed means and Winsorized variances, the Welch-James test with the usual least squares estimators for central tendency and variability and the Welch-James test using trimmed means and Winsorized variances. As hypothesized, we found that the latter approach provided excellent Type I error control, whereas the former two did not.

Discussion of the Papers by Atkinson, and Bates et al.

Discussion of the Papers by Atkinson, and Bates <i>et al.</i>

Some Design Problems in Crop Experimentation. III. Non-Orthogonality

SUMMARYIdeally each block of an experiment should be made up in the same way with respect to treatments, that is, the design should be ‘orthogonal’. In practice that can be difficult to achieve, especially if the blocks have been chosen to fit the fertility pattern of the field. Sometimes it is impossible, in which case each block will have to contain its own selection of treatments. A number of simple and useful possibilities exist.Whatever non-orthogonal design is chosen some of the contrasts of interest (perhaps all of them) will be evaluated less efficiently, but that can be compensated by the smaller error mean-square given by a better blocking system. Also, where blocks do differ in their content, comparing their means will provide additional information about treatment effects. Sometimes the information may be worth the trouble of recovery.Special attention is given in this paper to total balance (including balanced incomplete block designs), supplemented balance, square and rectangular lattices and alpha-designs. The reinforcement of a design is explained and the advantages considered.Problemas de diseño en la experimentación con cultivos. III

Efficient Experimental Design with Marketing Research Applications

The authors suggest the use of D-efficient experimental designs for conjoint and discrete-choice studies, discussing orthogonal arrays, nonorthogonal designs, relative efficiency, and nonorthogonal design algorithms. They construct designs for a choice study with asymmetry and interactions and for a conjoint study with blocks and aggregate interactions.

Spatial Interpolation for Photographic Reproduction of Color

This case study concerns reproduction of colors using a standard light source, color filters, and color paper. Statistically, the problem involves experimental design and inverse modeling of a trivariate response surface as a function of three control variables. The method of data collection dictates the use of nonorthogonal designs. The solution uses a lattice-structured quasi-mixture design and two-stage inverse spatial interpolation, exploiting the data's lattice structure and providing useful contour plots. The method is applicable for similarly structured data. The data and scenario are useful for illustrating and motivating a variety of modeling methods.

Spatial Interpolation for Photographic Reproduction of Color

This case study concerns reproduction of colors using a standard light source, color filters, and color paper. Statistically, the problem involves experimental design and inverse modeling of a trivariate response surface as a function of three control variables. The method of data collection dictates the use of nonorthogonal designs. The solution uses a lattice-structured quasi-mixture design and two-stage inverse spatial interpolation, exploiting the data's lattice structure and providing useful contour plots. The method is applicable for similarly structured data. The data and scenario are useful for illustrating and motivating a variety of modeling methods.

An Approach to the Analysis of Nonorthogonal Three-Way Designs

A method of analysis of designs eliminating heterogeneity in two directions (three-way designs) is proposed for situations where the row versus column classification is nonorthogonal. The method appears to be simpler than the existing ones when the number of “missing” cells is small relative to the number of treatments and the total number of possible cells. A numerical example is given.

Shorter Communication: Taguchi's Idle Column Method

The idle column method, proposed by Taguchi, enables a two-level orthogonal array to be used as the basis of a mixed-level fractional factorial experiment. The resulting designs are not orthogonal but can in some circumstances be described as nearly orthogonal. The method must be applied with care, because an apparently minor change in the technique produces a different kind of design. We illustrate these points with a simple example. The same example is used to compare the “orthogonal” analysis of nonorthogonal designs, as advocated by Taguchi, with the least squares method. Finally, we examine some alternative designs and try to identify the circumstances in which the idle-column method is likely to be most useful.

Taguchi's Idle Column Method

The idle column method, proposed by Taguchi, enables a two-level array to be used as the basis of a mixed-level fractional factorial experiment. The resulting designs are not but can in some circumstances be described as nearly orthogonal. The method must be applied with care, because an apparently minor change in the technique produces a different kind of design. We illustrate these points with a simple example. The same example is used to compare the orthogonal analysis of nonorthogonal designs, as advocated by Taguchi, with the least squares method. Finally, we examine some alternative designs and try to identify the circumstances in which the idle-column method is likely to be most useful.

Complex linear models and cyclic designs

Many orthogonal factorial designs can be defined by abelian group morphisms. By juxtaposition of such designs, useful nonorthogonal designs can also be obtained, including the classical generalized cyclic designs, as well as a new kind of one replicate factorial block designs. Their efficiencies are easily computed by means of a complex reparametrization based on the irreducible characters of the groups involved. The theory extends to the “group generated” designs defined by Bailey and Rowley, in which the group is not necessarily abelian. In some cases, we give explicit formulas for the efficiencies of these latter designs.

The Non-Orthogonal Design of Experiments

The Non-Orthogonal Design of Experiments

THE ANALYSIS OF THE ADDITIVE MIXED MODEL FOR CLASSES OF NON ORTHOGONAL DESIGNS

Tests for fixed and random effects can be difficult to derive for nonorthogonal designs with mixed models. However, extensions of the intrablock and inter-block analyses of Balanced Incomplete Block Designs can often be obtained. Here we derive the extensions for the broad class of Group Divisible Designs. Decompositions of the design space are used to develop exact tests for fixed and random effects in the additive mixed model with random block effects. Conditions on the design which permit the standard use of the intra-block and inter-block test statistics are given. Important subclasses of Group Divisible Designs include Equireplicate Variance Balanced Block Designs and Group Divisible Partially Balanced Incomplete Block Designs with Two Associate Classes. These two subclasses are also examined. An example from the literature of an experiment on fruit trees is used to illustrate the methods.

Efficient designs for comparing test treatments with a control

SUMMARY The work of Pearce is studied in relation to the recent research on designs for control versus test comparisons. A condition for a nonorthogonal design to be most efficient is given, and a lower bound to the average variance of supplemented contrasts is obtained.

Non-orthogonal designs for measuring dispersion effects in sequential factor screening experiments using search linear models

“Dispersion” effects are considered in addition to “Location” effects of factors in the inferential procedure of sequential factor screening experiments with m factors each at two levels under search linear models. Search designs in measuring Dispersion and Location effects of factors are presented for both stage one and stage two of factor screening experiments with 4 ≤ m ≤ 10.

The Effect of Nonorthogonality on the Dependence ofFRatios Sharing a Common Denominator

Abstract The problem of testing two or more hypotheses about cell means in an analysis of experimental data using F ratios sharing a common denominator mean squared error has been considered by Ghosh (1955), Ramachandran (1956), Krishnaiah (1964), and Hurlburt and Spiegel (1976). Hurlburt and Spiegel considered the case in which the numerator sums of squares of two F statistics are independent and the denominator sums of squares identical. In this article we extend Hurlburt and Spiegel's results to allow dependence between the numerator sums of squares. This can occur when testing row and column effects in nonorthogonal designs, with the degree of dependence varying with the degree of nonorthogonality. The Type I error in testing differences in row effects conditioned on rejecting no difference in column effects is a function of the degrees of freedom and the extent to which the design is nonorthogonal. These relations can be partially explained in terms of the correlation coefficient between the numerato...

The analysis of non-orthogonal designs with many classifications

The least squares analysis of non-orthogonal designs with many classifications is considered. A unified simpler approach than the existing methods is derived and simple expressions for the various sums of squares are given. The paper also generalizes the canonical forms of Pearce Jeffers (1971) for the adjusted treatment sum of squares and the error sum of squares in block designs to designs with several non-orthogonal classifications.

Nonorthogonal Analysis of Variance using a Generalized Conjugate-Gradient Algorithm

Abstract A method is developed that computes an exact nonorthogonal analysis of variance using cell means. The method is iterative and does not require that the non-orthogonal design matrix be stored or formed. At each stage in the process, a balanced analysis of variance problem must be solved. A monotonicity property for the estimates of the regression sum of squares is derived that could be used to minimize iteration in hypothesis testing. An application of the algorithm to the solution of analysis of covariance problems is also given.