# lqs

0th

Percentile

##### 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.

Keywords
robust, models
##### Usage
lqs(x, ...)## S3 method for class 'formula':
lqs(formula, data, \dots,
method = c("lts", "lqs", "lms", "S", "model.frame"),
subset, na.action, model = TRUE,
x.ret = FALSE, y.ret = FALSE, contrasts = NULL)## S3 method for class '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
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.
##### 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 [object Object],[object Object],[object Object]

##### Value

• An object of class "lqs". This is a list with components
• critthe value of the criterion for the best solution found, in the case of method == "S" before IWLS refinement.
• singcharacter. A message about the number of samples which resulted in singular fits.
• coefficientsof the fitted linear model
• bestonethe indices of those points fitted by the best sample found (prior to adjustment of the intercept, if requested).
• fitted.valuesthe fitted values.
• residualsthe residuals.
• scaleestimate(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.

##### Note

There seems no reason other than historical to use the lms and lqs options. LMS estimation is of low efficiency (converging at rate $n^{-1/3}$) whereas LTS has the same asymptotic efficiency as an M estimator with trimming at the quartiles (Marazzi, 1993, p.201). LQS and LTS have the same maximal breakdown value of (floor((n-p)/2) + 1)/n attained if floor((n+p)/2) <= quantile="" <="floor((n+p+1)/2). The only drawback mentioned of LTS is greater computation, as a sort was thought to be required (Marazzi, 1993, p.201) but this is not true as a partial sort can be used (and is used in this implementation).

 Adjusting the intercept for each trial fit does need the residuals to be sorted, and may be significant extra computation if n is large and p small.

 Opinions differ over the choice of psamp. Rousseeuw and Hubert (1997) only consider p; Marazzi (1993) recommends p+1 and suggests that more samples are better than adjustment for a given computational limit.

 The computations are exact for a model with just an intercept and adjustment, and for LQS for a model with an intercept plus one regressor and exhaustive search with adjustment. For all other cases the minimization is only known to be approximate.

##### 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.

predict.lqs

• lqs
• lqs.formula
• lqs.default
• lmsreg
• ltsreg
##### Examples
set.seed(123) # make reproducible
lqs(stack.loss ~ ., data = stackloss)
lqs(stack.loss ~ ., data = stackloss, method = "S", nsamp = "exact")
Documentation reproduced from package MASS, version 7.3-35, License: GPL-2 | GPL-3

### Community examples

Looks like there are no examples yet.