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1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.5. Quantitative Techniques

1.3.5.14.

Anderson-Darling Test
Purpose:
Test for Distributional Adequacy 		The Anderson-Darling test (Stephens, 1974) is used to test if a sample of data came from a population with a specific distribution. It is a modification of the Kolmogorov-Smirnov (K-S) test and gives more weight to the tails than does the K-S test. The K-S test is distribution free in the sense that the critical values do not depend on the specific distribution being tested. The Anderson-Darling test makes use of the specific distribution in calculating critical values. This has the advantage of allowing a more sensitive test and the disadvantage that critical values must be calculated for each distribution. Currently, tables of critical values are available for the normal, lognormal, exponential, Weibull, extreme value type I, and logistic distributions. We do not provide the tables of critical values in this Handbook (see Stephens 1974, 1976, 1977, and 1979) since this test is usually applied with a statistical software program that will print the relevant critical values.

The Anderson-Darling test is an alternative to the chi-square and Kolmogorov-Smirnov goodness-of-fit tests.

Definition The Anderson-Darling test is defined as:
H0: The data follow a specified distribution.
Ha: The data do not follow the specified distribution
Test Statistic: The Anderson-Darling test statistic is defined as
    A**2 = -N - S
where

S=SUM(i=1 to N)((2*i - 1)/N)*[LN(F(Y(i)) + LN(1 - F(Y(N+1-i))]

F is the cumulative distribution function of the specified distribution. Note that the Yi are the ordered data.

Significance Level: alpha
Critical Region: The critical values for the Anderson-Darling test are dependent on the specific distribution that is being tested. Tabulated values and formulas have been published (Stephens, 1974, 1976, 1977, 1979) for a few specific distributions (normal, lognormal, exponential, Weibull, logistic, extreme value type 1). The test is a one-sided test and the hypothesis that the distribution is of a specific form is rejected if the test statistic, A, is greater than the critical value.

Note that for a given distribution, the Anderson-Darling statistic may be multiplied by a constant (which usually depends on the sample size, n). These constants are given in the various papers by Stephens. In the sample output below, this is the "adjusted Anderson-Darling" statistic. This is what should be compared against the critical values. Also, be aware that different constants (and therefore critical values) have been published. You just need to be aware of what constant was used for a given set of critical values (the needed constant is typically given with the critical values).

Sample Output
 		Dataplot generated the following output for the Anderson-Darling test. 1,000 random numbers were generated for a normal, double exponential, Cauchy, and lognormal distribution. In all four cases, the Anderson-Darling test was applied to test for a normal distribution. When the data were generated using a normal distribution, the test statistic was small and the hypothesis was accepted. When the data were generated using the double exponential, Cauchy, and lognormal distributions, the statistics were significant, and the hypothesis of an underlying normal distribution was rejected at significance levels of 0.10, 0.05, and 0.01.

The normal random numbers were stored in the variable Y1, the double exponential random numbers were stored in the variable Y2, the Cauchy random numbers were stored in the variable Y3, and the lognormal random numbers were stored in the variable Y4. 

       ***************************************
       **  anderson darling normal test y1  **
       ***************************************
  
  
               ANDERSON-DARLING 1-SAMPLE TEST
               THAT THE DATA CAME FROM A NORMAL DISTRIBUTION
  
 1. STATISTICS:
       NUMBER OF OBSERVATIONS                =     1000
       MEAN                                  =   0.4359940E-02
       STANDARD DEVIATION                    =    1.001816
  
       ANDERSON-DARLING TEST STATISTIC VALUE =   0.2565918
       ADJUSTED TEST STATISTIC VALUE         =   0.2576117
  
 2. CRITICAL VALUES:
       90         % POINT    =   0.6560000
       95         % POINT    =   0.7870000
       97.5       % POINT    =   0.9180000
       99         % POINT    =    1.092000
  
 3. CONCLUSION (AT THE 5% LEVEL):
       THE DATA DO COME FROM A NORMAL DISTRIBUTION.
  
  
       ***************************************
       **  anderson darling normal test y2  **
       ***************************************
  
  
               ANDERSON-DARLING 1-SAMPLE TEST
               THAT THE DATA CAME FROM A NORMAL DISTRIBUTION
  
 1. STATISTICS:
       NUMBER OF OBSERVATIONS                =     1000
       MEAN                                  =   0.2034888E-01
       STANDARD DEVIATION                    =    1.321627
  
       ANDERSON-DARLING TEST STATISTIC VALUE =    5.826050
       ADJUSTED TEST STATISTIC VALUE         =    5.849208
  
 2. CRITICAL VALUES:
       90         % POINT    =   0.6560000
       95         % POINT    =   0.7870000
       97.5       % POINT    =   0.9180000
       99         % POINT    =    1.092000
  
 3. CONCLUSION (AT THE 5% LEVEL):
       THE DATA DO NOT COME FROM A NORMAL DISTRIBUTION.
  
  
       ***************************************
       **  anderson darling normal test y3  **
       ***************************************
  
  
               ANDERSON-DARLING 1-SAMPLE TEST
               THAT THE DATA CAME FROM A NORMAL DISTRIBUTION
  
 1. STATISTICS:
       NUMBER OF OBSERVATIONS                =     1000
       MEAN                                  =    1.503854
       STANDARD DEVIATION                    =    35.13059
  
       ANDERSON-DARLING TEST STATISTIC VALUE =    287.6429
       ADJUSTED TEST STATISTIC VALUE         =    288.7863
  
 2. CRITICAL VALUES:
       90         % POINT    =   0.6560000
       95         % POINT    =   0.7870000
       97.5       % POINT    =   0.9180000
       99         % POINT    =    1.092000
  
 3. CONCLUSION (AT THE 5% LEVEL):
       THE DATA DO NOT COME FROM A NORMAL DISTRIBUTION.
  
  
       ***************************************
       **  anderson darling normal test y4  **
       ***************************************
  
  
               ANDERSON-DARLING 1-SAMPLE TEST
               THAT THE DATA CAME FROM A NORMAL DISTRIBUTION
  
 1. STATISTICS:
       NUMBER OF OBSERVATIONS                =     1000
       MEAN                                  =    1.518372
       STANDARD DEVIATION                    =    1.719969
  
       ANDERSON-DARLING TEST STATISTIC VALUE =    83.06335
       ADJUSTED TEST STATISTIC VALUE         =    83.39352
  
 2. CRITICAL VALUES:
       90         % POINT    =   0.6560000
       95         % POINT    =   0.7870000
       97.5       % POINT    =   0.9180000
       99         % POINT    =    1.092000
  
 3. CONCLUSION (AT THE 5% LEVEL):
       THE DATA DO NOT COME FROM A NORMAL DISTRIBUTION.
Interpretation of the Sample Output The output is divided into three sections.
  1. The first section prints the number of observations and estimates for the location and scale parameters.

  2. The second section prints the upper critical value for the Anderson-Darling test statistic distribution corresponding to various significance levels. The value in the first column, the confidence level of the test, is equivalent to 100(1-alpha). We reject the null hypothesis at that significance level if the value of the Anderson-Darling test statistic printed in section one is greater than the critical value printed in the last column.

  3. The third section prints the conclusion for a 95% test. For a different significance level, the appropriate conclusion can be drawn from the table printed in section two. For example, for = 0.10, we look at the row for 90% confidence and compare the critical value 0.656 to the Anderson-Darling test statistic (for the normal data) 0.256. Since the test statistic is less than the critical value, we do not reject the null hypothesis at the alpha = 0.10 level.

As we would hope, the Anderson-Darling test accepts the hypothesis of normality for the normal random numbers and rejects it for the 3 non-normal cases.

The output from other statistical software programs may differ somewhat from the output above.

Questions The Anderson-Darling test can be used to answer the following questions:
  • Are the data from a normal distribution?
  • Are the data from a log-normal distribution?
  • Are the data from a Weibull distribution?
  • Are the data from an exponential distribution?
  • Are the data from a logistic distribution?
Importance Many statistical tests and procedures are based on specific distributional assumptions. The assumption of normality is particularly common in classical statistical tests. Much reliability modeling is based on the assumption that the data follow a Weibull distribution.

There are many non-parametric and robust techniques that do not make strong distributional assumptions. However, techniques based on specific distributional assumptions are in general more powerful than non-parametric and robust techniques. Therefore, if the distributional assumptions can be validated, they are generally preferred.

Related Techniques Chi-Square goodness-of-fit Test
Kolmogorov-Smirnov Test
Shapiro-Wilk Normality Test
Probability Plot
Probability Plot Correlation Coefficient Plot
Case Study Airplane glass failure time data.
Software The Anderson-Darling goodness-of-fit test is available in some general purpose statistical software programs, including Dataplot.
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