General formulae for expectations, variances and covariances of the mean squares for staggered nested designs

Staggered nested experimental designs are the most popular class of unbalanced nested designs. Using a special notation which covers the particular structure of the staggered nested design, this paper systematically derives the canonical form for the arbitrary m-factors. Under the normality assumption for every random variable, a vector comprising m canonical variables from each experimental unit is normally independently and identically distributed. Every sum of squares used in the analysis of variance (ANOVA) can be expressed as the sum of squares of the corresponding canonical variables. Hence, general formulae for the expectations, variances and covariances of the mean squares are directly obtained from the canonical form. Applying the formulae, the explicit forms of the ANOVA estimators of the variance components and unbiased estimators of the ratios of the variance components are introduced in this paper. The formulae are easily applied to obtain the variances and covariances of any linear combinations of the mean squares, especially the ANOVA estimators of the variance components. These results are eff ectively applied for the standardization of measurement methods.     

Missing data in linear models with correlated errors

One common method for analyzing data in experimental designs when observations are missing was devised by Yates (1933), who developed his procedure based upon a suggestion by R. A. Fisher. Considering a linear model with independent, equi-variate errors, Yates substituted algebraic values for the missing data and then minimized the error sum of squares with respect to both the unknown parameters and the algebraic values. Yates showed that this procedure yielded the correct error sum of squares and a positively biased hypothesis sum of squares.

Others have elaborated on this technique. Chakrabarti (1962) gave a formal proof of Fisher's rule that produced a way to simplify the calculations of the auxiliary values to be used in place of the missing observations. Kshirsagar (1971) proved that the hypothesis sum of squares based on these values was biased, and developed an easy way to compute that bias. Sclove     

Central composite designs for estimating the optimum conditions for a second-order model

Central composite designs which maximize both the precision and the accuracy of estimates of the extremal point of a second-order response surface for fixed values of the model parameters are constructed. Two optimality criteria are developed, the one relating to precision and based on the sum of the first-order approximations to the asymptotic variances and the other to accuracy and based on the sum of squares of the second-order approximations to the asymptotic biases of the estimates of the coordinates of the extremal point. Exact and continuous central composite designs are introduced and in particular designs which place no restriction on the pattern of the weights, termed benchmark designs, and designs which comprise equally weighted factorial and equally weighted axial points, termed axial-factorial designs, are explored. Algebraic results proved somewhat elusive and the requisite designs are obtained by a mix of algebra and numeric calculation or simply numerically. An illustrative example is presented and some interesting features which emerge from that example are discussed.     

Distribution of the biased hypothesis sum of squares in linear models with missing observations

In a linear model with missing observations, one can substitute algebraic quantities and then minimize the error sum of squares for the augmented model. This gives the correct error sum of squares. But this method does not produce the correct hypothesis sum of squares for testing a linear hypothesis about the parameters. The sum of squares obtained is biased but practitioners still use it. The distribution of this biased sum of squares is derived in this paper and the consequences of using this biased sum of squares on the type I and II errors is examined.     

Generalized staggered nested designs for variance components estimation

Staggered nested experimental designs are the most popular class of unbalanced nested designs in practical fields. The most important features of the staggered nested design are that it has a very simple open-ended structure and each sum of squares in the analysis of variance has almost the same degrees of freedom. Based on the features, a class of unbalanced nested designs that is a generalization of the staggered nested design is proposed in this paper. Formulae for the estimation of variance components and their sums are provided. Comparing the variances of the estimators to the staggered nested designs, it is found that some of the generalized staggered nested designs are more efficient than the traditional staggered nested design in estimating some of the variance components and their sums. An example is provided for illustration.     

Minimum aberration and model robustness for two-level fractional factorial designs

The performance of minimum aberration two-level fractional factorial designs is studied under two criteria of model robustness. Simple sufficient conditions for a design to dominate another design with respect to each of these two criteria are derived. It is also shown that a minimum aberration design of resolution III or higher maximizes the number of two-factor interactions which are not aliases of main effects and, subject to that condition, minimizes the sum of squares of the sizes of alias sets of two-factor interactions. This roughly says that minimum aberration designs tend to make the sizes of the alias sets very uniform. It follows that minimum aberration is a good surrogate for the two criteria of model robustness that are studied here. Examples are given to show that minimum aberration designs are indeed highly efficient.     

Complete diallel crosses plans through balanced incomplete block designs

The present investigation involves the methods of construction of complete diallel cross plans using balanced incomplete block (BIB) designs. Furthermore, the analysis of complete diallel crosses plans are carried out to estimate the general combining ability of the ith line (i=1, r 2, r …, r v) where the intra- block analysis of the adjusted sum of squares for GCA and the unadjusted block sum of squares are also obtained, thereafter the relationship between the estimates of BIB design and the estimates of the GCA effect of CDC plan has been established. Moreover, it has also been shown that the complete diallel crosses design obtained through two BIB designs satisfying v1=b1= 4 5 1+3=v2=b2, r r1=2 5 1+1=r2=k1=k2 and 5 1= 5 2 are universally optimum. These results are further supported by a suitable example of each. However, the need of this study is to show that the analysis of the CDC plan is reducible to the analysis of generating the BIB design.     

Robust designs for nearly linear regression

In this paper we seek designs and estimators which are optimal in some sense for multivariate linear regression on cubes and simplexes when the true regression function is unknown. More precisely, we assume that the unknown true regression function is the sum of a linear part plus some contamination orthogonal to the set of all linear functions in the L₂ norm with respect to Lebesgue measure. The contamination is assumed bounded in absolute value and it is shown that the usual designs for multivariate linear regression on cubes and simplices and the usual least squares estimators minimize the supremum over all possible contaminations of the expected mean square error. Additional results for extrapolation and interpolation, among other things, are discussed. For suitable loss functions optimal designs are found to have support on the extreme points of our design space.     

Properties of a randomization test for multifactor comparisons of groups

Multivariate hypothesis testing in studies of vegetation is likely to be hindered by unrealistic assumptions when based on conventional statistical methods. This can be overcome by randomization tests. In this paper, the accuracy and power of a MANOVA randomization test are evaluated for one and two factors with interaction with simulated data from three distributions. The randomization test is based on the partitioning of sum of squares computed from Euclidean distances. In one-factor designs, sample size and variance inequality were evaluated. The results showed a high level of accuracy. The power curve was higher with normal distribution, lower with uniform, intermediate with lognormal and was sensitive to variance inequality. In two-factor designs, three methods of permutations and two statistics were compared. The results showed that permutation of the residuals with F _pseudo is accurate and can give good power for testing the interaction and restricted permutation for testing main factors.     

Properties of Designs in Experiments on Spatial Lattices

ABSTRACT

When spatial variation is present in experiments, it is clearly sensible to use designs with favorable properties under both generalized and ordinary least squares. This will make the statistical analysis more robust to misspecification of the spatial model than would be the case if designs were based solely on generalized least squares. In this article, treatment information is introduced as a way of studying the ordinary least squares properties of designs. The treatment information is separated into orthogonal frequency or polynomial components which are assumed to be independent under the spatial model. The well-known trend-resistant designs are those with no treatment information at the very low order frequency or polynomial components which tend to have the higher variances under the spatial model. Ideally, designs would be chosen with all the treatment information distributed at the higher-order components. However, the results in this article show that there are limits on how much trend resistance can be achieved as there are many constraints on the treatment information. In addition, appropriately chosen Williams squares designs are shown to have favorable properties under both ordinary and generalized least squares. At all times, the ordinary least squares properties of the designs are balanced against the generalized least squares objectives of optimizing neighbor balance.     

Percentage points and power calculations for outlier tests in a regression situation

The usefulness of an extra sum of squares statistics Q_K for detecting K outliers has been discussed previously in the context of two-way tables. (See Gentleman and Wilk, 1975a, 1975b; John and Draper 1978; and Draper and John, 1980,) That work is extended here to straight line regression situations arising from, and motivated by, a specific set of research data. Percentage points for the appropriate test statistics are obtained by simulation, and approximations for these percentage points are suggested. Power calculations made for various designs and outlier situations are briefly summarized.     

Balanced incomplete Latin square designs

Latin squares have been widely used to design an experiment where the blocking factors and treatment factors have the same number of levels. For some experiments, the size of blocks may be less than the number of treatments. Since not all the treatments can be compared within each block, a new class of designs called balanced incomplete Latin squares (BILS) is proposed. A general method for constructing BILS is proposed by an intelligent selection of certain cells from a complete Latin square via orthogonal Latin squares. The optimality of the proposed BILS designs is investigated. It is shown that the proposed transversal BILS designs are asymptotically optimal for all the row, column and treatment effects. The relative efficiencies of a delete-one-transversal BILS design with respect to the optimal designs for both cases are also derived; it is shown to be close to 100%, as the order becomes large.     

Functional analysis of variance for Hilbert-valued multivariate fixed effect models

This paper presents new results on functional analysis of variance for fixed effect models with correlated Hilbert-valued Gaussian error components. The geometry of the reproducing kernel Hilbert space of the error term is considered in the computation of the total sum of squares, the residual sum of squares, and the sum of squares due to the regression. Under suitable linear transformation of the correlated functional data, the distributional characteristics of these statistics, their moment generating and characteristic functions, are derived. Fixed effect linear hypothesis testing is finally formulated in the Hilbert-valued multivariate Gaussian context considered.     

Blocking in two-level non-regular fractional factorial designs

In this paper we consider screening experiments where a two-level fractional factorial design is to be used to identify significant factors in an experimental process and where the runs in the experiment are to occur in blocks of equal size. A simple method based on the foldover technique is given for constructing resolution IV orthogonal and non-orthogonal blocked designs and examples are given to illustrate the process.     

An extended sweep operator for the cross validation of variable selection in linear regression

In its application to variable selection in the linear model, cross-validation is traditionally applied to an individual model contained in a set of potential models. Each model in the set is cross-validated independently of the rest and the model with the smallest cross-validated sum of squares is selected. In such settings, an efficient algorithm for cross-validation must be able to add and to delete single points quickly from a mixed model. Recent work in variable selection has applied cross-validation to an entire process of variable selection, such as Backward Elimination or Stepwise regression (Thall, Simon and Grier, 1992). The cross-validated version of Backward Elimination, for example, divides the data into an estimation and validation set and performs a complete Backward Elimination on the estimation set, while computing the cross-validated sum of squares at each step with the validation set. After doing this process once, a different validation set is selected and the process is repeated. The final model selection is based on the cross-validated sum of squares for all Backward Eliminations. An optimal algorithm for this application of cross-validation need not be efficient in adding and deleting observations from a single model but must be efficient in computing the cross-validation sum of squares from a series of models using a common validation set. This paper explores such an algorithm based on the sweep operator.     

Random balanced resampling: A new method for estimating variance components in unbalanced designs

Variance components in factorial designs with balanced data are commonly estimated by equating mean squares to expected mean squares. For unbalanced data, the usual extensions of this approach are the Henderson methods, which require formulas that are rather involved. Alternatively, maximum likelihood estimation based on normality has been proposed. Although the algorithm for maximum likelihood is computationally complex, programs exist in some statistical packages. This article introduces a simpler method, that of creating a balanced data set by resampling from the original one. Revised formulas for expected mean squares are presented for the two-way case; they are easily generalized to larger factorial designs. The results of a number of simulation studies indicate that, in certain types of designs, the proposed method has performance advantages over both the Henderson Method I and maximum likelihood estimators.     

Mean squares in saturated fractional designs revisited

In a previous paper (Bissell, 1989) some suggestions were offered for interpreting mean squares in saturated fractional designs where no independent estimate of experimental error is available. One of the methods leads to a simple numerical test of homogeneity which provides an objective accompaniment to half-Normal plotting of effects (Daniel, 1959) in 2n designs or exponential plotting of mean squares (Bissell, 1989) in 3n designs. A table of percentage points for a convenient test statistic is provided in this paper.     

On balanced incomplete Latin square designs

Recently, balanced incomplete Latin square designs are introduced in the literature. We propose three methods of constructions of balanced incomplete Latin square designs. Particular classes of Latin squares namely Knut Vik designs, semi Knut Vik designs, and crisscross Latin squares play a key role in the construction.     

On sum composition of fractional factorial designs

Combinatorial extension and composition methods have been extensively used in the construction of block designs. One of the composition methods, namely the direct product or Kronecker product method was utilized by Chakravarti [1956] to produce certain types of fractional factorial designs. The present paper shows how the direct sum operation can be utilized in obtaining from initial fractional factorial designs for two separate symmetrical factorials a fractional factorial design for the corresponding asymmetrical factorial. Specifically, we provide some results which are useful in the construction of non-singular fractional factorial designs via the direct sum composition method. In addition a modified direct sum method is discussed and the consequences of imposing orthogonality are explored.     

Generalized and oxdinary least squakes with estimated and unequal variances

An assumption which is often violated in the application of experimental designs is equality of variances. There are several methods available for estimating the unequal variances. This paper covers incorporating different estimators of the variances with the ordinary least squares and generalized least squares. A Monte Carlo study provides more insight into the behavior of these procedures. For some small sample sizes, the incorporations with the ordinary least squares perform satisfactorily, but with the generalized least squares they do not.