# Noncentral distribution

The term noncentral distribution refers to a transformation of a probability distribution, determined by a non-centrality parameter (σ) that is not equal to zero; an NCP of zero results in a central distribution [1]. Noncentral distributions can be thoughts of as generalizations of standard or “central” distributions or as infinite mixtures of their central counterparts.; For any null hypothesis and associated test statistic, the probability distribution under an alternative hypothesis is a non-central probability distribution of the same statistic under the null hypothesis.

## Noncentral distribution families

There are several families of probability distributions to which the term “noncentral distribution” may apply, including the Beta distribution, chi distribution, Chi-squared distribution, F-Distribution and Student’s t-distribution. For example, the standard (central) chi-squared distribution is the distribution of a sum of squared independent normal distributions with mean 0 and variance 1. The noncentral chi-squared distribution generalizes this to any normal distribution, with arbitrary mean and variance.

Noncentral distribution representations can be complicated to define. For example, a general representation of the noncentral t-distribution, given by Johnson and Kotz [2]

where

• I(x|ν,δ) = the incomplete beta function with parameters ν and δ,
• δ = the NCP,
• ν = degrees of freedom.

Student’s t distribution models the t-statistic

Where

• x̄ = the sample mean,
• s = sample standard deviation
• n = sample size.

If the population has a mean of zero, then the above t-statistic has a non-centrality parameter of

which is the normalized difference between the sample and population means (normalization means to adjust values measured on different scales to a common scale). The noncentral t distribution defines the probability of a correctly rejected false null hypothesis (with mean μ) when the population actually has a mean of μ0. It delivers the statistical power of the t-test, with increased power when the difference between μ0. – μ increases and when the sample size (n) increases [4].

## Uses of non-central distributions

The non-central distribution is used primarily to calculate statistical power. Additionally, it can also be used to calculate confidence intervals on the standardized effect size Cohen’s d [5].

When the distribution shifts because the alternate hypothesis is true, a non-central distribution occurs. The non-central distribution is formally defined by a non-centrality parameter (NCP), which measures “…the degree to which a null hypothesis is false” [6]. Increasing NCPs result in higher statistical power by shifting the distribution to the right. Consequently, a larger percentage of the curve can be found to the right of the critical value for a given alpha level.

When it comes to elementary statistics, you will mainly work with the central variations of distributions, which are special cases of the generalized distributions where σ = 0. These arise when the difference tested is null (i.e., the null hypothesis is true), resulting in the tables most commonly found in the back of statistics textbooks. For instance, the F distribution arises from the ratio of χ2 random variables divided by degrees of freedom. The central F distribution is a particular version of the non-central F distribution, where the numerator is a central chi-square random variable.

## References

[1]  Dodge, Y. (2003). The Oxford Dictionary of Statistical Terms, Oxford University Press.

[2] Johnson, N.L., Kotz, S., Balakrishnan N. (1995). Continuous univariate distributions, Volume 2 (2nd Edition). Wiley. ISBN 0-471-58494-0

[3] HilberTraum, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

[4] Cumming, G, How the noncentral distribution got its hump. Retrieved May 28, 2021 from: https://www.stat.auckland.ac.nz/~iase/publications/17/C106.pdf

[5] Kirk, R. (2012). Experimental Design: Procedures for Behavioral Sciences. SAGE Publications.

[6] Noncentral t distribution. Retrieved May 19, 2023 from: https://www.mathworks.com/help/stats/noncentral-t-distribution.html

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