On the distributions of components sums of squares in one-way unbalanced random model under heter06ene0us error variances

Exact sampling distributions of sums of squares in the unbalanced one-way random model are obtained under heterogeneous error variances. These distributions are used to investigate the effect of heteroscedasticity and unbalancedness on the probability of negative estimate of the group variance component. The computed results reveal that heteroscedasticity affects the probability of negative estimate in all situations of group sizes. Further, the probability decreases with heterogeneity of error variances for balanced situations and increases with variability among group size for equal error variances case.     

Maximum likelihood estimation of variance components in a completely random bib design

ABSTRACT

The purposes of this paper are to abstract from a number of articles variance component estimation procedures which can be used for completely random balanced incomplete block designs, to develop an iterated least squares (ITLS) computing algorithm for calculating maximum likelihood estimates, and to compare these procedures by use of simulated experiments. Based on the simulated experiments, the estimated mean square errors of the ITLS estimates are generally less than*those for previously proposed analysis of variance and symmetric sums estimators.     

Parametric Proportional Odds Frailty Models

We define a parametric proportional odds frailty model to describe lifetime data incorporating heterogeneity between individuals. An unobserved individual random effect, called frailty, acts multiplicatively on the odds of failure by time t. We investigate fitting by maximum likelihood and by least squares. For the latter, the parametric survivor function is fitted to the nonparametric Kaplan–Meier estimate at the observed failure times. Bootstrap standard errors and confidence intervals are obtained for the least squares estimates. The models are applied successfully to simulated data and to two real data sets. Least squares estimates appear to have smaller bias than maximum likelihood.     

On unweighted sums of squares in unbalanced analysis of variance

The well-known method of unweighted sums of squares (USSs) is examined using, as an example, a random two-way classification model with interaction. In particular, a better motivation is given to the association between the harmonic mean of the cell frequencies and the USSs. Furthermore, a procedure is developed for determining the adequacy of the USSs as approximate balanced analysis of variance (ANOVA) sums of squares. This procedure is easy to apply and provides a better insight into the effects of design and model’s variance components on such an approximation. The proposed methodology can be extended to higher-order models and other types of sums of squares.     

Proportional subclass numbers in two-factor ANOVA

The between-within split of total sum of squares in one-way analysis of variance (ANOVA) is intuitively appealing and computationally simple, whether balanced or not. In the balanced two-factor setting, the same heuristic and computations apply to analyse treatment sum of squares into main effects and interaction effects sums of squares. Accomplishing the same in unbalanced settings is more difficult, requiring development of tests of general linear hypotheses. However, textbooks treat unbalanced settings with proportional subclasss numbers (psn) as essentially equivalent to balanced settings. It is shown here that, while psn permit an ANOVA-like partition of sums of squares, test statistics for main effects of the two factors generally test the wrong hypotheses when the model includes interaction effects.     

Multilevel regression mixture analysis

Summary.  A two-level regression mixture model is discussed and contrasted with the conventional two-level regression model. Simulated and real data shed light on the modelling alternatives. The real data analyses investigate gender differences in mathematics achievement from the US National Education Longitudinal Survey. The two-level regression mixture analyses show that unobserved heterogeneity should not be presupposed to exist only at level 2 at the expense of level 1. Both the simulated and the real data analyses show that level 1 heterogeneity in the form of latent classes can be mistaken for level 2 heterogeneity in the form of the random effects that are used in conventional two-level regression analysis. Because of this, mixture models have an important role to play in multilevel regression analyses. Mixture models allow heterogeneity to be investigated more fully, more correctly attributing different portions of the heterogeneity to the different levels.     

Serial correlation or random subject effects

In longitudinal data analysis with random subject effects, there is often within subject serial correlation and possibly unequally spaced observations. This serial correlation can be partially confounded with the random between subject effects. In real data, it is often not clear whether there is serial correlation, random subject effects or both. Using inference based on the likelihood function, it is not always possible to identify the correct model, especially in small samples. However, it is important that some effort be made to attempt to find a good model rather than just making assumptions. This often means trying models with random coefficients, with serial correlation, and with both. Model selection criteria such as likelihood ratio tests and Akaike's Information Criterion (AIC) can be used. The problem of modelling serial correlation with unequally spaced observations is addressed. A real data example is presented where there is an apparent heterogeneity of variances, possible serial correlation and between subject random effects. In this example, it turns out that the random subject effects explains both the serial correlation and the variance heterogeneity.     

On LSE in regression model for long-range dependent random fields on spheres

We study the asymptotic behaviour of least squares estimators (LSE) in regression models for long-range dependent random fields observed on spheres. The LSE can be given as a weighted functional of long-range dependent random fields. It is known that in this scenario the limits can be non-Gaussian. We derive the limit distribution and the corresponding rate of convergence for the estimators. The results were obtained under rather general assumptions on the random fields. Simulation studies were conducted to support theoretical findings.     

Sparse Approximations of Fractional Matérn Fields

We consider fast lattice approximation methods for a solution of a certain stochastic non‐local pseudodifferential operator equation. This equation defines a Matérn class random field. We approximate the pseudodifferential operator with truncated Taylor expansion, spectral domain error functional minimization and rounding approximations. This allows us to construct Gaussian Markov random field approximations. We construct lattice approximations with finite‐difference methods. We show that the solutions can be constructed with overdetermined systems of stochastic matrix equations with sparse matrices, and we solve the system of equations with a sparse Cholesky decomposition. We consider convergence of the truncated Taylor approximation by studying band‐limited Matérn fields. We consider the convergence of the discrete approximations to the continuous limits. Finally, we study numerically the accuracy of different approximation methods with an interpolation problem.     

Impact of Stratification Imbalance on Probability of Type I Error

The impact of ignoring the stratification effect on the probability of a Type I error is investigated. The evaluation is in a clinical setting where the treatments may have different response rates among the strata. Deviation from the nominal probability of a Type I error, α, depends on the stratification imbalance and the heterogeneity in the response rates; it appears that the latter has a larger impact. The probability of a Type I error is depicted for cases in which the heterogeneity in the response rate is present but there is no stratification imbalance. Three-dimensional graphs are used to demonstrate the simultaneous impact of heterogeneity in response rates and of stratification imbalance.     

Consequences of effect size heterogeneity for meta-analysis: a Monte Carlo study

In this article we use Monte Carlo analysis to assess the small sample behaviour of the OLS, the weighted least squares (WLS) and the mixed effects meta-estimators under several types of effect size heterogeneity, using the bias, the mean squared error and the size and power of the statistical tests as performance indicators. Specifically, we analyse the consequences of heterogeneity in effect size precision (heteroskedasticity) and of two types of random effect size variation, one where the variation holds for the entire sample, and one where only a subset of the sample of studies is affected. Our results show that the mixed effects estimator is to be preferred to the other two estimators in the first two situations, but that WLS outperforms OLS and mixed effects in the third situation. Our findings therefore show that, under circumstances that are quite common in practice, using the mixed effects estimator may be suboptimal and that the use of WLS is preferable.     

Generalized Method of Moments Estimation When a Parameter Is on a Boundary

This article establishes the asymptotic distributions of generalized method of moments (GMM) estimators when the true parameter lies on the boundary of the parameter space. The conditions allow the estimator objective function to be nonsmooth and to depend on preliminary estimators. The boundary of the parameter space may be curved and/or kinked. The article discusses three examples: (1) instrumental variables (IV) estimation of a regression model with nonlinear equality and/or inequality restrictions on the parameters; (2) method of simulated moments estimation of a multinomial discrete response model with some random coefficient variances equal to 0, some random effect variances equal to 0, or some measurement error variances equal to 0; and (3) semiparametric least squares estimation of a partially linear regression model with nonlinear equality and/or inequality restrictions on the parameters.     

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.     

The present status of confidence interval estimation on variance components in balanced and unbalanced random models

Much research has been conducted to develop confidence Intervals on linear combinations and ratios of variance components in balanced and unbalanced random models.This paper first presents confidence intervals on functions of variance components in balanced designs.These results assume that classical analysis of variance sums of squares are independent and have exact scaled chi-squared distributions.In unbalanced designs, either one or both of these assumptions are violated, and modifications to the balanced model intervals are required.We report results of some recent work that examines various modifications for some particular unbalanced designs.     

Analysis of short-term systematic measurement error variance for the difference of paired data without repetition of measurement

The variance of short-term systematic measurement errors for the difference of paired data is estimated. The difference of paired data is determined by subtracting the measurement results of two methods, which measure the same item only once without measurement repetition. The unbiased estimators for short-term systematic measurement error variances based on the one-way random effects model are not fit for practical purpose because they can be negative. The estimators, which are derived for balanced data as well as for unbalanced data, are always positive but biased. The basis of these positive estimators is the one-way random effects model. The biases, variances, and the mean squared errors of the positive estimators are derived as well as their estimators. The positive estimators are fit for practical purpose.     

Multitaper testing of spectral lines and the detection of the solar rotation frequency and its harmonics

We develop statistical testing methodology for closely spaced spectral lines based on multitaper spectrum estimates and apply it to a time series of solar magnetic field magnitude data recorded by the Ulysses spacecraft. The test is formulated through complex-valued weighted least squares. Using this test in combination with the test for well-separated lines, we can accurately detect the solar equatorial rotation frequency of the sun, and its harmonics, from data recorded by Ulysses's magnetometer at solar mid-latitudes; this is a potentially important result for solar physicists, since it has implications for the structure and dynamics of magnetic fields in the solar corona.     

Performance of the weighted least squares apprdach to categorical data analysis

The weighted least squares method of Grizzle, Starmer and Koch is a common and convenient way to analyze categorical data. Tests of significance based on this method are typically performed by calculating a quadratic form which is asymptotically distrib-uted as a chi-square random variable under an appropriate null hy- pothesis.

This paper explores the validity of this distributional approximation for small samples in terms of the accuracy and power associated with this chi-square statistic. We report simulation results from the weighted least squares method applied to balanced factorial experiments, The simulations which constitute the majorpart of this work also provide insight into the relative performance of several response functions.     

Reducing errors-in-variables bias in linear regression using compact genetic algorithms

A new technique is devised to mitigate the errors-in-variables bias in linear regression. The procedure mimics a 2-stage least squares procedure where an auxiliary regression which generates a better behaved predictor variable is derived. The generated variable is then used as a substitute for the error-prone variable in the first-stage model. The performance of the algorithm is tested by simulation and regression analyses. Simulations suggest the algorithm efficiently captures the additive error term used to contaminate the artificial variables. Regressions provide further credit to the simulations as they clearly show that the compact genetic algorithm-based estimate of the true but unobserved regressor yields considerably better results. These conclusions are robust across different sample sizes and different variance structures imposed on both the measurement error and regression disturbances.     

Improved estimation of variance components in balanced hierarchical mixed models

The problem of simultaneous estimation of variance components is considered for a balanced hierarchical mixed model under a sum of squared error loss. A new class of estimators is suggested which dominate the usual sensible estimators. These estimators shrink towards the geometric mean of the component mean squares that appear in the ANOVA table. Numerical results are tabled to exhibit the improvement in risk under a simple model.     

Discrete Multicolour Random Mosaics with an Application to Network Extraction

We introduce a class of random fields that can be understood as discrete versions of multicolour polygonal fields built on regular linear tessellations. We focus first on a subclass of consistent polygonal fields, for which we show Markovianity and solvability by means of a dynamic representation. This representation is used to design new sampling techniques for Gibbsian modifications of such fields, a class which covers lattice‐based random fields. A flux‐based modification is applied to the extraction of the field tracks network from a Synthetic Aperture Radar image of a rural area.