Using the chinese remainder theorem in constructing confounded designs for mixed factorial experiments

A procedure for constructing confounded designs for mixed factorial experiments derived from the Chinese Remainder Theorem is presented. The present procedure as well as others, all through use of modular arithmetic, are compared.     

Confounding in mixed factorial experiments through isomorphisms

The derivation of a simpie mexhoa for confounding in mixed factorial experiments from an isomorphism of finite abelian groups is presented. The theoretical bases of confounding procedures that use modular arithmetic for such experiments are compared.     

AN ALGORITHM FOR CHOOSING THE SAMPLING TIMES FOR AN EXPERIMENT ON A CONTINUOUS PROCESS

Saunders & Eccleston (1992) and Saunders, Eccleston & Spessa (1992) developed an approach to the design of factorial experiments on continuous processes that allows for the correlation present in such processes. Their methods concentrated on identifying the order of application of treatments in such experiments, assuming that the spacing between experiments is constant. On a continuous process, there is no necessity to maintain equally spaced sampling times. This paper gives an algorithm for choosing the optimal sampling times for a factorial experiment aimed at estimating a particular parameter or set of parameters. It is shown that in practical situations the optimal sampling times give considerable improvements in the accuracy of the parameter estimates.     

Sample size determination for 2k-r experiments with a binomial response

This paper provides closed form expressions for the sample size for two-level factorial experiments when the response is the number of defectives. The sample sizes are obtained by approximating the two-sided test for no effect through tests for the mean of a normal distribution, and borrowing the classical sample size solution for that problem. The proposals are appraised relative to the exact sample sizes computed numerically, without appealing to any approximation to the binomial distribution, and the use of the sample size tables provided is illustrated through an example.     

Performance of the weighted least squares apprdach to categorical data analysis

The weighted least squares method of Grizzle, Starmer and Koch is a common and convenient way to analyze categorical data. Tests of significance based on this method are typically performed by calculating a quadratic form which is asymptotically distrib-uted as a chi-square random variable under an appropriate null hy- pothesis.

This paper explores the validity of this distributional approximation for small samples in terms of the accuracy and power associated with this chi-square statistic. We report simulation results from the weighted least squares method applied to balanced factorial experiments, The simulations which constitute the majorpart of this work also provide insight into the relative performance of several response functions.     

Orthogonal three-level parallel flats designs for user-specified resolution

Orthogonal factorial and fractional factorial designs are very popular in many experimental studies, particularly the two-level and three-level designs used in screening experiments. When an experimenter is able to specify the set of possibly nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design belonging to the class of parallel flats designs, that has a smaller run-size than a suitable design from the class of classical fractional factorial designs belonging to the class of single flat designs. Sri-vastava and Li (1996) proved a fundamental theorem of orthogonal s-level, s being a prime, designs of parallel flats type for the user-specified resolution. They also tabulated a series of orthogonal designs for the two-level case. No orthogonal designs for three-level case are available in their paper. In this paper, we present a simple proof for the theorem given in Srivastava and Li (1996) for the three-level case. We also give a dual form of the theorem, which is more useful for developing an algorithm for construction of orthogonal designs. Some classes of three-level orthogonal designs with practical run-size are given in the paper.     

Subset selection for normal means in multi-factor experiments

The procedure of Gupta [1956], [1965] for selecting a random sized subset of k ≧ 2 normal populations which contains the population with the largest population mean when the populations have a common variance is generalized to multi-factor experiments. Two-factor experiments with equal replication on each factor-level combination are discussed in detail. The cases of zero and non-zero interactions between factor levels are considered. For the two-factor, zero interaction case with a common number of observations at each factor-level combination, a table of constants necessary to implement the procedure is provided for experiments having selected levels per factor; the constants are equi-coordinate upper percentage points of a multivariate Student t distribution.     

Generalized confounded row–column designs

Generalized Confounded Row–Column (GCRC) designs for factorial experiments have been introduced and methods of constructing GCRC designs have been discussed. Fractionally replicated GCRC designs have also been constructed. The designs obtained ensure balancing with respect to estimable effects.     

Efficiency of bailey's trigonometric levels in factorial experiments

Bailey has shown that choice of certain trigonometlk levels for factors in a symmetrical confounded factorial design is more efficient for quantitative treatments. This paper introduces certain incidence matrices associated with the flats of different pencils of such designs to obtain an explicit expression for the efficiency and also gives a simpler derivation of Bailey's results.     

Generalized modulus-ratio tests for analysis of factorial designs with zero degrees of freedom for error

A method of statistical analysis of single replicate and fractional factorial designs requiring no estimate of error variance is given. By comparison of the relative magnitudes of independent effect .estimates, effects corresponding to relatively large effect estimates may be asserted to be nonzero. The procedure maintains a prespecified experimentwise error rate for a general class of modulus-ratio statistics.     

Characterization of a class cf discrete distributions by properties of their moment distributions

The concept of a moment distribution is here applied to the class of factorial series distributions introduced by the author (197U). Moment distributions turn out to have a simple interpretation in this case and often take on a simple form. A few examples are given to illustrate this fact     

Optimal designs for asymmetrical factorial paired comparison experiments

Three forms of a general null hypothesis Ho on the factorial parameters of a general asymmetrical factorial paired comparison experiment are considered. A class of partially balanced designscorresponding to each form of H₀ is constructed and the A,D and ioptimal design, minimizing the trace, determinant and largest eigenvalue of a defined covariance matrix of related maximumlikelihoodestimators, in that class is determined. Moreover, the optimal design in each class maximizes the noncentrality parameter λ² of the asymptotic noncentral chi-square distribution of the likelihood ratiostatistic -2 log λ for testing H_o under defined local alternatives. These results apply directly to symmetrical factorial paired comparison experiments as special casesExamples are given forillustrating applications of the developed results     

Three Parallel Flats Designs for Two-level Factorial Experiments

This paper investigates the properties of the class of three parallel flats designs for two-level factorial experiments. It shows that the designs constructed from this class of designs can have a very simple correlation structure. The correlation of any pair of best linear unbiased estimators of factorial effects is 0, ⅓ or ¼. Furthermore, the designs obtained also have high D-efficiency. Finally, a class of designs is generated with run-size N = 12 to illustrate the use of the theorem.     

Design two-level factorial experiments in blocks of size four

Recently, many researchers have devoted themselves to the investigation on the number of replicates needed for experiments in blocks of size two. In practice, experiments in blocks of size four might be more useful than those in blocks of size two. To estimate the main effects and two-factor interactions from a two-level factorial experiment in blocks, we might need many replicates. This article investigates designs with the least number of replicates for factorial experiments in blocks of size four. The methods to obtain such designs are presented.     

Balanced Factorial Designs for cDNA Microarray Experiments

Balanced factorial designs are introduced for cDNA microarray experiments. Single replicate designs obtained using the classical method of confounding are shown to be particularly useful for deriving suitable balanced designs for cDNA microarrays. Classical factorial designs obtained using methods other than the method of confounding are also shown to be useful. The paper provides a systematic method of deriving designs for microarray experiments as opposed to algorithmic and ad-hoc methods and generalizes several of the microarray designs given recently in the literature.     

Incomplete row-column designs with factorial treatment structure for estimating main effects with full efficiency

In this paper, we propose two methods of constructing row-column designs for factorial experiments. The constructed designs have orthogonal factorial structure with balance and permits estimation of main effects with full efficiency.     

AN ALGORITHM FOR THE DESIGN OF 2? FACTORIAL EXPERIMENTS ON CONTINUOUS PROCESSES

Saunders & Eccleston (1992) presented an approach to the design of 2-level factorial experiments for continuous processes. It determined sets of contrasts between the observations that could be well estimated, and then selected a design so that those contrasts estimated the parameters of interest. This paper shows that a well-estimated contrast must have a large number of changes of sign or level, and also be ‘paired’ in a particular sense. It develops an algorithm which constructs designs that must have a large number of changes of sign, evenly spread among the contrasts and optimal or near optimal. When such designs exist they are often preferable to those produced by the reverse foldover algorithm of Cheng & Steinberg (1991).     

Clustering Effects in Unreplicated Factorial Experiments

The analysis of unreplicated factorial designs concentrates much attention since there are no degrees of freedom left to estimate the error variance. In this article, we propose clustering the factorial estimates in two groups, one containing the active effects and one containing the inactive effects. The powerfulness of the proposed method is revealed via a comparative simulation study.     

Three-factor symmetrical balanced designs with full efficiency for main effects

Some methods for constructing balanced design for 3-factor symmetrical factorial experiments in which all the main effects are completely unconfounded by using balanced arrays and BIB designs are proposed. The method is flexible in terms of selecting block size.