Daniel Voss

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.

Resolving the mixed models controversy

Abstract Two standard mixed models with interactions are discussed. When each is viewed in the context of superpopulation models, the mixed models controversy is resolved. The tests suggested by the expected mean squares under the constrained-parameters model are correct for testing the main effects and interactions under both the unconstrained-and constrained-parameters models.

Complete Block Designs

In the presence of controllable nuisance factors, the device of blocking divides the experimental material into homogeneous blocks in such a way that treatments can be compared under similar conditions. This chapter describes complete block designs (including randomized block designs), together with block design models, model assumption checks, multiple comparisons, sample size calculations, and analysis of variance. The analyses of complete block designs are illustrated through two real experiments, one having factorial treatment combinations. The use of R and SAS software is described.

Response Surface Methodology

Experiments for fitting a predictive model involving several continuous variables are known as response surface experiments. The objectives of response surface methodology include the determination of variable settings for which the mean response is optimized and the estimation of the response surface in the vicinity of this good location. The first part this chapter discusses first-order designs and first-order models, including lack of fit and the path of steepest ascent to locate the optimum. The second part of the chapter introduces second-order designs and models for exploring the vicinity of the optimum location. The application of response surface methodology is demonstrated through a real experiment. The concepts introduced in this chapter are illustrated through the use of SAS and R software.

Analysis of Orthogonal Saturated Designs

This chapter provides a review of special methods for analyzing data from screening experiments conducted using regular fractional factorial designs. The methods considered are robust to the presence of multiple nonzero effects. Of special interest are methods that try to adapt effectively to the unknown number of nonzero effects. Emphasis is on the development of adaptive methods of analysis of orthogonal saturated designs which rigorously control Type I error rates of tests or confidence levels of confidence intervals under standard linear model assumptions. The robust, adaptive method of Lenth (1989) is used to illustrate the basic problem. Then nonadaptive and adaptive robust methods of testing and confidence interval estimation known to control error rates are introduced and illustrated. While the focus is on Type I error rates and orthogonal saturated designs, Type II error rates, nonorthogonal designs and supersaturated designs are also discussed briefly.

Simultaneous confidence intervals in the analysis of orthogonal saturated designs

Abstract We present the first known method of constructing exact simultaneous confidence intervals for the analysis of orthogonal, saturated factorial designs. Given m independent, normally distributed, unbiased estimators of treatment contrasts, if there is an independent chi-squared estimator of error variance, then simultaneous confidence intervals based on the Studentized maximum modulus distribution are exact under all parameter configurations. In this paper, an analogous method is developed for the case of an orthogonal saturated design, for which the treatment contrasts are independently estimable but there is no independent estimator of error variance. Lacking an independent estimator of the error variance, the smallest sums of squares of effect estimators are pooled. The simultaneous confidence intervals are based on a probability inequality, for which the simultaneous confidence coefficient is achieved in the null case.

First-Order Deletion Designs and the Construction of Efficient Nearly Orthogonal Factorial Designs in Small Blocks

Abstract Single replicate factorial designs for incomplete block experiments are obtained by first constructing a single replicate preliminary design in incomplete blocks for the same number of factors but an excessive number of levels of the first factor, then deleting the excess treatment combinations to obtain a deletion design. Any single replicate preliminary design yields a single replicate deletion design. Furthermore, if the preliminary design is orthogonal, then the resulting deletion design is shown to be nearly orthogonal and, under certain reduced models, to provide efficient estimation of lower-order effects and in some cases an orthogonal analysis. For example, a 2×32 deletion design is constructed in three blocks of size 6, the 2 df confounded being the sum of interaction effects of F 2 and F 3 and second-order interaction effects. If second-order interactions are assumed negligible, then the deletion design provides efficient estimation of interactions between F 2 and F 3 and an orthogonal...

On generalizations of the classical method of confounding to asymmetric factorial experiments

A Kronecker product structure is identified for a particular class of asymmetric factorial designs in blocks, including the classes of designs generated by several of the generalizations of the classical method in the literature. The Kronecker product structure is utilized to establish orthogonal factorial structure for the class of designs and to identify a Principle of Generalized Interaction.

Exact confidence intervals in analysis of nonorthogonal saturated designs

SYNOPTIC ABSTRACT Let m×1 = (i) ~ Nm (θ, Σ) have an m-variate normal distribution, where Σ = A′Aσ2, A′A is a known, nondiagonal positive definite matrix, and σ is unknown. The objective is to construct an exact confidence interval for each effect θi, the ith component of θ. For a saturated design, there are no error degrees of freedom from which to compute an independent estimator of the error variance component σ2. However, under effect sparsity, the smaller effect estimates can be used to provide comparable information with which to construct confidence intervals for the effects. Voss (1999) provided a method for the construction of exact individual confidence intervals for each effect θi in the analysis of orthogonal saturated factorial experiments, for which the covariance matrix Σ is diagonal. We extend his results to the case of saturated designs, for which Σ is not diagonal. Such nonorthogonality can be planned, in order to keep the number of observations small, or it may be the unplanned consequence of lost observations. In the latter case, A is a random matrix, so our results are conditional.

Inferences for Contrasts and Treatment Means

There are many types of comparisons among treatments that can be undertaken in the analysis of an experiment. Contrasts for pairwise comparisons, treatment-versus-control comparisons, trends, and difference of averages are introduced and examined in detail in this chapter. Confidence intervals and hypothesis testing for these contrasts are developed for the one-way analysis of variance model. The necessity for a multiplicity adjustment when examining more than one contrast is explained, and the Bonferroni, Scheffe, Tukey, and Dunnett methods of multiple comparisons are described. The chapter provides a detailed discussion of the calculation of sample sizes using the width of confidence intervals. The concepts introduced in this chapter are illustrated through a real experiment and with the use of SAS and R software.