noncentrality

Noncentrality refers to the degree to which a distribution departs from a central, or null, distribution under an alternative hypothesis. In statistics, this idea is quantified by the noncentrality parameter, typically denoted by lambda (λ). The noncentrality parameter summarizes the size of the true effect relative to variability and is central to power calculations for hypothesis tests that rely on noncentral distributions.

A key instance is the noncentral chi-square distribution. If X is the sum of squared standard normal variables with nonzero means, X follows a noncentral chi-square distribution with k degrees of freedom and noncentrality parameter λ, where λ measures the magnitude of the means relative to the standard deviations (often λ = sum of squared standardized means). Under the null hypothesis, all means are zero, so λ = 0 and the distribution reduces to the central chi-square. Larger λ implies greater departure from the null and shifts the distribution away from the central case.

Noncentrality also appears in the noncentral t and noncentral F distributions. The noncentral t distribution arises when the true mean differs from the null value, and its noncentrality parameter δ is proportional to the true effect size divided by the standard error. The noncentral F distribution, used in analyses of variance and regression, has a noncentrality parameter that depends on the true effect sizes and the experimental design. In all cases, λ or δ quantify how strongly the data under the alternative support rejecting the null.

Applications of noncentrality include power analysis and sample size planning, where the probability of detecting a true effect is computed using the corresponding noncentral distribution. The concept is also used to compare tests, construct confidence intervals, and interpret results in terms of effect size and study design.