scipy.special.fdtri#

scipy.special.fdtri(dfn, dfd, p, out=None) = <ufunc 'fdtri'>#

The p-th quantile of the F-distribution.

This function is the inverse of the F-distribution CDF, fdtr, returning the x such that fdtr(dfn, dfd, x) = p.

Parameters:

dfnarray_like: First parameter (positive float).
dfdarray_like: Second parameter (positive float).
parray_like: Cumulative probability, in [0, 1].
outndarray, optional: Optional output array for the function values

Returns:

xscalar or ndarray: The quantile corresponding to p.

See also

fdtr: F distribution cumulative distribution function
fdtrc: F distribution survival function
scipy.stats.f: F distribution

Notes

The computation is carried out using the relation to the inverse regularized beta function, \(I^{-1}_x(a, b)\). Let \(z = I^{-1}_p(d_d/2, d_n/2).\) Then,

\[x = \frac{d_d (1 - z)}{d_n z}.\]

If p is such that \(x < 0.5\), the following relation is used instead for improved stability: let \(z' = I^{-1}_{1 - p}(d_n/2, d_d/2).\) Then,

\[x = \frac{d_d z'}{d_n (1 - z')}.\]

Wrapper for the Cephes [1] routine fdtri.

The F distribution is also available as scipy.stats.f. Calling fdtri directly can improve performance compared to the ppf method of scipy.stats.f (see last example below).

References

[1]

Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

Examples

fdtri represents the inverse of the F distribution CDF which is available as fdtr. Here, we calculate the CDF for df1=1, df2=2 at x=3. fdtri then returns 3 given the same values for df1, df2 and the computed CDF value.

>>> import numpy as np
>>> from scipy.special import fdtri, fdtr
>>> df1, df2 = 1, 2
>>> x = 3
>>> cdf_value =  fdtr(df1, df2, x)
>>> fdtri(df1, df2, cdf_value)
3.000000000000006

Calculate the function at several points by providing a NumPy array for x.

>>> x = np.array([0.1, 0.4, 0.7])
>>> fdtri(1, 2, x)
array([0.02020202, 0.38095238, 1.92156863])

Plot the function for several parameter sets.

>>> import matplotlib.pyplot as plt
>>> dfn_parameters = [50, 10, 1, 50]
>>> dfd_parameters = [0.5, 1, 1, 5]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(dfn_parameters, dfd_parameters,
...                            linestyles))
>>> x = np.linspace(0, 1, 1000)
>>> fig, ax = plt.subplots()
>>> for parameter_set in parameters_list:
...     dfn, dfd, style = parameter_set
...     fdtri_vals = fdtri(dfn, dfd, x)
...     ax.plot(x, fdtri_vals, label=rf"$d_n={dfn},\, d_d={dfd}$",
...             ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$x$")
>>> title = "F distribution inverse cumulative distribution function"
>>> ax.set_title(title)
>>> ax.set_ylim(0, 30)
>>> plt.show()

../../_images/scipy-special-fdtri-1_00_00.png

The F distribution is also available as scipy.stats.f. Using fdtri directly can be much faster than calling the ppf method of scipy.stats.f, especially for small arrays or individual values. To get the same results one must use the following parametrization: stats.f(dfn, dfd).ppf(x)=fdtri(dfn, dfd, x).

>>> from scipy.stats import f
>>> dfn, dfd = 1, 2
>>> x = 0.7
>>> fdtri_res = fdtri(dfn, dfd, x)  # this will often be faster than below
>>> f_dist_res = f(dfn, dfd).ppf(x)
>>> f_dist_res == fdtri_res  # test that results are equal
True