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1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.5. Quantitative Techniques
1.3.5.17. Detection of Outliers

1.3.5.17.3.

Generalized ESD Test for Outliers

Purpose:
Detection of Outliers
The generalized (extreme Studentized deviate) ESD test (Rosner 1983) is used to detect one or more outliers in a univariate data set that follows an approximately normal distribution.

The primary limitation of the Grubbs test and the Tietjen-Moore test is that the suspected number of outliers, k, must be specified exactly. If k is not specified correctly, this can distort the conclusions of these tests. On the other hand, the generalized ESD test (Rosner 1983) only requires that an upper bound for the suspected number of outliers be specified.

Definition Given the upper bound, r, the generalized ESD test essentially performs r separate tests: a test for one outlier, a test for two outliers, and so on up to r outliers.

The generalized ESD test is defined for the hypothesis:

H0: There are no outliers in the data set
Ha: There are up to r outliers in the data set
Test Statistic: Compute

    \( R_i = \frac{\mbox{max}_i |x_i - \bar{x}|}{s} \)

with \(\bar{x}\) and s denoting the sample mean and sample standard deviation, respectively.

Remove the observation that maximizes \(|x_{i} - \bar{x}|\) and then recompute the above statistic with n - 1 observations. Repeat this process until r observations have been removed. This results in the r test statistics R1, R2, ..., Rr.

Significance Level: α
Critical Region: Corresponding to the r test statistics, compute the following r critical values

    \( \lambda_{i} = \frac{(n-i) \, t_{p, \, n-i-1}}{\sqrt{(n-i-1+t_{p, \, n-i-1}^{2}) (n-i+1)}} \hspace{.2in} i = 1, 2, \ldots, r \)

where tp,ν is the 100p percentage point from the t distribution with ν degrees of freedom and

    \( p = 1-\frac{\alpha}{2(n-i+1)} \)

The number of outliers is determined by finding the largest i such that Ri > λi.

Simulation studies by Rosner indicate that this critical value approximation is very accurate for n ≥ 25 and reasonably accurate for n ≥ 15.

Note that although the generalized ESD is essentially Grubbs test applied sequentially, there are a few important distinctions:

  • The generalized ESD test makes approriate adjustments for the critical values based on the number of outliers being tested for that the sequential application of Grubbs test does not.

  • If there is significant masking, applying Grubbs test sequentially may stop too soon. The example below identifies three outliers at the 5 % level when using the generalized ESD test. However, trying to use Grubbs test sequentially would stop at the first iteration and declare no outliers.
Generalized ESD Test Example The Rosner paper gives an example with the following data.

         -0.25 0.68 0.94 1.15 1.20 1.26 1.26
          1.34 1.38 1.43 1.49 1.49 1.55 1.56
          1.58 1.65 1.69 1.70 1.76 1.77 1.81
          1.91 1.94 1.96 1.99 2.06 2.09 2.10
          2.14 2.15 2.23 2.24 2.26 2.35 2.37
          2.40 2.47 2.54 2.62 2.64 2.90 2.92
          2.92 2.93 3.21 3.26 3.30 3.59 3.68
          4.30 4.64 5.34 5.42 6.01
      

As a first step, a normal probability plot was generated

Normal Probability Plot of the Data

This plot indicates that the normality assumption is questionable.

Following the Rosner paper, we test for up to 10 outliers:

 
      H0:  there are no outliers in the data
      Ha:  there are up to 10 outliers in the data

      Significance level:  α = 0.05
      Critical region:  Reject H0 if Ri > critical value
 
      Summary Table for Two-Tailed Test
      ---------------------------------------
            Exact           Test     Critical  
        Number of      Statistic    Value, λi  
      Outliers, i      Value, Ri          5 %  
      ---------------------------------------
              1          3.118          3.158  
              2          2.942          3.151  
              3          3.179          3.143 * 
              4          2.810          3.136  
              5          2.815          3.128  
              6          2.848          3.120  
              7          2.279          3.111  
              8          2.310          3.103  
              9          2.101          3.094  
             10          2.067          3.085  
 
For the generalized ESD test above, there are essentially 10 separate tests being performed. For this example, the largest number of outliers for which the test statistic is greater than the critical value (at the 5 % level) is three. We therefore conclude that there are three outliers in this data set.
Questions The generalized ESD test can be used to answer the following question:
  1. How many outliers does the data set contain?
Importance Many statistical techniques are sensitive to the presence of outliers. For example, simple calculations of the mean and standard deviation may be distorted by a single grossly inaccurate data point.

Checking for outliers should be a routine part of any data analysis. Potential outliers should be examined to see if they are possibly erroneous. If the data point is in error, it should be corrected if possible and deleted if it is not possible. If there is no reason to believe that the outlying point is in error, it should not be deleted without careful consideration. However, the use of more robust techniques may be warranted. Robust techniques will often downweight the effect of outlying points without deleting them.

Related Techniques Several graphical techniques can, and should, be used to help detect outliers. A simple normal probability plot, run sequence plot, a box plot, or a histogram should show any obviously outlying points. In addition to showing potential outliers, several of these graphics also help assess whether the data follow an approximately normal distribution.
Software Some general purpose statistical software programs support the generalized ESD test. Both Dataplot code and R code can be used to generate the analyses in this section.
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