Branch of statistics that is not based solely on parametrized families of probability distributions
Nonparametric statistics is a type of statistical analysis that makes minimal assumptions about the underlying distribution of the data being studied. Often these models are infinite-dimensional, rather than finite dimensional, as is parametric statistics.[1] Nonparametric statistics can be used for descriptive statisticsorstatistical inference. Nonparametric tests are often used when the assumptions of parametric tests are evidently violated.[2]
The term "nonparametric statistics" has been defined imprecisely in the following two ways, among others:
The first meaning of nonparametric involves techniques that do not rely on data belonging to any particular parametric family of probability distributions.
These include, among others:
Methods which are distribution-free, which do not rely on assumptions that the data are drawn from a given parametric family of probability distributions.
Statistics defined to be a function on a sample, without dependency on a parameter.
The discussion following is taken from Kendall's Advanced Theory of Statistics.[3]
Statistical hypotheses concern the behavior of observable random variables.... For example, the hypothesis (a) that a normal distribution has a specified mean and variance is statistical; so is the hypothesis (b) that it has a given mean but unspecified variance; so is the hypothesis (c) that a distribution is of normal form with both mean and variance unspecified; finally, so is the hypothesis (d) that two unspecified continuous distributions are identical.
It will have been noticed that in the examples (a) and (b) the distribution underlying the observations was taken to be of a certain form (the normal) and the hypothesis was concerned entirely with the value of one or both of its parameters. Such a hypothesis, for obvious reasons, is called parametric.
Hypothesis (c) was of a different nature, as no parameter values are specified in the statement of the hypothesis; we might reasonably call such a hypothesis non-parametric. Hypothesis (d) is also non-parametric but, in addition, it does not even specify the underlying form of the distribution and may now be reasonably termed distribution-free. Notwithstanding these distinctions, the statistical literature now commonly applies the label "non-parametric" to test procedures that we have just termed "distribution-free", thereby losing a useful classification.
The second meaning of non-parametric involves techniques that do not assume that the structure of a model is fixed. Typically, the model grows in size to accommodate the complexity of the data. In these techniques, individual variables are typically assumed to belong to parametric distributions, and assumptions about the types of associations among variables are also made. These techniques include, among others:
non-parametric regression, which is modeling whereby the structure of the relationship between variables is treated non-parametrically, but where nevertheless there may be parametric assumptions about the distribution of model residuals.
non-parametric hierarchical Bayesian models, such as models based on the Dirichlet process, which allow the number of latent variables to grow as necessary to fit the data, but where individual variables still follow parametric distributions and even the process controlling the rate of growth of latent variables follows a parametric distribution.
Non-parametric methods are widely used for studying populations that have 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 more general 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.
Non-parametric methods are sometimes considered simpler to use and more robust than parametric methods, even when the assumptions of parametric methods are justified. This is due to their more general nature, which may make them less susceptible to misuse and misunderstanding. Non-parametric methods can be considered a conservative choice, as they will work even when their assumptions are not met, whereas parametric methods can produce misleading results when their assumptions are violated.
The wider applicability and increased robustness of non-parametric tests comes at a cost: in cases where a parametric test's assumptions are met, non-parametric tests have less statistical power. In other words, a larger sample size can be required to draw conclusions with the same degree of confidence.
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.
Ahistogram is a simple nonparametric estimate of a probability distribution.
Non-parametric (ordistribution-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
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.
^Stuart A., Ord J.K, Arnold S. (1999), Kendall's Advanced Theory of Statistics: Volume 2A—Classical Inference and the Linear Model, sixth edition, §20.2–20.3 (Arnold).
^Conover, W.J. (1999), "Chapter 3.4: The Sign Test", Practical Nonparametric Statistics (Third ed.), Wiley, pp. 157–176, ISBN0-471-16068-7
