Greet Pison

Robust factor analysis

Our aim is to construct a factor analysis method that can resist the effect of outliers. For this we start with a highly robust initial covariance estimator, after which the factors can be obtained from maximum likelihood or from principal factor analysis (PFA). We find that PFA based on the minimum covariance determinant scatter matrix works well. We also derive the influence function of the PFA method based on either the classical scatter matrix or a robust matrix. These results are applied to the construction of a new type of empirical influence function (EIF), which is very effective for detecting influential data. To facilitate the interpretation, we compute a cutoff value for this EIF. Our findings are illustrated with several real data examples.

Small Sample Corrections for LTS and MCD

Abstract. The least trimmed squares estimator and the minimum covariance determinant estimator [6] are frequently used robust estimators of regression and of location and scatter. Consistency factors can be computed for both methods to make the estimators consistent at the normal model. However, for small data sets these factors do not make the estimator unbiased. Based on simulation studies we therefore construct formulas which allow us to compute small sample correction factors for all sample sizes and dimensions without having to carry out any new simulations. We give some examples to illustrate the effect of the correction factor.

Peripheral markers of serotonergic and noradrenergic function in post-pubertal, caucasian males with autistic disorder.

Some studies have suggested that disorders in the peripheral and central metabolism of serotonin (5-HT) and noradrenaline may play a role in the pathophysiology of autistic disorder. This study examines serotonergic and noradrenergic markers in a study group of 13 male, post-pubertal, caucasian autistic patients (age 12–18 y; I.Q. > 55) and 13 matched volunteers. [3H]-paroxetine binding Kd values were significantly higher in patients with autism than in healthy volunteers. Plasma concentrations of tryptophan, the precursor of 5-HT, were significantly lower in autistic patients than in healthy volunteers. There were no significant differences between autistic and normal children in the serum concentrations of 5-HT, or the 24-hr urinary excretion of 5-hydroxy-indoleacetic acid (5-HIAA), adrenaline, noradrenaline, and dopamine. There were no significant differences in [3H]-rauwolscine binding Bmax or Kd values, or in the serum concentrations of tyrosine, the precursor of noradrenaline, between both study groups. There were highly significant positive correlations between age and 24-hr urinary excretion of 5-HIAA and serum tryptophan. The results suggest that: 1) serotonergic disturbances, such as defects in the 5-HT transporter system and lowered plasma tryptophan, may play a role in the pathophysiology of autism; 2) autism is not associated with alterations in the noradrenergic system; and 3) the metabolism of serotonin in humans undergoes significant changes between the ages of 12 and 18 years.

Displaying a clustering with CLUSPLOT

In a bivariate data set it is easy to represent clusters, e.g. by manually circling them or separating them by lines. But many data sets have more than two variables, or they come in the form of inter-object dissimilarities. There exist methods to partition such a data set into clusters, but the resulting partition is not visual by itself. In this paper we construct a new graphical display called CLUSPLOT, in which the objects are represented as points in a bivariate plot and the clusters as ellipses of various sizes and shapes. The algorithm is implemented as an S-PLUS function. Several options are available, e.g. labelling of objects and clusters, drawing lines connecting clusters, and the use of color. We illustrate this new tool with several examples. c 1999 Elsevier Science B.V. All rights reserved.

Epidemiologic and phenomenological aspects of post-traumatic stress disorder: DSM-III-R diagnosis and diagnostic criteria not validated

The aim of this cohort study was: (i) to validate the diagnostic criteria for post-traumatic stress disorder (PTSD) of the DSM-III-R; and (ii) to examine the incidence rate of PTSD in a study population exposed to two different traumatic events, i.e. a fire in a hotel ball-room and a multiple collision car-crash on a Belgian highway. One hundred and eighty-five victims (130 fire and 55 car accident victims) were assessed between 7 and 9 months after the traumatic event using the Composite International Diagnostic Interview (CIDI), PTSD Module, a fully structured diagnostic interview for the assessment of PTSD according to DSM-III-R criteria. Twenty-three percent of the study population met DSM-III-R criteria for PTSD. By means of unsupervised and supervised multivariate statistical analyses we were unable to validate the three-factorial structure, i.e. criteria B, C and D, of the DSM-III-R PTSD diagnosis. The latter relies heavily on the C diagnostic criteria, which appear to be too restrictive. Women were more likely to develop symptoms of reexperience (B) and arousal (D) than men. There was a significantly higher incidence of criteria B, C and D, but not of PTSD, in fire than in car-accident victims. Between 42 and 57% of the victims developed the first PTSD symptoms on the day of the trauma; within the next week these incidence rates increased to 77.1, 57.8 and 73.5% for criteria B, C and D, respectively. In conclusion, this study was unable to demonstrate the validity of the diagnostic criteria for PTSD according to DSM-III-R. The present cohort study has defined a number of factors that may predict new occurrences of PTSD symptoms after a traumatic event, i.e. gender, type of trauma and time delay between the trauma and the assessment of the diagnostic criteria.

Fitting multiplicative models by robust alternating regressions

In this paper a robust approach for fitting multiplicative models is presented. Focus is on the factor analysis model, where we will estimate factor loadings and scores by a robust alternating regression algorithm. The approach is highly robust, and also works well when there are more variables than observations. The technique yields a robust biplot, depicting the interaction structure between individuals and variables. This biplot is not predetermined by outliers, which can be retrieved from the residual plot. Also provided is an accompanying robust R2-plot to determine the appropriate number of factors. The approach is illustrated by real and artificial examples and compared with factor analysis based on robust covariance matrix estimators. The same estimation technique can fit models with both additive and multiplicative effects (FANOVA models) to two-way tables, thereby extending the median polish technique.

Diagnostic Plots for Robust Multivariate Methods

Robust techniques for multivariate statistical methods—such as principal component analysis, canonical correlation analysis, and factor analysis—have been recently constructed. In contrast to the classical approach, these robust techniques are able to resist the effect of outliers. However, there does not yet exist a graphical tool to identify in a comprehensive way the data points that do not obey the model assumptions. Our goal is to construct such graphics based on empirical influence functions. These graphics not only detect the influential points but also classify the observations according to their robust distances. In this way the observations are divided into four different classes which are regular points, nonoutlying influential points, influential outliers, and noninfluential outliers. We thus gain additional insight in the data by detecting different types of deviating observations. Some real data examples will be given to show how these plots can be used in practice.

A robust version of principal factor analysis

Our aim is to construct a factor analysis method that can resist the effect of outliers. We start with a highly robust initial covariance estimator, after which the factors can be obtained from maximum likelihood or from principal factor analysis (PFA). We find that PFA based on the minimum covariance determinant scatter matrix works well. We also derive the influence function of the PFA method. A new type of empirical influence function (EIF) which is very effective for detecting influential data is constructed. If the data set contains fewer cases than variables, we estimate the factor loadings and scores by a robust interlocking regression algorithm.

Analyzing data with robust multivariate methods and diagnostic plots

Principal Component Analysis, Canonical Correlation Analysis and Factor Analysis (Johnson and Wichern 1998) are three different methods for analyzing multivariate data. Recently robust versions of these methods have been proposed by Croux and Haesbroeck (2000), Croux and Dehon (2001) and Pison et al. (2002) which are able to resist the effect of outliers. Influence functions for these methods are also present. However, there does not yet exist a graphical tool to display the results of the robust data analysis in a fast way. Therefore we now construct such a diagnostic tool based on empirical influence functions. These graphics will not only allow us to detect the influential points for the multivariate statistical method but also classify the observations according to their robust distances. In this way we can identify regular points, good (non-outlying) influential points, influential outliers, and non-influential outliers. We can downweigh the influential outliers in the classical estimation method to obtain reliable and efficient estimates of the model parameters. Some generated data examples will be given to show how these plots can be used in practice.

A hotelling test based on MCD

Hypothesis tests and confidence intervals based on the classical Hotelling T 2 statistic can be adversely affected by outliers. Therefore, we construct an alternative inference technique based on a statistic which uses the highly robust MCD estimator of Rousseeuw (1984) instead of the classical mean and covariance matrix. The distribution of this new statistic differs from the classical one. Similarly to the classical T 2 distribution, we obtain a multiple of a certain F-distribution. It is shown through a Monte Carlo study that this approximation is very accurate, both at the normal model and at contamination models.