MASS (version 7.3-58.3)

lqs: Resistant Regression


Fit a regression to the good points in the dataset, thereby achieving a regression estimator with a high breakdown point. lmsreg and ltsreg are compatibility wrappers.


lqs(x, ...)

# S3 method for formula lqs(formula, data, ..., method = c("lts", "lqs", "lms", "S", "model.frame"), subset, na.action, model = TRUE, x.ret = FALSE, y.ret = FALSE, contrasts = NULL)

# S3 method for default lqs(x, y, intercept = TRUE, method = c("lts", "lqs", "lms", "S"), quantile, control = lqs.control(...), k0 = 1.548, seed, ...)

lmsreg(...) ltsreg(...)


An object of class "lqs". This is a list with components


the value of the criterion for the best solution found, in the case of method == "S" before IWLS refinement.


character. A message about the number of samples which resulted in singular fits.


of the fitted linear model


the indices of those points fitted by the best sample found (prior to adjustment of the intercept, if requested).


the fitted values.


the residuals.


estimate(s) of the scale of the error. The first is based on the fit criterion. The second (not present for method == "S") is based on the variance of those residuals whose absolute value is less than 2.5 times the initial estimate.



a formula of the form y ~ x1 + x2 + ....


an optional data frame, list or environemnt from which variables specified in formula are preferentially to be taken.


an index vector specifying the cases to be used in fitting. (NOTE: If given, this argument must be named exactly.)


function to specify the action to be taken if NAs are found. The default action is for the procedure to fail. Alternatives include na.omit and na.exclude, which lead to omission of cases with missing values on any required variable. (NOTE: If given, this argument must be named exactly.)

model, x.ret, y.ret

logical. If TRUE the model frame, the model matrix and the response are returned, respectively.


an optional list. See the contrasts.arg of model.matrix.default.


a matrix or data frame containing the explanatory variables.


the response: a vector of length the number of rows of x.


should the model include an intercept?


the method to be used. model.frame returns the model frame: for the others see the Details section. Using lmsreg or ltsreg forces "lms" and "lts" respectively.


the quantile to be used: see Details. This is over-ridden if method = "lms".


additional control items: see Details.


the cutoff / tuning constant used for \(\chi()\) and \(\psi()\) functions when method = "S", currently corresponding to Tukey's ‘biweight’.


the seed to be used for random sampling: see .Random.seed. The current value of .Random.seed will be preserved if it is set..


arguments to be passed to lqs.default or lqs.control, see control above and Details.


Suppose there are n data points and p regressors, including any intercept.

The first three methods minimize some function of the sorted squared residuals. For methods "lqs" and "lms" is the quantile squared residual, and for "lts" it is the sum of the quantile smallest squared residuals. "lqs" and "lms" differ in the defaults for quantile, which are floor((n+p+1)/2) and floor((n+1)/2) respectively. For "lts" the default is floor(n/2) + floor((p+1)/2).

The "S" estimation method solves for the scale s such that the average of a function chi of the residuals divided by s is equal to a given constant.

The control argument is a list with components


the size of each sample. Defaults to p.


the number of samples or "best" (the default) or "exact" or "sample". If "sample" the number chosen is min(5*p, 3000), taken from Rousseeuw and Hubert (1997). If "best" exhaustive enumeration is done up to 5000 samples; if "exact" exhaustive enumeration will be attempted however many samples are needed.


should the intercept be optimized for each sample? Defaults to TRUE.


P. J. Rousseeuw and A. M. Leroy (1987) Robust Regression and Outlier Detection. Wiley.

A. Marazzi (1993) Algorithms, Routines and S Functions for Robust Statistics. Wadsworth and Brooks/Cole.

P. Rousseeuw and M. Hubert (1997) Recent developments in PROGRESS. In L1-Statistical Procedures and Related Topics, ed Y. Dodge, IMS Lecture Notes volume 31, pp. 201--214.

See Also



Run this code
set.seed(123) # make reproducible
lqs(stack.loss ~ ., data = stackloss)
lqs(stack.loss ~ ., data = stackloss, method = "S", nsamp = "exact")

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