Unbalanced Case Research Articles

Characteristics of Convergence Curve of the PE Estimate of Variance Ratio in the Balanced Case

The pseudoexpectation (PE) methods are an iterative procedure for estimating variance components in the general mixed linear model. The computational loads in the PE methods are relatively small and the estimates converge faster, although the methods are an approximation to the restricted maximum likelihood (REML) procedure. In the field of animal breeding, it has been considered that the PE methods can be used in estimating from large data sets including no or weak influence of selection, or in obtaining initial values to use in the REML estimation. This paper reports some basic characteristics of the convergence curve of the PE variance ratio estimate in the case of balanced data. In the specific balanced case, the PE and REML estimates are identical, but their convergence curves are different. The ratios of the difference between the consecutive estimates in the ith and the i-1th rounds to that in the i+1th and the ith rounds vary, for instance, in the REML estimation using the expectation-maximization algorithm, while those with the PE methods take a constant value regardless of initial values and rounds of iteration. Hence, the PE curve of convergence can be exactly represented using the result for a geometric series. Consequently, the PE curve is equivalent to the curve assumed underlyingly in the extrapolation techniques, or the techniques of directly predicting the final value such as the Aitken extrapolation and the common intercept approach. In the unbalanced case, the property of this kind of the PE curve is no longer retained, and the degrees of discrepancies among the PE and the extrapolation estimates would be dependent on the structure of the data to be analyzed. When one wishes to accelerate convergence in the REML analysis of unbalanced data, it is important to notice that the PE and the extrapolation techniques each have the advantages and disadvantages for the purpose.

Comparison of Meta-Analysis Versus Analysis of Variance of Individual Patient Data

Meta-analysis is a method of synthesizing the results of independent studies. We consider the case in which there are multiple treatments and a control, with the goal of estimating the relative effect of each treatment based on continuous outcomes. Even when all data are available, rather than only summary data, it has become common to use meta-analytic estimators of treatment contrasts. Alternatively, we could use a two-way analysis of variance model with no interaction in which one factor is study and one factor is treatment. For the unbalanced case, we obtain the surprising result that the standard meta-analysis estimates of treatment contrasts are identical to the least squares estimators of treatment contrasts in the linear model. Because a meta-analysis of individual patient data can be considerably more costly in terms of data retrieval than a meta-analysis of summary data, this equivalence provides for cost-efficient analysis.

Wald consistency and the method of sieves in REML estimation

We prove that for all unconfounded balanced mixed models of the analysis of variance, estimates of variance components parameters that maximize the (restricted) Gaussian likelihood are consistent and asymptotically normal--and this is true whether normality is assumed or not. For a general (nonnormal) mixed model, we show estimates of the variance components parameters that maximize the (restricted) Gaussian likelihood over a sequence of approximating parameter spaces (i.e., a sieve) constitute a consistent sequence of roots of the REML equations and the sequence is also asymptotically normal. The results do not require the rank p of the design matrix of fixed effects to be bounded. An example shows that, in some unbalanced cases, estimates that maximize the Gaussian likelihood over the full parameter space can be inconsistent, given the condition that ensures consistency of the sieve estimates.

Large Sample Tests for Binary Outcomes in Fixed-Dose Combination Drug Studies

Several test statistics are developed for testing the hypothesis that the combination of two drugs at a fixed-dose regimen is more effective than both of the single drugs used alone with respect to a dichotomous response variable. The response probability, logit, and arcsine-root scales are considered. The power function and the significance level are derived for large samples. For the sample size per group of 20 or greater, the power and type I error rate can be accurately calculated using the large sample power function when the response probability ranges from 0.2 to 0.8. These tests have similar power behaviors. In small samples, the large sample power functions of two of the tests can severely underestimate the type I error rate while overestimation can occur with one other test. The utilities of these tests are extended to unbalanced sample size cases. Generally speaking, there is a loss of power with unequal sample size allocation, but the loss is not severe.

A remark on testing covariance structure in a mixed model

Khuri (1989) tests for the intraclass covariance structure implied by the balanced two-way mixed analysis of variance model by computing wilks' likelihood ratio test statistic using the sample covariance matrix of the vectors of treatment means. In the unbalanced case he uses a linear transformation to augment the treatment-mean vectors to vectors which are expected to satisfy the intraclass structure, and then computes Wilks' statistic using these augmented vectors. We point out that the augmentation process is in fact equivalent to deleting observations until the design is balanced, so that the augmented test actually uses less information than that contained in the original sample means.

On balanced versus unbalanced computation trees

A great number of complexity classes between P and PSPACE can be defined via leaf languages for computation trees of nondeterministic polynomial-time machines. Jenner, McKenzie, and Therien (Proceedings of the 9th Conference on Structure in Complexity Theory, 1994) raised the issue of whether considering balanced or unbalanced trees makes any difference. For a number of leaf-language classes, coincidence of both models was shown, but for the very prominent example of leaf-language classes from the alternating logarithmic-time hierarchy the question was left open. It was only proved that in the balanced case these classes exactly characterize the classes from the polynomial-time hierarchy. Here, we show that balanced trees apparently make a difference: In the unbalanced case, a class from the logarithmic-time hierarchy characterizes the corresponding class from the polynomial-time hierarchy with a PP-oracle. Along the way, we get an interesting normal form for PP computations.

Saddlepoint approximations for P-values of some tests of covariance matrices

Test statistics for hypotheses about covariance matrices in multivariate analysis of variance have null distributions which are rather intractable to compute. Examples include the modified likelihood ratio statistic for testing (i) homogeneity of covariance matrices across normal populations (Bartlett-Box M-statistic), (ii) homogeneity of normal populations and (iii) simultaneous sphericity of covariances across normal populations. This paper provides two saddlepoint approximations for the null distributions of each of the above mentioned test statistics. These approximations result in extremely accurate p-value computations in all settings including the more difficult unbalanced sample size case. The computations are fast and easily programmed.

Some Tests for Variance Components Using GeneralizedpValues

The concept of a generalized p value, introduced by Tsui and Weerahandi, is applied for testing hypotheses in two situations, testing the significance of a variance component in a general balanced mixed model when an exact F test does not exist and comparing randomeffects variance components in two independent balanced mixed models. Extensions to the unbalanced cases are also indicated. The proposed test is compared with the test based on the Satterthwaite approximation through their simulated Type I error probabilities. The simulations indicate that the test based on the generalized p value hasType I error probabilities less than the chosen significance level most of the time, whereas the Type I error probabilities of the Satterthwaite approximate test can be much larger than the significance level. The results are illustrated using two examples.

An Introduction to Statistical Methods and Data Analysis

An Introduction to Statistical Methods and Data Analysis

Effect of transverse load‐resisting elements on inelastic earthquake response of eccentric‐plan buildings

AbstractThe accurate evaluation of code torsional provisions for plan‐eccentric structures exhibiting inelastic response relies on the adoption of appropriate systems defining both the torsionally balanced (reference) and torsionally unbalanced cases. Whilst a considerable number of analytical studies of this problem have been presented in the literature, inconsistencies have arisen in their conclusions. It is evident from a review of previous studies that one factor contributing significantly to these discrepancies arises in the definition of the structural layout. An issue of particular importance is whether the transverse load‐resisting elements oriented perpendicular to the assumed (lateral) direction of earthquake loading should, for purposes of realism, be included in model definitions. Given the diverse approaches in the existing literature, clarification of this issue is required in order to advance the understanding of inelastic torsional response behaviour and to assist the interpretation and comparison of previous studies. This paper aims to provide such clarification, based on analyses of a series of models defined rigorously according to code design provisions. Such models have been subjected to both uni‐ and bi‐directional ground motion input. It is concluded that for the flexible‐edge element, accurate estimates of additional ductility demand arising from torsional effects may be obtained from uni‐directional models (in which both the transverse elements and the corresponding earthquake component are neglected) only for medium‐period to long‐period systems. Such estimates may be over‐conservative for short‐period systems, which constitute a large proportion of systems for which code static torsional provisions are utilized. It is further concluded that models incorporating the transverse elements but analysed under uni‐directional lateral loading may underestimate by up to 100% the torsional effects in such systems, but are reasonably accurate for medium‐ and long‐period structures.

Improved Estimators of Variance Components with Smaller Probability of Negativity

SUMMARY A linear model with two variance components is considered, one variance component (say, σ 1 2 ≥ 0) corresponding to a random effect, and a second variance component (say, σ 2 > 0) corresponding to the experimental errors. A class of invariant quadratic estimators (IQEs) is characterized, having uniformly smaller mean-squared error (MSE), and uniformly smaller probability of negativity, compared with the analysis-of-variance (ANOVA) estimator of σ 1 2. It turns out that for balanced models, among IQEs of σ 1 2 with uniformly smaller MSE than its ANOVA estimator, there exists an IQE with the smallest probability of being negative, uniformly in the parameters. The results are applied to the balanced one-way classification model. Numerical computations show that the MSE improvement achievable through the use of the proposed estimators is quite significant. It is noted that, for the nonnegative estimation of σ 1 2, truncation of one of the proposed estimators at 0 provides a satisfactory solution. Extension of the results to the unbalanced case, and to the general mixed model with balanced data, is also indicated.

Letter: Prediction of the Outage Performance of an Unbalanced Orbital Diversity Earth‐Space System (Log‐Normal Case)

AbstractDiversity schemes are foreseen for earth to satellite links operating at frequencies above 10 GHz to overcome the high rain induced attenuation. In this paper, the orbital diversity configuration, which uses two satellites and an earth receiving site, is discussed.For this reason, a recently suggested model for the prediction of the outage performance of a balanced orbital diversity system is extended to include the more complicated but practical unbalanced case. The lognormal distribution and the convective raincells are assumed to be representative of the point rainfall statistics and the spatial structure of the rainfall, respectively. Numerical results are compared with experimental data of this kind, taken from an operated unbalanced system located at Fucino, Italy. The agreement was found to be quite encouraging.

On the inadmissibility of empirical averages as estimators in ranked set sampling

Abstract A ranked set sample (rss) consists entirely of independently distributed order statistics, and can occur naturally in many experimental settings, including problems in reliability. When each ranked set from which an order statistic is drawn is of the same size, and when the statistic of each fixed order is sampled the same number of times, the ranked set sample is said to be balanced. The mean X rss of a balanced ranked set sample is known to be an unbiased estimator of the population mean μ , and its variance has been shown to be less than or equal to that of the mean X srs of a simple random sample(srs) of the same size. Similarly, it has been shown that the empirical distribution function (edf) F rss of a balanced ranked set sample is superior to the edf F srs based on a simple random sample of the same size. The present study investigates the admissibility of X rss and F rss with the main results being as follows. When μ is either a scale or location parameter of the underlying distribution, it is possible to construct estimators which uniformly improve upon X rss relative to squared error loss. In the general nonparametric estimation problem, F rss is shown to be inadmissible relative to integrated squared error loss. Our alternative estimators have an additional virtue: in contrast with X rss and F rss , the new estimators extend naturally to the unbalanced case, and enjoy certain optimality properties in this broader setting.

Limitations of the rank transform procedure: A study of repeated measures designs, Part II

The applicability of the transform (RT) procedure in two-way generalized repeated measures designs were each individual receives each now treatment but only one column treatment is studied. All of the common testing problems in balanced and unbalanced designs are examined. The asymptotic version of the rank transformation (Akitras, 1990) is used to identify valid RT statistics and to obtain their asymptotic properties. The two valid statistics are for the hypothesis of no row effect (H 0: all α i + γ ij = 0), and for the hypothesis of no column effect (H 0: all β j + γ ij = 0). For the hypothesis of no row effect, the error covariance matrix is allowed to depend on the column treatment but the statistic is valid only in the balanced case. It is pointed out that the validity of this statistics is due to the robustness of the F-statistic, which is also shown, to model violations incured by the rank transformation. For the hypothesis of no column effect, the statistic is shown to be valid even in the unbalanced case but the covariance matrix is assumed constant. Further, it is shown that the RT procedure is not valid for testing for main effects or for testing for interaction.

DYNAMICS OF FULLY CONNECTED NEURAL NETWORKS WITH SIGN CONSTRAINTS

We consider fully connected neural networks near saturation, trained by a modified Edinburgh algorithm, with sign constraints on the synaptic couplings. We study the domains of attraction of the stored patterns for both the balanced and the unbalanced case (excess of positive over negative constraints). A comparison with the dilute network is also included.

Ranking and Estimation of Related Means in the Presence of a Covariate—A Bayesian Approach

Abstract Choosing the largest of several means can be a demanding problem, especially in the presence of a covariate. A hierarchical Bayesian approach to ranking and selection, as well as estimation of related means in the presence of a covariate, is considered. For the multiple slopes model we compute, in addition to the posterior means and standard deviations of the parameters, the posterior probabilities that each mean, at a given value of the covariate, is the largest. The vector of posterior probabilities thus obtained provides an easily understandable answer to the selection problem. Although calculation of the posterior probabilities may involve four-dimensional numerical integration in the difficult unbalanced design and unknown variance case, an efficient Monte Carlo method of evaluation has been developed and is given in the article. By reanalyzing a well-known data set on the breaking strength and thickness of starch films, we illustrate how our Bayesian approach produces meaningful conclusions...

The dilemma of negative analysis of variance estimators of intraclass correlation

At least two common practices exist when a negative variance component estimate is obtained, either setting it to zero or not reporting the estimate. The consequences of these practices are investigated in the context of the intraclass correlation estimation in terms of bias, variance and mean squared error (MSE). For the one-way analysis of variance random effects model and its extension to the common correlation model, we compare five estimators: analysis of variance (ANOVA), concentrated ANOVA, truncated ANOVA and two maximum likelihood-like (ML) estimators. For the balanced case, the exact bias and MSE are calculated via numerical integration of the exact sample distributions, while a Monte Carlo simulation study is conducted for the unbalanced case. The results indicate that the ANOVA estimator performs well except for designs with family size n = 2. The two ML estimators are generally poor, and the concentrated and truncated ANOVA estimators have some advantages over the ANOVA in terms of MSE. However, the large biases may make the concentrated and truncated ANOVA estimators objectionable when intraclass correlation (ϱ) is small. Bias should be a concern when a pooled estimate is obtained from the literature since ϱ<0.05 in many genetic studies.

Unbalanced, capacitatedp-median problems on a chain graph with a continuum of link demands

This article is concerned with the problem of locating p capacitated facilities on a chain graph and simultaneously determining the allocation of their supplies in order to satisfy a continuum of demand. The demand is characterized by some weighted probability density functions defined on the chain graph. The objective is to minimize the total (expected) trnasportation cost. This location-allocation problem is also referred to as the capacitated p-median problem on a chain graph. Two unbalanced cases of this problem are considered, namely, the overcapacitated case when total supply exceeds total demand and the deficit capacity case when total supply is less than total demand. Both these problems are nonconvex and are shown to be NP-hard even if the demand density function is piecewise uniform and positive. We provide a first-order characterization of optimality for these two problems and prescribe an enumerative algorithm based on a partitioning of the dual space in order to optimally solve these problems. An extension of these algorithms for solving the unbalanced, capacitated 2-median problem on a tree graph is also given.

One-sided tolerance limits for unbalanced one-way anova andomc effects model

In this article, we investigate techniques for constructing tolerance limits such that the probability is γ that at least p proportion of the population would exceed that limit. We consider the unbalanced case and study the behavior of the limit as a function of ni 's (where ni is the number of observations in the ith batch), as well as that of the variance ratio. To construct the tolerance limits we use the approximation given in Thomas and Hultquist (1978). We also discuss the procedure for constructing the tolerance limits when the variance ratio is unknown. An example is given to illustrate the results.

Generalized Wilcoxon Rank Tests and Sensitivity Curves

In this paper we use the sensitivity curves of TUKEY (1977) and the change of decision point (cdp) (a modified version of the breakdown point of YLVISAKER, 1977), supplemented by simulation studies to acquire knowledge about sensitivity in generalized Wilcoxon rank test statistics. Sensitivity depends on balanced or unbalanced sample size cases, censoring, combinations of failure distributions and sources of errors in the data. It is important to consider the quality of the data, and the results show that cdp and some properties of the sensitivity curves may serve as a hint when selecting a test statistic and when making a decision for a given test statistic.