# Statistical tests

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**Statistical tests** are mathematical techniques for statistical hypothesis testing to establish the probability of the results obtained from the experimental and control samples being from statistically different populations and not due to chance factors alone.

## Nonparametric Statistical tests

In statistics, the term **non-parametric statistics** covers a range of topics:

*distribution free*methods which do not rely on assumptions that the data are drawn from a given probability distribution. As such it is the opposite of parametric statistics. It includes non-parametric statistical models, inference and statistical tests.*non-parametric statistic*can refer to a statistic (a function on a sample) whose interpretation does not depend on the population fitting any parametrized distributions. Statistics based on the ranks of observations are one example of such statistics and these play a central role in many non-parametric approaches.*non-parametric regression*refers to modeling where 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.

### 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, for data on an *ordinal scale.*

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 *nonparametric* 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.

- A histogram is a simple nonparametric estimate of a probability distribution
- Kernel density estimation provides better estimates of the density than histograms.
- Nonparametric regression and semiparametric regression methods have been developed based on kernels, splines, and wavelets.
- Data Envelopment Analysis provides efficiency coefficients similar to those obtained by Multivariate Analysis without any distributional assumption.

### 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

- Anderson–Darling test
- Cliff's delta
- Cochran's Q
- Cohen's kappa
- Efron–Petrosian test
- Friedman two-way analysis of variance by ranks
- Kendall's tau
- Kendall's W
- Kolmogorov–Smirnov test
- Kruskal-Wallis one-way analysis of variance by ranks
- Kuiper's test
- Mann–Whitney U or Wilcoxon rank sum test
- median test
- Pitman's permutation test
- Rank products
- Siegel–Tukey test
- Spearman's rank correlation coefficient
- Van Elteren stratified Wilcoxon rank sum test
- Wald–Wolfowitz runs test
- Wilcoxon signed-rank test.

### General references

- Corder, G.W. & Foreman, D.I, "Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach", Wiley (2009) (ISBN: 9780470454619)
- Gibbons, Jean Dickinson and Chakraborti, Subhabrata, "Nonparametric Statistical Inference", 4th Ed. CRC (2003) (ISBN: 0824740521)
- (1998)
*Robust nonparametric statistical methods*, First, xiv+467 pp., London: Edward Arnold. Template:MR - Wasserman, Larry, "All of Nonparametric Statistics", Springer (2007) (ISBN: 0387251456)

### See also

- Parametric statistics
- Resampling (statistics)
- Robust statistics
- Particle filter for the general theory of sequential Monte Carlo methods