Two non-parametric, analysis-of-means-type tests for homogeneity of variances

After a brief review of the literature, two non-parametric tests for homogeneity of variances are presented. The first test is based on the analysis of means for ranks, which is a non-parametric version of the analysis of means (ANOM) that uses ranks as input for an ANOM test. The second test uses inverse normal scores of the ranks of scale transformations of the observations as input to the ANOM. Both homogeneity of variances tests can be presented in a graphical form, which makes it easy for practitioners to assess the practical and the statistical significance. A Monte Carlo study is used to show that these tests have power comparable with that of well-known robust tests for homogeneity of variances.     

Analysis of means: a generalized approach using R

Papers on the analysis of means (ANOM) have been circulating in the quality control literature for decades, routinely describing it as a statistical stand-alone concept. Therefore, we clarify that ANOM should rather be regarded as a special case of a much more universal approach known as multiple contrast tests (MCTs). Perceiving ANOM as a grand-mean-type MCT paves the way for implementing it in the open-source software R. We give a brief tutorial on how to exploit R's versatility and introduce the R package ANOM for drawing the familiar decision charts. Beyond that, we illustrate two practical aspects of data analysis with ANOM: firstly, we compare merits and drawbacks of ANOM-type MCTs and ANOVA F-test and assess their respective statistical powers, and secondly, we show that the benefit of using critical values from multivariate t-distributions for ANOM instead of simple Bonferroni quantiles is oftentimes negligible.     

A graphical model selection tool for mixed models

ABSTRACT

Model selection can be defined as the task of estimating the performance of different models in order to choose the most parsimonious one, among a potentially very large set of candidate statistical models. We propose a graphical representation to be considered as an extension to the class of mixed models of the deviance plot proposed in the literature within the framework of classical and generalized linear models. This graphical representation allows, once a reduced number of models have been selected, to identify important covariates focusing only on the fixed effects component, assuming the random part properly specified. Nevertheless, we suggest also a standalone figure representing the residual random variance ratio: a cross-evaluation of the two graphical representations will allow to derive some conclusions on the random part specification of the model and a more accurate selection of the final model.     

Graphical evaluation of the adequacy of the method of unweighted means

A graphical technique is introduced to assess the adequacy of the method of unweighted means in providing approximate F -tests for an unbalanced random model. These tests are similar to those obtained under a balanced ANOVA. The proposed technique is simple and can easily be used to determine the effects of imbalance and values of the variance components on the adequacy of the approximation. The one-way and two-way random models are used to illustrate the proposed methodology. Extensions to higher-order models are also mentioned.     

Estimated estimating equations: semiparametric inference for clustered and longitudinal data

Summary.  We introduce a flexible marginal modelling approach for statistical inference for clustered and longitudinal data under minimal assumptions. This estimated estimating equations approach is semiparametric and the proposed models are fitted by quasi-likelihood regression, where the unknown marginal means are a function of the fixed effects linear predictor with unknown smooth link, and variance–covariance is an unknown smooth function of the marginal means. We propose to estimate the nonparametric link and variance–covariance functions via smoothing methods, whereas the regression parameters are obtained via the estimated estimating equations. These are score equations that contain nonparametric function estimates. The proposed estimated estimating equations approach is motivated by its flexibility and easy implementation. Moreover, if data follow a generalized linear mixed model, with either a specified or an unspecified distribution of random effects and link function, the model proposed emerges as the corresponding marginal (population-average) version and can be used to obtain inference for the fixed effects in the underlying generalized linear mixed model, without the need to specify any other components of this generalized linear mixed model. Among marginal models, the estimated estimating equations approach provides a flexible alternative to modelling with generalized estimating equations. Applications of estimated estimating equations include diagnostics and link selection. The asymptotic distribution of the proposed estimators for the model parameters is derived, enabling statistical inference. Practical illustrations include Poisson modelling of repeated epileptic seizure counts and simulations for clustered binomial responses.     

Graphical Display of Means

A graphical procedure for the display of treatment means that enables one to determine the statistical significance of the observed differences is presented. It is shown that the widely used least significant difference and honestly significant difference statistics can be used to construct plots in which any two means whose uncertainty intervals do not overlap are significantly different at the assigned probability level. It is argued that these plots, because of their straightforward decision rules, are more effective than those that show the observed means with standard errors or confidence limits. Several examples of the proposed displays are included to illustrate the procedure.     

A New Procedure to Monitor the Mean of a Quality Characteristic

The Shewhart, Bonferroni-adjustment, and analysis of means (ANOM) control charts are typically applied to monitor the mean of a quality characteristic. The Shewhart and Bonferroni procedure are utilized to recognize special causes in production process, where the control limits are constructed by assuming normal distribution for known parameters (mean and standard deviation), and approximately normal distribution regarding to unknown parameters. The ANOM method is an alternative to the analysis of variance method. It can be used to establish the mean control charts by applying equicorrelated multivariate non central t distribution. In this article, we establish new control charts, in phases I and II monitoring, based on normal and t distributions having as a cause a known (or unknown) parameter (standard deviation). Our proposed methods are at least as effective as the classical Shewhart methods and have some advantages.     

Comparison of ANOVA-F and ANOM tests with regard to type I error rate and test power

A Monte Carlo simulation was conducted to compare the type I error rate and test power of the analysis of means (ANOM) test to the one-way analysis of variance F-test (ANOVA-F). Simulation results showed that as long as the homogeneity of the variance assumption was satisfied, regardless of the shape of the distribution, number of group and the combination of observations, both ANOVA-F and ANOM test have displayed similar type I error rates. However, both tests have been negatively affected from the heterogeneity of the variances. This case became more obvious when the variance ratios increased. The test power values of both tests changed with respect to the effect size (Δ), variance ratio and sample size combinations. As long as the variances are homogeneous, ANOVA-F and ANOM test have similar powers except unbalanced cases. Under unbalanced conditions, the ANOVA-F was observed to be powerful than the ANOM-test. On the other hand, an increase in total number of observations caused the power values of ANOVA-F and ANOM test approach to each other. The relations between effect size (Δ) and the variance ratios affected the test power, especially when the sample sizes are not equal. As ANOVA-F has become to be superior in some of the experimental conditions being considered, ANOM is superior in the others. However, generally, when the populations with large mean have larger variances as well, ANOM test has been seen to be superior. On the other hand, when the populations with large mean have small variances, generally, ANOVA-F has observed to be superior. The situation became clearer when the number of the groups is 4 or 5.     

Graphical Network Models for International Financial Flows

The late-2000s financial crisis stressed the need to understand the world financial system as a network of countries, where cross-border financial linkages play a fundamental role in the spread of systemic risks. Financial network models, which take into account the complex interrelationships between countries, seem to be an appropriate tool in this context. To improve the statistical performance of financial network models, we propose to generate them by means of multivariate graphical models. We then introduce Bayesian graphical models, which can take model uncertainty into account, and dynamic Bayesian graphical models, which provide a convenient framework to model temporal cross-border data, decomposing the model into autoregressive and contemporaneous networks. The article shows how the application of the proposed models to the Bank of International Settlements locational banking statistics allows the identification of four distinct groups of countries, that can be considered central in systemic risk contagion.     

A note on transformed likelihood approach in linear dynamic panel models

The transformed likelihood approach to estimation of fixed effects dynamic panel data models is shown to present very good inferential properties but it is not directly implemented in the most diffused statistical software. The present paper aims at showing how a simple model reformulation can be adopted to describe the problem in terms of classical linear mixed models. The transformed likelihood approach is based on the first differences data transformation, the following results derive from a convenient reformulation in terms of deviations from the first observations. Given the invariance to data transformation, the likelihood functions defined in the two cases coincide. Resulting in a classical random effect linear model form, the proposed approach significantly improves the number of available estimation procedures and provides a straightforward interpretation for the parameters. Moreover, the proposed model specification allows to consider all the estimation improvements typical of the random effects model literature. Simulation studies are conducted in order to study the robustness of the estimation method to mean stationarity violation.     

Extreme values identification in regression using a peaks-over-threshold approach

The problem of heavy tail in regression models is studied. It is proposed that regression models are estimated by a standard procedure and a statistical check for heavy tail using residuals is conducted as a tool for regression diagnostic. Using the peaks-over-threshold approach, the generalized Pareto distribution quantifies the degree of heavy tail by the extreme value index. The number of excesses is determined by means of an innovative threshold model which partitions the random sample into extreme values and ordinary values. The overall decision on a significant heavy tail is justified by both a statistical test and a quantile–quantile plot. The usefulness of the approach includes justification of goodness of fit of the estimated regression model and quantification of the occurrence of extremal events. The proposed methodology is supplemented by surface ozone level in the city center of Leeds.     

Toward an integrated Bayesian network approach to measurement error detection and correction

In this article, the quality of data produced by national statistical institutes and by governmental institutions is considered. In particular, the problem of measurement error is analyzed and an integrated Bayesian network decision support system based on non-parametric Bayesian networks is proposed for its detection and correction. Non-parametric Bayesian networks are graphical models expressing dependence structure via bivariate copulas associated to the edges of the graph. The network structure and the misreport probability are estimated using a validation sample. The Bayesian network model is proposed to decide: (i) which records have to be corrected; (ii) the kind and amount of correction to be adopted. The proposed correction procedure is applied to the Banca d’Italia Survey on Household Income and Wealth and, specifically, the bond amounts are analyzed. Finally, the sensitivity of the conditional distribution of the true value random variable given the observed one to different evidence configurations is studied.     

On probit versus logit dynamic mixed models for binary panel data

In this paper, we consider inferences in a binary dynamic mixed model. The existing estimation approaches mainly estimate the regression effects and the dynamic dependence parameters either through the estimation of the random effects or by avoiding the random effects technically. Under the assumption that the random effects follow a Gaussian distribution, we propose a generalized quasilikelihood (GQL) approach for the estimation of the parameters of the dynamic mixed models. The proposed approach is computationally less cumbersome than the exact maximum likelihood (ML) approach. We also carry out the GQL estimation under two competitive, namely, probit and logit mixed models, and discuss both the asymptotic and small-sample behaviour of their estimators.     

A shared parameter model of longitudinal measurements and survival time with heterogeneous random-effects distribution

Typical joint modeling of longitudinal measurements and time to event data assumes that two models share a common set of random effects with a normal distribution assumption. But, sometimes the underlying population that the sample is extracted from is a heterogeneous population and detecting homogeneous subsamples of it is an important scientific question. In this paper, a finite mixture of normal distributions for the shared random effects is proposed for considering the heterogeneity in the population. For detecting whether the unobserved heterogeneity exits or not, we use a simple graphical exploratory diagnostic tool proposed by Verbeke and Molenberghs [34] to assess whether the traditional normality assumption for the random effects in the mixed model is adequate. In the joint modeling setting, in the case of evidence against normality (homogeneity), a finite mixture of normals is used for the shared random-effects distribution. A Bayesian MCMC procedure is developed for parameter estimation and inference. The methodology is illustrated using some simulation studies. Also, the proposed approach is used for analyzing a real HIV data set, using the heterogeneous joint model for this data set, the individuals are classified into two groups: a group with high risk and a group with moderate risk.     

APPLICATIONS OF ROBUST ESTIMATION FOR VARIANCE COMPONENTS MODELS:THE DETECTION OF MAJOR GENE EFFECTS IN FINGER RIDGE COUNTS IN NORMAL AND FRAGILE X FAMILIES

A diagnostic technique is proposed to detect major gene effects and other systematic departures from a model for the trait means in the presence of outliers. The technique is based on the examination of residuals from fitting variance components models to quantitative pedigree data using robust statistical procedures. The approach is demonstrated using the total ridge count and ridge count of the middle finger from 54 extended families affected with the Fragile X syndrome, and a sample of 217 normal pedigrees.     

Robust I-sample analysis of means type randomization tests for variances

Four Analysis of Means (ANOM) type randomization tests for testing the equality of I variances are presented. Randomization techniques for testing statistical hypotheses can be used when parametric tests are inappropriate. Suppose that I independent samples have been collected. Randomization tests are based on shuffles or rearrangements of the (combined) sample. Putting each of the I samples "in a bowl" forms the combined sample. Drawing samples "from the bowl" forms a shuffle. Shuffles can be made with replacement (bootstrap shuffling) or without replacement (permutation shuffling). The tests that are presented offer two advantages. They are robust to non-normality and they allow the user to graphically present the results via a decision chart similar to a Shewhart control chart. The decision chart facilitates easy assessment of both statistical and practical significance. A Monte Carlo study is used to identify robust randomization tests that exhibit excellent power when compared to other robust tests.     

Graphical technique for comparing designs for random models

Methods for comparing designs for a random (or mixed) linear model have focused primarily on criteria based on single-valued functions. In general, these functions are difficult to use, because of their complex forms, in addition to their dependence on the model's unknown variance components. In this paper, a graphical approach is presented for comparing designs for random models. The one-way model is used for illustration. The proposed approach is based on using quantiles of an estimator of a function of the variance components. The dependence of these quantiles on the true values of the variance components is depicted by plotting the so-called quantile dispersion graphs (QDGs), which provide a comprehensive picture of the quality of estimation obtained with a given design. The QDGs can therefore be used to compare several candidate designs. Two methods of estimation of variance components are considered, namely analysis of variance and maximum-likelihood estimation.     

Harmonic mean approach to uunbalanced random effects models under heteroscedasticity

In this paper we consider unbalanced random effects models under heteroscedastic variances. By using' the harmonic mean approach, it is shown that the problems are analogous to those from balanced random effects models under horaoscedastic variances. Thus, by using the harmonic mean approach, statistical inferences about variance components are derived by using procedures from balanced models under homoscedastic variances. Laguerre polynomial expansion is used to approximate the sampling distributions of relevant statistics.     

A graphical method for testing the equality of several variances

SUMMARY The problem of testing the equality of several variances arises in many areas. For testing the equality of variances, several tests are available in the literature which demonstrate only the statistical significance of the variances. In this paper, a graphical method is presented for testing the equality of variances. This method simultaneously demonstrates the statistical and engineering significance. Two examples are given to illustrate the proposed graphical method, and the conclusions obtained are compared with the existing tests.     

Bayes linear analysis for graphical models: The geometric approach to local computation and interpretive graphics

This paper concerns the geometric treatment of graphical models using Bayes linear methods. We introduce Bayes linear separation as a second order generalised conditional independence relation, and Bayes linear graphical models are constructed using this property. A system of interpretive and diagnostic shadings are given, which summarise the analysis over the associated moral graph. Principles of local computation are outlined for the graphical models, and an algorithm for implementing such computation over the junction tree is described. The approach is illustrated with two examples. The first concerns sales forecasting using a multivariate dynamic linear model. The second concerns inference for the error variance matrices of the model for sales, and illustrates the generality of our geometric approach by treating the matrices directly as random objects. The examples are implemented using a freely available set of object-oriented programming tools for Bayes linear local computation and graphical diagnostic display.