This chapter introduces algorithms for the monitoringMultiple regression approach to robust statistical analysis and compares their performance in the analyses of several sets of data. The Forward SearchForward search (FSFS) is introduced in Sect. 4.1. This algorithm fits subsets of the data of increasing size in such a way that the most outlying observations are included towards the end of the search. To monitor other forms of robust regression, we estimate the parameters using, typically, a grid of 50 values of bdp or eff. Monitoring plots of residualsResiduals, parameter estimatesParameter estimates, and t-tests from very robust regression to LS are introduced in Sect. 4.6; these plots are enriched by brushing and linking to other plots. Frequently, the plots from monitoring residualsResiduals show an abrupt change from robust to LS analyses. In Sect. 4.7, we introduce the empirical bdp defining the most efficient robust estimator for each dataset, thus overcoming the arbitrariness of the conventional approach to robust statistics. Many examples are given in Sect. 4.9 for a variety of estimators. The chapter concludes in Sect. 4.11 with a generalized Bayesian Information Criterion (BIC) for model choice which, via the mean shift outlierOutlier modelMean shift outlier model, makes it possible to compare models in which different numbers of observations have been deleted.