Robust row-column designs for complete diallel cross experiments

This paper investigates the robustness of variance-balanced row-column designs for complete diallel cross experiments for estimating the comparisons among the general combining ability parameters against the loss of observations. A necessary and sufficient condition of robustness as per connectedness criterion is obtained. The robustness of optimal row-column designs of Gupta and Choi (1998) has been investigated for the loss of any m(≥1) observations in a column and for the loss of any two observations in the design. The study of robustness has also been conducted as per A-efficiency criterion.     

Choice of optimal initial designs in sequential experiments

Combined-optimal designs (Li and Lin, 2003) are obviously the best choices for the initial designs if we partition the experiment into two parts with equal size to obtain some information about the process, especially for the case not considering the blocking factor. In this paper, the definition of combined-optimal design is extended to the case when blocking factor is significant, and this new class of designs is called blocked combined-optimal designs. Some general results are obtained which relate 2^k−p_III initial designs with their complementary designs when , where n=2^k−p. By applying these results, we are able to characterize 2^k−p_III combined-optimal designs or blocked combined-optimal designs in terms of their complementary designs. It is also proved that both 2^k−p_III combined-optimal and blocked combined-optimal designs are not minimum aberration designs when and n−1−k > 2. And some combined-optimal and blocked combined-optimal designs with 16 and 32 runs are constructed for illustration. 2000 Mathematics Subject Classifications: 62K15, 62K05     

Minimax robust designs for field experiments

Experimental designs for field experiments are useful in planning agricultural experiments, environmental studies, etc. Optimal designs depend on the spatial correlation structures of field plots. Without knowing the correlation structures exactly in practice, we can study robust designs. Various neighborhoods of covariance matrices are introduced and discussed. Minimax robust design criteria are proposed, and useful results are derived. The generalized least squares estimator is often more efficient than the least squares estimator if the spatial correlation structure belongs to a small neighborhood of a covariance matrix. Examples are given to compare robust designs with optimal designs. The work was partially supported by research grants from the Natural Science and Engineering Research Council of Canada.     

Optimal designs for a mixed interference model

This paper generalizes Kunert and Martin’s (Ann Stat 28:1728–1742, 2000) method for finding optimal designs under a fixed interference model, to find optimal designs under a mixed interference model. The results are based on the properties of information matrices in fixed and mixed models given in Markiewicz (J Stat Plan Inference 59:127–137, 1997). The method is applied to find a design which is optimal for any given variances of random neighbor effects. Research partially supported by the KBN Grant Number 5 P03A 041 21.     

Optimal extended complete block designs for dependent observations

Optimal designs under general dependence structures are usually difficult to specify theoretically or find algorithmically. However, they can sometimes be found for a specific dependence structure and a particular parameter value. In this paper, a class of generalized binary block designs with t treatments and b blocks of size k>t is considered. Each block consists of h consecutive complete blocks and, at the end, an incomplete block of size k−ht (if k > ht). For a suitable number of blocks, a universally optimal design is found for a first-order stationary autoregressive process with positive correlations. Optimal generalized binary designs and balanced block designs are also considered. Some constructions for a universally optimal design are described. A negative dependence parameter, and some other dependence structures, are also considered.     

Optimal nested row-column designs for blocks of size 2×2 under dependence

This paper examines nested 2 ×2 row-column designs when within-block observations are assumed to be dependent. The model considered has fixed block effects, which may also include row and/or column effects. Optimal binary and non-binary designs, constructed from semi-balanced arrays, are given under both generalised and ordinary least squares estimation. It is shown that binary designs are optimal when dependence is low. In general, however, the optimal designs are highly specific to the correlation values. Received: October 1999     

Optimality and efficiency of small chessboard designs for correlated errors

Optimal p × q row–column designs are obtained via complete enumeration of all possible designs for two treatments in some fixed effects models with errors specified by a doubly geometric covariance structure. This is done, in part, by a computer search, for a finite set of sizes of the correlation coefficients and in cases where p and q are small enough to make such a search feasible.     

Optimality of balanced designs for minimum norm quadratic unbiased estimation of variance components

This paper develops an approximate theory for the optimality of balanced designs under minimum norm quadratic unbiased estimation of variance components in one-way classified data.     

Estimation and optimal designs for linear Haar-wavelet models

This paper gives an analytical expression for the best linear unbiased estimator (BLUE) of the unknown parameters in the linear Haar-wavelet model. From the analytical expression, we solve for the eigenvalues of the covariance matrix of the BLUE in analytical form. Further, we use these eigenvalues to construct some conventional discrete optimal designs for the model. The equivalences among these optimal designs are demonstrated and some examples are also given.      

Totally positive regression: E-optimal designs

E-optimality of approximate designs in linear regression models is paired with a dual problem of nonlinear Chebyshev approximation. When the regression functions form a totally positive system, then the information matrices of designs for subparameters turn out to be “almost” totally positive, a property which allows to solve the nonlinear Chebyshev problem. Thereby we obtain explicit formulae for E-optimal designs in terms of equi-oscillating generalized polynomials. The considerations unify and generalize known results on E-optimality for particular regression setups.     

Optimal design of nonlinear experiments with parameter constraints

The nonlinear regression model with N observations y _i=η(x _i,θ) +ε_i, and with the parameter θ subject to q nonlinear constraints C _j (θ)=0; j=1, …,q, is considered. As an example, the spline regression with unknown nodes is taken. Expressions for the variances (variance matrices) of the LSE are discussed. Because of the complexity of these expressions, and the singularity of the variance matrix of the LSE for θ, the optimality criteria and their properties, in particular the convexity and the equivalence theorem are considered from different aspects. Also the possibility of restriction to designs with limited values of measures of nonlinearity is mentioned. Research supported by the VEGA-grant of the Slovak grant agency No. 1/7295/20.     

Eight-run two-level factorial designs under dependence

Dependent observations commonly arise in factorial experiments. Apart from main-effects two-level designs formed by the Cheng & Steinberg reverse foldover algorithm, which are known to be very efficient designs under dependence using the D-criterion, little is known about other designs, models and criteria, and the range of possible behaviour. In this paper, we investigate in detail 8-run two-level designs. Received February 1998     

Integral stochastic optimal design criteria in linear models

Within the framework of classical linear regression model integral optimal design criteria of stochastic nature are considered and their properties are established. Their limit behaviour generalizes that of the distance stochastic optimality criterion. As an example a line fit model is taken. Acknowledgement. I would like to thank the referees for their constructive comments which improved the paper.     

Two ge neral classes of search designs for factor screening experiments with factors at three levels

In this paper, we are presenting general classes of factor screening designs for identifying a few important factors from a list of m (≥ 3) factors each at three levels. A design is a subset of 3^m possible runs. The problem of finding designs with small number of runs is considered here. A main effect plan requires at least (2m + 1) runs for estimating the general mean, linear and quadratic effects of m factors. An orthogonal main effect plan requires, in addition, the number of runs as a multiple of 9. For example, when m=5, a main effect plan requires at least 11 runs and an orthogonal main effect plan requires 18 runs. Two general factor screening designs presented here are nonorthogonal designs with (2m− 1) runs. These designs, called search designs permit us to search for and identify at most two important factors out of m factors under the search linear model introduced in Srivastava (1975). For example, when m=5, the two new plans given in this paper have 9 runs, which is a significant improvement over an orthogonal main effect plan with 18 runs in terms of the number of runs and an improvement over a main effect plan with at least 11 runs. We compare these designs, for 4≤m≤ 10, using arithmetic and geometric means of the determinants, traces, and maximum characteristic roots of certain matrices. Two designs D1 and D2 are identical for m=3 and this design is an optimal design in the class of all search designs under the six criteria discussed above. Designs D1 and D2 are also identical for m=4 under some row and column permutations. Consequently, D1 and D2 are equally good for searching and identifying one important factor out of m factors when m=4. The design D1 is marginally better than the design D2 for searching and identifying one important factor out of m factors when m=5, … , 10. The design D1 is marginally better than the D2 for searching and identifying two important factors out of m factors when m=5, 7, 9. The design D2 is somewhat better than the design D1 for m=6, 8. For m=10, D1 is marginally better than D2 w.r.t. the geometric mean and D2 is marginally better than D1 w.r.t. the arithmetic mean of the maximum characteristic roots.     

Optimal designs under two related models for interference

We consider two related models for interference: one having different directional neighbour effects and one with the same neighbour effects. We show that optimal designs for one model can be obtained from optimal designs for the other model. Acknowledgement. We are grateful to H. Kushner for advance notice of his work. SP was supported by the Isfahan University of Technology, Iran.     

On the role of labor supply for the optimal size of Social Security

The paper studies the welfare effects of a Social Security system in a stylized overlapping generations economy with random production and capital accumulation. Different welfare concepts including long run optimality, social optimality, and time consistency are employed to determine the optimal size of the system. When labor supply is exogenous, a unique contribution level can be identified which is optimal according to all three concepts. When labor supply is endogenous, however, this result generically fails to hold and the long-run optimal solution is only constrained socially optimal while the time-consistent policy may even lead to an inefficient equilibrium.     

D-optimal weighing designs for six objects

For all integers m≥6, we determine the maximum value of det X ^T X, where X is an m×6 (0, 1)-matrix, and exhibit (D-optimal) matrices X for which the maximum occurs. For D-optimal matrices X, the uniqueness of the Gram matrix X ^T X is discussed. Received: May 2000     

路面合成坡度与超高渐变率的选用

根据《公路路线设计规范》(JTGD20-2006)和工程设计实践,分析了超高过渡处路面合成坡度对超高渐变率的选用影响,并提出检验方法,供设计参考。     

Multiple aspect testing for case-control designs

When testing for the equality of two distributions in a case-control design with treatment effects presumed to act possibly on more than one aspect, different tests may be properly considered for testing for different features of a null hypothesis, leading to the multiple aspect testing issue. Two different aspects are therefore of interest: the location-aspect, based on the comparison of location indexes, and the distributional-aspect, based on the comparison of the empirical distribution functions. A simulation study shows that the combined testing procedure exhibits a good robust overall performance, and an application in biomedical research is also presented.     

Constructing optimal randomized response designs with consideration for the level of privacy protection

This paper considers the problem of procuring truthful responses to estimate the proportion of qualitative characteristics. In order to protect the respondent's privacy, various techniques of generating randomized response rather than direct response are available in the literature. But the theory concerning them is generally developed with no attention to the required level of privacy protection. Illustrating two randomization devices, we show how optimal randomized response designs may be achieved. The optimal designs of forced response procedure as well as BHARGAVA and SINGH [Statistica (2000) Vol. 6, pp 315–321] procedure are shown to be special cases. In addition, the equivalent designs of optimal WARNER [Journal of the American Statistical Association (1965) Vol. 60, pp. 63–69] procedure are considered as well. It is also shown that stratification with proportional allocation will be helpful for improving the estimation efficiency.