9 Robust Factor Analysis: Methods and Applications

The word robustness is frequently used in the literature, and is often stated with completely di®erent meaning. In this contribution robustness means to reduce the in°uence of \unusual" observations on statistical estimates. Such observations are frequently denoted as outliers, and are often thought to be extreme values caused by measurement or transcription errors. However, the notion of outliers also includes observations (or groups of observations) which are inconsistent with the remaining data set. The judgment whether observations are declared as outliers or as \inliers" is sometimes subjective, and robust statistics should serve as a tool for an objective decision. In terms of a statistical model, robust statistics could be de¯ned as follows \In a broad informal sense, robust statistics is a body of knowledge, partly formalized intòtheory of robustness', relating to deviations from idealized assumptions in statistics. statistics is aimed at yielding reliable results in cases where classical assumptions like normality, independence or linearity are violated. Real data sets almost always include outliers. Sometimes they are harmless and do not change the results if they are included in the analysis or deleted beforehand. However, they can have a major in°uence on the results, and completely alter the statistical estimates. Deleting such observations before analyzing the data would be a way out, but this implies that the outliers can indeed be identi¯ed, which is not trivial, especially in higher dimensions (see Section 9.2.3). Another way to reduce the in°uence of outliers is to ¯t the majority of the data, which is assumed to be the \good" part of data points. The majority ¯t is done by introducing a weight function for downweighting outliers, with weights equal to 0 and 1 or taken at the continuous scale. This process where outliers are not excluded beforehand but downweighted in the analysis is called a robust procedure. The outliers can be identi¯ed afterwards by looking at the values of the weight function or by inspecting the residuals which are large in the robust analysis. In either case, outliers should not simply be deleted or ignored. Rather, an important task is to ask what has caused these outliers. They have to be analyzed and interpreted because they often contain very important information on data quality or unexpected behavior of some observations.