Two-way Layout Research Articles

Best estimation of variance components with arbitrary kurtosis in two-way layouts mixed models

Under the assumptions that a linear model corresponding to two-way layouts is the mixed model without interaction, necessary and sufficient conditions are derived under which the minimum norm invariant quadratic unbiased estimators have uniformly minimum variance among all unbiased invariant quadratic estimators for data with arbitrary kurtosis. The conditions are expressed in terms of the incidence matrix N and C-matrix of the design. Under the model considered in this paper, the results obtained here strengthen those by Anderson et al. (1984), where data are required to satisfy a special balance condition.

A comparative study of some rank tests for interaction

Statistical methods based on ranking after alignment are known to be versatile and powerful techniques. Although the theoretical properties of these methods are well-known, no empirical results are available on these procedures. Two versions of the F-statistic analogue of the aligned rank method for interaction in two-way layouts are studied along with the classical F-test and the rank transform test that has been the focus of several studies in recent years. Based on extensive simulation studies we conclude that the aligned rank tests should generally be preferred in application. We further show that none of these tests performs well for the Cauchy distribution. We show that the problem with inflation in the Type I error rate of the rank transform statistic (Blair et al., 1987), can be overcome by the use of the normal scores.

Analysis of Two-Way Layouts.

Analysis of Two-Way Layouts.

New method for improving accuracy of 24-hour continuous intragastric pH-metry. Reflections on physiological and pharmacological studies.

Continuous 24-hr intragastric pH-metry was prospectively performed in 801 subjects with different clinical conditions using two pH electrodes placed closely adjacent. The aim was to assess the in situ repeatability of the test and to verify whether the removal of artifacts, interference, and noise usually superimposed onto the fundamental signal recorded by the measuring apparatus improves the clinical usefulness of experimental data. The following debugging/filtering procedure was used: first, pH recordings of each channel were amended separately from artifacts, then they underwent 7 min windowed median interference debugging, and finally Wiener noise filtering was applied. Afterwards, the 24-hr mean pH profile was obtained in each subject by averaging the pH tracings of the two channels every minute (1440 data points/24 hr). The efficiency of this procedure was assessed at each step by evaluating the difference among groups using the O'Brien test, a distribution-free nonparametric method well-suited for evaluating differences among groups allocated onto a two-way layout. The differences among groups calculated from raw pH data of the single channels can be very misleading, in that it is possible to find that they are significant on one channel and not significant on the other channel. Conversely, the significance of the differences among groups increases progressively at each step of the above debugging/filtering procedure applied to raw pH profiles of each channel. Seven minutes was shown to be the most suitable time lag for windowed median removal of interference.(ABSTRACT TRUNCATED AT 250 WORDS)

C410. a distribution-free test for homoscedasticity in a two-way layout

C410. a distribution-free test for homoscedasticity in a two-way layout

Similarity-based testing for community pattern: the two-way layout with no replication

The large, sparse arrays of species counts arising in both field and experimental community studies do not lend themselves to standard statistical tests based on multivariate normality. Instead, a valid and more revealing approach uses informal display methods, such as clustering or multi-dimensional scaling ordination (MDS), based on a biologically-motivated definition of pairwise similarity of samples in terms of species composition. Formal testing methods are still required, however, to establish that real assemblage differences exist between sites, times, experimental treatments, pollution states, etc. Earlier work has described a range of Manteltype permutation or randomisation procedures, making no distributional assumptions, which are termed ANOSIM tests because of their dependence only on (rank) similarities and the analogy to one and two-way ANOVA. This paper extends these tests to cover an important practical case, previously unconsidered, that of a two-way layoutwithout replication. Such cases arise for single samples (or pseudo-replicates) taken in a baseline monitoring survey of several sites over time, or a mesocosm experiment in which “treatments” are replicated only once within each experimental “block”. Significance tests are given for the overall presence of a treatment (or time) effect, based on a measure of concordance between rank similarities of samples within each block (or site); the role of the two factors can be reversed to obtain a test for block effects. As in the analogous univariate ANOVA test, the method relies on absence or relative weakness of treatment x block “interactions”. Its scope is illustrated with data from two experimental and two field studies, involving meiofaunal communities from soft-sediment and macro-algal habitats. It is seen also to accommodate a modest derree of missing data. Whilst the failure to replicate adequately is not encouraged—a richer inference is available with genuine replication—the paper does provide a limited way forward for hypothesis testing in the absence of replicates.

Strictly positive estimators for variance components

The paper is concerned with the problem of simultaneous estimation of the variance components in the balanced two-way layout random effects normal model under the usual quadratic loss function. It is demonstrated that a class of simple estimators suggested by DasGupta (1986) includes strictly positive estimators that are better than, or comparable with, all scale preserving and location invariant estimators now available. Numerical results are tabled to exhibit the risk of the considered estimators.

Testing for Homogeneous Variations in Periodic Data

When analyzing periodic data, there are not one but two intervals to be considered. Often these intervals correspond to seasons and years. Specifically, one expects relationships to exist (a) between observations for successive seasons in a particular year and (b) between the observations for the same season in successive years. Periodic data of this type may be tabulated in s rows due to s seasons and n columns due to n years, where s is the periodicity of the data. When the column variances are equal, the periodic pattern of the data generally remains stable over the years. On the other hand, when the variances are unequal, the periodic pattern changes over the years. The usual Bartlett (1937) test for the equality of n variances may be highly misleading because of the correlation among observations within rows as well as columns. In the present paper, we develop an asymptotically most powerful test for testing the homogeneity of column variances in a two-way layout assuming that observations follow a seasonal ARMA process. A modification to the standard Bartlett test is also proposed which accounts for the autocorrelations. The modified Bartlett test is simple to compute and it has asymptotically %2 distribution. The tests are illustrated by a numerical example.

Robust Bayesian Methods in Simple ANOVA Models.

We develop a robust Bayesian inference procedure for the simple analysis of variance (ANOVA) model. In the ANOVA model of a two-way layout crossover study, the problem of partial prior robustness can be reduced to componentwise variations of positive-definite diagonal matrices. The feasible ellipsoid (a hyperinterval), the intersection, and the union of HPD regions (HiFi region) are derived.

On rank statistics: an approach via metrics on the permutation group

Abstract This paper investigates a general construction that generates many rank test statistics, for most of the standard nonparametric hypothesis testing situations. The construction uses metrics on permutations in a new way: the induced rank test statistics is a distance between two sets of permutations. These sets of permutations have group-theoretic descriptions, and the resulting rank statistic metrizes a quotient space of the permutation group. This construction is applied systematically to the two-sample and multi-sample location problems, the two-way layout problem, the one-sample location problem, the two-sample dispersion problem with equal medians, and the problem of testing for trend. The construction gives rise to many familiar rank test statistics for these six hypothesis testing situations, and also produces some plausible new test statistics. Additionally, the method enables one to ‘extend’ rank statistics to other hypothesis testing situations, in the sense that it creates families of statistics, for the six given settings, that are linked by their use of the same metric. Some connections with standard nonparametric theory are discussed. In the course of implementing the general construction, certain combinatorial and group-theoretic properties of metrics on permutations are studied, that play key roles in the computation of the induced rank statistics.

Nonparametric rank based main effects test procedures for the two-way layout in the presence of interaction

In spite of an extensive literature dealing with rank approaches to testing for main effects when an additive model is assumed, virtually no consideration has been given to developing nonparametric main effects test procedures that adjust for the possible presence of interaction in a non-additive model. This problem has been briefly discussed in Iman (1974), but has otherwise been ignored in the literature. In this paper, we approach the two-way layout setting through an arbitrary non-additive model. We propose a technique for adjusting the data to account for the possible presence of interaction prior to conducting a rank test for either of the main effects. These adjustments involve both alignment and ranking of the data and utilize the basic properties of Latin squares. We discuss the results of a Monte Carlo simulation study of the relative powers of this new procedure and other nonparametric procedures designed only for an additive model.

Score tests in the two - way layout of counts

For the balanced two-way layout of a count response variable Y classified by fixed or random factors A and B, we address the problems of (i) testing for individual and interactive effects on Y of two fixed factors, and (ii) testing for the effect of a fixed factor in the presence of a random factor and conversely. In case (i), we assume independent Poisson responses with µij= E(Y| A=i,B=j) = αiβjγij corresponding respectively to the multiplicative interactive and non-interactive cases. For case (ii) with factor A random, we derive a multivariate gamma-Poisson model by mixing on the random variable associated with each level of A. In each case Neyman C(α) score tests are derived. We present simulation results,and apply the interaction test to a data set, to evaluate and compare the size and power of the score test for interaction between two fixed factors, the competing Poisson-based likelihood ratio test, and the F-tests based on the assumptions that √Y+1 or log(Y+1) are approximately normal. Our results ...

Statistical inference based on aligned ranks for two-way MANOVA with interaction

Multiresponse experiments in two-way layouts with interactions, having equal number of observations per cell, are considered. Robust procedures based on aligned ranks for statistical inference of interactions, main effects and an overall mean response in the models are proposed. Large sample properties of the proposed tests, estimators and confidence regions as the cell size tends to infinity are investigated. For the univariate case, it is found that the asymptotic relative efficiencies (ARE's) of the proposed procedures relative to classical procedures agree with the ARE-results of the two-sample rank test relative to the t-test. In addition, robustness due to Huber (1981, Robust Statistics, Wiley, New York) can be drawn.

A standardized test for the error components model with the two-way layout

This paper proposed a standardized test for the error components model with the twoway layout. The proposed test statistic has mean zero and variance one in finite samples under the null hypothesis, and will improve the accuracy of the size of the test.

Age, Cohort and Period in Life-Span Research: A Three-Way Analysis with Logically Missing Cells.

A solution is shown for addressing the age x cohort x period issue in life-span research. Previous solutions have utilized only portions of the data in a given two-way layout. The proposed solution uses all data, which for at least two of the three two-way layouts involves missing cells. The method can be used for repeated measure designs, or designs in which new subjects are measured at each period. The design allows the assessment of each main effect and each two-way interaction. The hypotheses tested are explicitly shown, together with the linear models that accomplish this testing.

A note on the rank transform for interactions

SUMMARY The asymptotic properties of the rank transform statistic for testing for interaction in a balanced two-way classification are studied. Necessary and sufficient conditions are obtained for the asymptotic distribution of this rank transform statistic to be chi-squared under the null hypothesis of no interaction. It is shown that the rank transform test statistic for interaction is asymptotically chi-squared, divided by its degrees of freedom, when there are exactly two levels of both main effects or when there is only one main effect. In the latter case, the test detects a nested effect instead of an interaction. In all other two-way layouts, there exist values for the main effects such that, under the null hypothesis of no interaction, the expected value of the rank transform test statistic goes to infinity as the sample size increases.

Nonparametric tests for interaction and group differences in a two-way layout

Nonparametric tests of group differences and interaction across strata are developed in which the null hypotheses for these tests are expressed as functions of rho i = P(X > Y) + 1/2P(X = Y), where X refers to a random observation from one group and Y refers to a random observation from the other group within stratum i. The estimator r of the parameter rho is shown to be a useful way to summarize and examine data for ordinal and continuous data.

Analysis of two-way layout of count data with negative binomial variation

A number of methods has been proposed for dealing with single-factor or factorial experiments when the requirements for performing the normal theory analysis of variance procedure are not satisfied. This paper suggests the use of the likelihood ratio statistic for testing the main effects and the interaction between the factors in two-way layout of count data following negative binomial distributions with a common dispersion parameter. The likelihood ratio statistic for testing the equality of the dispersion parameters of several groups of count data is also derived. The methods is illustrated by an example concerning the study of spatial and temporal variation of bacterial counts.

Characterizations of two-way layouts from the point of view of variance component estimation in the corresponding mixed linear models

Under the assumption that a linear model corresponding to a two-way layout is the mixed model without interaction, complete characterizations are derived of the layouts that assure the invariant unbiased estimability of every linear function of two variance components involved and of designs that assure the existence of the uniformly best invariant quadratic unbiased estimator for every estimable function of variance components.

The Analysis of Nonadditivity in Two-Way Analysis of Variance

Abstract Singular value decompositions are used to study interactions in two-way layouts. In contrast to traditional treatments, emphasis is on description of the nature of the interactions rather than on tests of whether they do or do not exist. The first model considered assumes that the matrix of interactions has rank 1, so in a singular value decomposition of the interactions, only one singular value is not 0. Given that this model holds and interactions are present, normal approximations are obtained under conditions described for the estimates of this singular value and for the corresponding scores for row categories and column categories. Thus approximate confidence intervals can be obtained for the parameters in this singular value decomposition of interactions. In addition, under this model, approximate t and F tests can be constructed for hypotheses that assign specified values to the row and column scores in the singular value decomposition. Results are illustrated in a reanalysis of data previ...