# Parametric statistical tests

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**Parametric statistics** is a branch of statistics that assumes data come from a type of probability distribution and makes inferences about the parameters of the distribution.^{[1]} Most well-known elementary statistical methods are parametric.^{[2]} and include:

- Analysis of variance (ANOVA)
- ANCOVA (Analysis of Covariance)
- F test
- Introduction to factor analysis
- MANOVA (Multivariate Analysis of Variance)
- Pearson product-moment correlation coefficient
- Principal components analysis
- Regression analysis
- T test

These tests are mathematical procedures for statistical hypothesis testing which assume that the distributions of the variables being assessed belong to known parametrized families of probability distributions. In that case we speak of parametric model.

For example, analysis of variance assumes that the underlying distributions are normally distributed and that the variances of the distributions being compared are similar. The Pearson product-moment correlation coefficient assumes normality.

Generally speaking parametric methods make more assumptions than non-parametric methods. If those extra assumptions are correct, parametric methods can produce more accurate and precise estimates. They are said to have more statistical power. However, if those assumptions are incorrect, parametric methods can be very misleading. For that reason they are often not considered robust. On the other hand, parametric formulae are often simpler to write down and faster to compute. In some, but definitely not all, cases their simplicity makes up for their non-robustness, especially if care is taken to examine diagnostic statistics.^{[3]}

Because parametric statistics require a probability distribution, they are not distribution-free.^{[4]}

## Example Edit

Suppose we have a sample of 99 test scores with a mean of 100 and a standard deviation of 10. If we assume all 100 test scores are random samples from a Normal distribution we predict there is a 1% chance that the 100^{th} test score will be higher than 123.65 (that is the mean plus 2.365 standard deviations) assuming that the 100^{th} test score comes from the same distribution as the others. The Normal family of distributions all have the same shape and are *parameterized* by mean and standard deviation. That means if you know the mean and standard deviation, and that the distribution is Normal, you know the probability of any future observation. Parametric statistical methods are used to compute the 2.365 value above, given 99 independent observations from the same Normal distribution.

A non-parametric estimate of the same thing is the maximum of the first 99 scores. We don't need to assume anything about the distribution of test scores to reason that before we gave the test it was equally likely that the highest score would be any of the first 100. Thus there is a 1% chance that the 100^{th} is higher than any of the 99 that preceeded it.

While parametric techniques are robust – that is, they often retain considerable power to detect differences or similarities even when these assumptions are violated – some distributions violate the assumptions so markedly that a non-parametric alternative is more likely to detect a difference or similarity.

## History Edit

Statistician Jacob Wolfowitz coined the statistical term "parametric" in order to define its opposite in 1942:

"Most of these developments have this feature in common, that the distribution functions of the various stochastic variables which enter into their problems are assumed to be of known functional form, and the theories of estimation and of testing hypotheses are theories of estimation of and of testing hypotheses about, one or more parameters. . ., the knowledge of which would completely determine the various distribution functions involved. We shall refer to this situation. . .as the parametric case, and denote the opposite case, where the functional forms of the distributions are unknown, as the non-parametric case."^{[5]}

## See alsoEdit

## ReferencesEdit

- ↑ Seymour Geisser and Wesley M. Johnson, Modes of Parametric Statistical Inference, John Wiley & Sons (2006), ISBN: 978-0471667261
- ↑ D. R. Cox, Principles of Statistical Inference, Cambridge University Press (2006), ISBN: 978-0521685672
- ↑ David Freedman, Statistical Models: Theory and Practice, Cambridge University Press (2000), ISBN: 978-0521671057
- ↑ David C. Hoaglin, Frederick Mosteller and John Tukey, Understanding Robust and Exploratory Data Analysis , Wiley-Interscience (2000), ISBN: 978-0471384915
- ↑ J. Wolfowitz, Annals of Mathematical Statistics XIII, p. 264 (1942)

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