MASS (version 7.3-47)

lqs:

Description

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.

Usage

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(…)

Arguments

formula

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

data

data frame from which variables specified in formula are preferentially to be taken.

subset

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

na.action

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.

contrasts

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

x

a matrix or data frame containing the explanatory variables.

y

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

intercept

should the model include an intercept?

method

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.

quantile

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

control

additional control items: see Details.

k0

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

seed

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.

Value

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

crit

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

sing

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

coefficients

of the fitted linear model

bestone

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

fitted.values

the fitted values.

residuals

the residuals.

scale

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.

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

psamp:

the size of each sample. Defaults to p.

nsamp:

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.

adjust:

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

References

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

predict.lqs

Examples

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