What is the Anderson-Darling statistic?

The Anderson-Darling statistic measures how well the data follow a particular distribution. For a particular data set and distribution, the better the distribution fits the data, the smaller this statistic will be. However, Anderson-Darling statistics that are calculated for different distributions may not be directly comparable. Therefore, slight differences in Anderson-Darling values may not have practical relevance. Use probability plots and other information to compare the fit of different distributions.

What is the adjusted Anderson-Darling statistic?

Minitab calculates an adjusted Anderson-Darling statistic for the Distribution ID Plot and for Reliability/Survival analyses. These adjusted Anderson-Darling statistics are represented in the output as Anderson-Darling (adj) or AD*. The adjustment to the traditional Anderson-Darling statistic makes the value of the statistic depend on the method used to plot the points on a probability plot. The traditional Anderson-Darling statistic always uses the Kaplan-Meier plot point method.

P-values for the adjusted Anderson-Darling statistic cannot be calculated for arbitrarily or multiply-censored data. For consistency, Minitab does not print any p-values for reliability, even when they are available.

Note

Minitab provides a p-value for the unadjusted Anderson-Darling statistic with the Individual Distribution Identification analysis in Quality Tools. That analysis always uses the Kaplan-Meier plot point method on uncensored data. For detailed information about how the adjusted Anderson-Darling statistic is calculated in a reliability analysis, go to Methods and formulas for goodness-of-fit measures in Parametric Distribution Analysis (Right Censoring) and click "Anderson-Darling statistic".

Why do Individual Distribution Identification and Reliability/Survival give different Anderson-Darling statistics?

The Anderson-Darling statistic reported from Graph > Probability Plot or Stat > Quality Tools > Individual Distribution Identification is not adjusted. However, the Anderson-Darling statistic reported from the commands in the Stat > Reliability/Survival menu is adjusted to handle censored data and different plot-point methods.

To compare the two different Anderson-Darling statistics, use the maximum likelihood estimation method and use the Kaplan-Meier method for calculating the plot points.

Note

Even when the data are uncensored, the adjusted Anderson-Darling statistic will not necessarily yield the same result as the non-adjusted Anderson-Darling statistic for small samples. However, for large sample sizes, the two approaches yield similar results.

Example of comparing Anderson-Darling values

In these results, the Weibull distribution has the lowest Anderson-Darling statistic of 6.056. However, the value of the Anderson-Darling statistic is close to that for the lognormal and normal distributions.

Goodness-of-Fit Anderson-Darling Distribution (adj) Weibull 6.048 Lognormal 6.165 Exponential 7.639 Normal 6.076

If you were to examine a probability plot and imagine a step function that connects the points, you would see that the Weibull, lognormal, and normal distributions have similar areas between the steps and the fitted line. The exponential distribution would have a much larger area.

Therefore, in these results, the difference between the Anderson-Darling values for the lognormal and Weibull is likely not a practical difference. The estimates of the percentiles from the different distributions could be similar. The difference between the exponential and Weibull Anderson-Darling values likely is a practical difference.

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