Nonparametric statistics

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Nonparametric statistics is the branch of statistics that is not based solely on parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on either being distribution-free or having a specified distribution but with the distribution's parameters unspecified. Nonparametric statistics includes both descriptive statistics and statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are violated.[1]

Definitions

The term "nonparametric statistics" has been imprecisely defined in the following two ways, among others.

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Applications and purpose

Non-parametric methods are widely used for studying populations that take on a ranked order (such as movie reviews receiving one to four stars). The use of non-parametric methods may be necessary when data have a ranking but no clear numerical interpretation, such as when assessing preferences. In terms of levels of measurement, non-parametric methods result in ordinal data.

As non-parametric methods make fewer assumptions, their applicability is much wider than the corresponding parametric methods. In particular, they may be applied in situations where less is known about the application in question. Also, due to the reliance on fewer assumptions, non-parametric methods are more robust.

Another justification for the use of non-parametric methods is simplicity. In certain cases, even when the use of parametric methods is justified, non-parametric methods may be easier to use. Due both to this simplicity and to their greater robustness, non-parametric methods are seen by some statisticians as leaving less room for improper use and misunderstanding.

The wider applicability and increased robustness of non-parametric tests comes at a cost: in cases where a parametric test would be appropriate, non-parametric tests have less power. In other words, a larger sample size can be required to draw conclusions with the same degree of confidence.

Non-parametric models

Non-parametric models differ from parametric models in that the model structure is not specified a priori but is instead determined from data. The term non-parametric is not meant to imply that such models completely lack parameters but that the number and nature of the parameters are flexible and not fixed in advance.

Methods

Non-parametric (or distribution-free) inferential statistical methods are mathematical procedures for statistical hypothesis testing which, unlike parametric statistics, make no assumptions about the probability distributions of the variables being assessed. The most frequently used tests include

 2 independent samples are drawn from the same distribution
  • Kuiper's test: tests whether a sample is drawn from a given distribution, sensitive to cyclic variations such as day of the week
  • Logrank test: compares survival distributions of two right-skewed, censored samples
  • Mann–Whitney U or Wilcoxon rank sum test: tests whether two samples are drawn from the same distribution, as compared to a given alternative hypothesis.
  • McNemar's test: tests whether, in 2 × 2 contingency tables with a dichotomous trait and matched pairs of subjects, row and column marginal frequencies are equal
  • Median test: tests whether two samples are drawn from distributions with equal medians
  • Pitman's permutation test: a statistical significance test that yields exact p values by examining all possible rearrangements of labels
  • Rank products: detects differentially expressed genes in replicated microarray experiments
  • Siegel–Tukey test: tests for differences in scale between two groups
  • Sign test: tests whether matched pair samples are drawn from distributions with equal medians
  • Spearman's rank correlation coefficient: measures statistical dependence between two variables using a monotonic function
  • Squared ranks test: tests equality of variances in two or more samples
  • Tukey–Duckworth test: tests equality of two distributions by using ranks
  • Wald–Wolfowitz runs test: tests whether the elements of a sequence are mutually independent/random
  • Wilcoxon signed-rank test: tests whether matched pair samples are drawn from populations with different mean ranks

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History

Early nonparametric statistics include the median (13th century or earlier, use in estimation by Edward Wright, 1599; see Template:Slink) and the sign test by John Arbuthnot (1710) in analyzing the human sex ratio at birth (see Template:Slink).[2][3]

See also

Notes

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General references

  • Bagdonavicius, V., Kruopis, J., Nikulin, M.S. (2011). "Non-parametric tests for complete data", ISTE & WILEY: London & Hoboken. Template:Isbn.
  • Template:Cite book
  • Gibbons, Jean Dickinson; Chakraborti, Subhabrata (2003). Nonparametric Statistical Inference, 4th Ed. CRC Press. Template:Isbn.
  • Template:Cite book also Template:Isbn.
  • Hollander M., Wolfe D.A., Chicken E. (2014). Nonparametric Statistical Methods, John Wiley & Sons.
  • Sheskin, David J. (2003) Handbook of Parametric and Nonparametric Statistical Procedures. CRC Press. Template:ISBN
  • Wasserman, Larry (2007). All of Nonparametric Statistics, Springer. Template:Isbn.
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