Individual Distribution Identification

Data Fitted with 14 Distributions and 2 Transformations - Goodness-Of-Fit Tests

  

The goodness-of-fit test includes Anderson-Darling test statistics (AD), corresponding p-values (P), and likelihood ratio test p-values to assess whether a distribution fits your data.

You can use Anderson-Darling test to examine whether a chosen distribution fits your data well.

Use the p-value (P) to determine which distribution fits your data well:

1    Identify the p-value for the distribution you want to evaluate.

2    Compare the p-value to your a-level. A commonly used a-level is 0.05.

·    A p-value greater than or equal to a suggests that the distribution is a good fit.

·    A p-value less than a suggests that the distribution is not a good fit.

For several distributions, Minitab offers the standard version as well as a version with an extra parameter. In these cases, use the LRT P to determine whether adding the extra parameter significantly improves the fit over the distribution without the extra parameter. A LRT P value less than 0.05 suggests that the improvement is significant.

The LRT P value is also useful for 3-parameter distributions for which there is no established method for calculating the p-value. In these cases, it is advisable to first examine the p-value for the corresponding two-parameter distribution. Then look at the LRT P for the 3-parameter distribution to determine whither the three-parameter distribution is significantly better than the two-parameter distribution. However, it may be advisable to choose a distribution which has a calculated p-value and a similar AD value.

Example Output

Goodness of Fit Test

 

Distribution                 AD       P  LRT P

Normal                    0.754   0.046

Box-Cox Transformation    0.398   0.353

Lognormal                 0.650   0.085

3-Parameter Lognormal     0.341       *  0.017

Exponential              20.614  <0.003

2-Parameter Exponential   1.684   0.014  0.000

Weibull                   1.442  <0.010

3-Parameter Weibull       0.230  >0.500  0.000

Smallest Extreme Value    1.656  <0.010

Largest Extreme Value     0.394  >0.250

Gamma                     0.702   0.071

3-Parameter Gamma         0.268       *  0.006

Logistic                  0.726   0.034

Loglogistic               0.659   0.050

3-Parameter Loglogistic   0.432       *  0.027

Johnson Transformation    0.124   0.986

 

Interpretation

Goodness-of-fit statistics for the Box-Cox transformation (AD = 0.398, P = 0.353) and the Johnson transformation (AD = 0.124, P = 0.986) suggest that a normal distribution is a good fit for both methods of transformed calcium data. The 3-parameter Weibull distribution (AD = 0.230 and P > 0.500) fits the data well.

The LRT P values for the 3-parameter lognormal (0.017), 3-parameter Weibull (0.000), 3-parameter gamma (0.006), and 3-parameter loglogistic (0.027) suggest that these distributions significantly improve the fit compared to their 2-parameter counterparts.

The goodness-of-fit test results for the 14 distributions and 2 transformations suggest that more than one distribution results in a p-value greater than 0.05. The 3-parameter Weibull (P > 0.500), the largest extreme value (P > 0.250), Box-Cox transformation (P = 0.353) and Johnson transformation (P = 0.986) fit the calcium data better than the other distributions.

Among extremely close p-values, select either a distribution:

·    that you have used previously for a similar data set

·    based on capability statistics

Choose the one that is most conservative.