 Dataplot Vol 1 Vol 2

# TIETJEN-MOORE TEST

Name:
TIETJEN-MOORE TEST
Type:
Analysis Command
Purpose:
Perform a Tietjen-Moore test for outliers.
Description:
The Tietjen-Moore test is a generalization of the Grubbs test that can be used to test for multiple outliers in a univariate data set. As with the Grubbs test, the Tietjen-Moore test assumes that the underlying distribution follows an approximately normal distribution.

It is important to note that the Tietjen-Moore test requires that the suspected number of outliers be specified exactly. If this is not known, it is recommended that the generalized extreme studentized deviate test be used instead (this test only requires an upper bound on the number of suspected outliers).

More formally, the Tietjen-Moore test can be defined as follows.

 H0: There are no outliers in the data set Ha: There are exactly k outliers in the data set Test Statistic: Sort the n data points from smallest to the largest so that yi denotes the ith largest data value. The test statistic for the k largest points is $$L_{k} = \frac{\sum_{i=1}^{n-k}{(y_{i} - \bar{y}_{k})^{2}}} {\sum_{i=1}^{n}{(y_{i} - \bar{y})^{2}}}$$ with $$\bar{y}$$ denoting the sample mean for the full sample and $$\bar{y}_{k}$$ denoting the sample mean with the largest k points removed. The test statistic for the k smallest points is $$L_{k} = \frac{\sum_{i=k+1}^{n}{(y_{i} - \bar{y}_{k})^{2}}} {\sum_{i=1}^{n}{(y_{i} - \bar{y})^{2}}}$$ with $$\bar{y}$$ denoting the sample mean for the full sample and $$\bar{y}_{k}$$ denoting the sample mean with the smallest k points removed. To test for outliers in both tails, compute the absolute residuals $$r_{i} = |y_{i} - \bar{y}|$$ and then let zi denote the sorted absolute residuals. The test statistic for this case is $$E_{k} = \frac{\sum_{i=k+1}^{n}{(z_{i} - \bar{z}_{k})^{2}}} {\sum_{i=1}^{n}{(z_{i} - \bar{z})^{2}}}$$ with $$\bar{z}$$ denoting the sample mean of the absolute residuals for the full sample and $$\bar{z}_{k}$$ denoting the sample mean of the absolute residuals with the largest k points removed. Significance Level: $$\alpha$$ Critical Region: The critical region for the Tietjen-Moore test is determined by simulation. The simulation is performed by generating a standard normal random sample of size n and computing the Tietjen-Moore test statistic. Typically, 10,000 random samples are used. The value of the Tietjen-Moore statistic obtained from the data are compared to this reference distribution. Dataplot performs this simulation dynamically. The critical values obtained may differ slightly from the critical values given in the Tietjen-Moore paper.

It is recommended that formal outlier tests be complemented with graphical methods. For example, a normal probability plot can be used to determine if the normality assumption is reasonable and also to determine an appropriate value for k.

Syntax 1:
TIETJEN-MOORE TEST <y>             <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable being tested;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax checks for outliers in both tails.

Syntax 2:
TIETJEN-MOORE MINIMUM TEST <y>       <SUBSET/EXCEPT/FOR qualification> where <y> is the response variable being tested;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax checks the minimum values for outliers.

Syntax 3:
TIETJEN-MOORE MAXIMUM TEST <y>       <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable being tested;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax checks the maximum values for outliers.

Syntax 4:
TIETJEN-MOORE MULTIPLE TEST <y1> ... <yk>
<SUBSET/EXCEPT/FOR qualification>
where <y1> ... <yk> is a list of up to k response variables;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax can also be used with the MINIMUM and MAXIMUM version of the tests. This syntax performs a Tietjen-Moore test on , then on , and so on. Up to 30 response variables may be specified.

Note that the syntax

TIETJEN-MOORE MULTIPLE TEST Y1 TO Y4

is supported. This is equivalent to

TIETJEN-MOORE MULTIPLE TEST Y1 Y2 Y3 Y4
Syntax 5:
TIETJEN-MOORE REPLICATED TEST <y> <x1> ... <xk>
<SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
<x1> ... <xk> is a list of up to k group-id variables;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax can also be used with the MINIMUM and MAXIMUM version of the tests. This syntax peforms a cross-tabulation of <x1> ... <xk> and performs a Tietjen-Moore test for each unique combination of cross-tabulated values. For example, if X1 has 3 levels and X2 has 2 levels, there will be a total of 6 Tietjen-Moore tests performed.

Up to six group-id variables can be specified.

Note that the syntax

TIETJEN-MOORE REPLICATED TEST Y X1 TO X4

is supported. This is equivalent to

TIETJEN-MOORE REPLICATED TEST Y X1 X2 X3 X4
Examples:
TIETJEN-MOORE TEST Y1
TIETJEN-MOORE TEST Y1 LABID
TIETJEN-MOORE MULTIPLE TEST Y1 Y2 Y3
TIETJEN-MOORE REPLICATED TEST Y X1 X2
TIETJEN-MOORE TEST Y1 SUBSET TAG > 2
TIETJEN-MOORE MINIMUM TEST Y1
TIETJEN-MOORE MAXIMUM TEST Y1
Note:
The number of outliers to test for is specified with the command

LET NOUTLIER = <value>
Note:
Masking and swamping are two issues that can affect outlier tests.

Masking can occur when we specify too few outliers in the test. For example, if we are testing for a single outlier when there are in fact two (or more) outliers, these additional outliers may influence the value of the test statistic enough so that no points are declared as outliers.

On the other hand, swamping can occur when we specify too many outliers in the test. For example, if we are testing for two outliers when there is in fact only a single outlier, both points may be declared outliers.

The possibility of masking and swamping are an important reason why it is useful to complement formal outlier tests with graphical methods. Graphics can often help identify cases where masking or swamping may be an issue.

Also, masking is one reason that trying to apply a single outlier test sequentially can fail. If there are multiple outliers, masking may cause the outlier test for the first outlier to return a conclusion of no outliers (and so the testing for any additional outliers is not done).

The Tietjen-Moore test is used to check for exactly k outliers. If k is not specified correctly, the results of the Tietjen-Moore test can be distorted.

If determining k is not obvious, then the generalized extreme studentized deviate tests may be preferred since this test only requires that an upper bound on the suspected number of outliers be specified.

Note:
Tests for outliers are dependent on knowing the distribution of the data. The Tietjen-Moore test assumes that the data come from an approximately normal distribution. For this reason, it is strongly recommended that the Tietjen-Moore test be complemented with a normal probability plot. If the data are not approximately normally distributed, then the Tietjen-Moore test may be detecting the non-normality of the data rather than the presence of outliers.
Note:
You can specify the number of digits in the Tietjen-Moore output with the command

SET WRITE DECIMALS <value>
Note:
The TIETJEN-MOORE TEST command automatically saves the following parameters:

 STATVAL = the value of the test statistic PVAL = the p-value of the test statistic CUTOFF0 = the 0 percent point of the reference distribution CUTOFF50 = the 50 percent point of the reference distribution CUTOFF75 = the 75 percent point of the reference distribution CUTOFF90 = the 90 percent point of the reference distribution CUTOFF95 = the 95 percent point of the reference distribution CUTOFF975 = the 97.5 percent point of the reference distribution CUTOFF99 = the 99 percent point of the reference distribution

If the MULTIPLE or REPLICATED option is used, these values will be written to the file "dpst1f.dat" instead.

Note:
In addition to the TIETJEN-MOORE TEST command, the following commands can also be used:

LET A = TIETJEN-MOORE Y
LET A = TIETJEN-MOORE MINIMUM Y
LET A = TIETJEN-MOORE MAXIMUM Y

In addition to the above LET command, built-in statistics are supported for 20+ different commands (enter HELP STATISTICS for details).

Default:
None
Synonyms:
MULTIPLE TIETJEN MOORE is a synonym for TIETJEN MOORE MULTIPLE
REPLICATION TIETJEN MOORE is a synonym for TIETJEN MOORE REPLICATION
Related Commands:
 GRUBB TEST = Perform a Grubbs outlier test. EXTREME STUDENTIZED DEVIATE TEST = Perform a extreme studentized deviate outlier test. DIXON TEST = Perform a Dixon outlier test. ANDERSON DARLING TEST = Perform an Anderson Darling normality test. WILKS SHAPIRO NORMALITY TEST = Perform a Wilks Shapiro normality test. HISTOGRAM = Generate a histogram. PROBABILITY PLOT = Generates a probability plot. BOX PLOT = Generate a box plot.
Reference:
Gary Tietjen and Roger Moore (August 1972), "Some Grubbs-Type Statistics for the Detection of Several Outliers", Technometrics, Vol. 14, No. 3, pp. 583-597.
Applications:
Outlier Detection
Implementation Date:
2009/11
Program:

.  Following example from Tietjen-Moore paper
-1.40
-0.44
-0.30
-0.24
-0.22
-0.13
-0.05
0.06
0.10
0.18
0.20
0.39
0.48
0.63
1.01
END OF DATA
.
.  First generate a normal probability plot
.
LABEL CASE ASIS
TITLE CASE ASIS
TITLE OFFSET 5
.
Y1LABEL Data
X1LABEL Theoretical
TITLE Normal Probability Plot of Semi-Diameters of Venus
LINE BLANK
CHAR CIRCLE
CHAR FILL ON
LET H = 1.2
LET W = H*0.75
CHAR HW H W
.
NORMAL PROBABILITY PLOT Y
.
.  Now perform the Tietjen-Moore test
.
SET WRITE DECIMALS 5
LET NOUTLIER = 2
TIETJEN-MOORE Y

The following output is generated Tietjen-Moore Test for Multiple Outliers: Two-Sided Case
(Assumption: Normality)

Response Variable: Y

H0: There are no outliers
Ha: The     2 most extreme points are outliers
Potential Outlier Value Tested:                      -1.39999
Potential Outlier Value Tested:                       0.06000

Summary Statistics:
Number of Observations:                                    15
Sample Minimum:                                      -1.39999
Sample Maximum:                                       1.01000
Sample Mean:                                          0.01800
Sample SD:                                            0.55094

Tietjen-Moore Test Statistic Value:                   0.29199
CDF Value:                                            0.96560
P-Value                                               0.03440

Percent Points of the Reference Distribution
-----------------------------------
Percent Point               Value
-----------------------------------
0.0    =          0.064
1.0    =          0.237
2.5    =          0.277
5.0    =          0.315
10.0    =          0.362
25.0    =          0.435
50.0    =          0.508
100.0    =          0.755

Conclusions (Lower 1-Tailed Test)
----------------------------------------------
Alpha    CDF   Critical Value     Conclusion
----------------------------------------------
10%    10%            0.362      Reject H0
5%     5%            0.315      Reject H0
2.5%   2.5%            0.277      Accept H0
1%     1%            0.237      Accept H0

*Critical Values Based on    10000 Monte Carlo Simulations


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Date created: 09/09/2010
Last updated: 11/04/2015