Noncentral F Noncentrality λ Solver

Back-solve the noncentrality parameter λ of the noncentral F distribution from a critical value x, lower probability p, and degrees of freedom d1, d2.

Input

Enter a critical value x, a lower probability p, and the numerator and denominator degrees of freedom d1, d2 to back-solve the noncentrality λ of the noncentral F distribution by bisection.

Critical value x

A positive value at which the noncentral F distribution is evaluated

Lower probability p

Target lower probability greater than 0 and less than 1

Degrees of freedom d1 (numerator)

Numerator degrees of freedom (positive)

Degrees of freedom d2 (denominator)

Denominator degrees of freedom (positive)

Result

Noncentrality λ giving lower probability 0.5 at x ＝ 3.84

9.16225556

DF d1

DF d2

Achieved lower probability

0.5

Upper probability (power)

0.5

Central lower probability (λ ＝ 0)

0.97458262

Reference effect size f

0.61786782

Noncentrality λ versus lower probability

The dashed line marks the solved λ ＝ 9.16225556.

How it works

The noncentral F cumulative distribution can be written as a Poisson-weighted mixture of central F distributions (regularized incomplete beta). The weight w_j is e^(-λ/2) times (λ/2)^j divided by j factorial.
Increasing the noncentrality λ shifts the distribution to the right, so the lower probability at a fixed critical value x decreases monotonically. This lets us back-solve for the λ that matches the target lower probability p using bisection.
If the target lower probability p exceeds the lower probability at λ equals 0 (the central F distribution), no λ can reach it, so the case is reported as unreachable.
The upper probability equals 1 minus the lower probability and represents the chance of exceeding the critical value, which corresponds to ANOVA statistical power.
The reference effect size is recovered from the approximation λ equals f squared times (d1 plus d2 plus 1), solved for f. Treat it as a rough guide.
All special functions (log gamma and regularized incomplete beta) are implemented from scratch with no external library dependencies.

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