< List of probability distributions
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.
See also:
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