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def power(test, rvs, n_observations, *, significance=0.01, vectorized=None, n_resamples=10000, batch=None, kwargs=None)
Simulate the power of a hypothesis test under an alternative hypothesis.
Parameters
----------
test : callable
Hypothesis test for which the power is to be simulated.
`test` must be a callable that accepts a sample (e.g. ``test(sample)``)
or ``len(rvs)`` separate samples (e.g. ``test(samples1, sample2)`` if
`rvs` contains two callables and `n_observations` contains two values)
and returns the p-value of the test.
If `vectorized` is set to ``True``, `test` must also accept a keyword
argument `axis` and be vectorized to perform the test along the
provided `axis` of the samples.
Any callable from `scipy.stats` with an `axis` argument that returns an
object with a `pvalue` attribute is also acceptable.
rvs : callable or tuple of callables
A callable or sequence of callables that generate(s) random variates
under the alternative hypothesis. Each element of `rvs` must accept
keyword argument ``size`` (e.g. ``rvs(size=(m, n))``) and return an
N-d array of that shape. If `rvs` is a sequence, the number of callables
in `rvs` must match the number of elements of `n_observations`, i.e.
``len(rvs) == len(n_observations)``. If `rvs` is a single callable,
`n_observations` is treated as a single element.
n_observations : tuple of ints or tuple of integer arrays
If a sequence of ints, each is the sizes of a sample to be passed to `test`.
If a sequence of integer arrays, the power is simulated for each
set of corresponding sample sizes. See Examples.
significance : float or array_like of floats, default: 0.01
The threshold for significance; i.e., the p-value below which the
hypothesis test results will be considered as evidence against the null
hypothesis. Equivalently, the acceptable rate of Type I error under
the null hypothesis. If an array, the power is simulated for each
significance threshold.
kwargs : dict, optional
Keyword arguments to be passed to `rvs` and/or `test` callables.
Introspection is used to determine which keyword arguments may be
passed to each callable.
The value corresponding with each keyword must be an array.
Arrays must be broadcastable with one another and with each array in
`n_observations`. The power is simulated for each set of corresponding
sample sizes and arguments. See Examples.
vectorized : bool, optional
If `vectorized` is set to ``False``, `test` will not be passed keyword
argument `axis` and is expected to perform the test only for 1D samples.
If ``True``, `test` will be passed keyword argument `axis` and is
expected to perform the test along `axis` when passed N-D sample arrays.
If ``None`` (default), `vectorized` will be set ``True`` if ``axis`` is
a parameter of `test`. Use of a vectorized test typically reduces
computation time.
n_resamples : int, default: 10000
Number of samples drawn from each of the callables of `rvs`.
Equivalently, the number tests performed under the alternative
hypothesis to approximate the power.
batch : int, optional
The number of samples to process in each call to `test`. Memory usage is
proportional to the product of `batch` and the largest sample size. Default
is ``None``, in which case `batch` equals `n_resamples`.
Returns
-------
res : PowerResult
An object with attributes:
power : float or ndarray
The estimated power against the alternative.
pvalues : ndarray
The p-values observed under the alternative hypothesis.
Notes
-----
The power is simulated as follows:
- Draw many random samples (or sets of samples), each of the size(s)
specified by `n_observations`, under the alternative specified by
`rvs`.
- For each sample (or set of samples), compute the p-value according to
`test`. These p-values are recorded in the ``pvalues`` attribute of
the result object.
- Compute the proportion of p-values that are less than the `significance`
level. This is the power recorded in the ``power`` attribute of the
result object.
Suppose that `significance` is an array with shape ``shape1``, the elements
of `kwargs` and `n_observations` are mutually broadcastable to shape ``shape2``,
and `test` returns an array of p-values of shape ``shape3``. Then the result
object ``power`` attribute will be of shape ``shape1 + shape2 + shape3``, and
the ``pvalues`` attribute will be of shape ``shape2 + shape3 + (n_resamples,)``.
Examples
--------
Suppose we wish to simulate the power of the independent sample t-test
under the following conditions:
- The first sample has 10 observations drawn from a normal distribution
with mean 0.
- The second sample has 12 observations drawn from a normal distribution
with mean 1.0.
- The threshold on p-values for significance is 0.05.
>>> import numpy as np
>>> from scipy import stats
>>> rng = np.random.default_rng(2549598345528)
>>>
>>> test = stats.ttest_ind
>>> n_observations = (10, 12)
>>> rvs1 = rng.normal
>>> rvs2 = lambda size: rng.normal(loc=1, size=size)
>>> rvs = (rvs1, rvs2)
>>> res = stats.power(test, rvs, n_observations, significance=0.05)
>>> res.power
0.6116
With samples of size 10 and 12, respectively, the power of the t-test
with a significance threshold of 0.05 is approximately 60% under the chosen
alternative. We can investigate the effect of sample size on the power
by passing sample size arrays.
>>> import matplotlib.pyplot as plt
>>> nobs_x = np.arange(5, 21)
>>> nobs_y = nobs_x
>>> n_observations = (nobs_x, nobs_y)
>>> res = stats.power(test, rvs, n_observations, significance=0.05)
>>> ax = plt.subplot()
>>> ax.plot(nobs_x, res.power)
>>> ax.set_xlabel('Sample Size')
>>> ax.set_ylabel('Simulated Power')
>>> ax.set_title('Simulated Power of `ttest_ind` with Equal Sample Sizes')
>>> plt.show()
Alternatively, we can investigate the impact that effect size has on the power.
In this case, the effect size is the location of the distribution underlying
the second sample.
>>> n_observations = (10, 12)
>>> loc = np.linspace(0, 1, 20)
>>> rvs2 = lambda size, loc: rng.normal(loc=loc, size=size)
>>> rvs = (rvs1, rvs2)
>>> res = stats.power(test, rvs, n_observations, significance=0.05,
... kwargs={'loc': loc})
>>> ax = plt.subplot()
>>> ax.plot(loc, res.power)
>>> ax.set_xlabel('Effect Size')
>>> ax.set_ylabel('Simulated Power')
>>> ax.set_title('Simulated Power of `ttest_ind`, Varying Effect Size')
>>> plt.show()
We can also use `power` to estimate the Type I error rate (also referred to by the
ambiguous term "size") of a test and assess whether it matches the nominal level.
For example, the null hypothesis of `jarque_bera` is that the sample was drawn from
a distribution with the same skewness and kurtosis as the normal distribution. To
estimate the Type I error rate, we can consider the null hypothesis to be a true
*alternative* hypothesis and calculate the power.
>>> test = stats.jarque_bera
>>> n_observations = 10
>>> rvs = rng.normal
>>> significance = np.linspace(0.0001, 0.1, 1000)
>>> res = stats.power(test, rvs, n_observations, significance=significance)
>>> size = res.power
As shown below, the Type I error rate of the test is far below the nominal level
for such a small sample, as mentioned in its documentation.
>>> ax = plt.subplot()
>>> ax.plot(significance, size)
>>> ax.plot([0, 0.1], [0, 0.1], '--')
>>> ax.set_xlabel('nominal significance level')
>>> ax.set_ylabel('estimated test size (Type I error rate)')
>>> ax.set_title('Estimated test size vs nominal significance level')
>>> ax.set_aspect('equal', 'box')
>>> ax.legend(('`ttest_1samp`', 'ideal test'))
>>> plt.show()
As one might expect from such a conservative test, the power is quite low with
respect to some alternatives. For example, the power of the test under the
alternative that the sample was drawn from the Laplace distribution may not
be much greater than the Type I error rate.
>>> rvs = rng.laplace
>>> significance = np.linspace(0.0001, 0.1, 1000)
>>> res = stats.power(test, rvs, n_observations, significance=0.05)
>>> print(res.power)
0.0587
This is not a mistake in SciPy's implementation; it is simply due to the fact
that the null distribution of the test statistic is derived under the assumption
that the sample size is large (i.e. approaches infinity), and this asymptotic
approximation is not accurate for small samples. In such cases, resampling
and Monte Carlo methods (e.g. `permutation_test`, `goodness_of_fit`,
`monte_carlo_test`) may be more appropriate.
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