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In today’s digital era, establishing an online presence and maintaining a well-structured website is vitalfor companies to remain competitive in their respective markets. A crucial aspect of online success liesin strategically selecting the right words to optimize customer engagement and search engine visibility.However, this process is often time-consuming, involving extensive analysis of a company’s website aswell as its competitors’. This thesis focuses on developing an efficient binary classification approachto identify key words and phrases extracted from multiple company websites. The data set used forthis solution consists of approximately 92,000 scraped samples, primarily comprising non-key samples.Various features were extracted, and a word embedding model was employed to assess each sample’srelevance to its specific industry and topic. The logistic regression, decision tree and random forestalgorithms were all explored and implemented as different solutions to the classification problem. Theresults indicated that the logistic regression model excelled in retaining keywords but was less effectivein eliminating non-keywords. Conversely, the tree-based methods demonstrated superior classificationof keywords, albeit at the cost of misclassifying a few keywords. Overall, the random forest approachoutperformed the others, achieving a result of 76 percent in recall and 20 percent in precision whenpredicting key samples. In summary, this thesis presents a solution for classifying words and phrasesfrom company websites into key and non-key categories, and the developed methodology could offervaluable insights for companies seeking to enhance their website optimization strategies.

In this paper we endeavour to provide a largely non-technical description of the issues surrounding unbalanced factorial ANOVA and review the arguments made for and against the use of Type I, Type II and Type III sums of squares. Though the issue of which is the 'best' approach has been debated in the literature for decades, to date confusion remains around how the procedures differ and which is most appropriate. We ultimately recommend use of the Type II sums of squares for analysis of main effects because when no interaction is present it tests meaningful hypotheses and is the most statistically powerful alternative.

A balanced ANOVA design provides an unambiguous interpretation of the F-tests, and has more power than an unbalanced design. In earlier literature, multiple imputation was proposed to create balance in unbalanced designs, as an alternative to Type-III sum of squares. In the current simulation study we studied four pooled statistics for multiple imputation, namely D₀, D₁, D₂, and D₃ in unbalanced data, and compared them with Type-III sum of squares. Statistics D₁ and D₂ generally performed best regarding Type-I error rates, and had power rates closest to that of Type-III sum of squares. Additionally, for the interaction, D₁ produced power rates higher than Type-III sum of squares. For multiply imputed datasets D₁ and D₂ may be the best methods for pooling the results in multiply imputed datasets, and for unbalanced data, D₁ might be a good alternative to Type-III sum of squares regarding the interaction.

Retraction(s) for this article:Methodology, 16(4), 354–355, https://doi.org/10.5964/meth.6073 A new version is available:Methodology, 17(1), 39–57, https://doi.org/10.5964/meth.6085 Original abstract:In earlier literature, multiple imputation was proposed to create balance in unbalanced designs, as an alternative to Type III sum of squares in two-way ANOVA. In the current simulation study we studied four pooled statistics for multiple imputation, namely D₀, D₁, D₂, and D₃ in unbalanced data, and compared these statistics with Type III sum of squares. Statistics D₀ and D₂ generally performed best regarding Type-I error rates, and had power rates closest to that of Type III sum of squares. However, none of the statistics produced power rates higher than Type III sum of squares. The results lead to the conclusion that for multiply imputed datasets D₀ and D₂ may be the best methods for pooling the results of multiparameter estimates in multiply imputed datasets, and that for unbalanced data, Type III sum of square is to be preferred over using multiple imputation in obtaining ANOVA results.

In the case of extremely unbalanced data, the results of the traditional classification algorithm are very unbalanced, and most samples are often divided into the categories of majority samples, so the accuracy of judgment of the minority classes will be reduced. In this paper, we propose a classification algorithm for unbalanced data based on RSM and binomial undersampling. We use RSM’s random part features rather than all each classifier to make each training classifier reduce the dimensions, and dimension reduction makes relatively minority class samples indirectly lift. Using the above characteristics of the RSM to reduce dimension can solve the problem that unbalanced data classification in the minority class samples is too little, and it can also find the important attribute of variables to make the model have the ability of explanation. Experiments show that our algorithm has high classification accuracy and model interpretation ability when classifying unbalanced data.

Graduate school of Bogor Agricultural University (SPs-IPB) stated that not all students of IPB master program successfully complete their studies. This becomes an evaluation for IPB to be more selective in choosing students in the future. This study aims to model the success classification of IPB master students in 2011 to 2015. The classification method used is rotation forest. The percentage of students who graduated is very large compared to those who did not pass, this can cause the evaluation value different. SMOTE (Synthetic Minority Oversampling Technique) is one of method to handle such unbalanced data by generating artificial data. The ROC (Receiver Operating Characteristic) curve is built to see the optimum cut off value. There are two classification models, they are rotation forest models before and after handled by SMOTE. The comparison results show that the rotation forest model after SMOTE with cut off value 0.6 is the best model. This model can increase the sensitivity value more than 50% although the accuracy and specificity value decreased compared to the modeling before SMOTE.

When faced with unbalanced data, it is often necessary to estimate the necessary missing values before the application of the analysis of variance technique. Previous studies have shown that different designs require different missing value estimators. With the introduction of some relatively new statistical designs, it has become expedient to derive missing value estimators for such designs. In this study, least squares estimators of missing values in a three-factor nested-factorial design are derived. Properties of the estimators are equally determined. A numerical example is given to show the application of the theoretical results obtained in this paper. Our empirical results establish the appropriateness of the missing value estimation method presented in this study.

Species distribution modelling (SDM) gained importance on biodiversity distribution and conservation studies worldwide, including prioritizing areas for public policies and international treaties. Useful for large-scale approaches and species distribution estimates, it is a plus considering that a minor fraction of the planet is adequately sampled. However, minimizing errors is challenging, but essential, considering the uses and consequences of such models. In situ validation of the SDM outputs should be a key-step—in some cases, urgent. Bioacoustics can be used to validate and refine those outputs, especially if the focal species’ vocalizations are conspicuous and species-specific. This is the case of echolocating bats. Here, we used extensive acoustic monitoring (>120 validation points over an area of >758,000 km2, and producing >300,000 sound files) to validate MaxEnt outputs for six neotropical bat species in a poorly-sampled region of Brazil. Based on in situ validation, we evaluated four threshold-dependent theoretical evaluation metrics’ ability in predicting models’ performance. We also assessed the performance of three widely used thresholds to convert continuous SDMs into presence/absence maps. We demonstrated that MaxEnt produces very different outputs, requiring a careful choice on thresholds and modeling parameters. Although all theoretical evaluation metrics studied were positively correlated with accuracy, we empirically demonstrated that metrics based on specificity-sensitivity and sensitivity-precision are better for testing models, considering that most SDMs are based on unbalanced data. Without independent field validation, we found that using an arbitrary threshold for modelling can be a precarious approach with many possible outcomes, even after getting good evaluation scores. Bioacoustics proved to be important for validating SDMs for the six bat species analyzed, allowing a better refinement of SDMs in large and under-sampled regions, with relatively low sampling effort. Regardless of the species assessing method used, our research highlighted the vital necessity of in situ validation for SDMs.

In order to establish a nondestructive detection model for mouldy core disease in apples, 265 samples of healthy and suspected disease fruits were tested for 11 dielectric parameters in the 100hz - 3.98Mhz frequency. After dissection, the 17 diseased apples were observed. The Extra-Tree algorithm was used to extract the characteristics of samples. It found that the parameters of loss factor, conductance, negative impedance phase, parallel resistance, series resistance, and admittance in the low-frequency band of 100hz-1580hz had strong discriminability. As the number of healthy fruits and diseased fruits are unbalanced, using the SMOTE and other sampling techniques. Then, six classifiers were used to build the discriminant models. Experiments show that SMOTE-Tomek Link technique maintains the data distribution better, and GBDT classifier has the highest accuracy of detection, up to 95.97%. Thus, verifying the effectiveness of apple mouldy core disease nondestructive detection model based on unbalanced dielectric data.

A test procedure for testing homogeneity of location parameters against simple ordered alternative is proposed for k(k ≥ 2) members of two parameter exponential distribution under unbalanced data and heteroscedasticity of the scale parameters. The relevant one-sided and two-sided simultaneous confidence intervals (SCIs) for all k(k − 1)/2 ordered pairwise differences of location parameters are also proposed. Simulation-based study revealed that the proposed procedure is better than the recently proposed procedure in terms of power, coverage probability, and average volume of SCIs. The implementation of proposed procedure is demonstrated through real life data.

Unanticipated events often occur in the development of an experiment, especially in field experiments, often causing data loss. Through the present study, we sought to verify whether there is a difference in the result of the analysis of variance (ANOVA) test for unbalanced data in the Completely Randomized Design (DIC) when the inclusion of data obtained from the data imputation technique was performed of Predictive Average. Experiments were simulated in the DIC with 5 treatments and 10 repetitions, generating complete databases. From each database, 10% of the plots were removed and after the imputation method was applied, comparing the ANOVA results in each step. The imputation yielded acceptable results, but not better than those obtained when performing the specific ANOVA test for unbalanced data.

The expected mean squares for unbalanced mixed effect interactive model were derived using Brute Force Method. From the expected mean squares, there are no obvious denominators for testing for the main effects when the factors are mixed. An expression for F-test for testing for the main effects was derived which was proved to be unbiased.

Convergence problems often arise when complex linear mixed-effects models are fitted. Previous simulation studies (see, e.g. [Buyse M, Molenberghs G, Burzykowski T, Renard D, Geys H. The validation of surrogate endpoints in meta-analyses of randomized experiments. Biostatistics. 2000;1:49–67, Renard D, Geys H, Molenberghs G, Burzykowski T, Buyse M. Validation of surrogate endpoints in multiple randomized clinical trials with discrete outcomes. Biom J. 2002;44:921–935]) have shown that model convergence rates were higher (i) when the number of available clusters in the data increased, and (ii) when the size of the between-cluster variability increased (relative to the size of the residual variability). The aim of the present simulation study is to further extend these findings by examining the effect of an additional factor that is hypothesized to affect model convergence, i.e. imbalance in cluster size. The results showed that divergence rates were substantially higher for data sets with unbalanced cluster sizes – in particular when the model at hand had a complex hierarchical structure. Furthermore, the use of multiple imputation to restore ‘balance’ in unbalanced data sets reduces model convergence problems. Financial support from the IAP research network #P7/06 of the Belgian Government (Belgian Science Policy) is gratefully acknowledged. Wim Van der Elst acknowledges funding from the European Seventh Framework programme FP7 2007 − 2013 under grant agreement Nr. 602552. Geert Molenberghs acknowledges funding from Intel, Janssen Pharmaceutica and by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT).

Cultivars have to be evaluated under different crop management systems across agro-ecosystems and years using multi-environment trials (MET) before releasing them to the market. Frequently, data collected in METs are arranged according to cultivar (G), management (M), location, (L) and year (Y) combinations in a four-way G x M x L x Y data table that is highly unbalanced for cultivars across locations and time. Therefore, we present the restricted maximum likelihood method (REML) for linear mixed models (LMM) with a factor analytic variance-covariance matrix for assessing cultivar adaptation to crop management systems and environments based on unbalanced datasets. Such a multi-environmental trial system has been in operation in Poland for winter wheat (Triticum aestivum L.) in the form of the Post-registration Variety Testing System (PVTS). This study aimed to illustrate the use of LMM in the analysis of unbalanced four-way G x M x L x Y data. LMM analysis provided adjusted means of grain yield for 51 winter wheat cultivars bred in different regions in Europe, tested across 18 trial locations and seven consecutive cropping seasons in two crop management intensities. The application of the four-way LMM with a factor analytic variance-covariance matrix is a complementary and effective tool for evaluating the unbalanced G x M x L x Y table. Cultivars tested had different adaptive responses to the Polish agro-ecosystems separately for each of the crop management intensities. Wide adaptation in both crop management systems was exhibited by cultivars Mulan and Jenga bred in Germany.

In this article, we propose a parametric bootstrap (PB) test for testing main, simple and interaction effects in heteroscedastic two-way MANOVA models under multivariate normality. The PB test is shown to be invariant under permutation-transformations, and affine-transformations, respectively. Moreover, the PB test is independent of the choice of weights used to define the parameters uniquely. The proposed test is compared with existing Lawley-Hotelling trace (LHT) and approximate Hotelling T 2 (AHT) tests by the invariance and the intensive simulations. Simulation results indicate that the PB test performs satisfactorily for various cell sizes and parameter configurations when the homogeneity assumption is seriously violated, and tends to outperform the LHT and AHT tests for moderate or larger samples in terms of power and controlling size. In addition, simulation results also indicate that the PB test does not lose too much power when the homogeneity assumption is actually valid or the model assumptions are approximately correct. We propose a parametric bootstrap (PB) test in heteroscedastic two-way MANOVA.The PB test is invariant under affine- and permutation-transformations.The PB test is independent of the choices of weights used to identify the parameters.

This article derives generalized prediction intervals for random effects in linear random-effects models. For balanced and unbalanced data in two-way layouts, models are considered with and without interaction. Coverage of the proposed generalized prediction intervals was estimated in a simulation study based on an agricultural field experiment. Generalized prediction intervals were compared with prediction intervals based on the restricted maximum likelihood (REML) procedure and the approximate methods of Satterthwaite and Kenward and Roger. The simulation study showed that coverage of generalized prediction intervals was closer to the nominal level 0.95 than coverage of prediction intervals based on the REML procedure.

In recent years, the focus on validating the statistical methods used in the field of neuroimaging has increased. While several papers have already highlighted the importance of non-parametric methods and especially permutation testing for general linear models (GLMs), it seems like the importance of validating classification results other than through cross-validation has taken a back seat. But classification, especially binary classification, is one of the most common tools in neuroimaging. Often permutations are not performed using the argument that they are too computationally expensive, especially for trainingintensive classifier as e.g. neural networks. In the following we want to re-visit the use of permutation tests for validating cross-validation results statistically and employ recent approximate permutation methods that reduce the number of permutations that need to be performed. We evaluate the feasibility of using full as well as approximate permutation methods in the extreme cases of small and unbalanced data sets. Our results indicate the applicability of a tail and Gamma approximation to perform permutation testing for binary classification tasks.

This research is to provide a solution of one-way ANOVA without using transformation when variances are heteroscedastic and group sizes are unequal. Parametric bootstrap test (Krishnamoorthy et al., 2007) has been shown to be competitive with many other methods when testing the equality of group means. We extend the parametric bootstrap algorithm to a multiple comparison procedure. Simulation results show that the parametric bootstrap approach works well for one-way ANOVA.

A system of regression equations for analyzing panel data with random heterogeneity in intercepts and coefficients, and unbalanced panel data is considered. A maximum likelihood (ML) procedure for joint estimation of all parameters is described. Since its implementation for numerical computation is complicated, simplified procedures are presented. The simplifications essentially concern the estimation of the covariance matrices of the random coefficients. The application and ‘anatomy’ of the proposed algorithm for modified ML estimation are illustrated by using panel data for output, inputs and costs for 111 manufacturing firms observed up to 22 years.

The classification of multi-class microarray datasets is much more difficult compared with the binary datasets because the former usually consist of unbalanced data with a smaller sample size in each class. Our paper focuses on the multi-class problem, and proposes a new method based on a class-overlapping measure, named as Minimum Class-Overlapping Error-Correcting Output Codes (MCO-ECOC). In this algorithm, important variables are selected through different filter methods firstly. Then, the class overlapping is measured in training sets, the algorithm searches all class splitting schemes, and select the one minimizing the class-overlapping measure. Each column of the coding matrix represents such a splitting scheme. And then all the coding matrixs are combined by eliminating the redundant columns to make the final ensemble system compact. Neural networks are used as binary classifiers. MCO-ECOC algorithm is applied to classify the different multi-class microarray datasets, and the output of each base learner are fused to produce the final decision based on the Hamming distance. The experimental results show that the performance of MCO-ECOC is significantly higher than those obtained by DECOC and Forest ECOC.

Active learning is innately very suitable for addressing class imbalance problems, as the actively selected training set is significantly less unbalanced than the real distribution. An improved active learning algorithm named CMIES (correcting mistakes in early stages) was proposed. It is based on the belief that a learner can learn more from correcting a mistake than ascertaining something uncertain. In early stages of active learning, the confidence of the classifier is very low. Even instance with the highest probability to be positive according to the current classifier is quite likely to be actually negative, especially with unbalanced data distribution. Correcting such a mistake is much more beneficial than learning from the most uncertain instance, resulting in extreme shrinkage of the version space and thus great improvement of the classification performance.

International audience; AbstractKey messageMultivariate mixed models can be used to combine complex data into a single-step analysis to improve prediction accuracy of open-pollinated seed lots for all attributes and candidates, and identify elite seed lots.ContextData available for genetic selection may be complex and unbalanced; however, utilisation of all available information for prediction of genetic value may improve prediction accuracy to better identify elite candidates for selection.AimsThis study aimed to develop, implement and evaluate a single-step multivariate mixed model for complex and unbalanced data and use the results to identify elite candidates.MethodsMultivariate mixed models were developed and applied to a case study of seed-orchard open-pollinated Eucalyptus dunnii families grown in progeny trials in Australia and Uruguay to identify elite seed lots for biofuel utilisation. This approach combined all available data across trials and ages and included models of spatial variation to predict OP seed lot values for growth and wood quality attributes. Predictions were used to estimate response to selection and correlations between breeding values of parents predicted from their own performance and seed lot values predicted from the progeny trials.ResultsPrediction accuracy was highest for a single-step multivariate model. Prediction of seed lot values using this model indicated that selection of the best 12.5 % resulted in a gain of 30 % in cellulose content, and breeding value of parents predicted from own performance was only weakly correlated with seed lot performance in progeny trials.ConclusionSingle-step multivariate approaches provide the most accurate prediction of genetic value for all attributes, for all candidates, and hence leads to greater selection response. In eucalypts, gain from selection using seed lot values predicted from progeny trials will be greater than from selection using breeding values of parents predicted from their own performance.

This paper suggests a method for getting sums of squares due to sources of variation under the assumption of two-way fixed effects model. The method used for calculating the quantities due to fixed-effects is based on the projections of an observation vector y on the column space generated by the model matrix X under the assumed model. The suggested method shows that the calculation of Type II sums of squares by projections is much easier than the classical Type II analysis.

In this article, we have developed a rank (intra-subject) based analysis of clinical trials with unbalanced repeated measures data. We assume that the errors within each patient are exchangeable and continuous random variables. This rank-based inference is valid when the unbalanced data are missing either completely at random or by design. A drop in dispersion test is developed for general linear hypotheses. A numerical example is given to illustrate the procedure.

It is necessary for enterprises to establish a customer churn management system. In this article, we take the heterogeneity into consideration and divide the churn people into two classes according to the data characteristics. Moreover, we try to modify the bias of multi-class unbalanced data classification. Then we propose a new method based on Real Adaboost for the problem. The proposed method takes the within-group error into consideration and creates another view of reweighing the cases. Empirical study on our sample data shows that the new method performs better than the other method.

The research on unbalanced data classification is a hot topic in recent years. LogitBoost algorithm is an adaptive algorithm that can get much higher prediction precision. But in the face of unbalanced data, this algorithm could produce a large minority class prediction error. In this article, we propose an improved LogitBoost algorithm named BLogitBoost, based on a stratified normalization method which normalizes within class sampling probability first, then normalizes between classes. The experiments on simulation data and empirical data show that the new algorithm can reduce the minority class prediction error significantly.

Matrix Pencil method is one of the promising method to estimate DOA in non-stationary, multi-path coherent environment. Not only the Matrix Pencil Method offers better resolution than the conventional approach using covariance matrix, but also it is computationally very efficient. In this paper, we presented an effect of unbalanced data weighting in the formulation of the Matrix Pencil method. A new formulation has been suggested to mitigate the effect of unbalanced data weighting. Numerical simulation demonstrated that the proposed method can successfully eliminate the problem of unbalanced data weighting.

This paper discusses how to get the sums of squares due to treatment factors when Type I Analysis is used by projections for the analysis of data under the assumption of a two-way ANOVA model. The suggested method does not need to calculate the residual sums of squares for the calculation of sums of squares. There-fore, the calculation is easier and faster than classical ANOVA methods. It also discusses how eigenvectors and eigenvalues of the projection matrices can be used to get the calculation of sums of squares. An example is given to illustrate the calculation procedure by projections for unbalanced data.

Construction of closed-form confidence intervals on linear combinations of variance components were developed generically for balanced data and studied mainly for one-way and two-way random effects analysis of variance models. The Satterthwaite approach is easily generalized to unbalanced data and modified to increase its coverage probability. They are applied on measures of assay precision in combination with (restricted) maximum likelihood and Henderson III Type 1 and 3 estimation. Simulations of interlaboratory studies with unbalanced data and with small sample sizes do not show superiority of any of the possible combinations of estimation methods and Satterthwaite approaches on three measures of assay precision. However, the modified Satterthwaite approach with Henderson III Type 3 estimation is often preferred above the other combinations.

SUMMARYOn-farm participatory varietal selection (PVS) trials are often of two types: mother trials (with all of the entries) and baby trials (each having one, or very few of the entries from the mother trials). We conducted PVS trials on 17 wheat varieties in 12 villages of four districts of Bangladesh over three years but the data were highly unbalanced. Both quantitative and qualitative traits were measured in the on-farm trials. The factors in the trials were both fixed effects (varieties and districts) and random (years and farmers). We used the residual or restricted maximum likelihood (REML) analysis for the mixed model for quantitative traits. For qualitative data on farmers' perceptions, logistic regression procedures were used that are equally applicable to balanced and unbalanced data sets. The REML analysis provided adjusted mean values for quantitative traits for all the varieties, for the mother and baby trials separately, using the data from all years and all locations. It identified varieties BAW 1006 and BAW 1008 that yielded 19–30% more than the control Kanchan and also had a higher 1000-grain weight, were at least as early to flower and had a high overall ranking by farmers in the mother trials. The logistic regression analysis of perception data agreed with the results of the REML analysis as these varieties were most preferred by farmers for grain yield, earlier maturity and better chapatti making quality. The less labour-intensive method of recording qualitative perceptions can usefully replace actual yield measurements, particularly when validated by other participatory measures such as intended and actual adoption. In 2005, BAW 1006 was released as BARI Gom 23 or Bijoy and BAW 1008 as BARI Gom 24 or Prodip for the whole of Bangladesh. The validity of the results of the REML analysis was confirmed by the high early adoption trends of the identified varieties. Since REML is an effective analysis for unbalanced PVS trial data using a mixed model, its wider use by researchers would increase the value of the PVS process.

The authors consider general estimators for the mean and variance parameters in the random effect model and in the transformation model for data with multiple levels of variation. They show that these estimators have different distributions under the two models unless all the variables have Gaussian distributions. They investigate the asymptotic properties of bootstrap procedures designed for the two models. They also report simulation results and illustrate the bootstraps using data on red spruce trees. Reechantillonnage de donnees a variation multiniveau Les auteurs s'interessent a des estimateurs generaux des parametres de moyenne et variance dans le modele a effets aleatoires et le modele de transformation pour des donnees a variation multiniveau. Ils montrent que la loi de ces estimateurs depend du modele sauf si toutes les variables sont gaussiennes. Ils explorent les proprietes asymptotiques de procedures bootstrap propres aux deux modeles. Ils presentent des resultats de simulation et illustrent l'emploi de ces bootstraps a l'aide de donnees sur l'epinette rouge.

Yang and Land (2006) and Yang (forthcoming-b) developed a mixed (fixed and random) effects model for the age–period–cohort (APC) analysis of micro data sets in the form of a series of repeated cross-section sample surveys that are increasingly available to demographers. The authors compare the fixedversus random-effects model specifications for APC analysis. They use data on verbal test scores from 15 cross sections of the General Social Survey (GSS), 1974 to 2000, for substantive illustrations. Strengths and weaknesses are identified for both the random- and fixed-effects formulations. However, under each of the two data conditions studied, the random-effects hierarchical APC model is the most appropriate specification. While additional analyses and comparisons of random- and fixed-effects APC models using other data sets are necessary before generalizations can be drawn, this finding is consistent with results from other methodological studies with unbalanced data designs.

Summary We consider a well-known controversy that stems from the use of two mixed models for the analysis of balanced experimental data with a fixed and a random factor. It essentially originates in the different statistics developed from such models for testing that the variance parameter associated to the random factor is null. The corresponding hypotheses are interpreted as that of null random factor main effects in the presence of interaction. The controversy is further complicated by different opinions regarding the appropriateness of such hypothesis. Assuming that this is a sensible option, we show that the standard test statistics obtained under both models are really directed at different hypotheses and conclude that the problem lies in the definition of the main effects and interactions. We use expected values as in the fixed effects case to resolve the controversy showing that under the most commonly used model, the test usually associated to the inexistence of the random factor main effects addresses a different hypothesis. We discuss the choice of models, and some further problems that occur in the presence of unbalanced data. Resume Nous considerons une controverse bien-connue provenant de l'emploi de deux modeles mixtes avec un facteur fixe et un facteur aleatoire pour l'analyse de donnees experimentales. Le probleme provient essentiellement des statistiques differentes developpees a partir de ces modeles pour tester que le parametre de variance associe au facteur aleatoire est nul. Les hypotheses correspondantes sont interpretees comme celles de l'inexistence de l'effet principal du facteur aeatoire en presence de l'interaction entre les deux facteurs. La controverse est rendue plus complexe encore par les differentes opinions sur la propriete de cette hypothese. En admettant que le choix est sense, nous montrons que les statistiques usuelles obtenues a partir des deux modeles a s'adressent reellement a des hypotheses differentes et nous arrivons a la conclusion que le probleme est cause par les definitions des effets principaux et de l'interaction. Nous utilisons des valeurs moyennes comme dans le cas de modeles a effets fixes pour resoudre la controverse et nous montrons que selon le modele plus utilise, le test generalement associe a l'inexistence de l'effet principal du facteur aleatoire en presence de l'interaction s'adresse, en realite, a une hypothese differente. Nous discutons du choix des modeles et d'autres problemes qui interviennent en presence de donnees non-equilibrees.

The additive genetic value for oil yield of 135 parents from several African populations tested in the first cycle of Indonesian Oil Palm Research Institute (IOPRI) breeding programme were estimated by best linear unbiased prediction (BLUP) using an unbalanced data set. These values were used to evaluate the possibility of reduction in generation selection time in an oil palm breeding programme. The ranks of parental additive genetic values obtained with early yield period and whole cycle was found to be consistent. These make that highly potential parents could be selected and recombined at a more precocious time in order to reduce the period between cycle. Since oil yield trait is mainly controlled under additive gene effect, the recombination should be done carefully for retaining as highly as possible this character in a parent to be improved.

In this study, the MIVQUE estimators of variance components for the one-way random model with unbalanced data are investigated. In order to compare the efficiency of MIVQUE estimators obtained by using three priori estimates, the Empirical Quantile Dispersion Graphs (EQDGs) are used. From the results of Monte-Carlo study, the MIVQUE estimator using ${\sigma}^2_{\alpha}\

A procedure for constructing two-sided beta-content, gamma-confidence tolerance intervals is proposed for general random effects models, in both balanced and unbalanced data scenarios. The proposed intervals are based on the concept of effective sample size and modified large sample methods for constructing confidence bounds on functions of variance components. The performance of the proposed intervals is evaluated via simulation techniques. The results indicate that the proposed intervals generally maintain the nominal confidence and content levels. Application of the proposed procedure is illustrated with a one-fold nested design used to evaluate the performance of a quantitative bioanalytical method.

Methods for analyzing unbalanced factorial designs can be traced back to Yates (1934). Today, most major statistical programs perform, by default, unbalanced ANOVA based on Type III sums of squares (Yates's weighted squares of means). As criticized by Nelder and Lane (1995), this analysis is founded on unrealistic models—models with interactions, but without all corresponding main effects. The Type II analysis (Yates's method of fitting constants) is usually not preferred because of the underlying assumption of no interactions. This argument is, however, also founded on unrealistic models. Furthermore, by considering the power of the two methods, it is clear that Type II is preferable.

This paper approachs the new estimator for variance components in mixed linear models with an arbitrary number of variance components, called sub-d. This estimator was deduced and tested in random 'one-way' and 'two-way' nested and crossed designs with balanced or unbalanced data, by Silva (2017); specifically, this paper aims to give the sub-d explicit formula for the two variance components in random 'one-way' designs, ensuring their existence through consistent theoretical results. In order to derive the explicit above announced formula, we propose and prove some robust algebraic results. A numerical example where both variance components are estimated is given.

Investigation of species abundance has become a vital component of many ecological monitoring studies. The primary objective of these studies is to understand how specific species are distributed across the study domain, as well as quantification of the sampling efficiency for detecting these species. To achieve these goals, preselected locations are sampled during scheduled visits, in which the number of species observed at each location is recorded. This results in spatially referenced replicated count data that are often unbalanced in structure and exhibit overdispersion. Motivated by the Baltimore Ecosystem Study, we propose Bayesian hierarchical binomial mixture models, including Binomial Conway-Maxwell Poisson (Bin-CMP) mixture models, that formally account for varying levels of spatial dispersion. Our proposed models also allow for variable selection of model covariates and grouping of dispersion parameters through the implementation of reversible jump Markov chain Monte Carlo methodology. Finally, using demographic covariates from the American Community Survey, we demonstrate the effectiveness of our approach through estimation of abundance for the American Robin (Turdus migratorius) in the Baltimore Ecosystem Study., Comment: Published at http://dx.doi.org/10.1214/14-AOAS801 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The plant breeding program of the National University of Colombia carried out 10 potato regional trials during 1998 and 1999 to evaluate 15 promising potato clones (Solanum tuberosum ssp.andigena). Genotype by environment interaction was analyzed using Kang’s methodology, which links yield performance and phenotypic stability. A MACRO was implemented using the SAS system to obtain yield-stability indices. Shukla’s variance was estimated by restricted maximum likelihood (REML), which allowed handling unbalanced data for both genotypes and replicates. In addition, a multivariate analysis methodology was developed, based on the yield-stability index. This methodology is useful when tuber yield is categorized by size and quality, which is usual in Colombian potato harvests.

This paper derives several Lagrange Multiplier tests for the unbalanced nested error component model. Economic data with a natural nested grouping include firms grouped by industry; or students grouped by schools. The LM tests derived include the joint test for both effects as well as the test for one effect conditional on the presence of the other. The paper also derives the standardized versions of these tests, their asymptotic locally mean most powerful version as well as their robust to local misspecification version. Monte Carlo experiments are conducted to study the performance of these LM tests.

The purpose of this study was to compare the ability of non-compartmental analysis and compartmental mixed effects modelling (MEM) to determine the existence and magnitude of exposure differences (i.e. exposure ratio estimates) between subsets of animals during destructive toxicokinetic studies. Data from five toxicokinetic studies of an experimental compound were analysed using a linear trapezoidal calculation of the area under the curve (non-compartmental analysis) or modelled using MEM. With the non-compartmental method the Bailer-Satterthwaite approximation was used to construct confidence intervals around the exposure estimates of each subset of animals and these were used to determine if exposure differed between the subsets. The MEM analyses were performed on the full datasets and on datasets with arbitrary reductions in the number of animal replicates. With MEM, additional model parameters were used to differentiate between subsets of animals, and were incorporated only if they were justified statistically. Estimates of the existence and magnitude of exposure differences between animal subsets were similar with the two techniques. The MEM analyses were influenced only marginally by substantial reductions in the number of animals studied and were less compromised by extremely limited or unbalanced data. These analyses show that MEM and non-compartmental methods are similarly effective at detecting exposure differences between animal subsets in toxicokinetic studies. Estimates provided by both methods were influenced by the degree of variance in the data. These results support the proposition that it may be possible to reduce the number of animals employed in toxicokinetic studies if MEM is used.

Multienvironment trials are often analyzed to assess the yield stability of genotypes. Most of the common stability measures correspond to parameters of a mixed model with fixed genotypes and random environments. Analysis within a mixed model framework allows unbalanced data to be handled. This note shows how to fit the most common stability models using the MIXED procedure of the SAS system.

The problem of estimation of variance components based on diallel model for unbalanced data has been addressed. The least squares approach to quadratic estimation has been adopted in obtaining the explicit solutions for the design and genetic components of variance.

Some general issues connected with components of variance are reviewed. These include matters of definition, of formal inference in a normal-theory context, of goodness of fit, and of simple methods for unbalanced data.

For a diallel model (TOPHAM, 1966) estimators of variance components are obtained by adopting the general approach outlined by SEELY (1969, 1970, 1971), SEELY and ZYSKIND (1971) and YUAN (1977) in the case of unbalanced data.

Analysis of variance (ANOVA) is a procedure frequently used for analyzing experimental and quasi-experimental data in psychology. Nonetheless, there is confusion which subtype to prefer for unbalanced data. Much of this confusion can be prevented when an adequate hypothesis is formulated at first. In the present paper this is done by using a theory of causal effects. This is the starting point for the following simulation study done on unbalanced two-way designs. Simulated data sets differed in the presence of an (average) effect, the degree of interaction, total sample size, stochasticity of subsample sizes and if there was confounding between the two factors (i.e. experimental vs. quasi-experimental design). Different subtypes of ANOVA as well as other competing procedures from the research on causal effects were compared with regard to type-I-error rate and power. Results suggest that different types of ANOVA should be used with care, especially in quasi-experimental designs and when there is interaction. Procedures developed within the research on causal effects are feasible alternatives that may serve better to answer meaningful hypotheses.

We consider inference for row effects in the presence of possible interactions in a two-way fixed effects model when the numbers of observations are themselves random variables. Let Nij be the number of observations in the ( i , j ) cell, π ij be the probability that a particular observation is in that cell and μ ij be the expected value of an observation in that cell. We assume that the { N ij } have a joint multinomial distribution with parameters n and { π ij } . Then μ ¯ i . = ∑ j π ij μ ij / ∑ j π ij is the expected value of a randomly chosen observation in the ith row. Hence, we consider testing that the μ ¯ i . are equal. With the { π ij } unknown, there is no obvious sum of squares and F-ratio computed by the widely available statistical packages for testing this hypothesis. Let Y ¯ i ‥ be the sample mean of the observations in the ith row. We show that Y ¯ i ‥ is an MLE of μ ¯ i . , is consistent and is conditionally unbiased. We then find the asymptotic joint distribution of the Y ¯ i ‥ and use it to construct a sensible asymptotic size α test of the equality of the μ ¯ i . and asymptotic simultaneous ( 1 − α ) confidence intervals for contrasts in the μ ¯ i . .