Goodnessoffittests

GoodnessoffitTests, a nonstandard spelling of goodness-of-fit tests, refer to a family of statistical methods used to assess whether observed data are consistent with a specified theoretical distribution. They evaluate how well a proposed model describes the data, as opposed to tests that compare different samples or test for association.

Common examples include the chi-squared goodness-of-fit test for categorical data; the Kolmogorov–Smirnov test and the Cramér–von Mises and Anderson–Darling tests for continuous data; and normality tests such as Shapiro–Wilk and Lilliefors. The choice depends on the data type and whether distribution parameters are fixed or estimated.

Procedure: state the null hypothesis that the data come from the reference distribution. Compute a test statistic that measures the discrepancy between observed and expected (or empirical) distributions, then obtain a p-value or critical value to decide whether to reject the null. In chi-squared tests, one may need to combine bins with low expected counts and adjust degrees of freedom for estimated parameters.

Assumptions and limitations: independence of observations and an appropriate reference distribution are assumed. Tests may have low power against certain alternatives, can be sensitive to sample size and parameter estimation, and a failure to reject does not prove the model is correct.

Applications are widespread across quality control, biology, psychology, meteorology, and finance. The concept dates to early 20th-century work of Pearson and others, with later developments such as the Lilliefors correction and the Anderson–Darling statistic expanding their usefulness.