# noncentral chi-square distribution

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## noncentral chi-square distribution

[‚nän¦sen·trəl ¦kī ‚skwer ‚dis·trə′byü·shən]
(statistics)
The distribution of the sum of squares of independent normal random variables, each with unit variance and nonzero mean; used to determine the power function of the chi-square test.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Results indicate a noncentral chi-square distribution for rows and columns of the GxE interaction matrix, which was also verified by the Kolmogorov-Smirnov test and Q-Q plot.
KEY WORDS / Genotype and Environment Contribution / GxE Interaction / Noncentral Chi-Square Distribution / Modified F Test /
To verify adhesion of the sum of squares of the simulated data to the noncentral chi-square distribution used of the Kolmogorov-Smirnov test.
has a noncentral chi-square distribution with v degrees of freedom and noncentrality parameter [lambda] = [[SIGMA].sub.i=1.sup.v][[delta].sub.i.sup.2].
If [X.sub.1] has a noncentral chi-square distribution then the distribution of X = [square root of [X.sub.1]] is referred to as noncentral chi distribution.
If [X.sub.1] has a noncentral chi-square distribution with [v.sub.1] degrees of freedom and noncentrality parameter [lambda], [X.sub.2] has a chi-square distribution with [v.sub.2] degrees of freedom, and [X.sub.1] and [X.sub.2] are independently distributed then
If [X.sub.1] has a noncentral chi-square distribution with [v.sub.1] degrees of freedom and noncentrality parameter [[lambda].sub.1], [X.sub.2] has a noncentral chi-square distribution with [v.sub.2] degrees of freedom and noncentrality parameter [[lambda].sub.2], and [X.sub.1] and [X.sub.2] are independently distributed then
If [X.sub.1] has the standard normal distribution and [X.sub.2] has an independent noncentral chi-square distribution with v degrees of freedom and noncentrality parameter [lambda], then
Johnson, Tables of Percentile Points of Noncentral Chi-Square Distribution, Mimeo Series 568, Institute of Statistics, University of North Carolina, Chapel Hill, NC (1968).
The statistical power of the chi-square test is evaluated from the noncentral chi-square distribution (Broffitt and Randles, 1977; Guenther, 1977; Weir and Cockerham, 1978).
Tables of the noncentral chi-square distribution yield the power of the test as a function of the noncentrality parameter, the degrees of freedom and the significance level (see Haynam et al., 1970).
The distribution function of the test statistic V' is closely approximated by the chi-square distribution with 2TW degrees of freedom in the noise case and by the noncentral chi-square distribution with 2TW degrees of freedom and noncentrality parameter [lamda] = [2E.sub.s./N.sub.01] in the signal case.

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