Noncentral Chi-Square Noncentrality λ Solver

Solve for the noncentrality parameter λ of the noncentral chi-square distribution from x, lower probability p, and degrees of freedom k. Useful for power analysis.

Input

Enter the critical value x, lower probability p, and degrees of freedom k to solve for the noncentrality λ satisfying the CDF F(x;k,λ)=p by bisection.

x (evaluation point, critical value)

A positive real number, the point at which the distribution is evaluated.

Lower probability p

Greater than 0 and less than 1. The target value of F(x;k,λ).

Degrees of freedom k

A positive real number, the degrees of freedom of the central chi-square component.

Result

Noncentrality λ for x=10, p=0.5, degrees of freedom k=3

7.96006402

Degrees of freedom k

Lower probability p

0.5

Mean k+λ

10.96006402

Variance 2(k+2λ)

37.84025606

Check F(x;k,λ)

0.5

Lower probability at λ=0

0.98143386

Lower probability F(x;k,λ) versus noncentrality λ

How it works

The noncentral chi-square CDF can be written as a Poisson-weighted mixture of central chi-square distributions: F(x;k,λ)=Σ e^(-λ/2)(λ/2)^j/j! × P(k/2+j, x/2), a sum of central lower probabilities weighted by Poisson probabilities.
This tool solves F(x;k,λ)=p for the noncentrality λ by bisection. Because F decreases monotonically in λ (larger λ shifts the distribution right and lowers the lower probability), bisection converges reliably.
If the entered lower probability p exceeds the lower probability at λ=0 (the central distribution), no solution exists. In that case lower p or increase x.
The mean of the noncentral distribution is k+λ and the variance is 2(k+2λ). In power analysis this is used to find the noncentrality matching a critical value x and a target power.
The incomplete gamma function is computed using complementary series and continued-fraction expansions. When the verification value closely matches the entered p, the solution has sufficient precision.

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