Calculations for GOF Methods

R-Squared

If and , then the parameters for are estimated using the following equations:

Kolmogorov-Smirnov Test

Calculating modified failure order for sample dataset

Where,
- Oj is the order of the jth failure
- Ij is the increment for the jth failure
- n is the total number of data points, both censored and uncensored
- Op is the order of previous failure
- c is the number of data points remaining in the data set, including the current data point
Note: When there are no censored data, the above equation can be simplified to Oj = j.
Calculating median rank from modified failure order

Median rank is the median value of the rank distribution. Rank distribution is the distribution of the probability of j^th ordered value x out of a sample n.

Incomplete beta function is used to find median rank of the above equation
Finding statistics of the KS test

Statistics refers to the maximum difference between median rank calculated from dataset and probability calculated from the reference distribution. This steps does not considers the censored data.
Calculating p-value from KS test statistics and sample size

Where,
Calculating significant level (alpha)

alpha = 1-confidence level
Failing the Goodness of Fit test

If the p-value <= alpha, Goodness of Fit test fails because there is statistically significant difference between the sample dataset and the reference distribution.