The Kolmogorov-Smirnov nonparametric test (K-S test) is applied to decide whether a sample comes from a specific distribution population, that is, to determine if two sets of data significantly differ, and it is based on a function of empirical distribution. As such, it is one of the many tests of goodness fit which assess if univariate data have a “hypothesized continuous probability distribution” (Salkind, 2010). Thus, the Kolmogorov-Smirnov test can be used to answer questions such as if data is from a distribution that is normal, log-normal, Weibull, exponential or logistic distribution.
The origin and Key attributes of Kolmogorov-Smirnov nonparametric test
Many of the procedures of statistics assume that data is normally distributed. However, K-S test is used to validate the application of those procedures as it can test the assumption that these errors are normally distributed, for instance, in a linear regression analysis. K-S test application requires that the outcomes to be independent and distributed identically and the sample to be unbiased, conditions which are violated in many degrees by spatially continuous features. Thus, K-S test was developed for ungrouped and uncensored data to test normal distribution (Salkind, 2010).
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One of the key features of K-S test is that its distribution of test statistics does not rely on the underlying function of the cumulative distribution being tested. Second, K-S test is an exact test, that is, unlike other nonparametric tests that rely on a sample size that is adequate for the estimations to be valid, K-S test does not depend on such sample size. Third, it makes no assumption about data distribution (Razali & Wah, 2011).
Compare and contrast the attributes of K-S tests with other nonparametric tests
K-S test is different from Mann-Whitney U test because it is used to test the cumulative distribution difference of two samples (shape and location) whereas Mann-Whitney U test is used to measure mean ranks discrepancy between the groups. Also, compared to other methods of nonparametric tests, K-S test does not make an assumption about data distribution whereas other methods do (Razali & Wah, 2011).
References
Razali, N. M., & Wah, Y. B. (2011). Power comparisons of shapiro-wilk, kolmogorov-smirnov, lilliefors and anderson-darling tests. Journal of statistical modeling and analytics , 2 (1), 21-33.
Salkind, N. J. (Ed.). (2010). Encyclopedia of research design (Vol. 1). Sage.
