Two-way Layout Research Articles

Simulated efficiencies of tests and estimators from general linear models analysis based on ranks: the two-way layout with interaction

Simulated efficiencies of tests and estimators from general linear models analysis based on ranks: the two-way layout with interaction

Exact F Tests in an ANOVA Procedure for Dependent Observations.

Several general correlation patterns are shown in this paper which give exact F tests in an ANOVA procedure with dependent observations. This paper presents the most general correlation patterns one can assume in a one-way and two-way layout and still have the F tests be valid. Exact F tests are given for various designs. These include the unbalanced ANOVA design, analysis of covariance, random effects models, and mixed models. Bartlett's test for homogeneity of variances is shown to be exact when the independence assumption is relaxed. An example is provided to illustrate how the general correlation can occur in an experimental design.

The F Statistic in the Two-Way Layout with Rank-Score Transformed Data

The F Statistic in the Two-Way Layout with Rank-Score Transformed Data

TheFStatistic in the Two-Way Layout With Rank–Score Transformed Data

Abstract The limiting null distribution of the usual F statistic for main effects in the two-way layout is shown to have the same limiting distribution when applied to ranks and scores based on ranks as when applied to normal data. The limit is taken as the cell size N increases without bound. The denominator of the F statistic times (J − 1)/J is shown to provide an unbiased and consistent estimator of the limit (as N → ∞) of N -1 times the variance of the treatment sum of scores when the null hypothesis is true. The test based on the scores statistic is demonstrated to be consistent for a wide class of fixed alternative hypotheses.

An asymptotic expansion for a one-sided rank test in a two-way layout

In the randomized block design with I blocks and two treatments, the within-block rank test is considered. It is found that the test is asymptotically efficient for a large number of observations per each cell and the asympotic expansion of that under the null hypothesis is obtained.

Quantile Estimation in Dependent Sequences

Standard nonparametric estimators of quantiles based on order statistics can be used not only when the data are i.i.d., but also when the data are drawn from a stationary, ϕ-mixing process of continuous random variables. However, when the random variables are highly positively correlated, the sample sizes needed for estimating extreme quantiles become computationally unmanageable. This paper gives a practical scheme, based on a maximum transformation in a two-way layout of the data, that reduces the sample size sufficiently to allow an experimenter to obtain a point estimate of an extreme quantile. The paper gives three schemes that lead to confidence interval estimates for the quantile. One uses a spectral analysis of the reduced sample. The other two, averaged group quantiles and nested group quantiles, are extensions of the method of batched means to quantile estimation. These two schemes give even greater data compaction than the first scheme. None of the schemes requires that the process being simulated is regenerative.

An approach to defining the pattern of interaction effects in a two-way layout

A method is given to classify rows and columns into subgroups so that additivity holds within each of the subtables made of the grouped rows or the grouped columns. The least squares estimators of the cell means are easily obtained for the resulting linear model together with their variances. An estimator of the error varianceσ2 is given when there is only one observation per cell. A treatment of an ordered table is also given.

Multiple Comparisons in a Mixed Model

Abstract In this note we give a simple proof of the result that, in the symmetric version of Scheffe's mixed model for a balanced two-way layout, an exact T procedure for pair-wise comparisons between the levels of the fixed factor can be based on the interaction mean square as the Studentizing factor. Such a proof does not seem to be available in the literature and will hopefully remove the confusion caused by some textbooks that incorrectly prescribe the error mean square as the Studentizing factor.

On Distribution-Free Rank Tests for Two-Way Layouts

Abstract A class of distribution-free statistics using scores based on ranks for testing main effects in two-way layouts, with no interactions and more than one observation per cell, is discussed. Asymptotic distributions are developed under the null and alternative hypotheses for the case in which cell size is fixed and the number of blocks is allowed to become large, and as cell size becomes large with the number of blocks fixed. Efficiencies are considered under both asymptotic situations. We show that under the case in which the number of blocks becomes large, it is possible to choose a set of scores that is optimal in the sense of achieving the highest possible asymptotic relative efficiency for a given distribution.

On Distribution-Free Rank Tests for Two-Way Layouts

On Distribution-Free Rank Tests for Two-Way Layouts

Measuring Agreement for Multinomial Data

A kappa-like statistic is proposed as a measure of agreement for a set of multinomial random variables arrayed in a two-way layout. This statistic is shown to arise either via the chancecorrection of a certain proportion of agreement or via an analysis of variance for a two-way layout. The asymptotic standard error is derived under the assumption of complete independence among the random variables.

Some Classes of Optimality Criteria and Optimal Designs for Complete Two-Way Layouts

For a given class of linear models in standard form an optimal experimental design is to be chosen for estimating some linear functions of the unknown parameters. An optimality criterion is defined to be a real function of the covariance matrices of the Gauss-Markov estimators. Conditions which are imposed on the criteria are monotonicity, quasiconvexity or quasiconcavity, and invariance or order-invariance. A characterization of the $D$-criterion by order-invariance is included which strengthens a result of P. Whittle. In the main part of the paper optimal designs for the usual two-way layouts in ANOVA are computed for large classes of optimality criteria. Some related optimization problems are solved with the technique of majorization of vectors in the sense of Schur.

An Analysis of Transformations Revisited

Abstract Following Box and Cox (1964), we assume that a transform Z i = h(Yi , λ) of our original data {Yi } satisfies a linear model. Consistency properties of the Box-Cox estimates (MLE's) of λ and the parameters in the linear model, as well as the asymptotic variances of these estimates, are considered. We find that in some structured models such as transformed linear regression with small to moderate error variances, the asymptotic variances of the estimates of the parameters in the linear model are much larger when the transformation parameter λ is unknown than when it is known. In some unstructured models such as transformed one-way analysis of variance with moderate to large error variances, the cost of not knowing λ is moderate to small. The case where the error distribution in the linear model is not normal but actually unknown is considered, and robust methods in the presence of transformations are introduced for this case. Asymptotics and simulation results for the transformed additive two-way layout show that much is gained by this robustification of the Box-Cox methods when the ratios of the means to the error standard deviation are moderate to large. However, the performance of all Box-Cox type procedures is unstable and highly dependent on the parameters of the model in structured models with small to moderate error variances.

An Analysis of Transformations Revisited

Abstract Following Box and Cox (1964), we assume that a transform Z i = h(Yi , λ) of our original data {Yi } satisfies a linear model. Consistency properties of the Box-Cox estimates (MLE's) of λ and the parameters in the linear model, as well as the asymptotic variances of these estimates, are considered. We find that in some structured models such as transformed linear regression with small to moderate error variances, the asymptotic variances of the estimates of the parameters in the linear model are much larger when the transformation parameter λ is unknown than when it is known. In some unstructured models such as transformed one-way analysis of variance with moderate to large error variances, the cost of not knowing λ is moderate to small. The case where the error distribution in the linear model is not normal but actually unknown is considered, and robust methods in the presence of transformations are introduced for this case. Asymptotics and simulation results for the transformed additive two-way layout show that much is gained by this robustification of the Box-Cox methods when the ratios of the means to the error standard deviation are moderate to large. However, the performance of all Box-Cox type procedures is unstable and highly dependent on the parameters of the model in structured models with small to moderate error variances.

Maximum likelihood estimation in a two-way analysis of variance with correlated errors in one classification

A two-way analysis of variance model with correlated errors in one classification is discussed. It is assumed that the p measurements in each row have a general covariance matrix Σ The maximum likelihood estimates of the row and column parameters as well as of the covariance matrix are obtained and their asymptotic distribution are discussed. Finally, the likelihood ratio test statistics for the various hypotheses in a two-way layout are given.

Subset Selection Procedures With Application to Motor Vehicle Fatality Data in a Two-Way layout

In this paper, the origin of selection and ranking problems is discussed. The two basic approaches to the selection problem—the indifference zone approach and the subset selection approach—are reviewed briefly. Gupta's subset selection procedure is applied to motor vehicle fatality data which fit into a two-way layout.

Subset Selection Procedures with Application to Motor Vehicle Fatality Data in a Two-Way Layout

Subset Selection Procedures with Application to Motor Vehicle Fatality Data in a Two-Way Layout

Two Monte Carlo Studies of Silverstein's Nonparametric Multiple Comparison Tests

Two Monte Carlo studies were done to find the Type 1 error rates for Silverstein's nonparametric pairwise multiple-comparison tests for a one- and two-way layout with ranked data. Silverstein's tests had excellent experimentwise error rates but did not do as well as Wilcoxon's tests and the stepped-down sign test when pairwise comparisons were performed after significant over-all tests. Silverstein's tests were shown to be equivalent to recently proposed tests by Levy.

Analysis of Interaction Effects in a Two-way Layout without Replication

Analysis of Interaction Effects in a Two-way Layout without Replication

Carbonization of Redactive Ethylation Product of Yubari Coal Under the High Pressure

The present report deals with carbonization of ethylated product of Yubari coal at 400 and 500°C, <20 and 1000 kg/Cm2 of nitrogen for 5 h. in order to obtain the needle like coke.Temperature and pressure effects to the ethylated product were at first investigated, then compared with those to the natural coal. The variance analyses (two-way layout) were applied to the various data such as the specific extinction coefficient K of carbonization products at each IR absorption band and the solubility of them in benzene and pyridine, and then it was found that benzene extract yield of carbonization products were only affected by temperature differences between 400 and 500°C, while pyridine extract yield affected by these interaction as well as temperature and pressure (significance at 1%, respectively)Microscopic observation for carbonization products under the normal pressure of methylated- and ethylated products from Yubari coal and Taiheiyo coal, showed that mesophase did not exist in them. These findings suggested that alkyl groups introduced evolved up to ca. 370°C and did not contribute the f ormation of mesophase. Furthermore, mean structure of pyridine extracts of carbonization products were also discussed.