While the F distribution characterizes how the F test statistic is distributed when the null hypothesis is assumed to be true, the noncentral F distribution instead shows how the F test statistic is distributed when the alternative hypothesis is assumed to be true (i.e. when the null hypothesis is assumed to be false). As such it is useful in calculating the power of the usual F tests (ANOVA, regression, etc.).
Definition 1: The noncentral F distribution, abbreviated as F(k1, k2, λ) has the cumulative distribution function F(x), written as Fk1,k2,λ(x) when necessary, where k1, k2 = the degrees of freedom and non-negative λ = the noncentrality parameter.
when x ≥ 0, where Ir(a,b) is the distribution function of the beta distribution
Iq(a,b) = BETADIST(q, a, b)
When x < 0, the noncentral F distribution is F(x) = 0.
Observation: The probability density function (pdf) of the noncentral F distribution can be calculated as follows:
where B(a, b) is the beta function
and Γ(k) is the gamma function. B(a, b) can be calculated in Excel by the formula
=EXP(GAMMALN(a)+GAMMALN(b)−GAMMALN(a + b))
Real Statistics Functions: The following functions are provided in the Real Statistics Pack:
NF_DIST(x, df1, df2, λ, cum, iter, prec). If cum = TRUE then the value of the noncentral F distribution F(df1, df2, λ) at x is returned, while if cum = FALSE then the value of the pdf at x is returned.
NF_INV(p, df1, df2, λ, iter, iter0, prec) = the inverse of the cdf of the noncentral F distribution F(df1, df2, λ) at p, i.e. the value of x such that NF_DIST(x, df1, df2, λ, TRUE, iter, prec) = p.
NF_NCP(p, df1, df2, x, iter, iter0, prec) = the value of the noncentrality parameter λ such the cdf of the noncentral F distribution F(df1, df2, λ) at x is p, i.e. NF_DIST(x, df1, df2, λ, TRUE, iter, prec) = p.
BETA(x, y) = beta function at x, y
Here iter = the upper limit in the infinite sum (default 1,000) and iter0 = the number of iterations used to calculate NF_INV or NF_NCP (default 40). Also the calculation of the infinite sum for the noncentral F distribution stops when the level of precision exceeds prec (default 0.000000001).
Observation: The following chart shows the graphs of the noncentral F distribution with 5, 10 degrees of freedom for λ = 0, 1, 5, 10, 20. Note that when λ = 0, the distribution is the central F distribution, i.e. F(k1, k2, 0) = F(k1, k2).
Figure 1 – Noncentral F pdf by noncentrality parameter