A single outlier can drag an ordinary least-squares line anywhere, and the standard errors will not warn you. Robust regression gives a slope that reflects the bulk of the data. Magic Stat runs the Theil–Sen and Siegel repeated-median fits used by R's mblm, and Huber M-estimation when there is more than one predictor.
Least squares works by squaring residuals, so a point far from the line has a disproportionate say. Robust regression replaces that with medians or with weights that shrink the influence of large residuals. The fitted line then tracks the majority of the data instead of being pulled toward a few unusual observations.
For a single predictor Magic Stat offers the two classic estimators: the Theil–Sen slope, the median of all pairwise slopes, and the Siegel repeated median, the median of the medians of slopes within each point. Both come with a robust scale (the MAD) and, following R's mblm, a Wilcoxon signed-rank test and confidence interval for the slope and intercept.
Theil–Sen and repeated median are inherently bivariate — there is no canonical multivariate version. When you have several predictors, Magic Stat uses M-estimation with Huber weights, the same approach as R's MASS::rlm. It is a different method, and it is the honest one to use there.
summary.mblm and confint.mblm.MASS::rlm.Robust method (one predictor): Repeated median (Siegel — the R mblm default) · Theil–Sen (classic).
Several predictors: M-estimation with Huber weights (tuning constant 1.345, the R default) — the Theil–Sen family does not extend to multiple predictors.
No. It changes how much each point influences the fit, using medians or down-weighting. The observations stay in the data and in the figure; they simply do not dictate the line.
Both are resistant; the repeated median has a higher breakdown point, meaning it tolerates a larger fraction of extreme points before the fit is distorted. Repeated median is the default in R's mblm, and it is a reasonable default here too.
Theil–Sen and repeated median are bivariate methods and are not extended to several predictors. Magic Stat fits a robust model by M-estimation with Huber weights instead — the same approach as R's MASS::rlm.
The robust bivariate estimators do not produce a conventional coefficient of determination, and reporting one would be misleading. What is reported is the slope, its robust scale, and a non-parametric test and interval for it.
No. If you use R, note that mblm and MASS::rlm are the references these results are built to match; for specialised robust methods beyond these, R's ecosystem is broader.
Linear regression → Nonlinear regression → Non-parametric tests →