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.     

A note on the robustness of PBIBD(2)s against breakdown in the event of observation loss

Robustness against design breakdown following observation loss is investigated for Partially Balanced Incomplete Block Designs with two associate classes (PBIBD(2)s). New results are obtained which add to the body of knowledge on PBIBD(2)s. In particular, using an approach based on the E‐value of a design, all PBIBD(2)s with triangular and Latin square association schemes are established as having optimal block breakdown number. Furthermore, for group divisible designs not covered by existing results in the literature, a sufficient condition for optimal block breakdown number establishes that all members of some design sub‐classes have this property.     

A more powerful test for incomplete block designs

We studied the behavior of a statistic that tests for treatment effects in incomplete block designs, Simulation was used to estimate the first four moments of a test statistic that combines intra- and inter-block estimates of treatment contrasts. For a wide range of alternative hypotheses, the moments are remarkably close to those of an F distribution with one degree of freedom in the numerator, and the denominator degrees of freedom equal to that of the mean square error in the usual intra-block analysis, The noncentrality parameter depends upon the block and unit variances and the eigenvalues of the incidence matrix of the design, The power of the test statistic can be estimated from standard power function charts.     

Cubic lattices

In some experiments, the problem is to compare many unstructured treatments in small blocks, the classical example being the study of new plant varieties on variable land. A common method is to use lattice designs, i.e. block designs based upon rows and columns of a square format, with ficrther replicates being formed, required, from orthogonal squares applied to the format. It has been known for some time that cubes can be used instead; this paper sets out to explore the possibilities. There are two cases. In one case, the blocks are formed from the planes of the cube and, in the other case, from its lines. The cubic lattice basically has three replicates-one from each dimension-but, two or four replicates are required, a design can be found by omitting or duplicating one of the dimensions. Where standard treatments need to be introduced, -a useful device is to reinforce, i.e. supplement each block with additional plots of standards, with each block of a replicate being supplemented in the same way. These possibilities are examined. It emerges that cubic lattices with two or three replicates usefully extend the range of available designs, but that those with four replicates are disappointing. However, there is the alternative of using designs based upon Latin cubes. This matter is not taken far but it is shown that, where the Latin cube exists, it gives a better design. A quick way of calculating an approximate analysis of variance is given, which is applicable in a wide range of cases.     

Hyper-Graeco-Latin Squares and Fractional Factorial Designs

A Latin square of order s is an arrangement of the s letters in an s × s square so that every letter appears exactly once in every row and exactly once in every column. Copeland and Nelson (2000) used two examples to show that a Latin square can be chosen such that it corresponds to a fractional factorial design. In this article, we are going to study this topic more precisely. Furthermore, we will explore the relationship between fractional factorial designs and hyper-Graeco-Latin squares in general, where s is a prime or a power of a prime.     

Pairwise orthogonal F-rectangle designs

The concept of pairwise orthogonal Latin square design is applied to r row by c column experiment designs which are called pairwise orthogonal F-rectangle designs. These designs are useful in designing successive and/or simulataneous experiments on the same set of rc experimental units, in constructing codes, and in constructing orthogonal arrays. A pair of orthogonal F-rectangle designs exists for any set of v treatment (symbols), whereas no pair of orthogonal Latin square designs of order two and six exists; one of the two construction methods presented does not rely on any previous knowledge about the existence of a pair of orthogonal Latin square designs, whereas the second one does. It is shown how to extend the methods to r=pv row by c=qv column designs and how to obtain t pairwise orthogonal F-rectangle design. When the maximum possible number of pairwise orthogonal F-rectangle designs is attained the set is said to be complete. Complete sets are obtained for all v for which v is a prime power. The construction method makes use of the existence of a complete set of pairwise orthogonal Latin square designs and of an orthogonal array with vⁿ columns, (vⁿ−1)/(v−1) rows, v symbols, and of strength two.     

Near-Balanced Incomplete Block Designs,With an Application to Poster Competitions

Judging scholarly posters creates a challenge to assign the judges efficiently. If there are many posters and few reviews per judge, the commonly used balanced incomplete block design is not a feasible option. An additional challenge is an unknown number of judges before the event. We propose two connected near-balanced incomplete block designs that both satisfy the requirements of our setting: one that generates a connected assignment and balances the treatments and another one that further balances pairs of treatments. We describe both fixed and random effects models to estimate the population marginal means of the poster scores and rationalize the use of the random effects model. We evaluate the estimation accuracy and efficiency, especially the winning chance of the truly best posters, of the two designs in comparison with a random assignment via simulation studies. The two proposed designs both demonstrate accuracy and efficiency gain over the random assignment.     

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.     

Missing values in replicated Latin squares

Designs based on any number of replicated Latin squares are examined for their robustness against the loss of up to three observations randomly scattered throughout the design. The information matrix for the treatment effects is used to evaluate the average variances of the treatment differences for each design in terms of the number of missing values and the size of the design. The resulting average variances are used to assess the overall robustness of the designs. In general, there are 16 different situations for the case of three missing values when there are at least three Latin square replicates in the design. Algebraic expressions may be determined for all possible configurations, but here the best and worst cases are given in detail. Numerical illustrations are provided for the average variances, relative efficiencies, minimum and maximum variances and the frequency counts, showing the effects of the missing values for a range of design sizes and levels of replication.     

Experimental designs and their efficiencies for spatially correlated observations in two dimensions

Optimality of experimental designs for spatially correlated observations is investigated.come two dimensional correlation structures are discussed and an attempt has been made to find optimal or nearly optimal design for each sitution.The solution lend to designs similar to that used for repeated measurements.The relative efficiency of the proposed designs in comparison to randomized latin square designs is tabulated for some cases.     

Nonparametric rank-based test procedures for non-additive models in the two-way layout I. no replications

One of the major unresolved problems in the area of nonparametric statistics is the need for satisfactory rank-based test procedures for non-additive models in the two-way layout, especially when there is only one observation on each combination of the levels of the experimental factors. In this paper we consider an arbitrary non-additive model for the two-way layout with n levels of each factor. We utilize both alignment and ranking of the data together with basic properties of Latin squares to develop rank tests for interaction (non-additivity). Our technique involves first aligning within one of the main effects, ranking within the other main effects (columns and rows) and then adding the resulting ranks within “interaction bands” corresponding to orthogonal partitions of the interaction for the model, as denoted by the letters of an n × n Latin square. A Friedman-type statistic is then computed on the resulting sums. This is repeated for each of (n?1) mutually orthogonal Latin squares (thus accounting for all the interaction degrees of freedom). The resulting (n?1) Friedman-type statistics are finally combined to obtain an overall test statistic. The necessary null distribution tables for applying the proposed test for non-additivity are presented and we discuss the results of a Monte Carlo simulation study of the relative powers of this new procedure and other (parametric and nonparametric) procedures designed to detect interaction in a two-way layout with one observation per cell.     

Analysis of binary data in the randomized complete block design

The technique proposed by Shah and Claypool (1984) is extended here for the randomized complete block design with one binary observation per cell. In addition, it provides an al- ternative derivation of the distribution of Cochran's Q sta- tistic which is straightforward.     

Block designs robust against the presence of positional effects

The problem considered is to find optimum designs for treatment effects in a block design (BD) setup, when positional effects are also present besides treatment and block effects, but they are ignored while formulating the model. In the class of symmetric balanced incomplete block designs, the Youden square design is shown to be optimal in the sense of minimizing the bias term in the mean squared error (MSE) of the best linear unbiased estimators of the full set of orthonormal treatment contrasts, irrespective of the value of the positional effects.     

Low-order spatially-distinct latin squares

The order three to five spatially-distinct Latin squares, and the order three to six spatially-distinct Latin square treatment designs are listed. Some statistical results are given. Designs for 4, 5 and 6 treatments that were found previously to be robust to a linear by linear interacrion are shown to be optimal. Designs with good neighbour balanced are also considered.     

Smoothing fertility trends in agricultural field experiments

Fertility trend within blocks and local variations are the major obstacles to estimate cultivar contrasts in agricultural field trials. This paper examines methods of smoothing fertility trends in field trials using the P-spline. We begin by smoothing trend within block and for each block, and proceeds to demonstrate how it can be extended to smooth trends in trials with two-dimensional setting. We propose simultaneous modelling of trends and local variation. We use Papadakis [J.S. Papadakis, Comparison de differentes methds d'expermentation phytotechnique, Rev. Argen. Agronom. 7 (1940), pp. 297–362.] and kriged covariate to model local variation. We emphasize on the benefit of using P-spline to compromise between parametric and non-parametric approaches. Data sets from wheat and barley trials, designed as randomized complete block design and row-column, are analyzed. We set out a simple strategy of choosing between additive model and two-dimensional setting. We explore different estimation methods and offer some generalizations. The results show importance of the P-spline in modelling trend and the need to choose between additive and two-dimensional settings.     

Sweeping,alignment and the analysis of unbalanced data

The terms sweeping and alignment refer to the same process. Sweeping/alignment is used by data analysts as a technique for describing the effects of a model factor (e.g., treatments in a randomized block design) after the effects of nuisance parameters (e.g., blocks) have been removed from the data. In this paper sweeping/alignment is used as the basis for developing tests of factors in unbalanced experimental design models. Formulas are presented for treatment effects in randomized block designs with missing observations, and for interaction and main effects in unbalanced two-way factorial designs with empty cells.     

Permutation techniques for analyzing multi-response data from randomized block experiments

A class of permutation techniques is presented for the randomized block design. This class is specifically devised for analyses involving multivariate data. A numerical example illustrates an application based on multivariate data. Many well known techniques are special cases of this class. Among these special cases are (i) the permutation version of the classical univariate technique for randomized blocks which 1s associated with analysis of variance, (ii) the Friedman randomized block test, (iii) one-sample matched-pair tests, (iv) the Pearson correlation measure, and (v) the Spearman rank correlation and foot-rule measures. Furthermore, variations and multivariate versions among this class suggest a variety of new techniques which have not received any previous attention.     

A Reinforced Randomized Block Design with Correlated Errors

In the analysis of variance the errors are assumed to be independent which may not always be true. In this article, a restricted randomized block design under compound autocorrelated structure is considered. A method of analysis for such a randomized block design has been developed. It provides confidence ellipsoid and confidence interval of a set of estimable functions and an estimable function of treatment parameters, respectively. Analogous to Scheffe's method, the multiple comparison technique of judging all possible treatment contrasts has been derived. In the process, a method of estimation of unknown parameters has been also developed. An example (simulated data) illustrates this approach.     

Systematic latin squares

In order to properly utilize restricted randomization in the selection of t × t Latin squares it is necessary to have some idea of the various types of systematic Latin squares that should be removed from the admissible sets. The best known systematic squares are the diagonal squares and the Knut Vik squares. When t is not a prime number there are various other types of diagonal and balanced Latin squares. Eleven types of 4 × 4 Latin squares, each of them being systematic, are identified, displayed, and their properties indicated. Eight types of systematic 6 × 6 Latin squares are also identified and displayed. The effect of removing systematic squares from the admissible sets of Latin squares is discussed. Recommendations are made on when a restricted randomization procedure is to be preferred to a full randomization procedure in the selection of a random t × t Latin square.     

A clinical application of expert system methodology

In recent years, several expert systems have been developed for practical applications in applied statistical methodologies. Existing expert systems in statistics have explored several areas, e.g. the determination of appropriate statistical tests, regression analysis, and determination of the ‘best’ experimental design for industrial screening experiments. We present here the DESIGN EXPERT which is a prototype expert system for the design of complex statistical experiments. It is intended for scientific investigators and statisticians who must design and analyze complex experiments, e.g. multilevel medical experiments with nested factors, repeated measures, and both fixed and random eflects. This system is ‘expert’ in the sense that it is capable of the following:(i) recognize specific types of complex experimental designs, based on the application of inference rules to non-technical information supplied by the user; (ii) encode the obtained and inferred information in a flexible general-purpose internal representation, for use by other program modules; (iii) generate analysis of variance tables for the recognized design and an appropriate BMDP runfile for data analysis, using the encoded information. DESIGN EXPERT is capable of recognizing randomized block designs, including lattice designs within embedded Latin squares, cross-over designs, split plots, nesting, repeated measures and covariates. It is written in an experimental programming language developed specifically for research in artificial intelligence.