Regression III: Robust regressions
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Outliers
Tests for outlier detections
Robust regressions
Breakdown point
Least trimmed squares
M-estimators
Outliers
Outliers are observations that deviate from all others
significantly. They may occur by accident or they
may be results of measurement errors. Presence of
outliers may lead to misleading results. In the
example shown on figure without outlier mean value
is -0.41 and with outlier it is -0.01. If we want to test
the hypothesis: H0:mean=0 then without outlier we
conclude that H0 can be rejected (p-value is 0.009),
however with outlier we cannot reject null-hypothesis
(p-value is 0.98).
Analysis and dealing with outliers is an important
ingredient of modern statistical analysis. Sometimes
careful analysis of outliers, their removal or weighting
down may change the conclusions considerably.
With outliers
Outlier
Without outlier
Dealing with outliers
For simple cases such as mean, variance calculations one way of dealing with
outliers is using trimmed data. For example in case above mean without outlier is .41, with outlier -0.09 and trimmed mean with 10% removed is -0.33. Trimmed
mean gives “better” than mean based on all data with outliers.
Often mean and other statistics are used for testing some hypothesis. In these cases
using non-parameteric (wilcox.test) tests may be better alternatives to t.test
(wilcox.test gives p-value 0.09 and t.test gives p-value 0.98).
For simple cases like mean, covariance calculations and test based on them usual
approach is to use rank of observations instead of their values. Obviously when rank
is used the power of tests will be reduced, however better conclusions could be more
reliable.
Before carrying out analysis and tests it is always good idea to visualise data and
explore to see if there are outliers.
Outlier detection: Grubbs’ test
It may be a good idea to test for outliers and remove them if possible before starting to
analyse the data and doing hypothesis testing. One of the techniques for doing this is
Grubbs’ test. It tests
H0: there is no outlier versus H1: there is an outlier
To do this Grubbs suggested using the statistic:
G = (max(y)-mean(y))/sd(y)
to test if the maximum value is outlier
G = (mean(y)-mean(y))/sd(y)
to test if the minimum value is outlier and
G = max(|yi-mean(y)|)/sd(y)
to test if maximum or minimum is outlier. There are versions for two outliers also.
Obviously test statistics will depend on the number of observations.
Grubbs’ test (grubbs.test) is available from the package outliers. It is not part of the
standard R distributions.
Outlier detection: Grubbs’ test
Applying grubbs.test for the example we get:
Grubbs test for one outlier
data: del1
G = 2.9063, U = 0.0709, p-value = 9.858e-06
alternative hypothesis: highest value 4 is an outlier
Since p-value is very small we can reject null hypothesis that there is no outlier.
Once we are sure that there is an outlier then we remove it and carry out tests and/or analysis.
Outlier detection: Simulated distribution
If outliers package is not available then we can generate distribution for the statistics
given above. Let us write for one of them (H0: maximum value is not outlier):
outdist = function(nsample,n){
ff = vector(length=n)
for (i in 1:n){rr=rnorm(nsample);ff[i]=max(rr-mean(rr))/sd(rr)}
ff
}
Now we can generate the distribution of the statistics for samples of different sizes. For
example the distribution for sample of sizes 10,15 and 20 are shown on the figure. To
generate the figure the following set of commands are used
oo10 = outdist(10,10000)
curve(ecdf(oo10)(x),from=0,to=7,lwd=3)
oo15 = outdist(15,10000)
curve(ecdf(oo15)(x),from=0,to=7,lwd=3)
oo20 = outdist(20,10000)
curve(ecdf(oo20)(x),from=0,to=7,lwd=3)
Obviously as the sample size increases probability that
large values will be genuinely observed will increase. That is why as sample size
increases the distribution shifts to the right.
Outlier detection: Simulated distribution
Once we have the desired distributions we can calculate test and detect outliers. For
example for the case we considered the sample size is 11. Let us generate empirical
cumulative probability distribution and use it for outlier detection:
oo11=outdist(11,10000)
ec11 = ecdf(oo11)
st = max(del1-mean(del1))/sd(del1)
1-ec11(st)
This sequence of commands will produce p-values. If the maximum value is 4 then pvalue is 0, if the maximum value is 2 then p-value is 0.001 and when maximum value is
1 then p-value is 0.04. We can reject null-hypothesis for first and the second case,
however for the third case we should be careful.
Breakdown point
Breakdown point of an estimator is a fraction of a sample if changed arbitrarily
that does not affect the estimation significantly. For example for mean value if we
change one point arbitrarily we can change the mean value as much as we want.
Let us take an example:
-0.8 -0.6 -0.3 0.1 -1.1 0.2 -0.3 -0.5 -0.5 -0.3 4.0
The sample size is 11, the mean value is -0.09. If we change the last value to 100
then the mean value becomes 8.72. So breakdown point of mean is 0.
Another limiting case is median. Median of the above sample is -0.3. If we change
one value and make it extremely large then median will not change much. For
example if we change the last value to -100 then median will become -0.5.
Breakdown point for median is 0.5, i.e. more than 50% of the sample should be
changed arbitrarily to change the median arbitrarily. Breakdown point 0.5 is the
theoretical limit.
Efficiency of estimators with high breakdown point is usually worse than those
with lower breakdown point. In other words variances of estimators with high
breakdown point are larger.

Outliers and regression
Regression: no outliers
Let us remind us the form of the least-squares
equations for regressions. Again x is a vector of
input (predictor) parameters, β is a vector of
parameters, y is output, the number of sample
points is n.
n
(y
i
 g(x i ,  ))2 ==> min
i1
As we know in special case when g(x,β) =β, and
β is a single value then least-squares estimation
gives mean value of y. We can consider above
estimation as an extension of mean value
estimation. Breakdown point of this estimation is
0, so least-squares is very sensitive to outliers.
There are several approaches to deal with
outliers in regression analysis. We will consider
only two of them: 1) least-trimmed squares; 2)
M-estimators
Regression: with an outlier
Outlier
Least trimmed squares
Least trimmed squares works iteratively.
1) Set up initial values for the model parameters (for
example using simple least squares method implemented in
lm)
2) Calculate squared residuals ri2=(yi-g(xi,β))2
3) Sort squared residuals
4) Remove fraction of observations for which squared
residuals are large
5) Minimise least squares using these observations only
6) Repeat 2)-6) until convergence achieved.
The function lts in R does LTS and several others as a
special case of LTS (least median and least quantile
squares). The number of used obsevations for different
methods are different. For least-median it is [(n+1)/2], for
least quantile [(n+p+1)/2] and for LTS it is [n/2]+[(p+1)/2],
where [] is the integer part of the argument
The result of default lqs
Robust M-estimators
An extension of least-squares to deal with outliers is written as:
n
f ( )   (y i  g(x i ,  ))  min
i1
Form the function ρ defines various
forms of robust M-estimators. When ρ(z)=z2 it
becomes simple least-squares.
Let us first this function.
To minimise this function let us use Gauss-Newton method.

To use this method we need the first and second derivatives (more precisely an
approximation for the second derivative)
n
f
g
  '(y i  g(x i ,  ))


i1
n
2 f
g g n
 2g


''(y

g(x
,

))


'(y

g(x
,

))
 i
 i
i
i
 T  i1
 T  i1
 T 
Where ρ’, ρ’’ are the first and the second derivative of ρ. In Gauss-Newton methods
the second term of the second derivative equation is usually ignored. Usually
ρ’=ψ and ρ’’=w notations are used. If we look at the equations we can see that it
looks like an extension of least-squares equations. The minimisation of the
function is done iteratively using iteratively reweighted least squares (IRLS or
IWLS).
ψ function is an influence function. Analysis of values of this function at the
observations may help to understand outliers in the data and how are dealt with.
Forms of robust regression
Robust M-estimators are usually chosen so
that to make contribution of gradients for large
residuals small, in other words to weight down
large deviations. They can be chosen either
using ρ or ψ.
Basic idea behind robust estimators is: For
small differences behaviour of the function
should be similar to that of least squares and
for large deviations contributions should be
weighted down. Different functions differ by
degree of weighting.
Example of ρ and ψ (Geman
and Mcclure function)
Forms of robust regression
Most popular forms of robust estimators are:
1) Huber
x 2 /2,

(x)  
k
k(| x |  )


2
| x | k
ot herwise
1) Tukey’s bisquare
c 2
x
 (1 (1 ( ) 2 ) 2 )
(x)   6
c
c 2 /6


| x | c
otherwise
1) Geman and Mcclure

x2
(x)  2
c  x2
1) Welsch
2
2
2
c
(x) 
(1 ex / c )
2
2) t-distribution (actually it is a little bit modified form of –log t distribution)

c2
(x)  log(1 x 2 /c 2 )
2
Robust estimators
The result of robust regression with Huber
function is very good. In practice choice of
the function will depend on the number and
severity of outliers.
Robust estimators: Dagnostic plots
Boxplots for residuals and weighted residuals
R commands for robust estimation
lqs – least trimmed, least median estimation
rlm – robust linear model estimation using M-estimator
outliers package may be helpful.
References
1)
2)
3)
4)
P. J. Huber (1981) “Robust Statistics”.
F. R. Hampel, E. M. Ronchetti, P. J. Rousseeuw and W. A. Stahel
(1986) “Robust Statistics: The Approach based on Influence
Functions”
A. Marazzi (1993) “Algorithms, Routines and S Functions for
Robust Statistics”,
W. N. and Ripley, B. D. (2002) “Modern Applied Statistics with S”