Ana M. Pires

Effects of inbreeding on population mean performance and observational variances in Eucalyptus globulus

Abstract• Mean performance and variances were studied in self (SELF), open pollinated (OP) and unrelated polymix (POL) crosses of common parentage in Eucalyptus globulus.• Inbreeding depression for survival (SURV) and basal area per hectare (BAH) was the highest reported for a SELF eucalypt population, increasing with age to reach 74 and 77%, respectively, over 10 years.• Inbreeding depression in the OP was 36% for SURV and 32% for BAH at age 10 years, and estimates of outcrossing rate from BAH were stable across ages, averaging 0.56. In contrast, OP inbreeding depression for stem diameter (DBH) of survivors decreased with age and few selfs appeared to survive to 10 years.• There was more variation in DBH between and within SELF than POL families, with variance ratios consistent with rare and partially recessive deleterious alleles causing inbreeding depression.• The OP variances were initially more similar to the SELF population but converged to the POL population after 10 years.• It is argued that when outcrossing rates are low, as in the present case, inbreeding depression will be a significant force countering local adaptation in forest trees.

On optimal reject rules and ROC curves

In this paper we make the connection between two approaches for supervised classification with a rejection option. The first approach is due to Tortorella and is based on ROC curves and the second is a generalisation of Chows optimal rule.

Projection-pursuit approach to robust linear discriminant analysis

Discriminant analysis plays an important role in multivariate statistics as a prediction and classification method. It has been successfully applied in many fields of work and research. As it happens with other multivariate methods, discriminant analysis is highly vulnerable to the presence of outliers that commonly occur in many real world data sets. The lack of robustness of the classical estimators on which the linear discriminant function is based is a severe disadvantage and several authors have worked to find efficient ways to prevent the damage that outliers can cause. This paper focuses on the projection-pursuit approach to discriminant analysis. The projection-pursuit estimators are described and theoretical properties are deduced and their relevance is highlighted. These include Fisher consistency, affine equivariance, partial influence functions and asymptotic distributions. Application to real data and a simulation study reveal the robustness of the projection-pursuit approach. In both analyses the data relates to a large number of variables, a situation that is becoming common when new technology is applied to data gathering.

Detection of Outliers in Multivariate Data: A Method Based on Clustering and Robust Estimators

Outlier identification is important in many applications of multivariate analysis. Either because there is some specific interest in finding anomalous observations or as a pre-processing task before the application of some multivariate method, in order to preserve the results from possible harmful effects of those observations. It is also of great interest in supervised classification (or discriminant analysis) if, when predicting group membership, one wants to have the possibility of labelling an observation as “does not belong to any of the available groups”. The identification of outliers in multivariate data is usually based on Mahalanobis distance. The use of robust estimates of the mean and the covariance matrix is advised in order to avoid the masking effect (Rousseeuw and Leroy, 1985; Rousseeuw and von Zomeren, 1990; Rocke and Woodruff, 1996; Becker and Gather, 1999). However, the performance of these rules is still highly dependent of multivariate normality of the bulk of the data. The aim of the method here described is to remove this dependence.

Verification of QTL linked markers for propagation traits in Eucalyptus

In a previous study, several quantitative trait loci (QTLs) affecting vegetative propagation traits were detected in a hybrid cross between Eucalyptus tereticornis and Eucalyptus globulus. The objective of this work was to confirm stable QTL linked markers (detected in different years) for propagation traits in an independent set of the same segregating population and in two related crosses involving the original E. globulus parent. Phenotypic averages of groups of individuals carrying alternative allelic forms of the stable QTL linked markers were statistically tested for significant differences. Adventitious rooting and petrification marker–trait associations, detected previously in the E. tereticornis parent, were verified in an independent sample of the original progeny. In the E. globulus parent, the QTL linked marker was only verified in one related genetic background. Verification was possible only for high-effect QTL linked markers. This study highlights the importance of sample size in QTL detection for low-heritability traits.

Robust Bootstrap with Non Random Weights Based on the Influence Function

Abstract The existence of outliers in a sample is an obvious problem which can become worse when the usual bootstrap is applied, because some resamples may have a higher contamination level than the initial sample. Bootstrapping using robust estimators may be a solution to this problem. However, in many instances, this will not be enough because it can lead to several complications, such as: (i) the breakdown point for the whole procedure may be small even when based on an estimator with a high breakdown point; (ii) mathematical difficulties; (iii) very high computation time. In this paper, we suggest a modification of the bootstrap procedure in order to solve these problems which consists of forming each bootstrap sample by resampling with different probabilities so that the potentially more harmful observations have smaller probabilities of selection. The aim is to protect the whole procedure against a given number of arbitrary outliers. As an illustration, we consider point and interval estimation for the correlation coefficient. We use Monte Carlo methods to compare this method with another robust bootstrap procedure, the winsorized bootstrap, recently suggested by Singh [Singh, K. (1998). Breakdown theory for bootstrap quantiles. Ann. Statist. 26:1719–1732].

M-regression, false discovery rates and outlier detection with application to genetic association studies

Robust multiple linear regression methods are valuable tools when underlying classical assumptions are not completely fulfilled. In this setting, robust methods ensure that the analysis is not significantly disturbed by any outlying observation. However, knowledge of these observations may be important to assess the underlying mechanisms of the data. Therefore, a robust outlier test is discussed, together with an adequate false discovery rate correction measure, to be used in the context of multiple linear regression with categorical explanatory variables. The methodology focuses on genetic association studies of quantitative traits, though it has much broader applications. The method is also compared to a benchmark rule from the literature and its good performance is validated by a simulation study and a real data example from a candidate gene study.

Sample size for estimating a binomial proportion: comparison of different methods

The poor performance of the Wald method for constructing confidence intervals (CIs) for a binomial proportion has been demonstrated in a vast literature. The related problem of sample size determination needs to be updated and comparative studies are essential to understanding the performance of alternative methods. In this paper, the sample size is obtained for the Clopper–Pearson, Bayesian (Uniform and Jeffreys priors), Wilson, Agresti–Coull, Anscombe, and Wald methods. Two two-step procedures are used: one based on the expected length (EL) of the CI and another one on its first-order approximation. In the first step, all possible solutions that satisfy the optimal criterion are obtained. In the second step, a single solution is proposed according to a new criterion (e.g. highest coverage probability (CP)). In practice, it is expected a sample size reduction, therefore, we explore the behavior of the methods admitting 30% and 50% of losses. For all the methods, the ELs are inflated, as expected, but the coverage probabilities remain close to the original target (with few exceptions). It is not easy to suggest a method that is optimal throughout the range (0, 1) for p. Depending on whether the goal is to achieve CP approximately or above the nominal level different recommendations are made.

Detecting influential observations in principal components and common principal components

Detecting outlying observations is an important step in any analysis, even when robust estimates are used. In particular, the robustified Mahalanobis distance is a natural measure of outlyingness if one focuses on ellipsoidal distributions. However, it is well known that the asymptotic chi-square approximation for the cutoff value of the Mahalanobis distance based on several robust estimates (like the minimum volume ellipsoid, the minimum covariance determinant and the S-estimators) is not adequate for detecting atypical observations in small samples from the normal distribution. In the multi-population setting and under a common principal components model, aggregated measures based on standardized empirical influence functions are used to detect observations with a significant impact on the estimators. As in the one-population setting, the cutoff values obtained from the asymptotic distribution of those aggregated measures are not adequate for small samples. More appropriate cutoff values, adapted to the sample sizes, can be computed by using a cross-validation approach. Cutoff values obtained from a Monte Carlo study using S-estimators are provided for illustration. A real data set is also analyzed.

Paternity analysis in Excel

Paternity analysis using microsatellite information is a well-studied subject. These markers are ideal for parentage studies and fingerprinting, due to their high-discrimination power. This type of data is used to assign paternity, to compute the average selfing and outcrossing rates and to estimate the biparental inbreeding. There are several public domain programs that compute all this information from data. Most of the time, it is necessary to export data to some sort of format, feed it to the program and import the output to an Excel book for further processing. In this article we briefly describe a program referred from now on as Paternity Analysis in Excel (PAE), developed at IST and IBET (see the acknowledgments) that computes paternity candidates from data, and other information, from within Excel. In practice this means that the end user provides the data in an Excel sheet and, by pressing an appropriate button, obtains the results in another Excel sheet. For convenience PAE is divided into two modules. The first one is a filtering module that selects data from the sequencer and reorganizes it in a format appropriate to process paternity analysis, assuming certain conventions for the names of parents and offspring from the sequencer. The second module carries out the paternity analysis assuming that one parent is known. Both modules are written in Excel-VBA and can be obtained at the address (www.math.ist.utl.pt/~fmd/pa/pa.zip). They are free for non-commercial purposes and have been tested with different data and against different software (Cervus, FaMoz, and MLTR).