Resistant Regression
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
1 2 3 4 5 6 7 8 9 10 11 12 13 14  lqs(x, ...)
## S3 method for class '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)
## Default S3 method:
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 
data 
data frame from which variables specified in

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

model, x.ret, y.ret 
logical. If 
contrasts 
an optional list. See the 
x 
a matrix or data frame containing the explanatory variables. 
y 
the response: a vector of length the number of rows of 
intercept 
should the model include an intercept? 
method 
the method to be used. 
quantile 
the quantile to be used: see 
control 
additional control items: see 
k0 
the cutoff / tuning constant used for chi()
and psi() functions when 
seed 
the seed to be used for random sampling: see 
... 
arguments to be passed to 
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 ismin(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
.
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 
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 
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((np)/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 L1Statistical Procedures and Related Topics, ed Y. Dodge, IMS Lecture Notes volume 31, pp. 201–214.
See Also
predict.lqs
Examples
1 2 3  set.seed(123) # make reproducible
lqs(stack.loss ~ ., data = stackloss)
lqs(stack.loss ~ ., data = stackloss, method = "S", nsamp = "exact")
