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Chhikara, R. S.

Feiveson, A. H.

English

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for aq useA t WW. `
0^+n-  1Q1 5_.
JSC-14594
NASA C94
A STATISTICAL TEST PROCEDURE FOR DETECTING
14ULTIPLE OUTLIERS IN A DATA SET
Job Order 73-715
Ref: 642-7234
(E80-10195) A STATISTICAL TEST PROCEDURE
FOR DETECTING MULTIPLE OUTLIERS IN A DATA
SET (Lockheed Electronics Co.) 18 p
HC A02 /MF A01	 CSCL 05B
G3/43
N80-28792
Unclas
00195
Prepared By
Lockheed electronics Company, Inc.
Systems and Services Division
Houston, Texas
Contract NAS 9-15200
For
EARTH OBSERVATIONS DIVISION
SPACE AND LIFE SCIENCES DIRECTORATE
National Aeronautics and Space Administration
L YNDON B. JOHNSON SPACE CENTER
Houston, Texas
November 1978
LEC-12910
JSC-14594
A STATISTICAL TEST PROCEDURE FOR DETECTING
MULTIPLE OUTLIERS IN A DATA SET
Job Order 73-715
Ref: 642-7234
I	
PREPARED BY
R. S. Chhikara
Earth Observations Development and Evaluation Department
Research, Test, and Evaluation Branch
APPROVED BY
LEC
	
NASA
v
B. L. Carroll, Manager	 D. Erickson, Chief
Earth Observations Development and 	 Research, Test, and Evaluation Branch
Evaluation Department
Prepared By
Lockheed Electronics Company, Inc.
For
Earth Observations Division
Space and Life Sciences Directorate
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
LYNDON B. JOHNSON SPACE CENTER
HOUSTON, TEXAS
November 1978
LEC-12910
A STATISTICAL TEST PROCEDURE FOR DETECTING
MULTIPLE OUTLIERS IN A DATA SET
1. INTRODUCTION
f-1
Data contamination is a fairly common problem. When analyzing data, it is
sometimes desirable to examine whether or not observations have come from
the same distribution and it detect the potential outliers in the data. So
far, any solution to this problem has been limited to the detection of a
specified number of outliers. The number of outliers is generally unknown
and cannot be specified in advance. The major drawbacks of testing for a
fixed number of outliers are discussed in reference 1.
The sample size is an important influence on the number of observations
likely to be outliers. It is reasonable to think of outliers as a minority,
hence not more than 50 percent of a set of observations can be outliers.
It has been suggested (ref. 1) that a certain percentage of the data
should be considered for potential outliers; however, the percentage should
be variable rather than fixed. For example, it is reasonable to consider
3 potential outliers in a data set of 10 observations, but it is unrealistic
to expect 30 outliers out of a data set of 100 observations. In the latter
case, the outlier detection problem becomes one of discrimination between
two or more classes of data.
Several test statistics have been suggested for the significance test for
detecting outliers (refs. 1 through 4). Among all the suggested tests, the
extreme studentized deviate (ESD) test procedure is most favored, as it is
shown to have more power against certain, alternatives for the number of
outliers and their distributions, and this procedure is computationally
simple. The problem of obtaining the joint distribution of the ESD test
statistics for the multiple outliers detection is still intractable; however,
certain percentage points in the case of testing for the existence of
specifically one or two outliers have been obtained for the test statistics
using the Monte Carlo technique (ref. 1).
In the present Monte Carlo analysis, Rosner's table (ref. 1) is extended to
include percentage points for the 5-percent significance level for testing
as many as 19 outliers in a data set for which the primary distribution is
normal. Also given are certain empirically developed relationships which
can be used to easily obtain the percentage points of the ESD test statistics
for various combinations of sample size and the hypothetical number-of
outliers.
a
a
a
2
2. THE TEST PROCEDURE
The test procedure based on ESD statistics is described in this section.
Following Rosner (ref. 1), who has given a general formulation of many
outlier test procedures, let X 1 , X20 •••, Xn be a data set with k possible
outliers. Consider the sequence of subsets A0 9 Al , •••, Ak , where
AO	
IxlP X2 , •••, Xn! and Ai+1 = Ai - t X (i) 1, i = 1, 2, 
000
,k-1 where X(i)
is defined by
X(i) - Y(Ai )	 MAX Xj - k(Ai)
X j eAi
X(Ai ) = n I---r 	X.
XjeAi
Thus, AnCA n-I ••• CA I CA0 and A i is obtained by deleting fro,-,, A i _ 1 the data
point farthest away from the mean of Ai _ 1 . The test statistic t(AM)
applied to assess the significance of the most outlying observation in Ai
is defined by
MAXi ^Xj - X(Ai)^
(1)t(Ai+l) =
cAX.
	 S i
where
S2 (A i ) = n - i	 1	 lX j - X(A i )^ 2	 i = 0, 1, 2,	 R k - 1XjeAi
Considering Ho , the hypothesis of no outlier present in the data, the
significance test procedures is to reject the hypothesis and declare that
the data contains some outliers if
t(Ai ) > Ni, for some i c{l, 2, •--, k}	 (2)
1
3
where a l , A2 9 •••, Ak are determined by
Prob U [t(A i ) > A i] HD = a	 (3)
i=1
with a as the desired significance level. Clearly, the determination of Xi
requires knowing the joint distribution of t(A l ), t(A2 )
0
 •••, t(Ak ) under H0.
Moreover, the solution of equation (3) forms a hyperplane and thus there is
no unique Set of 
Ai 
satisfying equation (3). To achieve a unique solution,
another requirement besides equation (3) is introduced by considering critical
regions to have a fixed significance level 5 in each dimension. As such,
the procedure is to find s and A ik (0), i = 1, 2, •••, k so that, given H0,
Prob{t(A i ) > A ik (0 4 _ 0	 (4)
and
Prob U l
t(A i ) > Aik(S)l = a	 (5)i=1
If t(Ad < a ik (R) simultaneously for all i t no outliers in the data are
k
declared. Otherwise, if U It(A i ) > a ik (S), holds and
i=1
M =	 MAX
	 ti: t(A i ) > Aik(S))i=1,2,000,k
then X (0) , X (l) , •••, X (m-1) are declared as outliers, i.e., the data points
excluded to form Am are to be declared as outliers.
It may be pointed outthat though s in equation (4) is chosen independent of
i t it can be considered different for different outliers. Unless there is a
specific reason, it is appropriate to attach equal significance to different
outliers and to have the same S for all i in equation (4).
i'	 4
3. DETERMINATION OF CRITICAL VALUES FOR t(Ai)
The joint distribution of t(A i ), i = i, 2, •••, k, is needed to obtain the
critical values of aik (a) for equation (5). No exact derivation of the
distribution is possible; instead, the critical values for a 5-percent
significance level are evaluated by the Monte Carlo procedure. For each
sample size n 3(1)7, 10(5) 3 IU) 100 9 1000 samples of ordered normally
distributed observations were generated. Considering detection of k outliers,
where k = 1, 2, --•, [Zj < 19, the empirical distribution of t(A i ) was
obtained with intervals or probability of size 0.001, enabling us to find a
and Xik(a) such that
Prob[t(Ai ) > Xik (01 = a ; for i = 1, 2 0 •••, k
and
k
Prob U [t(A i' ) > Xik(a)I = 0.05i=1
The critical values of aik (R) are presented in table I. For k = 2, these
values are also given in reference 1. A comparison between the values
given here and corresponding values given in table 7, reference 1, indicates
a difference at the second decimal place. Since a larger number of samples
are used in computations by Rosner (ref. 1), the critical values given in
his paper for the case of k = 2 should be regarded as being more accurate.
In order to obtain the critical values for the 5-percent significance
level for any combination (n, k), where n < 100 and k < 19, a model was
empirically developed for Aik (
 
a) as a function of n and k. A least-square
fit led to the following equation for approximating the critical values.
ikn - Aik + Bik 1og( 'f - k) ; i	
1, 2, ..., k	 (6)
where values for A ik and Bik are given 'in tables II and III, respectively.
Although equation (6) can be used to obtain approximate critical values, the
relative difference of an approximated value from the actual could be as
high as 5 percent. When the outliers are outstanding, it is safe to use
5
z
i
equation (6) to approximate the critical values of the significance test.
On the other hand, more accurate critical values may need to be computed
and used'to detect outliers that are not so outstanding,,, Considering that
errors in the distributional assumption could invalidate the test even if its
exact critical values are known, equation (6) can be regarded as a practical
solution to the problem of determining the critical values of a significance
test at the 5-percent level for detecting as many as 19 outliers in a data
set.
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4. REFERENCES
1. Rosner, Bernard: On the Detection of Many Outliers. Technometrics, 17,
1975, pp. 221-227.
2. Grubbs, Frank E.: Sample Criteria c: Testing Outlying Observations.
Annals Math. Statist., 21, 1950 1, dap. ; ? 7-58.
3. McMillan, R. G.: Tests for One or Two Outliers in Normal Samples With
Unknown Variance. Technometrics, 49, 1971, pp. 87-100.
4. Tietjen, Gary L.; and Moore, Roger H.: Some Grubbs-Type Statistics
for the Detection of Several Outliers. Technometrics, 14, 1972,
pp. 583-598.
15
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A statistical test procedure for detecting multiple outliers in a data set

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