Returns the mean, variance, standard deviation, skewness, and excess
kurtosis of a noncentral F distribution with df_1 numerator and
df_2 denominator degrees of freedom and noncentrality parameter
ncp. The noncentral F is the reference distribution of the
F statistic when an effect is present, so its moments describe the
sampling behavior of \(R^2\), eta squared, and the omnibus F test
under the alternative. A central F (ncp = 0) is the special
case.
Value
A data.frame (class dmar_tbl) in
term / value layout with the mean, variance,
sd, skewness, and excess_kurtosis (any of which may be
NA when df_2 is too small), followed by the df_1,
df_2, and ncp that produced them.
Details
Writing the noncentral F as
\(F = (X_1/\nu_1)/(X_2/\nu_2)\) with \(X_1 \sim \chi^2_{\nu_1}(\lambda)\)
a noncentral chi square and \(X_2 \sim \chi^2_{\nu_2}\) independent, the
raw moments are
$$\mathrm{E}[F^k] = \Bigl(\tfrac{\nu_2}{\nu_1}\Bigr)^k
\mathrm{E}[X_1^k]\, \prod_{i=1}^{k}\frac{1}{\nu_2 - 2i},
\qquad \nu_2 > 2k,$$
where the noncentral chi square moments \(\mathrm{E}[X_1^k]\) follow from
its cumulants \(\kappa_n = 2^{n-1}(n-1)!\,(\nu_1 + n\lambda)\). The mean
exists for \(\nu_2 > 2\), the variance for \(\nu_2 > 4\), the skewness
for \(\nu_2 > 6\), and the excess kurtosis for \(\nu_2 > 8\); a moment
whose denominator degrees of freedom condition is not met is returned as
NA. The mean reduces to the familiar
\(\nu_2(\nu_1 + \lambda)/[\nu_1(\nu_2 - 2)]\), and the variance to
\(2(\nu_2/\nu_1)^2[(\nu_1 + \lambda)^2 + (\nu_1 + 2\lambda)(\nu_2 - 2)] /
[(\nu_2 - 2)^2(\nu_2 - 4)]\).
References
Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions (Vol. 2, 2nd ed., Chapter 30). Wiley.
See also
moments_nct for the noncentral t;
conf_limits_ncf for the noncentral F confidence
limits used in effect size intervals; df for the
density.
Other noncentral distribution moments:
moments_nc_chisq(),
moments_nct()
Author
Ken Kelley kkelley@nd.edu
Examples
# A noncentral F with 3 and 40 df and noncentrality 8.
moments_ncf(df_1 = 3, df_2 = 40, ncp = 8)
#> term value
#> mean 3.86
#> variance 5.77
#> sd 2.4
#> skewness 1.34
#> excess_kurtosis 3.12
#> df_1 3
#> df_2 40
#> ncp 8
# ncp = 0 is the central F: mean df_2 / (df_2 - 2).
moments_ncf(df_1 = 3, df_2 = 40)
#> term value
#> mean 1.05
#> variance 0.841
#> sd 0.917
#> skewness 1.98
#> excess_kurtosis 6.62
#> df_1 3
#> df_2 40
#> ncp 0
# The variance is undefined for four or fewer denominator df.
moments_ncf(df_1 = 2, df_2 = 4, ncp = 5)
#> term value
#> mean 7
#> variance <NA>
#> sd <NA>
#> skewness <NA>
#> excess_kurtosis <NA>
#> df_1 2
#> df_2 4
#> ncp 5