Unbalanced Model Research Articles

Developments from analysis of variance through to generalized linear models and beyond

The collaborations between statisticians and biologists during the 100 years since AAB was founded have led to a very impressive list of statistical techniques, whose use now goes well beyond agriculture and biology. One example is the maximum likelihood methodology for probit analysis, arising from the collaboration between Sir Ronald Fisher and Chester Bliss. Others include analysis of variance, design of experiments, generalized linear models and the residual, or restricted, maximum likelihood (REML) algorithm for fitting unbalanced linear mixed models.

Bayes factor consistency for unbalanced ANOVA models

In practical situations, most experimental designs often yield unbalanced data which have different numbers of observations per unit because of cost constraints, missing data, etc. In this paper, we consider the Bayesian approach to hypothesis testing or model selection under the one-way unbalanced fixed-effects analysis-of-variance (ANOVA) model. We adopt Zellner's g-prior with the beta-prime distribution for g, which results in an explicit closed-form expression of the Bayes factor without integral representation. Furthermore, we investigate the model selection consistency of the Bayes factor under three different asymptotic scenarios: either the number of units goes to infinity, the number of observations per unit goes to infinity, or both go to infinity. The results presented extend some existing ones of the Bayes factor for the balanced ANOVA models in the literature.

Within-brand and cross-brand word-of-mouth for sequential multi-innovation diffusions

Competition between rival brands within the same category gives rise to special competition/substitution effects of great interest to involved firms. In homogeneous product categories, the model structure should allow distinguishing between within-brand and cross-brand word-of-mouth effects. In this paper, we propose a model that, besides separating these two imitative sources, also allows for a change in the parameter values of the first entrant as soon as the second one enters the market. We prove that our model has a closed-form solution, allowing parameters to be estimated with sales data. Moreover, we compare our model with other unbalanced models, both from a theoretical point of view and from an empirical one, by assessing their performance on pharmaceutical drug data.

The Cliff and Ord Test for Spatial Correlation of the Disturbances in Unbalanced Panel Models

In many cases, only an unbalanced panel data set with some observations missing at random is available. This note derives a Cliff and Ord test for spatially autocorrelated disturbances for such data. In a small Monte Carlo simulation exercise, the performance of the proposed test is similar to its balanced counterpart. In almost all simulation experiments, the test is properly sized. Naturally, the lower the power of the test, the higher the share of missing data.

Parametric bootstrap tests for unbalanced nested designs under heteroscedasticity

In this article, we consider the two-factor unbalanced nested design model without the assumption of equal error variance. For the problem of testing ‘main effects’ of both factors, we propose a parametric bootstrap (PB) approach and compare it with the existing generalized F (GF) test. The Type I error rates of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test performs better than the GF test. The PB test performs very satisfactorily even for small samples while the GF test exhibit poor Type I error properties when the number of factorial combinations or treatments goes up. It is also noted that the same tests can be used to test the significance of the random effect variance component in a two-factor mixed effects nested model under unequal error variances.

Master production scheduling and sequencing at mixed-model assembly lines in the automotive industry

The customization of final products in the automotive industry involves a large number of optional parts and leads to a huge variety of operation times at the various stations of the assembly line. The master production scheduling problem (MPS) for high-variant mixed-model assembly lines is to assign the individual customer-defined models of a basic product type to short-term production periods while anticipating the negative impacts of an unbalanced model sequence at the lower planning level. We propose a mathematical model formulation for the MPS and develop heuristic solution procedures that attempt to minimize the workload variability. Specifically, these procedures anticipate decisions on the mixed-model sequence and the resulting work overload at stations which has to be balanced by the assignment of utility workers. Furthermore, an integrated planning approach for solving the MPS and the production sequencing problem is proposed.

Generalized Point Estimation with Application to Small Response Estimation

Motivated by a number of drawbacks of classical methods of point estimation, we generalize the definitions of point estimation, and address such notions as unbiasedness and estimation under constraints. The utility of the extension is shown by deriving more reliable estimates for small coefficients of regression models, and for variance components and random effects of mixed models. The extension is in the spirit of generalized confidence intervals introduced by Weerahandi (1993) and should encourage much needed further research in point estimation in unbalanced models, multi-variate models, non normal models, and nonlinear models.

Inferences on the reliability in balanced and unbalanced one-way random models

In this article, the hypothesis testing and interval estimation for the reliability parameter are considered in balanced and unbalanced one-way random models. The tests and confidence intervals for the reliability parameter are developed using the concepts of generalized p-value and generalized confidence interval. Furthermore, some simulation results are presented to compare the performances between the proposed approach and the existing approach. For balanced models, the simulation results indicate that the proposed approach can provide satisfactory coverage probabilities and performs better than the existing approaches across the wide array of scenarios, especially for small sample sizes. For unbalanced models, the simulation results show that the two proposed approaches perform more satisfactorily than the existing approach in most cases. Finally, the proposed approaches are illustrated using two real examples.

Interval estimation of variance ratio in non-normal unbalanced one-way random models

Confidence intervals based on the harmonic mean method are proposed for estimating the ratio of variance components and/or the intraclass correlation coefficient (ρ) in unbalanced one-way random models without the normality assumptions. Since the proposed procedure is heavily dependent on the estimation of the kurtosis of the underlying distributions, bias-corrected estimators of kurtosis are proposed as well. Several asymptotic results concerning the proposed procedure are given along with simulation results to assess its performance in finite sample size situations. The proposed intervals are also compared with the corresponding confidence intervals based on the arithmetic mean method and were found to effectively maintain the nominal probability of coverage, except for leptokurtic distributions with fairly large kurtosis where the intervals tend to be liberal. According to the simulation results, the proposed harmonic mean intervals are recommended for use in practice. However, while computing the harmonic mean intervals, the bias-corrected estimators of kurtosis should be used when it is anticipated that ρ is small, but for large ρ the empirically corrected estimators of kurtosis should be adopted. The procedure is illustrated using a real data set.

On Computing Maximum-Likelihood Estimates of the Unbalanced Two-Way Random-Effects Model

This article applies the ECME algorithm to derive an easily implemented iterative feasible generalized least squares procedure for calculating maximum-likelihood estimates of the parameters of the unbalanced two-way random-effects model. The algorithm increases the log-likelihood monotonically and the fitted variance components are guaranteed to be non negative. This article applies the algorithm in an example.

Determinants of scale efficiency in the Brazilian 3PL industry: a 10-year analysis

This paper focuses on the Brazilian third-party logistics (3PL) sector which, facing increasing competition, offers clients a wide variety of services/IT technologies in the quest to bolster efficiency. The main research objective is to determine which variables significantly impact the 3PLs' scale efficiency by applying a two-stage DEA model. Based on an unbalanced panel model, secondary data from the annual study published by Revista Tecnologística (years 2001–2010) were collected and analysed. The required inputs and outputs for the analysis were identified, as were the contextual variables that can affect the scale efficiency of the 3PLs. The results support findings in the literature regarding the role coordination processes play in logistics performance. Managerial implications for both shippers and 3PLs are discussed.

Unbalanced distribution power-flow model and analysis of wind turbine generating systems

SUMMARY The paper presents an unbalanced three-phase power-flow model for wind turbine generating systems (WTGSs). The model takes into account voltage unbalance factor which exists at the point of common coupling. The developed model is integrated with the unbalanced forward/backward sweep power-flow method. The model comprises of three main components: they are the wind turbine, induction generator, and interface transformer to the grid. Due to their design symmetry, the generator and the transformer are modeled using symmetrical sequence networks. The results show that the developed model has robust convergence characteristics. The solution of the IEEE unbalanced radial feeders shows that the injected powers per phase due to the WTGS are not equal and strongly dependent on the voltage unbalance factor. The results also show that the simplified models based on positive sequence network lead to inaccurate overall power-flow solution. Copyright © 2012 John Wiley & Sons, Ltd.

Approximate interval for the between-group variance under heteroscedasticity

The paper presents an approximate confidence interval for the between-group variance in a one-way heteroscedastic random-effects model. Derivation of the interval is inspired by the ideas in Williams [A confidence intervals for variance components, Biometrika 49 (1962), pp. 278–281] for the homoscedastic case. Simulations suggest that the proposed interval performs better than or comparably to the other available procedures as studied in Wimmer and Witkovský [Between group variance component interval estimation for the unbalanced heteroscedastic one-way random effects model, J. Stat. Comput. Simul. 73 (2003), pp. 333–346], Hartung and Knapp [On confidence intervals for the among-group variance in the one-way random effects model with unequal error variances, J. Statist. Plann. Inference 127 (2005), pp. 157–177], Li [Comparison of confidence intervals on between group variance in unbalanced heteroscedastic one-way random models, Comm. Statist. Simulation Comput. 36 (2007), pp. 381–390]. Moreover, it is relatively easy to compute.

North European logistics flows with RIC: actor and national economy analysis

Emerging market trade of Sweden and Finland is growing, particularly so with Russia, India and China (RIC). However, trade is far from balanced: Russian exports dominate all trade statistics, and trade deficits swell. Chinese trade in the beginning of the last decade was very beneficial, but rapidly it changed to deficits, and Chinese manufacturing dominance. Indian relationship is still developing in manufacturing sector, since Indian economy has its strengths in IT and services. Research uses longitudinal survey data to examine logistics flow development between RIC and North Europe. Our findings show that in the near future situation stays put with China: North European companies continue to source and/or manufacturing products from China. Russia remains as ‘a great opportunity’ in the forthcoming future as it has done so also earlier. Indian logistics flows are starting to replicate Chinese unbalanced model, but lack volume.

Measurement Based on the "Group Linear Regression Method" of Harmonic Impedance

Harmonic pollution monitoring and control is the key to an accurate estimate of harmonic impedance, and thus an accurate assessment of public connection points of the harmonic emission level. This paper analyzes the link under the star-shaped and triangular three-phase asymmetrical power systems derived established a three-phase unbalanced power system harmonic impedance matrix estimation model. Model is based on a large number of systems and load the public connection point of the simulation data, the real and imaginary part of the harmonic impedance by opening and using grouping multiple linear regression method, on the system side harmonic impedance matrix to estimate the value of each element. Simulation and error analysis shows that the model obtained is more accurate and effective.

Sequential market entries and competition modelling in multi-innovation diffusions

The diffusion of innovations for simultaneous processes cannot take into account and properly explain systematic perturbations due to competition-substitution effects if they are examined one by one. A first aspect in simultaneous competing diffusions is the distinction between simultaneous market entries ( synchronic competition) and sequential entries ( diachronic competition). In the latter case, the beginning of competition may upset the first entrant’s diffusion. A second important aspect in multiple competition is represented by the choice to model the word-of-mouth effect either at the category level ( balanced model) or at the brand level, separating the within-brand effect from the cross-brand one ( unbalanced model). In this paper, balanced models are studied, and we propose a model that allows for a change in the parameter values of the first entrant as soon as the second one enters the market. The resulting differential system has a closed-form solution that enables, through sales data, an empirical validation of the assumptions underlying the model structure, improving the forecasting accuracy. An application to pharmaceutical drug competition is discussed.

The relative contribution of band number to phylogenetic accuracy in AFLP data sets

We examined the effect of increasing the number of sampled amplified fragment length polymorphism (AFLP) bands to reconstruct an accurate and well-supported AFLP-based phylogeny. In silico AFLP was performed using simulated DNA sequences evolving along balanced and unbalanced model trees with recent, uniform and ancient radiations and average branch lengths (from the most internal node to the tip) ranging from 0.02 to 0.05 substitutions per site. Trees were estimated by minimum evolution (ME) and maximum parsimony (MP) methods from both DNA sequences and virtual AFLP fingerprints. The comparison of the true tree with the estimated AFLP trees suggests that moderate numbers of AFLP bands are necessary to recover the correct topology with high bootstrap support values (i.e. >70%). Fewer numbers of bands are necessary for shorter tree lengths and for balanced than for unbalanced tree topologies. However, branch length estimation was rather unreliable and did not improve substantially after a certain number of bands were sampled. These results hold for different levels of genome coverage and number of taxa analysed. In silico AFLP using bacterial genomic DNA sequences recovered a well-supported tree topology that mirrored an empirical phylogeny based on a set of 31 orthologous gene sequences when as few as 263 AFLP bands were scored. These results suggest that AFLPs may be an efficient alternative to traditional DNA sequencing for accurate topology reconstruction of shallow trees when not very short ancestral branches exist.

Blood Pressure and Heart Rate Variability Complexity Analysis in Pregnant Women with Hypertension

Background. In this work, we perform a comparative analysis of blood pressure and heart rate variability complexity during pregnancy between normal, hypertensive, and preeclamptic women. Methods and Results. A total of 563 short electrocardiographic (10 min) records were obtained from 217 pregnant women (135 normal, 55 hypertensive, and 27 preeclamptic) during several gestational ages in sitting position. We used a mixed unbalanced model for the longitudinal statistical analysis and besides the conventional spectral analysis, we applied Lempel–Ziv complexity, sample entropy, approximated entropy, and detrended fluctuation analysis in the complexity measurement. Conclusions. The obtained results revealed significant differences between pathological and normal states with important considerations related to pregnancy adaptability and evolution as well as the relationship of complexity and blood pressure with factors such as maternal age, familial history of diabetes or hypertension, and parity.

Approximate tests in unbalanced two-way random models without interaction

In the presence of non-normality, we consider testing for the significance of the variance components in the unbalanced two-way random model without interaction. The approximate test is based on the F-statistic for this model. The asymptotic distribution of the F-statistic is derived as the number of treatments tends to infinity while the number of observations for a treatment in any block takes value from a finite set of positive integers. Robustness of the approximate test is given.

A note on fiducial generalized pivots for in one-way heteroscedastic ANOVA with random effects

The paper deals with generalized confidence intervals for the between-group variance in one-way heteroscedastic (unbalanced) ANOVA with random effects. The approach used mimics the standard one applied in mixed linear models with two variance components, where interval estimators are based on a minimal sufficient statistic derived after an initial reduction by the principle of invariance. A minimal sufficient statistic under heteroscedasticity is found to resemble its homoscedastic counterpart and further analogies between heteroscedastic and homoscedastic cases lead us to two classes of fiducial generalized pivots for the between-group variance. The procedures suggested formerly by Wimmer and Witkovský [Between group variance component interval estimation for the unbalanced heteroscedastic one-way random effects model, J. Stat. Comput. Simul. 73 (2003), pp. 333–346] and Li [Comparison of confidence intervals on between group variance in unbalanced heteroscedastic one-way random models, Comm. Statist. Simulation Comput. 36 (2007), pp. 381–390] are found to belong to these two classes. We comment briefly on some of their properties that were not mentioned in the original papers. In addition, properties of another particular generalized pivot are considered.