Results for small n and moderate p show that the size and power of robust tests of regression models can be far from the nominal asymptotic valuesAsymptotic value. This chapter uses simulation to investigate the properties of outlierOutlier tests for moderate sample sizes. Section 5.1 compares the size of outlierOutlier tests for 30 very robust estimators, which leads to the selection of five estimators for comparison in the next sections: S, MM, LTSLeast Trimmed Squares (LTS), and LTSr, the reweighted LTS estimator, together with FSFS. Section 5.2 compares the sizes of the five test for n from 100 to 1,000 and Sect. 5.3 compares the average power (the proportion of outliersOutlier correctly identified) over a range of values of the shift in the contamination. Section 5.4 describes a parametric framework for comparing robust regression estimatorsRegression estimators. In this a set of outliersOutlier is moved along some trajectory in the space of y and X and the effect on inferences calculated; typically biasBias and variance of parameter estimatesParameter estimates and average power. In all these comparisons, all estimators, apart from FSFS, have fixed, very robust, settings. Monitoring the comparisons is provided in Sect. 5.6 by using the extended BIC (Sect.  4.11 ) to determine the robustnessRobustness level at which the properties of the various estimators are assessed.