Search

The Box-Cox power transformation family for non-negative responses in linear models has a long and interesting history in both statistical practice and theory, which we summarize. The relationship between generalized linear models and log transformed data is illustrated. Extensions investigated include the transform both sides model and the Yeo-Johnson transformation for observations that can be positive or negative. The paper also describes an extended Yeo-Johnson transformation that allows positive and negative responses to have different power transformations. Analyses of data show this to be necessary. Robustness enters in the fan plot for which the forward search provides an ordering of the data. Plausible transformations are checked with an extended fan plot. These procedures are used to compare parametric power transformations with nonparametric transformations produced by smoothing.

We consider a classical regression model contaminated by multiple outliers arising simultaneously from mean-shift and variance-inflation mechanisms—which are generally considered as alternative. Identifying multiple outliers leads to computational challenges in the usual variance-inflation framework. We propose the use of robust estimation techniques to identify outliers arising from each mechanism, and we rely on restricted maximum likelihood estimation to accommodate variance-inflated outliers into the model. Furthermore, we introduce diagnostic plots which help to guide the analysis. We compare classical and robust methods with our novel approach on both simulated and real data.

Unlike the Box-Cox transformation, that of Yeo and Johnson for the response of a linear model can be applied when the observations are not constrained to be positive. We study the extended Yeo–Johnson transformation in which positive and negative observations can be transformed with different parameter values. The procedure is illustrated for data with many outliers. The data are cleaned with a robust method, the forward search, and the obtained transformations compared with the results from two nonparametric transformation methods based on data smoothing.

Producción Científica, A new methodology for constrained parsimonious model-based clustering is introduced, where some tuning parameter allows to control the strength of these constraints. The methodology includes the 14 parsimonious models that are often applied in model-based clustering when assuming normal components as limit cases. This is done in a natural way by filling the gap among models and providing a smooth transition among them. The methodology provides mathematically well-defined problems and is also useful to prevent us from obtaining spurious solutions. Novel information criteria are proposed to help the user in choosing parameters. The interest of the proposed methodology is illustrated through simulation studies and a real-data application on COVID data., Ministerio de Economía y Competitividad (grant MTM2017-86061-C2-1-P), Junta de Castilla y León - FEDER (grants VA005P17 and VA002G18), CRoNoS COST y el proyecto “Estadísticas para la detección de fraudes, con aplicaciones para datos comerciales y estados financieros ”de la Universidad de Parma (grant IC1408), Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCLE

We review the sampling and results of the radiocarbon dating of the archaeological cloth known as the Shroud of Turin, in the light of recent statistical analyses of both published and raw data. The statistical analyses highlight an inter-laboratory heterogeneity of the means and a monotone spatial variation of the ages of subsamples that suggest the presence of contaminants unevenly removed by the cleaning pretreatments. We consider the significance and overall impact of the statistical analyses on assessing the reliability of the dating results and the design of correct sampling. These analyses suggest that the 1988 radiocarbon dating does not match the current accuracy requirements. Should this be the case, it would be interesting to know the accurate age of the Shroud of Turin. Taking into account the whole body of scientific data, we discuss whether it makes sense to date the Shroud again.

Summary We analyse data on the performance of investment funds, 99 out of 309 of which report a loss, and on the profitability of 1405 firms, 407 of which report losses. The problem in both cases is to use regression to predict performance from sets of explanatory variables. In one case, it is clear from scatter plots of the data that the negative responses have a lower variance than the positive responses and a different relationship with the explanatory variables. Because the data include negative responses, the Box–Cox transformation cannot be used. We develop a robust version of an extension to the Yeo–Johnson transformation which allows different transformations for positive and negative responses. Tests and graphical methods from our robust analysis enable the detection of outliers, the assessment of values of the two transformation parameters and the building of simple regression models. Performance comparisons are made with non-parametric transformations.

Mycotoxins are secondary metabolites produced by pathogenic fungi. They are found in a variety of different products, such as spices, cocoa, and cereals, and they can contaminate fields before and/or after harvest and during storage. Mycotoxins negatively impact human and animal health, causing a variety of adverse effects, ranging from acute poisoning to long-term effects. Given a large number of mycotoxins (currently more than 300 are known), it is impossible to use in vitro/in vivo methods to detect the potentially harmful effects to human health of all of these. To overcome this problem, this work aims to present a new robust computational approach, based on a combination of in silico and statistical methods, in order to screen a large number of molecules against the nuclear receptor family in a cost and time-effective manner and to discover the potential endocrine disruptor activity of mycotoxins. The results show that a high number of mycotoxins is predicted as a potential binder of nuclear receptors. In particular, ochratoxin A, zearalenone, α- and β-zearalenol, aflatoxin B1, and alternariol have been shown to be putative endocrine disruptors chemicals for nuclear receptors.

Deciding the number of clusters k is one of the most difficult problems in clus- ter analysis. For this purpose, complexity-penalized likelihood approaches have been introduced in model-based clustering, such as the well known BIC and ICL crite- ria. However, the classi cation/mixture likelihoods considered in these approaches are unbounded without any constraint on the cluster scatter matrices. Constraints also prevent traditional EM and CEM algorithms from being trapped in (spurious) local maxima. Controlling the maximal ratio between the eigenvalues of the scatter matrices to be smaller than a xed constant c 1 is a sensible idea for setting such constraints. A new penalized likelihood criterion which takes into account the higher model complexity that a higher value of c entails, is proposed. Based on this criterion, a novel and fully automated procedure, leading to a small ranked list of optimal (k; c) couples is provided. A new plot called \car-bike" which provides a concise summary of the solutions is introduced. The performance of the procedure is assessed both in empirical examples and through a simulation study as a function of cluster overlap. Supplemental materials for the article are available online., Spanish Ministerio de Economía y Competitividad, grant MTM2017-86061-C2-1-P, and by Consejería de Educación de la Junta de Castilla y León and FEDER, grant VA005P17 and VA002G18.

Misinvoicing is a major tool in fraud including money laundering. We develop a method of detecting the patterns of outliers that indicate systematic mis‐pricing. As the data only become available year by year, we develop a combination of very robust regression and the use of ‘cleaned’ prior information from earlier years, which leads to early and sharp indication of potentially fraudulent activity that can be passed to legal agencies to institute prosecution. As an example, we use yearly imports of a specific seafood into the European Union. This is only one of over one million annual data sets, each of which can currently potentially contain 336 observations. We provide a solution to the resulting big data problem, which requires analysis with the minimum of human intervention.

The forward search is a method of robust data analysis in which outlier free subsets of the data of increasing size are used in model fitting; the data are then ordered by closeness to the model. Here the forward search, with many random starts, is used to cluster multivariate data. These random starts lead to the diagnostic identification of tentative clusters. Application of the forward search to the proposed individual clusters leads to the establishment of cluster membership through the identification of non-cluster members as outlying. The method requires no prior information on the number of clusters and does not seek to classify all observations. These properties are illustrated by the analysis of 200 six-dimensional observations on Swiss banknotes. The importance of linked plots and brushing in elucidating data structures is illustrated. We also provide an automatic method for determining cluster centres and compare the behaviour of our method with model-based clustering. In a simulated example with 8 clusters our method provides more stable and accurate solutions than model-based clustering. We consider the computational requirements of both procedures.

Trimming principles play an important role in robust statistics. However, their use for clustering typically requires some preliminary information about the contamination rate and the number of groups. We suggest a fresh approach to trimming that does not rely on this knowledge and that proves to be particularly suited for solving problems in robust cluster analysis. Our approach replaces the original K‐population (robust) estimation problem with K distinct one‐population steps, which take advantage of the good breakdown properties of trimmed estimators when the trimming level exceeds the usual bound of 0.5. In this setting, we prove that exact affine equivariance is lost on one hand but, on the other hand, an arbitrarily high breakdown point can be achieved by “anchoring” the robust estimator. We also support the use of adaptive trimming schemes, in order to infer the contamination rate from the data. A further bonus of our methodology is its ability to provide a reliable choice of the usually unknown number of groups.

Heteroskedastic regression data are modelled using a parameterized variance function. This procedure is robustified using a method with high breakdown point and high efficiency, which provides a direct link between observations and the weights used in model fitting. This feature is vital for the application, the analysis of international trade data from the European Union. Heteroskedasticity is strongly present in such data, as are outliers. A further example shows that the new method outperforms ordinary least squares with heteroskedasticity robust standard errors, even when the form of heteroskedasticity is mis-specified. A discussion of computational matters concludes the paper. An appendix presents the new scoring algorithm for estimation of the parameters of heteroskedasticity. Generalizes the standard model for heteroskedasticity in non-robust regression.Flexibility of the robust model shown on complex international trade data.Outperforms conventional "heteroskedastic robust" standard errors.Linked graphics provide insight into importance of individual observations.Provides publicly available Matlab code for very robust heteroskedastic regression.

The objective of this paper was to test whether investing activity in the futures markets of different commodities (grains, sugar, coffee, cotton, cocoa, livestock) could be identified as a source of the increasing level and volatility of agricultural commodity prices. The causal link between trading activity and market factors (returns, volatility) can be investigated using weekly data, usually derived from the Commitment of Traders Reports released by the US Commodity Futures Trading Commission (CFTC), or daily data expressed as the ratio of volume to open interest (VOIR). To increase the power of the estimation process and investigate the role of causal variables to determine the trends of all the market factors, the authors tested the estimates obtained by seemingly unrelated regression (SUR). One innovation is represented by the evaluation of the inverse relationships between market factors and causal variables. The market factors were also tested as causal variables, avoiding giving priority to only one part of the relationship according to Granger's causality. The lack of significance revealed by the Granger causality test on weekly models could be due to the inappropriate frequency of the information. The ratio of volume to open interest in futures contracts performs better than other parameters extensively adopted in the literature. The likely reason is that it depends on the daily frequency of this parameter, which provides statistical evidence of phenomena that include their effect in weekly intervals. The estimations for the daily model provide statistical evidence of a mutual relationship only between trading activity and realized volatility. No causal relationships were found for returns. The behaviour of all 12 futures markets examined is quite similar and uniform with respect to the scale of the coefficients and their temporal profile.

The “bag-plot” of Rousseeuw et al. (1990), which you exemplify in your Figure 8, provides a polygonal approximation to the unknown distribution. The robust bivariate boxplot introduced by Zani et al. (1998) provides a smoother approximation to the unknown distribution. We briefly recall some of its properties. Zani et al. (1998) use the peeling of convex hulls to, in the tradition of very robust statistics, find a region that contains approximately 50% of the data. Peeling of hulls continues until the first one is obtained which includes not more than 50% of the data (and asymptotically half the data as the sample size increases). The “50% hull” so found is smoothed using B-splines, with cubic pieces which use the vertices of the 50% hull to provide information about the location of the knots. The robust centre of the data is found as the bivariate mean of the observations lying with the 50% contour. Given the centre and the contour, the distance of any point from

Summary Motivated by the requirement of controlling the number of false discoveries that arises in several application fields, we study the behaviour of diagnostic procedures obtained from popular high-breakdown regression estimators when no outlier is present in the data. We find that the empirical error rates for many of the available techniques are surprisingly far from the prescribed nominal level. Therefore, we propose a simulation-based approach to correct the liberal diagnostics and reach reliable inferences. We provide evidence that our approach performs well in a wide range of settings of practical interest and for a variety of robust regression techniques, thus showing general appeal. We also evaluate the loss of power that can be expected from our corrections under different contamination schemes and show that this loss is often not dramatic. Finally, we detail some possible extensions that may further enhance the applicability of the method.

The frequentist forward search yields a flexible and informative form of robust regression. The device of fictitious observations provides a natural way to include prior information in the search. However, this extension is not straightforward, requiring weighted regression. Bayesian versions of forward plots are used to exhibit the presence of multiple outliers in a data set from banking with 1903 observations and nine explanatory variables which shows, in this case, the clear advantages from including prior information in the forward search. Use of observation weights from frequentist robust regression is shown to provide a simple general method for robust Bayesian regression.

Diagnostic tools must rely on robust high-breakdown methodologies to avoid distortion in the presence of contamination by outliers. However, a disadvantage of having a single, even if robust, summary of the data is that important choices concerning parameters of the robust method, such as breakdown point, have to be made prior to the analysis. The effect of such choices may be difficult to evaluate. We argue that an effective solution is to look at several pictures, and possibly to a whole movie, of the available data. This can be achieved by monitoring, over a range of parameter values, the results computed through the robust methodology of choice. We show the information gain that monitoring provides in the study of complex data structures through the analysis of multivariate datasets using different high-breakdown techniques. Our findings support the claim that the principle of monitoring is very flexible and that it can lead to robust estimators that are as efficient as possible. We also address through simulation some of the tricky inferential issues that arise from monitoring.

© 2018 The Author(s) Time series often contain outliers and level shifts or structural changes. These unexpected events are of the utmost importance in fraud detection, as they may pinpoint suspicious transactions. The presence of such unusual events can easily mislead conventional time series analysis and yield erroneous conclusions. A unified framework is provided for detecting outliers and level shifts in short time series that may have a seasonal pattern. The approach combines ideas from the FastLTS algorithm for robust regression with alternating least squares. The double wedge plot is proposed, a graphical display which indicates outliers and potential level shifts. The methodology was developed to detect potential fraud cases in time series of imports into the European Union, and is illustrated on two such series. ispartof: Econometrics and Statistics vol:9 pages:108-121 status: published

The Forward Search is a powerful general method for detecting anomalies in structured data, whose diagnostic power has been shown in many statistical contexts. However, despite the wealth of empirical evidence in favor of the method, only few theoretical properties have been established regarding the resulting estimators. We show that the Forward Search estimators are strongly consistent at the multivariate normal model. We also obtain their finite sample breakdown point. Our results put the Forward Search approach for multivariate data on a solid statistical ground, which formally motivates its use in robust applied statistics. Furthermore, they allow us to compare the Forward Search estimators with other well known multivariate high-breakdown techniques.

A striking feature of most applied statistical analyses is the use of methods that are well known to be sensitive to outliers or to other departures from the postulated model. Since data contamination is often the rule, rather than the exception, we investigate the reasons for this contradictory (and perhaps unintended) choice. We also provide empirical evidence, in a real-world regression problem concerning international trade, of the advantages of a new approach to data analysis based on monitoring. Our approach enhances the applicability of robust techniques and the interpretation of their results, thus yielding a positive step towards a reconciliation between robustness and applied statistics.

The forward search provides a flexible and informative form of robust regression. We describe the introduction of prior information into the regression model used in the search through the device of fictitious observations. The extension to the forward search is not entirely straightforward, requiring weighted regression. Forward plots are used to exhibit the effect of correct and incorrect prior information on inferences.

The problem of robust estimation and multivariate outlier detection of the term structure of default intensity is considered. Both the multivariate Vasicek and CIR models, embedding the Kalman filter algorithm in a forward search context, are used to estimate default intensity. The focus is not on the estimation of credit models including jumps, but on the automatic detection of masked multiple outliers in multivariate time series. Both simulated and real market credit spread time series are analyzed. In order to make inference on outliers, confidence envelopes which are virtually independent of the estimated parameters are introduced. The output is not only a unique default intensity term structure curve, as often used in the financial literature, but a robust confidence interval within which default intensity is likely to stay.

We present the FSDA (Forward Search for Data Analysis) toolbox, a new software library that extends MATLAB and its Statistics Toolbox to support a robust and efficient analysis of complex datasets, affected by different sources of heterogeneity. As the name of the library indicates, the project was born around the Forward Search approach, but it has evolved to include the main traditional robust multivariate and regression techniques, including LMS, LTS, MCD, MVE, MM and S estimation. To address problems where data deviate from typical model assumptions, tools are available for robust data transformation and robust model selection. When different views of the data are available, e.g. a scatterplot of units and a plot of distances of such units from a fitted model, FSDA links such views and offers the possibility to interact with them. For example, selections of objects in a plot are highlighted in the other plots. This considerably simplifies the exploration of the data in view of extracting information and detecting patterns. We show the potential of the FSDA in chemometrics using data from chemical and pharmaceutical problems, where the presence of outliers, multiple groups, deviations from normality and other complex structures is not an exceptional circumstance.

Summary Current demand for accountability and efficiency of healthcare organizations, combined with the greater availability of routine data on clinical care and outcomes, has led to an increased focus on statistical methods in healthcare regulation. We consider three different regulatory functions in which statistical analysis plays a vital role: rating organizations, deciding whom to inspect and continuous surveillance for arising problems. A common approach to data standardization based on (possibly overdispersed) Z-scores is proposed, although specific tools are used for assessing performance against a target, combining indicators when screening for inspection, and continuous monitoring using risk-adjusted sequential testing procedures. We pay particular attention to the problem of simultaneously monitoring over 200000 indicators for excess mortality, both with respect to the statistical issues surrounding massive multiplicity, and the organizational aspects of dealing with such a complex but high profile process.

The Forward Search is a powerful general method, incorporating flexible data-driven trimming, for the detection of outliers and unsuspected structure in data and so for building robust models. Starting from small subsets of data, observations that are close to the fitted model are added to the observations used in parameter estimation. As this subset grows we monitor parameter estimates, test statistics and measures of fit such as residuals. The paper surveys theoretical development in work on the Forward Search over the last decade. The main illustration is a regression example with 330 observations and 9 potential explanatory variables. Mention is also made of procedures for multivariate data, including clustering, time series analysis and fraud detection.

The forward search is a powerful general method for detecting multiple masked outliers and for determining their effect on inferences about models fitted to data. From the monitoring of a series of statistics based on subsets of data of increas- ing size we obtain multiple views of any hidden structure. One of the problems of the forward search has always been the lack of an automatic link among the great variety of plots which are monitored. Usually it happens that a lot of interesting fea- tures emerge unexpectedly during the progression of the forward search only when a specific combination of forward plots is inspected at the same time. Thus, the analyst should be able to interact with the plots and redefine or refine the links among them. In the absence of dynamic linking and interaction tools, the analyst risks to miss relevant hidden information. In this paper we fill this gap and provide the user with a set of new robust graphical tools whose power will be demonstrated on several regression problems. Through the analysis of real and simulated data we give a series of exam- ples where dynamic interaction with different "robust plots" is used to highlight the

We address the problem of seasonal adjustment of a nonlinear transformation of the original time series, measured on a ratio scale, which aims at enforcing two essential features: additivity and orthogonality of the components. The posterior mean and variance of the seasonally adjusted series admit an analytic finite representation only for particular values of the transformation parameter, e.g. for a fractional Box–Cox transformation parameter. Even if available, the analytical derivation can be tedious and difficult. As an alternative we propose to compute the two conditional moments of the seasonally adjusted series by means of numerical and Monte Carlo integration. The former is both fast and reliable in univariate applications. The latter uses the algorithm known as the ‘simulation smoother’ and it is most useful in multivariate applications. We present two case studies dealing with robust seasonal adjustment under the square root and the fourth root transformation. Our overall conclusion is that robust seasonal adjustment under transformations is feasible from the computational standpoint and that the possibility of transforming the scale ought to be considered as a further option for improving the quality of seasonal adjustment.

SummaryWe use the forward search to provide robust Mahalanobis distances to detect the presence of outliers in a sample of multivariate normal data. Theoretical results on order statistics and on estimation in truncated samples provide the distribution of our test statistic. We also introduce several new robust distances with associated distributional results. Comparisons of our procedure with tests using other robust Mahalanobis distances show the good size and high power of our procedure. We also provide a unification of results on correction factors for estimation from truncated samples.

It is well known that transformation of the response may improve the homogeneity and the approximate normality of the errors. Unfortunately, the estimated transformation and related test statistic may be sensitive to the presence of one, or several, atypical observations. In addition, it is important to remark that outliers in one transformed scale may not be atypical in another scale. Therefore, it is important to choose a transformation which does not depend on the presence of particular observations. In this article we suggest an efficient procedure based on a robust score test statistic which quantifies the effect of each observation on the choice of the transformation.

We combine the selection of a statistical model with the robust parameter estimation and diagnostic properties of the Forward Search. As a result we obtain procedures that select the best model in the presence of outliers. We derive distributional properties of our method and illustrate it on data on ozone concentration. The effect of outliers on the choice of a model is revealed. Although our example is for regression,the connection with AIC is stressed.

The forward search provides a series of robust parameter estimates based on increasing numbers of observations. The resulting series of robust Mahalanobis distances is used to cluster multivariate normal data. The method depends on envelopes of the distribution of the test statistics in forward plots. These envelopes can be found by simulation; flexible polynomial approximations to the envelopes are given. New graphical tools provide methods not only of detecting clusters but also of determining their membership. Comparisons are made with mclust and k-means clustering.

The Forward Search is used in an exploratory manner, with many random starts, to indicate the number of clusters and their membership in continuous data. The prospective clusters can readily be distinguished from background noise and from other forms of outliers. A confirmatory Forward Search, involving control on the sizes of statistical tests, establishes precise cluster membership. The method performs as well as robust methods such as TCLUST. However, it does not require prior specification of the number of clusters, nor of the level of trimming of outliers. In this way it is “user friendly”.

Three straightforward methods using truncated simple samples are described for approximating the pointwise distribution of the test statistic for multiple outliers in regression. The simulations also proved a powerful method of calibrating pointwise inferences for simultaneous tests for an unknown number of outliers. Analysis of data on fidelity cards reveals an unexpected group of outliers.

SUMMARY Monitoring the t-tests for individual regression coefficients in 'forward' search fails to identify the importance of observations to the significance of the individual regressors. This failure is due to the ordering of the data by the search. We introduce an added-variable test which has the desired properties since the projection leading to residuals destroys the effect of the ordering. An example illustrates the effect of several masked outliers on model selection. Comments are given on the related test for response transformations.

The fan plot of the score statistic for transformation during the forward search is a powerful tool for detecting masked outliers that indicate an incorrect transformation. We use simulation to investigate the distribution of this statistic during the search. Data structures where the normal approximation is, and is not, satisfactory are characterised.

There are several methods for obtaining very robust estimates of regression parameters that asymptotically resist 50% of outliers in the data. Differences in the behaviour of these algorithms depend on the distance between the regression data and the outliers. We introduce a parameter $\lambda$ that defines a parametric path in the space of models and enables us to study, in a systematic way, the properties of estimators as the groups of data move from being far apart to close together. We examine, as a function of $\lambda$, the variance and squared bias of five estimators and we also consider their power when used in the detection of outliers. This systematic approach provides tools for gaining knowledge and better understanding of the properties of robust estimators., Comment: Published in at http://dx.doi.org/10.1214/13-STS437 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Robust methods are little applied (although much studied by statisticians). We monitor very robust regression by looking at the behaviour of residuals and test statistics as we smoothly change the robustness of parameter estimation from a breakdown point of 50% to non-robust least squares. The resulting procedure provides insight into the structure of the data including outliers and the presence of more than one population. Monitoring overcomes the hindrances to the routine adoption of robust methods, being informative about the choice between the various robust procedures. Methods tuned to give nominal high efficiency fail with our most complicated example. We find that the most informative analyses come from S estimates combined with Tukey’s biweight or with the optimal $\rho$ functions. ¶ For our major example with 1,949 observations and 13 explanatory variables, we combine robust S estimation with regression using the forward search, so obtaining an understanding of the importance of individual observations, which is missing from standard robust procedures. We discover that the data come from two different populations. They also contain six outliers. ¶ Our analyses are accompanied by numerous graphs. Algebraic results are contained in two appendices, the second of which provides useful new results on the absolute odd moments of elliptically truncated multivariate normal random variables.

Summary. We suggest a simple robust method for the detection of atypical and influential observations in binomial data. Our technique is based on a forward search procedure which orders the observations from those most in agreement with a specified generalized linear model to those least in agreement with it. The effectiveness of the forward search estimator in detecting masked multiple outliers, and more generally in ordering binomial data, is shown by means of three data sets. Plots of diagnostic quantities during the forward search clearly show the effect of individual observations on residuals and test statistics. These examples reveal the strength of our method in showing the structure of the data in a way which is more simple and effective than it would be by using standard deletion diagnostic procedures.

SUMMARY This paper deals with the performance evaluation of investment funds. The goal is the monitoring of the pro"tability of the funds using a set of variables referred to as short- and medium-term performance, volatility and percentage of treasury bonds on total assets. The problem is the ordering of multivariate data and the search for the units lying far from the centroid. This question is related to the detection of multivariate outliers (Atkinson AC. Fast Very Robust Methods for the Detection of Multiple outliers. JASA 1994; 89:1329}1339.; Riani M, Zani S. An Iterative Method for the Detection of Multiple outliers Metron 1997; 55:101}117. In this analysis we use a variation of the robust-bivariate boxplot for each pair of variables suggested in a previous paper "rst on the whole of the observations and then on each distinct category of funds (i.e. stock, mixed (balanced) and bond funds). The purpose is to split the units into a few subsets: those inside the inner region close to the centroid and those lying outside the outer contour (that is the most extreme observations). In the middle we can "nd groups of observations ordered according to their distance from a robust centroid. We apply our method "rst to the original variables and then to principal components and canonical variates. An analysis of longitudinal data is also considered. Copyright ( 1999 John Wiley & Sons, Ltd.

In psychiatry, the assessment of outcome, to be relevant for clinical practice, should be multidimensional. This demands suitable statistical tools both to represent and to analyse the complex relationships between variables. This paper aims to provide guidelines for selecting graphical displays of multidimensional data that enable better capturing of the relationships between variables and summarize information available for individual cases as well as for groups of patients. Multivariate graphical techniques such as Chernoff faces, stars, parallel co-ordinates and Andrews plots are applied to a set of data drawn from the South Verona Outcome Project, a naturalistic study based on a multidimensional model for the assessment of outcomes, routinely conducted at the Institute of Psychiatry of the University of Verona, Italy. The assessment includes global functioning, psychopathology, social disability, subjective quality of life, and satisfaction with services, number of service contacts and days of hospitalization. The selected sequence of graphical displays allows a quick understanding of the multivariate data set not otherwise obtainable with numerical indexes and specifically enabled us to: (a) picture the characteristics of the data set on each single variable; (b) identify relationships between two or more variables; (c) identify peculiarities of individual patients and of group of patients. In psychiatry and in other areas of medicine a graphical approach to multidimensional data is a useful analytical tool for increasing the impact of outcome studies, in everyday clinical practice and in monitoring and evaluating of services. Copyright © 1998 Whurr Publishers Ltd.

In this paper we suggest a simple way of constructing a bivariate boxplot based on convex hull peeling and B-spline smoothing. The proposed method shows some advantages with respect to that suggested by Goldberg and Iglewicz (1992). Our approach leads to defining a natural inner region which is completely nonparametric and smooth. Furthermore it retains the correlation in the observations and adapts to differing spread of the data in the different directions. Theouter contour, which is based on a multiple of the distance of the inner region from the centre, is robust to the presence of clusters of outliers. We also show how the construction of a bivariate boxplot for each pair of variables can become a very useful tool for the detection of multivariate outliers.

Outliers can have a large influence on the model fitted to data. The models we consider are the transformation of data to approximate normality and also discriminant analysis, perhaps on transformed observations. If there are only one or a few outliers, they may often be detected by the deletion methods associated with regression diagnostics. These can be thought of as 'backwards' methods, as they start from a model fitted to all the data. However such methods become cumbersome, and may fail, in the presence of multiple outliers. We instead consider a 'forward' procedure in which very robust methods, such as least median of squares, are used to select a small, outlier free, subset of the data. This subset is increased in size using a search which avoids the inclusion of outliers. During the forward search we monitor quantities of interest, such as score statistics for transformation or, in discriminant analysis, misclassification probabilities. Examples demonstrate how the method very clearly reveals structure in the data and finds influential observations, which appear towards the end of the search. In our examples these influential observations can readily be related to patterns in the original data, perhaps after transformation.