2256 lines
96 KiB
Python
2256 lines
96 KiB
Python
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from __future__ import annotations
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import warnings
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import numpy as np
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from itertools import combinations, permutations, product
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from collections.abc import Sequence
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from dataclasses import dataclass
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import inspect
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from scipy._lib._util import check_random_state, _rename_parameter, rng_integers
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from scipy._lib._array_api import (array_namespace, is_numpy, xp_minimum,
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xp_clip, xp_moveaxis_to_end)
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from scipy.special import ndtr, ndtri, comb, factorial
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from ._common import ConfidenceInterval
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from ._axis_nan_policy import _broadcast_concatenate, _broadcast_arrays
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from ._warnings_errors import DegenerateDataWarning
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__all__ = ['bootstrap', 'monte_carlo_test', 'permutation_test']
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def _vectorize_statistic(statistic):
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"""Vectorize an n-sample statistic"""
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# This is a little cleaner than np.nditer at the expense of some data
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# copying: concatenate samples together, then use np.apply_along_axis
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def stat_nd(*data, axis=0):
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lengths = [sample.shape[axis] for sample in data]
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split_indices = np.cumsum(lengths)[:-1]
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z = _broadcast_concatenate(data, axis)
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# move working axis to position 0 so that new dimensions in the output
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# of `statistic` are _prepended_. ("This axis is removed, and replaced
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# with new dimensions...")
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z = np.moveaxis(z, axis, 0)
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def stat_1d(z):
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data = np.split(z, split_indices)
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return statistic(*data)
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return np.apply_along_axis(stat_1d, 0, z)[()]
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return stat_nd
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def _jackknife_resample(sample, batch=None):
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"""Jackknife resample the sample. Only one-sample stats for now."""
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n = sample.shape[-1]
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batch_nominal = batch or n
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for k in range(0, n, batch_nominal):
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# col_start:col_end are the observations to remove
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batch_actual = min(batch_nominal, n-k)
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# jackknife - each row leaves out one observation
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j = np.ones((batch_actual, n), dtype=bool)
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np.fill_diagonal(j[:, k:k+batch_actual], False)
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i = np.arange(n)
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i = np.broadcast_to(i, (batch_actual, n))
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i = i[j].reshape((batch_actual, n-1))
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resamples = sample[..., i]
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yield resamples
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def _bootstrap_resample(sample, n_resamples=None, random_state=None):
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"""Bootstrap resample the sample."""
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n = sample.shape[-1]
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# bootstrap - each row is a random resample of original observations
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i = rng_integers(random_state, 0, n, (n_resamples, n))
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resamples = sample[..., i]
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return resamples
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def _percentile_of_score(a, score, axis):
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"""Vectorized, simplified `scipy.stats.percentileofscore`.
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Uses logic of the 'mean' value of percentileofscore's kind parameter.
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Unlike `stats.percentileofscore`, the percentile returned is a fraction
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in [0, 1].
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"""
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B = a.shape[axis]
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return ((a < score).sum(axis=axis) + (a <= score).sum(axis=axis)) / (2 * B)
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def _percentile_along_axis(theta_hat_b, alpha):
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"""`np.percentile` with different percentile for each slice."""
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# the difference between _percentile_along_axis and np.percentile is that
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# np.percentile gets _all_ the qs for each axis slice, whereas
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# _percentile_along_axis gets the q corresponding with each axis slice
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shape = theta_hat_b.shape[:-1]
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alpha = np.broadcast_to(alpha, shape)
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percentiles = np.zeros_like(alpha, dtype=np.float64)
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for indices, alpha_i in np.ndenumerate(alpha):
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if np.isnan(alpha_i):
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# e.g. when bootstrap distribution has only one unique element
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msg = (
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"The BCa confidence interval cannot be calculated."
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" This problem is known to occur when the distribution"
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" is degenerate or the statistic is np.min."
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)
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warnings.warn(DegenerateDataWarning(msg), stacklevel=3)
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percentiles[indices] = np.nan
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else:
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theta_hat_b_i = theta_hat_b[indices]
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percentiles[indices] = np.percentile(theta_hat_b_i, alpha_i)
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return percentiles[()] # return scalar instead of 0d array
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def _bca_interval(data, statistic, axis, alpha, theta_hat_b, batch):
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"""Bias-corrected and accelerated interval."""
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# closely follows [1] 14.3 and 15.4 (Eq. 15.36)
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# calculate z0_hat
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theta_hat = np.asarray(statistic(*data, axis=axis))[..., None]
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percentile = _percentile_of_score(theta_hat_b, theta_hat, axis=-1)
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z0_hat = ndtri(percentile)
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# calculate a_hat
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theta_hat_ji = [] # j is for sample of data, i is for jackknife resample
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for j, sample in enumerate(data):
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# _jackknife_resample will add an axis prior to the last axis that
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# corresponds with the different jackknife resamples. Do the same for
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# each sample of the data to ensure broadcastability. We need to
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# create a copy of the list containing the samples anyway, so do this
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# in the loop to simplify the code. This is not the bottleneck...
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samples = [np.expand_dims(sample, -2) for sample in data]
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theta_hat_i = []
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for jackknife_sample in _jackknife_resample(sample, batch):
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samples[j] = jackknife_sample
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broadcasted = _broadcast_arrays(samples, axis=-1)
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theta_hat_i.append(statistic(*broadcasted, axis=-1))
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theta_hat_ji.append(theta_hat_i)
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theta_hat_ji = [np.concatenate(theta_hat_i, axis=-1)
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for theta_hat_i in theta_hat_ji]
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n_j = [theta_hat_i.shape[-1] for theta_hat_i in theta_hat_ji]
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theta_hat_j_dot = [theta_hat_i.mean(axis=-1, keepdims=True)
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for theta_hat_i in theta_hat_ji]
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U_ji = [(n - 1) * (theta_hat_dot - theta_hat_i)
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for theta_hat_dot, theta_hat_i, n
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in zip(theta_hat_j_dot, theta_hat_ji, n_j)]
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nums = [(U_i**3).sum(axis=-1)/n**3 for U_i, n in zip(U_ji, n_j)]
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dens = [(U_i**2).sum(axis=-1)/n**2 for U_i, n in zip(U_ji, n_j)]
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a_hat = 1/6 * sum(nums) / sum(dens)**(3/2)
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# calculate alpha_1, alpha_2
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z_alpha = ndtri(alpha)
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z_1alpha = -z_alpha
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num1 = z0_hat + z_alpha
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alpha_1 = ndtr(z0_hat + num1/(1 - a_hat*num1))
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num2 = z0_hat + z_1alpha
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alpha_2 = ndtr(z0_hat + num2/(1 - a_hat*num2))
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return alpha_1, alpha_2, a_hat # return a_hat for testing
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def _bootstrap_iv(data, statistic, vectorized, paired, axis, confidence_level,
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alternative, n_resamples, batch, method, bootstrap_result,
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random_state):
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"""Input validation and standardization for `bootstrap`."""
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if vectorized not in {True, False, None}:
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raise ValueError("`vectorized` must be `True`, `False`, or `None`.")
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if vectorized is None:
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vectorized = 'axis' in inspect.signature(statistic).parameters
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if not vectorized:
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statistic = _vectorize_statistic(statistic)
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axis_int = int(axis)
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if axis != axis_int:
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raise ValueError("`axis` must be an integer.")
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n_samples = 0
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try:
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n_samples = len(data)
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except TypeError:
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raise ValueError("`data` must be a sequence of samples.")
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if n_samples == 0:
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raise ValueError("`data` must contain at least one sample.")
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message = ("Ignoring the dimension specified by `axis`, arrays in `data` do not "
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"have the same shape. Beginning in SciPy 1.16.0, `bootstrap` will "
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"explicitly broadcast elements of `data` to the same shape (ignoring "
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"`axis`) before performing the calculation. To avoid this warning in "
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"the meantime, ensure that all samples have the same shape (except "
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"potentially along `axis`).")
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data = [np.atleast_1d(sample) for sample in data]
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reduced_shapes = set()
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for sample in data:
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reduced_shape = list(sample.shape)
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reduced_shape.pop(axis)
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reduced_shapes.add(tuple(reduced_shape))
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if len(reduced_shapes) != 1:
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warnings.warn(message, FutureWarning, stacklevel=3)
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data_iv = []
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for sample in data:
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if sample.shape[axis_int] <= 1:
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raise ValueError("each sample in `data` must contain two or more "
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"observations along `axis`.")
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sample = np.moveaxis(sample, axis_int, -1)
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data_iv.append(sample)
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if paired not in {True, False}:
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raise ValueError("`paired` must be `True` or `False`.")
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if paired:
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n = data_iv[0].shape[-1]
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for sample in data_iv[1:]:
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if sample.shape[-1] != n:
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message = ("When `paired is True`, all samples must have the "
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"same length along `axis`")
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raise ValueError(message)
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# to generate the bootstrap distribution for paired-sample statistics,
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# resample the indices of the observations
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def statistic(i, axis=-1, data=data_iv, unpaired_statistic=statistic):
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data = [sample[..., i] for sample in data]
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return unpaired_statistic(*data, axis=axis)
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data_iv = [np.arange(n)]
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confidence_level_float = float(confidence_level)
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alternative = alternative.lower()
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alternatives = {'two-sided', 'less', 'greater'}
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if alternative not in alternatives:
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raise ValueError(f"`alternative` must be one of {alternatives}")
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n_resamples_int = int(n_resamples)
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if n_resamples != n_resamples_int or n_resamples_int < 0:
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raise ValueError("`n_resamples` must be a non-negative integer.")
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if batch is None:
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batch_iv = batch
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else:
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batch_iv = int(batch)
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if batch != batch_iv or batch_iv <= 0:
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raise ValueError("`batch` must be a positive integer or None.")
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methods = {'percentile', 'basic', 'bca'}
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method = method.lower()
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if method not in methods:
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raise ValueError(f"`method` must be in {methods}")
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message = "`bootstrap_result` must have attribute `bootstrap_distribution'"
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if (bootstrap_result is not None
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and not hasattr(bootstrap_result, "bootstrap_distribution")):
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raise ValueError(message)
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message = ("Either `bootstrap_result.bootstrap_distribution.size` or "
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"`n_resamples` must be positive.")
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if ((not bootstrap_result or
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not bootstrap_result.bootstrap_distribution.size)
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and n_resamples_int == 0):
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raise ValueError(message)
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random_state = check_random_state(random_state)
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return (data_iv, statistic, vectorized, paired, axis_int,
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confidence_level_float, alternative, n_resamples_int, batch_iv,
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method, bootstrap_result, random_state)
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@dataclass
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class BootstrapResult:
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"""Result object returned by `scipy.stats.bootstrap`.
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Attributes
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----------
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confidence_interval : ConfidenceInterval
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The bootstrap confidence interval as an instance of
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`collections.namedtuple` with attributes `low` and `high`.
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bootstrap_distribution : ndarray
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The bootstrap distribution, that is, the value of `statistic` for
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each resample. The last dimension corresponds with the resamples
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(e.g. ``res.bootstrap_distribution.shape[-1] == n_resamples``).
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standard_error : float or ndarray
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The bootstrap standard error, that is, the sample standard
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deviation of the bootstrap distribution.
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"""
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confidence_interval: ConfidenceInterval
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bootstrap_distribution: np.ndarray
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standard_error: float | np.ndarray
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def bootstrap(data, statistic, *, n_resamples=9999, batch=None,
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vectorized=None, paired=False, axis=0, confidence_level=0.95,
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alternative='two-sided', method='BCa', bootstrap_result=None,
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random_state=None):
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r"""
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Compute a two-sided bootstrap confidence interval of a statistic.
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When `method` is ``'percentile'`` and `alternative` is ``'two-sided'``,
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a bootstrap confidence interval is computed according to the following
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procedure.
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1. Resample the data: for each sample in `data` and for each of
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`n_resamples`, take a random sample of the original sample
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(with replacement) of the same size as the original sample.
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2. Compute the bootstrap distribution of the statistic: for each set of
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resamples, compute the test statistic.
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3. Determine the confidence interval: find the interval of the bootstrap
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distribution that is
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- symmetric about the median and
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- contains `confidence_level` of the resampled statistic values.
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While the ``'percentile'`` method is the most intuitive, it is rarely
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used in practice. Two more common methods are available, ``'basic'``
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('reverse percentile') and ``'BCa'`` ('bias-corrected and accelerated');
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they differ in how step 3 is performed.
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If the samples in `data` are taken at random from their respective
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distributions :math:`n` times, the confidence interval returned by
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`bootstrap` will contain the true value of the statistic for those
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distributions approximately `confidence_level`:math:`\, \times \, n` times.
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Parameters
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----------
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data : sequence of array-like
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Each element of `data` is a sample containing scalar observations from an
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underlying distribution. Elements of `data` must be broadcastable to the
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same shape (with the possible exception of the dimension specified by `axis`).
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.. versionchanged:: 1.14.0
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`bootstrap` will now emit a ``FutureWarning`` if the shapes of the
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elements of `data` are not the same (with the exception of the dimension
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specified by `axis`).
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Beginning in SciPy 1.16.0, `bootstrap` will explicitly broadcast the
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elements to the same shape (except along `axis`) before performing
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the calculation.
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statistic : callable
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Statistic for which the confidence interval is to be calculated.
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`statistic` must be a callable that accepts ``len(data)`` samples
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as separate arguments and returns the resulting statistic.
|
||
|
|
If `vectorized` is set ``True``,
|
||
|
|
`statistic` must also accept a keyword argument `axis` and be
|
||
|
|
vectorized to compute the statistic along the provided `axis`.
|
||
|
|
n_resamples : int, default: ``9999``
|
||
|
|
The number of resamples performed to form the bootstrap distribution
|
||
|
|
of the statistic.
|
||
|
|
batch : int, optional
|
||
|
|
The number of resamples to process in each vectorized call to
|
||
|
|
`statistic`. Memory usage is O( `batch` * ``n`` ), where ``n`` is the
|
||
|
|
sample size. Default is ``None``, in which case ``batch = n_resamples``
|
||
|
|
(or ``batch = max(n_resamples, n)`` for ``method='BCa'``).
|
||
|
|
vectorized : bool, optional
|
||
|
|
If `vectorized` is set ``False``, `statistic` will not be passed
|
||
|
|
keyword argument `axis` and is expected to calculate the statistic
|
||
|
|
only for 1D samples. If ``True``, `statistic` will be passed keyword
|
||
|
|
argument `axis` and is expected to calculate the statistic along `axis`
|
||
|
|
when passed an ND sample array. If ``None`` (default), `vectorized`
|
||
|
|
will be set ``True`` if ``axis`` is a parameter of `statistic`. Use of
|
||
|
|
a vectorized statistic typically reduces computation time.
|
||
|
|
paired : bool, default: ``False``
|
||
|
|
Whether the statistic treats corresponding elements of the samples
|
||
|
|
in `data` as paired.
|
||
|
|
axis : int, default: ``0``
|
||
|
|
The axis of the samples in `data` along which the `statistic` is
|
||
|
|
calculated.
|
||
|
|
confidence_level : float, default: ``0.95``
|
||
|
|
The confidence level of the confidence interval.
|
||
|
|
alternative : {'two-sided', 'less', 'greater'}, default: ``'two-sided'``
|
||
|
|
Choose ``'two-sided'`` (default) for a two-sided confidence interval,
|
||
|
|
``'less'`` for a one-sided confidence interval with the lower bound
|
||
|
|
at ``-np.inf``, and ``'greater'`` for a one-sided confidence interval
|
||
|
|
with the upper bound at ``np.inf``. The other bound of the one-sided
|
||
|
|
confidence intervals is the same as that of a two-sided confidence
|
||
|
|
interval with `confidence_level` twice as far from 1.0; e.g. the upper
|
||
|
|
bound of a 95% ``'less'`` confidence interval is the same as the upper
|
||
|
|
bound of a 90% ``'two-sided'`` confidence interval.
|
||
|
|
method : {'percentile', 'basic', 'bca'}, default: ``'BCa'``
|
||
|
|
Whether to return the 'percentile' bootstrap confidence interval
|
||
|
|
(``'percentile'``), the 'basic' (AKA 'reverse') bootstrap confidence
|
||
|
|
interval (``'basic'``), or the bias-corrected and accelerated bootstrap
|
||
|
|
confidence interval (``'BCa'``).
|
||
|
|
bootstrap_result : BootstrapResult, optional
|
||
|
|
Provide the result object returned by a previous call to `bootstrap`
|
||
|
|
to include the previous bootstrap distribution in the new bootstrap
|
||
|
|
distribution. This can be used, for example, to change
|
||
|
|
`confidence_level`, change `method`, or see the effect of performing
|
||
|
|
additional resampling without repeating computations.
|
||
|
|
random_state : {None, int, `numpy.random.Generator`,
|
||
|
|
`numpy.random.RandomState`}, optional
|
||
|
|
|
||
|
|
Pseudorandom number generator state used to generate resamples.
|
||
|
|
|
||
|
|
If `random_state` is ``None`` (or `np.random`), the
|
||
|
|
`numpy.random.RandomState` singleton is used.
|
||
|
|
If `random_state` is an int, a new ``RandomState`` instance is used,
|
||
|
|
seeded with `random_state`.
|
||
|
|
If `random_state` is already a ``Generator`` or ``RandomState``
|
||
|
|
instance then that instance is used.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
res : BootstrapResult
|
||
|
|
An object with attributes:
|
||
|
|
|
||
|
|
confidence_interval : ConfidenceInterval
|
||
|
|
The bootstrap confidence interval as an instance of
|
||
|
|
`collections.namedtuple` with attributes `low` and `high`.
|
||
|
|
bootstrap_distribution : ndarray
|
||
|
|
The bootstrap distribution, that is, the value of `statistic` for
|
||
|
|
each resample. The last dimension corresponds with the resamples
|
||
|
|
(e.g. ``res.bootstrap_distribution.shape[-1] == n_resamples``).
|
||
|
|
standard_error : float or ndarray
|
||
|
|
The bootstrap standard error, that is, the sample standard
|
||
|
|
deviation of the bootstrap distribution.
|
||
|
|
|
||
|
|
Warns
|
||
|
|
-----
|
||
|
|
`~scipy.stats.DegenerateDataWarning`
|
||
|
|
Generated when ``method='BCa'`` and the bootstrap distribution is
|
||
|
|
degenerate (e.g. all elements are identical).
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Elements of the confidence interval may be NaN for ``method='BCa'`` if
|
||
|
|
the bootstrap distribution is degenerate (e.g. all elements are identical).
|
||
|
|
In this case, consider using another `method` or inspecting `data` for
|
||
|
|
indications that other analysis may be more appropriate (e.g. all
|
||
|
|
observations are identical).
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap,
|
||
|
|
Chapman & Hall/CRC, Boca Raton, FL, USA (1993)
|
||
|
|
.. [2] Nathaniel E. Helwig, "Bootstrap Confidence Intervals",
|
||
|
|
http://users.stat.umn.edu/~helwig/notes/bootci-Notes.pdf
|
||
|
|
.. [3] Bootstrapping (statistics), Wikipedia,
|
||
|
|
https://en.wikipedia.org/wiki/Bootstrapping_%28statistics%29
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
Suppose we have sampled data from an unknown distribution.
|
||
|
|
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> rng = np.random.default_rng()
|
||
|
|
>>> from scipy.stats import norm
|
||
|
|
>>> dist = norm(loc=2, scale=4) # our "unknown" distribution
|
||
|
|
>>> data = dist.rvs(size=100, random_state=rng)
|
||
|
|
|
||
|
|
We are interested in the standard deviation of the distribution.
|
||
|
|
|
||
|
|
>>> std_true = dist.std() # the true value of the statistic
|
||
|
|
>>> print(std_true)
|
||
|
|
4.0
|
||
|
|
>>> std_sample = np.std(data) # the sample statistic
|
||
|
|
>>> print(std_sample)
|
||
|
|
3.9460644295563863
|
||
|
|
|
||
|
|
The bootstrap is used to approximate the variability we would expect if we
|
||
|
|
were to repeatedly sample from the unknown distribution and calculate the
|
||
|
|
statistic of the sample each time. It does this by repeatedly resampling
|
||
|
|
values *from the original sample* with replacement and calculating the
|
||
|
|
statistic of each resample. This results in a "bootstrap distribution" of
|
||
|
|
the statistic.
|
||
|
|
|
||
|
|
>>> import matplotlib.pyplot as plt
|
||
|
|
>>> from scipy.stats import bootstrap
|
||
|
|
>>> data = (data,) # samples must be in a sequence
|
||
|
|
>>> res = bootstrap(data, np.std, confidence_level=0.9,
|
||
|
|
... random_state=rng)
|
||
|
|
>>> fig, ax = plt.subplots()
|
||
|
|
>>> ax.hist(res.bootstrap_distribution, bins=25)
|
||
|
|
>>> ax.set_title('Bootstrap Distribution')
|
||
|
|
>>> ax.set_xlabel('statistic value')
|
||
|
|
>>> ax.set_ylabel('frequency')
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
The standard error quantifies this variability. It is calculated as the
|
||
|
|
standard deviation of the bootstrap distribution.
|
||
|
|
|
||
|
|
>>> res.standard_error
|
||
|
|
0.24427002125829136
|
||
|
|
>>> res.standard_error == np.std(res.bootstrap_distribution, ddof=1)
|
||
|
|
True
|
||
|
|
|
||
|
|
The bootstrap distribution of the statistic is often approximately normal
|
||
|
|
with scale equal to the standard error.
|
||
|
|
|
||
|
|
>>> x = np.linspace(3, 5)
|
||
|
|
>>> pdf = norm.pdf(x, loc=std_sample, scale=res.standard_error)
|
||
|
|
>>> fig, ax = plt.subplots()
|
||
|
|
>>> ax.hist(res.bootstrap_distribution, bins=25, density=True)
|
||
|
|
>>> ax.plot(x, pdf)
|
||
|
|
>>> ax.set_title('Normal Approximation of the Bootstrap Distribution')
|
||
|
|
>>> ax.set_xlabel('statistic value')
|
||
|
|
>>> ax.set_ylabel('pdf')
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
This suggests that we could construct a 90% confidence interval on the
|
||
|
|
statistic based on quantiles of this normal distribution.
|
||
|
|
|
||
|
|
>>> norm.interval(0.9, loc=std_sample, scale=res.standard_error)
|
||
|
|
(3.5442759991341726, 4.3478528599786)
|
||
|
|
|
||
|
|
Due to central limit theorem, this normal approximation is accurate for a
|
||
|
|
variety of statistics and distributions underlying the samples; however,
|
||
|
|
the approximation is not reliable in all cases. Because `bootstrap` is
|
||
|
|
designed to work with arbitrary underlying distributions and statistics,
|
||
|
|
it uses more advanced techniques to generate an accurate confidence
|
||
|
|
interval.
|
||
|
|
|
||
|
|
>>> print(res.confidence_interval)
|
||
|
|
ConfidenceInterval(low=3.57655333533867, high=4.382043696342881)
|
||
|
|
|
||
|
|
If we sample from the original distribution 100 times and form a bootstrap
|
||
|
|
confidence interval for each sample, the confidence interval
|
||
|
|
contains the true value of the statistic approximately 90% of the time.
|
||
|
|
|
||
|
|
>>> n_trials = 100
|
||
|
|
>>> ci_contains_true_std = 0
|
||
|
|
>>> for i in range(n_trials):
|
||
|
|
... data = (dist.rvs(size=100, random_state=rng),)
|
||
|
|
... res = bootstrap(data, np.std, confidence_level=0.9,
|
||
|
|
... n_resamples=999, random_state=rng)
|
||
|
|
... ci = res.confidence_interval
|
||
|
|
... if ci[0] < std_true < ci[1]:
|
||
|
|
... ci_contains_true_std += 1
|
||
|
|
>>> print(ci_contains_true_std)
|
||
|
|
88
|
||
|
|
|
||
|
|
Rather than writing a loop, we can also determine the confidence intervals
|
||
|
|
for all 100 samples at once.
|
||
|
|
|
||
|
|
>>> data = (dist.rvs(size=(n_trials, 100), random_state=rng),)
|
||
|
|
>>> res = bootstrap(data, np.std, axis=-1, confidence_level=0.9,
|
||
|
|
... n_resamples=999, random_state=rng)
|
||
|
|
>>> ci_l, ci_u = res.confidence_interval
|
||
|
|
|
||
|
|
Here, `ci_l` and `ci_u` contain the confidence interval for each of the
|
||
|
|
``n_trials = 100`` samples.
|
||
|
|
|
||
|
|
>>> print(ci_l[:5])
|
||
|
|
[3.86401283 3.33304394 3.52474647 3.54160981 3.80569252]
|
||
|
|
>>> print(ci_u[:5])
|
||
|
|
[4.80217409 4.18143252 4.39734707 4.37549713 4.72843584]
|
||
|
|
|
||
|
|
And again, approximately 90% contain the true value, ``std_true = 4``.
|
||
|
|
|
||
|
|
>>> print(np.sum((ci_l < std_true) & (std_true < ci_u)))
|
||
|
|
93
|
||
|
|
|
||
|
|
`bootstrap` can also be used to estimate confidence intervals of
|
||
|
|
multi-sample statistics. For example, to get a confidence interval
|
||
|
|
for the difference between means, we write a function that accepts
|
||
|
|
two sample arguments and returns only the statistic. The use of the
|
||
|
|
``axis`` argument ensures that all mean calculations are perform in
|
||
|
|
a single vectorized call, which is faster than looping over pairs
|
||
|
|
of resamples in Python.
|
||
|
|
|
||
|
|
>>> def my_statistic(sample1, sample2, axis=-1):
|
||
|
|
... mean1 = np.mean(sample1, axis=axis)
|
||
|
|
... mean2 = np.mean(sample2, axis=axis)
|
||
|
|
... return mean1 - mean2
|
||
|
|
|
||
|
|
Here, we use the 'percentile' method with the default 95% confidence level.
|
||
|
|
|
||
|
|
>>> sample1 = norm.rvs(scale=1, size=100, random_state=rng)
|
||
|
|
>>> sample2 = norm.rvs(scale=2, size=100, random_state=rng)
|
||
|
|
>>> data = (sample1, sample2)
|
||
|
|
>>> res = bootstrap(data, my_statistic, method='basic', random_state=rng)
|
||
|
|
>>> print(my_statistic(sample1, sample2))
|
||
|
|
0.16661030792089523
|
||
|
|
>>> print(res.confidence_interval)
|
||
|
|
ConfidenceInterval(low=-0.29087973240818693, high=0.6371338699912273)
|
||
|
|
|
||
|
|
The bootstrap estimate of the standard error is also available.
|
||
|
|
|
||
|
|
>>> print(res.standard_error)
|
||
|
|
0.238323948262459
|
||
|
|
|
||
|
|
Paired-sample statistics work, too. For example, consider the Pearson
|
||
|
|
correlation coefficient.
|
||
|
|
|
||
|
|
>>> from scipy.stats import pearsonr
|
||
|
|
>>> n = 100
|
||
|
|
>>> x = np.linspace(0, 10, n)
|
||
|
|
>>> y = x + rng.uniform(size=n)
|
||
|
|
>>> print(pearsonr(x, y)[0]) # element 0 is the statistic
|
||
|
|
0.9954306665125647
|
||
|
|
|
||
|
|
We wrap `pearsonr` so that it returns only the statistic, ensuring
|
||
|
|
that we use the `axis` argument because it is available.
|
||
|
|
|
||
|
|
>>> def my_statistic(x, y, axis=-1):
|
||
|
|
... return pearsonr(x, y, axis=axis)[0]
|
||
|
|
|
||
|
|
We call `bootstrap` using ``paired=True``.
|
||
|
|
|
||
|
|
>>> res = bootstrap((x, y), my_statistic, paired=True, random_state=rng)
|
||
|
|
>>> print(res.confidence_interval)
|
||
|
|
ConfidenceInterval(low=0.9941504301315878, high=0.996377412215445)
|
||
|
|
|
||
|
|
The result object can be passed back into `bootstrap` to perform additional
|
||
|
|
resampling:
|
||
|
|
|
||
|
|
>>> len(res.bootstrap_distribution)
|
||
|
|
9999
|
||
|
|
>>> res = bootstrap((x, y), my_statistic, paired=True,
|
||
|
|
... n_resamples=1000, random_state=rng,
|
||
|
|
... bootstrap_result=res)
|
||
|
|
>>> len(res.bootstrap_distribution)
|
||
|
|
10999
|
||
|
|
|
||
|
|
or to change the confidence interval options:
|
||
|
|
|
||
|
|
>>> res2 = bootstrap((x, y), my_statistic, paired=True,
|
||
|
|
... n_resamples=0, random_state=rng, bootstrap_result=res,
|
||
|
|
... method='percentile', confidence_level=0.9)
|
||
|
|
>>> np.testing.assert_equal(res2.bootstrap_distribution,
|
||
|
|
... res.bootstrap_distribution)
|
||
|
|
>>> res.confidence_interval
|
||
|
|
ConfidenceInterval(low=0.9941574828235082, high=0.9963781698210212)
|
||
|
|
|
||
|
|
without repeating computation of the original bootstrap distribution.
|
||
|
|
|
||
|
|
"""
|
||
|
|
# Input validation
|
||
|
|
args = _bootstrap_iv(data, statistic, vectorized, paired, axis,
|
||
|
|
confidence_level, alternative, n_resamples, batch,
|
||
|
|
method, bootstrap_result, random_state)
|
||
|
|
(data, statistic, vectorized, paired, axis, confidence_level,
|
||
|
|
alternative, n_resamples, batch, method, bootstrap_result,
|
||
|
|
random_state) = args
|
||
|
|
|
||
|
|
theta_hat_b = ([] if bootstrap_result is None
|
||
|
|
else [bootstrap_result.bootstrap_distribution])
|
||
|
|
|
||
|
|
batch_nominal = batch or n_resamples or 1
|
||
|
|
|
||
|
|
for k in range(0, n_resamples, batch_nominal):
|
||
|
|
batch_actual = min(batch_nominal, n_resamples-k)
|
||
|
|
# Generate resamples
|
||
|
|
resampled_data = []
|
||
|
|
for sample in data:
|
||
|
|
resample = _bootstrap_resample(sample, n_resamples=batch_actual,
|
||
|
|
random_state=random_state)
|
||
|
|
resampled_data.append(resample)
|
||
|
|
|
||
|
|
# Compute bootstrap distribution of statistic
|
||
|
|
theta_hat_b.append(statistic(*resampled_data, axis=-1))
|
||
|
|
theta_hat_b = np.concatenate(theta_hat_b, axis=-1)
|
||
|
|
|
||
|
|
# Calculate percentile interval
|
||
|
|
alpha = ((1 - confidence_level)/2 if alternative == 'two-sided'
|
||
|
|
else (1 - confidence_level))
|
||
|
|
if method == 'bca':
|
||
|
|
interval = _bca_interval(data, statistic, axis=-1, alpha=alpha,
|
||
|
|
theta_hat_b=theta_hat_b, batch=batch)[:2]
|
||
|
|
percentile_fun = _percentile_along_axis
|
||
|
|
else:
|
||
|
|
interval = alpha, 1-alpha
|
||
|
|
|
||
|
|
def percentile_fun(a, q):
|
||
|
|
return np.percentile(a=a, q=q, axis=-1)
|
||
|
|
|
||
|
|
# Calculate confidence interval of statistic
|
||
|
|
ci_l = percentile_fun(theta_hat_b, interval[0]*100)
|
||
|
|
ci_u = percentile_fun(theta_hat_b, interval[1]*100)
|
||
|
|
if method == 'basic': # see [3]
|
||
|
|
theta_hat = statistic(*data, axis=-1)
|
||
|
|
ci_l, ci_u = 2*theta_hat - ci_u, 2*theta_hat - ci_l
|
||
|
|
|
||
|
|
if alternative == 'less':
|
||
|
|
ci_l = np.full_like(ci_l, -np.inf)
|
||
|
|
elif alternative == 'greater':
|
||
|
|
ci_u = np.full_like(ci_u, np.inf)
|
||
|
|
|
||
|
|
return BootstrapResult(confidence_interval=ConfidenceInterval(ci_l, ci_u),
|
||
|
|
bootstrap_distribution=theta_hat_b,
|
||
|
|
standard_error=np.std(theta_hat_b, ddof=1, axis=-1))
|
||
|
|
|
||
|
|
|
||
|
|
def _monte_carlo_test_iv(data, rvs, statistic, vectorized, n_resamples,
|
||
|
|
batch, alternative, axis):
|
||
|
|
"""Input validation for `monte_carlo_test`."""
|
||
|
|
axis_int = int(axis)
|
||
|
|
if axis != axis_int:
|
||
|
|
raise ValueError("`axis` must be an integer.")
|
||
|
|
|
||
|
|
if vectorized not in {True, False, None}:
|
||
|
|
raise ValueError("`vectorized` must be `True`, `False`, or `None`.")
|
||
|
|
|
||
|
|
if not isinstance(rvs, Sequence):
|
||
|
|
rvs = (rvs,)
|
||
|
|
data = (data,)
|
||
|
|
for rvs_i in rvs:
|
||
|
|
if not callable(rvs_i):
|
||
|
|
raise TypeError("`rvs` must be callable or sequence of callables.")
|
||
|
|
|
||
|
|
# At this point, `data` should be a sequence
|
||
|
|
# If it isn't, the user passed a sequence for `rvs` but not `data`
|
||
|
|
message = "If `rvs` is a sequence, `len(rvs)` must equal `len(data)`."
|
||
|
|
try:
|
||
|
|
len(data)
|
||
|
|
except TypeError as e:
|
||
|
|
raise ValueError(message) from e
|
||
|
|
if not len(rvs) == len(data):
|
||
|
|
raise ValueError(message)
|
||
|
|
|
||
|
|
if not callable(statistic):
|
||
|
|
raise TypeError("`statistic` must be callable.")
|
||
|
|
|
||
|
|
if vectorized is None:
|
||
|
|
try:
|
||
|
|
signature = inspect.signature(statistic).parameters
|
||
|
|
except ValueError as e:
|
||
|
|
message = (f"Signature inspection of {statistic=} failed; "
|
||
|
|
"pass `vectorize` explicitly.")
|
||
|
|
raise ValueError(message) from e
|
||
|
|
vectorized = 'axis' in signature
|
||
|
|
|
||
|
|
xp = array_namespace(*data)
|
||
|
|
|
||
|
|
if not vectorized:
|
||
|
|
if is_numpy(xp):
|
||
|
|
statistic_vectorized = _vectorize_statistic(statistic)
|
||
|
|
else:
|
||
|
|
message = ("`statistic` must be vectorized (i.e. support an `axis` "
|
||
|
|
f"argument) when `data` contains {xp.__name__} arrays.")
|
||
|
|
raise ValueError(message)
|
||
|
|
else:
|
||
|
|
statistic_vectorized = statistic
|
||
|
|
|
||
|
|
data = _broadcast_arrays(data, axis, xp=xp)
|
||
|
|
data_iv = []
|
||
|
|
for sample in data:
|
||
|
|
sample = xp.broadcast_to(sample, (1,)) if sample.ndim == 0 else sample
|
||
|
|
sample = xp_moveaxis_to_end(sample, axis_int, xp=xp)
|
||
|
|
data_iv.append(sample)
|
||
|
|
|
||
|
|
n_resamples_int = int(n_resamples)
|
||
|
|
if n_resamples != n_resamples_int or n_resamples_int <= 0:
|
||
|
|
raise ValueError("`n_resamples` must be a positive integer.")
|
||
|
|
|
||
|
|
if batch is None:
|
||
|
|
batch_iv = batch
|
||
|
|
else:
|
||
|
|
batch_iv = int(batch)
|
||
|
|
if batch != batch_iv or batch_iv <= 0:
|
||
|
|
raise ValueError("`batch` must be a positive integer or None.")
|
||
|
|
|
||
|
|
alternatives = {'two-sided', 'greater', 'less'}
|
||
|
|
alternative = alternative.lower()
|
||
|
|
if alternative not in alternatives:
|
||
|
|
raise ValueError(f"`alternative` must be in {alternatives}")
|
||
|
|
|
||
|
|
# Infer the desired p-value dtype based on the input types
|
||
|
|
min_float = getattr(xp, 'float16', xp.float32)
|
||
|
|
dtype = xp.result_type(*data_iv, min_float)
|
||
|
|
|
||
|
|
return (data_iv, rvs, statistic_vectorized, vectorized, n_resamples_int,
|
||
|
|
batch_iv, alternative, axis_int, dtype, xp)
|
||
|
|
|
||
|
|
|
||
|
|
@dataclass
|
||
|
|
class MonteCarloTestResult:
|
||
|
|
"""Result object returned by `scipy.stats.monte_carlo_test`.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
----------
|
||
|
|
statistic : float or ndarray
|
||
|
|
The observed test statistic of the sample.
|
||
|
|
pvalue : float or ndarray
|
||
|
|
The p-value for the given alternative.
|
||
|
|
null_distribution : ndarray
|
||
|
|
The values of the test statistic generated under the null
|
||
|
|
hypothesis.
|
||
|
|
"""
|
||
|
|
statistic: float | np.ndarray
|
||
|
|
pvalue: float | np.ndarray
|
||
|
|
null_distribution: np.ndarray
|
||
|
|
|
||
|
|
|
||
|
|
@_rename_parameter('sample', 'data')
|
||
|
|
def monte_carlo_test(data, rvs, statistic, *, vectorized=None,
|
||
|
|
n_resamples=9999, batch=None, alternative="two-sided",
|
||
|
|
axis=0):
|
||
|
|
r"""Perform a Monte Carlo hypothesis test.
|
||
|
|
|
||
|
|
`data` contains a sample or a sequence of one or more samples. `rvs`
|
||
|
|
specifies the distribution(s) of the sample(s) in `data` under the null
|
||
|
|
hypothesis. The value of `statistic` for the given `data` is compared
|
||
|
|
against a Monte Carlo null distribution: the value of the statistic for
|
||
|
|
each of `n_resamples` sets of samples generated using `rvs`. This gives
|
||
|
|
the p-value, the probability of observing such an extreme value of the
|
||
|
|
test statistic under the null hypothesis.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
data : array-like or sequence of array-like
|
||
|
|
An array or sequence of arrays of observations.
|
||
|
|
rvs : callable or tuple of callables
|
||
|
|
A callable or sequence of callables that generates random variates
|
||
|
|
under the null hypothesis. Each element of `rvs` must be a callable
|
||
|
|
that accepts keyword argument ``size`` (e.g. ``rvs(size=(m, n))``) and
|
||
|
|
returns an N-d array sample of that shape. If `rvs` is a sequence, the
|
||
|
|
number of callables in `rvs` must match the number of samples in
|
||
|
|
`data`, i.e. ``len(rvs) == len(data)``. If `rvs` is a single callable,
|
||
|
|
`data` is treated as a single sample.
|
||
|
|
statistic : callable
|
||
|
|
Statistic for which the p-value of the hypothesis test is to be
|
||
|
|
calculated. `statistic` must be a callable that accepts a sample
|
||
|
|
(e.g. ``statistic(sample)``) or ``len(rvs)`` separate samples (e.g.
|
||
|
|
``statistic(samples1, sample2)`` if `rvs` contains two callables and
|
||
|
|
`data` contains two samples) and returns the resulting statistic.
|
||
|
|
If `vectorized` is set ``True``, `statistic` must also accept a keyword
|
||
|
|
argument `axis` and be vectorized to compute the statistic along the
|
||
|
|
provided `axis` of the samples in `data`.
|
||
|
|
vectorized : bool, optional
|
||
|
|
If `vectorized` is set ``False``, `statistic` will not be passed
|
||
|
|
keyword argument `axis` and is expected to calculate the statistic
|
||
|
|
only for 1D samples. If ``True``, `statistic` will be passed keyword
|
||
|
|
argument `axis` and is expected to calculate the statistic along `axis`
|
||
|
|
when passed ND sample arrays. If ``None`` (default), `vectorized`
|
||
|
|
will be set ``True`` if ``axis`` is a parameter of `statistic`. Use of
|
||
|
|
a vectorized statistic typically reduces computation time.
|
||
|
|
n_resamples : int, default: 9999
|
||
|
|
Number of samples drawn from each of the callables of `rvs`.
|
||
|
|
Equivalently, the number statistic values under the null hypothesis
|
||
|
|
used as the Monte Carlo null distribution.
|
||
|
|
batch : int, optional
|
||
|
|
The number of Monte Carlo samples to process in each call to
|
||
|
|
`statistic`. Memory usage is O( `batch` * ``sample.size[axis]`` ). Default
|
||
|
|
is ``None``, in which case `batch` equals `n_resamples`.
|
||
|
|
alternative : {'two-sided', 'less', 'greater'}
|
||
|
|
The alternative hypothesis for which the p-value is calculated.
|
||
|
|
For each alternative, the p-value is defined as follows.
|
||
|
|
|
||
|
|
- ``'greater'`` : the percentage of the null distribution that is
|
||
|
|
greater than or equal to the observed value of the test statistic.
|
||
|
|
- ``'less'`` : the percentage of the null distribution that is
|
||
|
|
less than or equal to the observed value of the test statistic.
|
||
|
|
- ``'two-sided'`` : twice the smaller of the p-values above.
|
||
|
|
|
||
|
|
axis : int, default: 0
|
||
|
|
The axis of `data` (or each sample within `data`) over which to
|
||
|
|
calculate the statistic.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
res : MonteCarloTestResult
|
||
|
|
An object with attributes:
|
||
|
|
|
||
|
|
statistic : float or ndarray
|
||
|
|
The test statistic of the observed `data`.
|
||
|
|
pvalue : float or ndarray
|
||
|
|
The p-value for the given alternative.
|
||
|
|
null_distribution : ndarray
|
||
|
|
The values of the test statistic generated under the null
|
||
|
|
hypothesis.
|
||
|
|
|
||
|
|
.. warning::
|
||
|
|
The p-value is calculated by counting the elements of the null
|
||
|
|
distribution that are as extreme or more extreme than the observed
|
||
|
|
value of the statistic. Due to the use of finite precision arithmetic,
|
||
|
|
some statistic functions return numerically distinct values when the
|
||
|
|
theoretical values would be exactly equal. In some cases, this could
|
||
|
|
lead to a large error in the calculated p-value. `monte_carlo_test`
|
||
|
|
guards against this by considering elements in the null distribution
|
||
|
|
that are "close" (within a relative tolerance of 100 times the
|
||
|
|
floating point epsilon of inexact dtypes) to the observed
|
||
|
|
value of the test statistic as equal to the observed value of the
|
||
|
|
test statistic. However, the user is advised to inspect the null
|
||
|
|
distribution to assess whether this method of comparison is
|
||
|
|
appropriate, and if not, calculate the p-value manually.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
|
||
|
|
.. [1] B. Phipson and G. K. Smyth. "Permutation P-values Should Never Be
|
||
|
|
Zero: Calculating Exact P-values When Permutations Are Randomly Drawn."
|
||
|
|
Statistical Applications in Genetics and Molecular Biology 9.1 (2010).
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
|
||
|
|
Suppose we wish to test whether a small sample has been drawn from a normal
|
||
|
|
distribution. We decide that we will use the skew of the sample as a
|
||
|
|
test statistic, and we will consider a p-value of 0.05 to be statistically
|
||
|
|
significant.
|
||
|
|
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> from scipy import stats
|
||
|
|
>>> def statistic(x, axis):
|
||
|
|
... return stats.skew(x, axis)
|
||
|
|
|
||
|
|
After collecting our data, we calculate the observed value of the test
|
||
|
|
statistic.
|
||
|
|
|
||
|
|
>>> rng = np.random.default_rng()
|
||
|
|
>>> x = stats.skewnorm.rvs(a=1, size=50, random_state=rng)
|
||
|
|
>>> statistic(x, axis=0)
|
||
|
|
0.12457412450240658
|
||
|
|
|
||
|
|
To determine the probability of observing such an extreme value of the
|
||
|
|
skewness by chance if the sample were drawn from the normal distribution,
|
||
|
|
we can perform a Monte Carlo hypothesis test. The test will draw many
|
||
|
|
samples at random from their normal distribution, calculate the skewness
|
||
|
|
of each sample, and compare our original skewness against this
|
||
|
|
distribution to determine an approximate p-value.
|
||
|
|
|
||
|
|
>>> from scipy.stats import monte_carlo_test
|
||
|
|
>>> # because our statistic is vectorized, we pass `vectorized=True`
|
||
|
|
>>> rvs = lambda size: stats.norm.rvs(size=size, random_state=rng)
|
||
|
|
>>> res = monte_carlo_test(x, rvs, statistic, vectorized=True)
|
||
|
|
>>> print(res.statistic)
|
||
|
|
0.12457412450240658
|
||
|
|
>>> print(res.pvalue)
|
||
|
|
0.7012
|
||
|
|
|
||
|
|
The probability of obtaining a test statistic less than or equal to the
|
||
|
|
observed value under the null hypothesis is ~70%. This is greater than
|
||
|
|
our chosen threshold of 5%, so we cannot consider this to be significant
|
||
|
|
evidence against the null hypothesis.
|
||
|
|
|
||
|
|
Note that this p-value essentially matches that of
|
||
|
|
`scipy.stats.skewtest`, which relies on an asymptotic distribution of a
|
||
|
|
test statistic based on the sample skewness.
|
||
|
|
|
||
|
|
>>> stats.skewtest(x).pvalue
|
||
|
|
0.6892046027110614
|
||
|
|
|
||
|
|
This asymptotic approximation is not valid for small sample sizes, but
|
||
|
|
`monte_carlo_test` can be used with samples of any size.
|
||
|
|
|
||
|
|
>>> x = stats.skewnorm.rvs(a=1, size=7, random_state=rng)
|
||
|
|
>>> # stats.skewtest(x) would produce an error due to small sample
|
||
|
|
>>> res = monte_carlo_test(x, rvs, statistic, vectorized=True)
|
||
|
|
|
||
|
|
The Monte Carlo distribution of the test statistic is provided for
|
||
|
|
further investigation.
|
||
|
|
|
||
|
|
>>> import matplotlib.pyplot as plt
|
||
|
|
>>> fig, ax = plt.subplots()
|
||
|
|
>>> ax.hist(res.null_distribution, bins=50)
|
||
|
|
>>> ax.set_title("Monte Carlo distribution of test statistic")
|
||
|
|
>>> ax.set_xlabel("Value of Statistic")
|
||
|
|
>>> ax.set_ylabel("Frequency")
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
"""
|
||
|
|
args = _monte_carlo_test_iv(data, rvs, statistic, vectorized,
|
||
|
|
n_resamples, batch, alternative, axis)
|
||
|
|
(data, rvs, statistic, vectorized, n_resamples,
|
||
|
|
batch, alternative, axis, dtype, xp) = args
|
||
|
|
|
||
|
|
# Some statistics return plain floats; ensure they're at least a NumPy float
|
||
|
|
observed = xp.asarray(statistic(*data, axis=-1))
|
||
|
|
observed = observed[()] if observed.ndim == 0 else observed
|
||
|
|
|
||
|
|
n_observations = [sample.shape[-1] for sample in data]
|
||
|
|
batch_nominal = batch or n_resamples
|
||
|
|
null_distribution = []
|
||
|
|
for k in range(0, n_resamples, batch_nominal):
|
||
|
|
batch_actual = min(batch_nominal, n_resamples - k)
|
||
|
|
resamples = [rvs_i(size=(batch_actual, n_observations_i))
|
||
|
|
for rvs_i, n_observations_i in zip(rvs, n_observations)]
|
||
|
|
null_distribution.append(statistic(*resamples, axis=-1))
|
||
|
|
null_distribution = xp.concat(null_distribution)
|
||
|
|
null_distribution = xp.reshape(null_distribution, [-1] + [1]*observed.ndim)
|
||
|
|
|
||
|
|
# relative tolerance for detecting numerically distinct but
|
||
|
|
# theoretically equal values in the null distribution
|
||
|
|
eps = (0 if not xp.isdtype(observed.dtype, ('real floating'))
|
||
|
|
else xp.finfo(observed.dtype).eps*100)
|
||
|
|
gamma = xp.abs(eps * observed)
|
||
|
|
|
||
|
|
def less(null_distribution, observed):
|
||
|
|
cmps = null_distribution <= observed + gamma
|
||
|
|
cmps = xp.asarray(cmps, dtype=dtype)
|
||
|
|
pvalues = (xp.sum(cmps, axis=0, dtype=dtype) + 1.) / (n_resamples + 1.)
|
||
|
|
return pvalues
|
||
|
|
|
||
|
|
def greater(null_distribution, observed):
|
||
|
|
cmps = null_distribution >= observed - gamma
|
||
|
|
cmps = xp.asarray(cmps, dtype=dtype)
|
||
|
|
pvalues = (xp.sum(cmps, axis=0, dtype=dtype) + 1.) / (n_resamples + 1.)
|
||
|
|
return pvalues
|
||
|
|
|
||
|
|
def two_sided(null_distribution, observed):
|
||
|
|
pvalues_less = less(null_distribution, observed)
|
||
|
|
pvalues_greater = greater(null_distribution, observed)
|
||
|
|
pvalues = xp_minimum(pvalues_less, pvalues_greater) * 2
|
||
|
|
return pvalues
|
||
|
|
|
||
|
|
compare = {"less": less,
|
||
|
|
"greater": greater,
|
||
|
|
"two-sided": two_sided}
|
||
|
|
|
||
|
|
pvalues = compare[alternative](null_distribution, observed)
|
||
|
|
pvalues = xp_clip(pvalues, 0., 1., xp=xp)
|
||
|
|
|
||
|
|
return MonteCarloTestResult(observed, pvalues, null_distribution)
|
||
|
|
|
||
|
|
|
||
|
|
@dataclass
|
||
|
|
class PowerResult:
|
||
|
|
"""Result object returned by `scipy.stats.power`.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
----------
|
||
|
|
power : float or ndarray
|
||
|
|
The estimated power.
|
||
|
|
pvalues : float or ndarray
|
||
|
|
The simulated p-values.
|
||
|
|
"""
|
||
|
|
power: float | np.ndarray
|
||
|
|
pvalues: float | np.ndarray
|
||
|
|
|
||
|
|
|
||
|
|
def _wrap_kwargs(fun):
|
||
|
|
"""Wrap callable to accept arbitrary kwargs and ignore unused ones"""
|
||
|
|
|
||
|
|
try:
|
||
|
|
keys = set(inspect.signature(fun).parameters.keys())
|
||
|
|
except ValueError:
|
||
|
|
# NumPy Generator methods can't be inspected
|
||
|
|
keys = {'size'}
|
||
|
|
|
||
|
|
# Set keys=keys/fun=fun to avoid late binding gotcha
|
||
|
|
def wrapped_rvs_i(*args, keys=keys, fun=fun, **all_kwargs):
|
||
|
|
kwargs = {key: val for key, val in all_kwargs.items()
|
||
|
|
if key in keys}
|
||
|
|
return fun(*args, **kwargs)
|
||
|
|
return wrapped_rvs_i
|
||
|
|
|
||
|
|
|
||
|
|
def _power_iv(rvs, test, n_observations, significance, vectorized,
|
||
|
|
n_resamples, batch, kwargs):
|
||
|
|
"""Input validation for `monte_carlo_test`."""
|
||
|
|
|
||
|
|
if vectorized not in {True, False, None}:
|
||
|
|
raise ValueError("`vectorized` must be `True`, `False`, or `None`.")
|
||
|
|
|
||
|
|
if not isinstance(rvs, Sequence):
|
||
|
|
rvs = (rvs,)
|
||
|
|
n_observations = (n_observations,)
|
||
|
|
for rvs_i in rvs:
|
||
|
|
if not callable(rvs_i):
|
||
|
|
raise TypeError("`rvs` must be callable or sequence of callables.")
|
||
|
|
|
||
|
|
if not len(rvs) == len(n_observations):
|
||
|
|
message = ("If `rvs` is a sequence, `len(rvs)` "
|
||
|
|
"must equal `len(n_observations)`.")
|
||
|
|
raise ValueError(message)
|
||
|
|
|
||
|
|
significance = np.asarray(significance)[()]
|
||
|
|
if (not np.issubdtype(significance.dtype, np.floating)
|
||
|
|
or np.min(significance) < 0 or np.max(significance) > 1):
|
||
|
|
raise ValueError("`significance` must contain floats between 0 and 1.")
|
||
|
|
|
||
|
|
kwargs = dict() if kwargs is None else kwargs
|
||
|
|
if not isinstance(kwargs, dict):
|
||
|
|
raise TypeError("`kwargs` must be a dictionary that maps keywords to arrays.")
|
||
|
|
|
||
|
|
vals = kwargs.values()
|
||
|
|
keys = kwargs.keys()
|
||
|
|
|
||
|
|
# Wrap callables to ignore unused keyword arguments
|
||
|
|
wrapped_rvs = [_wrap_kwargs(rvs_i) for rvs_i in rvs]
|
||
|
|
|
||
|
|
# Broadcast, then ravel nobs/kwarg combinations. In the end,
|
||
|
|
# `nobs` and `vals` have shape (# of combinations, number of variables)
|
||
|
|
tmp = np.asarray(np.broadcast_arrays(*n_observations, *vals))
|
||
|
|
shape = tmp.shape
|
||
|
|
if tmp.ndim == 1:
|
||
|
|
tmp = tmp[np.newaxis, :]
|
||
|
|
else:
|
||
|
|
tmp = tmp.reshape((shape[0], -1)).T
|
||
|
|
nobs, vals = tmp[:, :len(rvs)], tmp[:, len(rvs):]
|
||
|
|
nobs = nobs.astype(int)
|
||
|
|
|
||
|
|
if not callable(test):
|
||
|
|
raise TypeError("`test` must be callable.")
|
||
|
|
|
||
|
|
if vectorized is None:
|
||
|
|
vectorized = 'axis' in inspect.signature(test).parameters
|
||
|
|
|
||
|
|
if not vectorized:
|
||
|
|
test_vectorized = _vectorize_statistic(test)
|
||
|
|
else:
|
||
|
|
test_vectorized = test
|
||
|
|
# Wrap `test` function to ignore unused kwargs
|
||
|
|
test_vectorized = _wrap_kwargs(test_vectorized)
|
||
|
|
|
||
|
|
n_resamples_int = int(n_resamples)
|
||
|
|
if n_resamples != n_resamples_int or n_resamples_int <= 0:
|
||
|
|
raise ValueError("`n_resamples` must be a positive integer.")
|
||
|
|
|
||
|
|
if batch is None:
|
||
|
|
batch_iv = batch
|
||
|
|
else:
|
||
|
|
batch_iv = int(batch)
|
||
|
|
if batch != batch_iv or batch_iv <= 0:
|
||
|
|
raise ValueError("`batch` must be a positive integer or None.")
|
||
|
|
|
||
|
|
return (wrapped_rvs, test_vectorized, nobs, significance, vectorized,
|
||
|
|
n_resamples_int, batch_iv, vals, keys, shape[1:])
|
||
|
|
|
||
|
|
|
||
|
|
def power(test, rvs, n_observations, *, significance=0.01, vectorized=None,
|
||
|
|
n_resamples=10000, batch=None, kwargs=None):
|
||
|
|
r"""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.
|
||
|
|
|
||
|
|
"""
|
||
|
|
tmp = _power_iv(rvs, test, n_observations, significance,
|
||
|
|
vectorized, n_resamples, batch, kwargs)
|
||
|
|
(rvs, test, nobs, significance,
|
||
|
|
vectorized, n_resamples, batch, args, kwds, shape)= tmp
|
||
|
|
|
||
|
|
batch_nominal = batch or n_resamples
|
||
|
|
pvalues = [] # results of various nobs/kwargs combinations
|
||
|
|
for nobs_i, args_i in zip(nobs, args):
|
||
|
|
kwargs_i = dict(zip(kwds, args_i))
|
||
|
|
pvalues_i = [] # results of batches; fixed nobs/kwargs combination
|
||
|
|
for k in range(0, n_resamples, batch_nominal):
|
||
|
|
batch_actual = min(batch_nominal, n_resamples - k)
|
||
|
|
resamples = [rvs_j(size=(batch_actual, nobs_ij), **kwargs_i)
|
||
|
|
for rvs_j, nobs_ij in zip(rvs, nobs_i)]
|
||
|
|
res = test(*resamples, **kwargs_i, axis=-1)
|
||
|
|
p = getattr(res, 'pvalue', res)
|
||
|
|
pvalues_i.append(p)
|
||
|
|
# Concatenate results from batches
|
||
|
|
pvalues_i = np.concatenate(pvalues_i, axis=-1)
|
||
|
|
pvalues.append(pvalues_i)
|
||
|
|
# `test` can return result with array of p-values
|
||
|
|
shape += pvalues_i.shape[:-1]
|
||
|
|
# Concatenate results from various nobs/kwargs combinations
|
||
|
|
pvalues = np.concatenate(pvalues, axis=0)
|
||
|
|
# nobs/kwargs arrays were raveled to single axis; unravel
|
||
|
|
pvalues = pvalues.reshape(shape + (-1,))
|
||
|
|
if significance.ndim > 0:
|
||
|
|
newdims = tuple(range(significance.ndim, pvalues.ndim + significance.ndim))
|
||
|
|
significance = np.expand_dims(significance, newdims)
|
||
|
|
powers = np.mean(pvalues < significance, axis=-1)
|
||
|
|
|
||
|
|
return PowerResult(power=powers, pvalues=pvalues)
|
||
|
|
|
||
|
|
|
||
|
|
@dataclass
|
||
|
|
class PermutationTestResult:
|
||
|
|
"""Result object returned by `scipy.stats.permutation_test`.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
----------
|
||
|
|
statistic : float or ndarray
|
||
|
|
The observed test statistic of the data.
|
||
|
|
pvalue : float or ndarray
|
||
|
|
The p-value for the given alternative.
|
||
|
|
null_distribution : ndarray
|
||
|
|
The values of the test statistic generated under the null
|
||
|
|
hypothesis.
|
||
|
|
"""
|
||
|
|
statistic: float | np.ndarray
|
||
|
|
pvalue: float | np.ndarray
|
||
|
|
null_distribution: np.ndarray
|
||
|
|
|
||
|
|
|
||
|
|
def _all_partitions_concatenated(ns):
|
||
|
|
"""
|
||
|
|
Generate all partitions of indices of groups of given sizes, concatenated
|
||
|
|
|
||
|
|
`ns` is an iterable of ints.
|
||
|
|
"""
|
||
|
|
def all_partitions(z, n):
|
||
|
|
for c in combinations(z, n):
|
||
|
|
x0 = set(c)
|
||
|
|
x1 = z - x0
|
||
|
|
yield [x0, x1]
|
||
|
|
|
||
|
|
def all_partitions_n(z, ns):
|
||
|
|
if len(ns) == 0:
|
||
|
|
yield [z]
|
||
|
|
return
|
||
|
|
for c in all_partitions(z, ns[0]):
|
||
|
|
for d in all_partitions_n(c[1], ns[1:]):
|
||
|
|
yield c[0:1] + d
|
||
|
|
|
||
|
|
z = set(range(np.sum(ns)))
|
||
|
|
for partitioning in all_partitions_n(z, ns[:]):
|
||
|
|
x = np.concatenate([list(partition)
|
||
|
|
for partition in partitioning]).astype(int)
|
||
|
|
yield x
|
||
|
|
|
||
|
|
|
||
|
|
def _batch_generator(iterable, batch):
|
||
|
|
"""A generator that yields batches of elements from an iterable"""
|
||
|
|
iterator = iter(iterable)
|
||
|
|
if batch <= 0:
|
||
|
|
raise ValueError("`batch` must be positive.")
|
||
|
|
z = [item for i, item in zip(range(batch), iterator)]
|
||
|
|
while z: # we don't want StopIteration without yielding an empty list
|
||
|
|
yield z
|
||
|
|
z = [item for i, item in zip(range(batch), iterator)]
|
||
|
|
|
||
|
|
|
||
|
|
def _pairings_permutations_gen(n_permutations, n_samples, n_obs_sample, batch,
|
||
|
|
random_state):
|
||
|
|
# Returns a generator that yields arrays of size
|
||
|
|
# `(batch, n_samples, n_obs_sample)`.
|
||
|
|
# Each row is an independent permutation of indices 0 to `n_obs_sample`.
|
||
|
|
batch = min(batch, n_permutations)
|
||
|
|
|
||
|
|
if hasattr(random_state, 'permuted'):
|
||
|
|
def batched_perm_generator():
|
||
|
|
indices = np.arange(n_obs_sample)
|
||
|
|
indices = np.tile(indices, (batch, n_samples, 1))
|
||
|
|
for k in range(0, n_permutations, batch):
|
||
|
|
batch_actual = min(batch, n_permutations-k)
|
||
|
|
# Don't permute in place, otherwise results depend on `batch`
|
||
|
|
permuted_indices = random_state.permuted(indices, axis=-1)
|
||
|
|
yield permuted_indices[:batch_actual]
|
||
|
|
else: # RandomState and early Generators don't have `permuted`
|
||
|
|
def batched_perm_generator():
|
||
|
|
for k in range(0, n_permutations, batch):
|
||
|
|
batch_actual = min(batch, n_permutations-k)
|
||
|
|
size = (batch_actual, n_samples, n_obs_sample)
|
||
|
|
x = random_state.random(size=size)
|
||
|
|
yield np.argsort(x, axis=-1)[:batch_actual]
|
||
|
|
|
||
|
|
return batched_perm_generator()
|
||
|
|
|
||
|
|
|
||
|
|
def _calculate_null_both(data, statistic, n_permutations, batch,
|
||
|
|
random_state=None):
|
||
|
|
"""
|
||
|
|
Calculate null distribution for independent sample tests.
|
||
|
|
"""
|
||
|
|
n_samples = len(data)
|
||
|
|
|
||
|
|
# compute number of permutations
|
||
|
|
# (distinct partitions of data into samples of these sizes)
|
||
|
|
n_obs_i = [sample.shape[-1] for sample in data] # observations per sample
|
||
|
|
n_obs_ic = np.cumsum(n_obs_i)
|
||
|
|
n_obs = n_obs_ic[-1] # total number of observations
|
||
|
|
n_max = np.prod([comb(n_obs_ic[i], n_obs_ic[i-1])
|
||
|
|
for i in range(n_samples-1, 0, -1)])
|
||
|
|
|
||
|
|
# perm_generator is an iterator that produces permutations of indices
|
||
|
|
# from 0 to n_obs. We'll concatenate the samples, use these indices to
|
||
|
|
# permute the data, then split the samples apart again.
|
||
|
|
if n_permutations >= n_max:
|
||
|
|
exact_test = True
|
||
|
|
n_permutations = n_max
|
||
|
|
perm_generator = _all_partitions_concatenated(n_obs_i)
|
||
|
|
else:
|
||
|
|
exact_test = False
|
||
|
|
# Neither RandomState.permutation nor Generator.permutation
|
||
|
|
# can permute axis-slices independently. If this feature is
|
||
|
|
# added in the future, batches of the desired size should be
|
||
|
|
# generated in a single call.
|
||
|
|
perm_generator = (random_state.permutation(n_obs)
|
||
|
|
for i in range(n_permutations))
|
||
|
|
|
||
|
|
batch = batch or int(n_permutations)
|
||
|
|
null_distribution = []
|
||
|
|
|
||
|
|
# First, concatenate all the samples. In batches, permute samples with
|
||
|
|
# indices produced by the `perm_generator`, split them into new samples of
|
||
|
|
# the original sizes, compute the statistic for each batch, and add these
|
||
|
|
# statistic values to the null distribution.
|
||
|
|
data = np.concatenate(data, axis=-1)
|
||
|
|
for indices in _batch_generator(perm_generator, batch=batch):
|
||
|
|
indices = np.array(indices)
|
||
|
|
|
||
|
|
# `indices` is 2D: each row is a permutation of the indices.
|
||
|
|
# We use it to index `data` along its last axis, which corresponds
|
||
|
|
# with observations.
|
||
|
|
# After indexing, the second to last axis of `data_batch` corresponds
|
||
|
|
# with permutations, and the last axis corresponds with observations.
|
||
|
|
data_batch = data[..., indices]
|
||
|
|
|
||
|
|
# Move the permutation axis to the front: we'll concatenate a list
|
||
|
|
# of batched statistic values along this zeroth axis to form the
|
||
|
|
# null distribution.
|
||
|
|
data_batch = np.moveaxis(data_batch, -2, 0)
|
||
|
|
data_batch = np.split(data_batch, n_obs_ic[:-1], axis=-1)
|
||
|
|
null_distribution.append(statistic(*data_batch, axis=-1))
|
||
|
|
null_distribution = np.concatenate(null_distribution, axis=0)
|
||
|
|
|
||
|
|
return null_distribution, n_permutations, exact_test
|
||
|
|
|
||
|
|
|
||
|
|
def _calculate_null_pairings(data, statistic, n_permutations, batch,
|
||
|
|
random_state=None):
|
||
|
|
"""
|
||
|
|
Calculate null distribution for association tests.
|
||
|
|
"""
|
||
|
|
n_samples = len(data)
|
||
|
|
|
||
|
|
# compute number of permutations (factorial(n) permutations of each sample)
|
||
|
|
n_obs_sample = data[0].shape[-1] # observations per sample; same for each
|
||
|
|
n_max = factorial(n_obs_sample)**n_samples
|
||
|
|
|
||
|
|
# `perm_generator` is an iterator that produces a list of permutations of
|
||
|
|
# indices from 0 to n_obs_sample, one for each sample.
|
||
|
|
if n_permutations >= n_max:
|
||
|
|
exact_test = True
|
||
|
|
n_permutations = n_max
|
||
|
|
batch = batch or int(n_permutations)
|
||
|
|
# cartesian product of the sets of all permutations of indices
|
||
|
|
perm_generator = product(*(permutations(range(n_obs_sample))
|
||
|
|
for i in range(n_samples)))
|
||
|
|
batched_perm_generator = _batch_generator(perm_generator, batch=batch)
|
||
|
|
else:
|
||
|
|
exact_test = False
|
||
|
|
batch = batch or int(n_permutations)
|
||
|
|
# Separate random permutations of indices for each sample.
|
||
|
|
# Again, it would be nice if RandomState/Generator.permutation
|
||
|
|
# could permute each axis-slice separately.
|
||
|
|
args = n_permutations, n_samples, n_obs_sample, batch, random_state
|
||
|
|
batched_perm_generator = _pairings_permutations_gen(*args)
|
||
|
|
|
||
|
|
null_distribution = []
|
||
|
|
|
||
|
|
for indices in batched_perm_generator:
|
||
|
|
indices = np.array(indices)
|
||
|
|
|
||
|
|
# `indices` is 3D: the zeroth axis is for permutations, the next is
|
||
|
|
# for samples, and the last is for observations. Swap the first two
|
||
|
|
# to make the zeroth axis correspond with samples, as it does for
|
||
|
|
# `data`.
|
||
|
|
indices = np.swapaxes(indices, 0, 1)
|
||
|
|
|
||
|
|
# When we're done, `data_batch` will be a list of length `n_samples`.
|
||
|
|
# Each element will be a batch of random permutations of one sample.
|
||
|
|
# The zeroth axis of each batch will correspond with permutations,
|
||
|
|
# and the last will correspond with observations. (This makes it
|
||
|
|
# easy to pass into `statistic`.)
|
||
|
|
data_batch = [None]*n_samples
|
||
|
|
for i in range(n_samples):
|
||
|
|
data_batch[i] = data[i][..., indices[i]]
|
||
|
|
data_batch[i] = np.moveaxis(data_batch[i], -2, 0)
|
||
|
|
|
||
|
|
null_distribution.append(statistic(*data_batch, axis=-1))
|
||
|
|
null_distribution = np.concatenate(null_distribution, axis=0)
|
||
|
|
|
||
|
|
return null_distribution, n_permutations, exact_test
|
||
|
|
|
||
|
|
|
||
|
|
def _calculate_null_samples(data, statistic, n_permutations, batch,
|
||
|
|
random_state=None):
|
||
|
|
"""
|
||
|
|
Calculate null distribution for paired-sample tests.
|
||
|
|
"""
|
||
|
|
n_samples = len(data)
|
||
|
|
|
||
|
|
# By convention, the meaning of the "samples" permutations type for
|
||
|
|
# data with only one sample is to flip the sign of the observations.
|
||
|
|
# Achieve this by adding a second sample - the negative of the original.
|
||
|
|
if n_samples == 1:
|
||
|
|
data = [data[0], -data[0]]
|
||
|
|
|
||
|
|
# The "samples" permutation strategy is the same as the "pairings"
|
||
|
|
# strategy except the roles of samples and observations are flipped.
|
||
|
|
# So swap these axes, then we'll use the function for the "pairings"
|
||
|
|
# strategy to do all the work!
|
||
|
|
data = np.swapaxes(data, 0, -1)
|
||
|
|
|
||
|
|
# (Of course, the user's statistic doesn't know what we've done here,
|
||
|
|
# so we need to pass it what it's expecting.)
|
||
|
|
def statistic_wrapped(*data, axis):
|
||
|
|
data = np.swapaxes(data, 0, -1)
|
||
|
|
if n_samples == 1:
|
||
|
|
data = data[0:1]
|
||
|
|
return statistic(*data, axis=axis)
|
||
|
|
|
||
|
|
return _calculate_null_pairings(data, statistic_wrapped, n_permutations,
|
||
|
|
batch, random_state)
|
||
|
|
|
||
|
|
|
||
|
|
def _permutation_test_iv(data, statistic, permutation_type, vectorized,
|
||
|
|
n_resamples, batch, alternative, axis, random_state):
|
||
|
|
"""Input validation for `permutation_test`."""
|
||
|
|
|
||
|
|
axis_int = int(axis)
|
||
|
|
if axis != axis_int:
|
||
|
|
raise ValueError("`axis` must be an integer.")
|
||
|
|
|
||
|
|
permutation_types = {'samples', 'pairings', 'independent'}
|
||
|
|
permutation_type = permutation_type.lower()
|
||
|
|
if permutation_type not in permutation_types:
|
||
|
|
raise ValueError(f"`permutation_type` must be in {permutation_types}.")
|
||
|
|
|
||
|
|
if vectorized not in {True, False, None}:
|
||
|
|
raise ValueError("`vectorized` must be `True`, `False`, or `None`.")
|
||
|
|
|
||
|
|
if vectorized is None:
|
||
|
|
vectorized = 'axis' in inspect.signature(statistic).parameters
|
||
|
|
|
||
|
|
if not vectorized:
|
||
|
|
statistic = _vectorize_statistic(statistic)
|
||
|
|
|
||
|
|
message = "`data` must be a tuple containing at least two samples"
|
||
|
|
try:
|
||
|
|
if len(data) < 2 and permutation_type == 'independent':
|
||
|
|
raise ValueError(message)
|
||
|
|
except TypeError:
|
||
|
|
raise TypeError(message)
|
||
|
|
|
||
|
|
data = _broadcast_arrays(data, axis)
|
||
|
|
data_iv = []
|
||
|
|
for sample in data:
|
||
|
|
sample = np.atleast_1d(sample)
|
||
|
|
if sample.shape[axis] <= 1:
|
||
|
|
raise ValueError("each sample in `data` must contain two or more "
|
||
|
|
"observations along `axis`.")
|
||
|
|
sample = np.moveaxis(sample, axis_int, -1)
|
||
|
|
data_iv.append(sample)
|
||
|
|
|
||
|
|
n_resamples_int = (int(n_resamples) if not np.isinf(n_resamples)
|
||
|
|
else np.inf)
|
||
|
|
if n_resamples != n_resamples_int or n_resamples_int <= 0:
|
||
|
|
raise ValueError("`n_resamples` must be a positive integer.")
|
||
|
|
|
||
|
|
if batch is None:
|
||
|
|
batch_iv = batch
|
||
|
|
else:
|
||
|
|
batch_iv = int(batch)
|
||
|
|
if batch != batch_iv or batch_iv <= 0:
|
||
|
|
raise ValueError("`batch` must be a positive integer or None.")
|
||
|
|
|
||
|
|
alternatives = {'two-sided', 'greater', 'less'}
|
||
|
|
alternative = alternative.lower()
|
||
|
|
if alternative not in alternatives:
|
||
|
|
raise ValueError(f"`alternative` must be in {alternatives}")
|
||
|
|
|
||
|
|
random_state = check_random_state(random_state)
|
||
|
|
|
||
|
|
return (data_iv, statistic, permutation_type, vectorized, n_resamples_int,
|
||
|
|
batch_iv, alternative, axis_int, random_state)
|
||
|
|
|
||
|
|
|
||
|
|
def permutation_test(data, statistic, *, permutation_type='independent',
|
||
|
|
vectorized=None, n_resamples=9999, batch=None,
|
||
|
|
alternative="two-sided", axis=0, random_state=None):
|
||
|
|
r"""
|
||
|
|
Performs a permutation test of a given statistic on provided data.
|
||
|
|
|
||
|
|
For independent sample statistics, the null hypothesis is that the data are
|
||
|
|
randomly sampled from the same distribution.
|
||
|
|
For paired sample statistics, two null hypothesis can be tested:
|
||
|
|
that the data are paired at random or that the data are assigned to samples
|
||
|
|
at random.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
data : iterable of array-like
|
||
|
|
Contains the samples, each of which is an array of observations.
|
||
|
|
Dimensions of sample arrays must be compatible for broadcasting except
|
||
|
|
along `axis`.
|
||
|
|
statistic : callable
|
||
|
|
Statistic for which the p-value of the hypothesis test is to be
|
||
|
|
calculated. `statistic` must be a callable that accepts samples
|
||
|
|
as separate arguments (e.g. ``statistic(*data)``) and returns the
|
||
|
|
resulting statistic.
|
||
|
|
If `vectorized` is set ``True``, `statistic` must also accept a keyword
|
||
|
|
argument `axis` and be vectorized to compute the statistic along the
|
||
|
|
provided `axis` of the sample arrays.
|
||
|
|
permutation_type : {'independent', 'samples', 'pairings'}, optional
|
||
|
|
The type of permutations to be performed, in accordance with the
|
||
|
|
null hypothesis. The first two permutation types are for paired sample
|
||
|
|
statistics, in which all samples contain the same number of
|
||
|
|
observations and observations with corresponding indices along `axis`
|
||
|
|
are considered to be paired; the third is for independent sample
|
||
|
|
statistics.
|
||
|
|
|
||
|
|
- ``'samples'`` : observations are assigned to different samples
|
||
|
|
but remain paired with the same observations from other samples.
|
||
|
|
This permutation type is appropriate for paired sample hypothesis
|
||
|
|
tests such as the Wilcoxon signed-rank test and the paired t-test.
|
||
|
|
- ``'pairings'`` : observations are paired with different observations,
|
||
|
|
but they remain within the same sample. This permutation type is
|
||
|
|
appropriate for association/correlation tests with statistics such
|
||
|
|
as Spearman's :math:`\rho`, Kendall's :math:`\tau`, and Pearson's
|
||
|
|
:math:`r`.
|
||
|
|
- ``'independent'`` (default) : observations are assigned to different
|
||
|
|
samples. Samples may contain different numbers of observations. This
|
||
|
|
permutation type is appropriate for independent sample hypothesis
|
||
|
|
tests such as the Mann-Whitney :math:`U` test and the independent
|
||
|
|
sample t-test.
|
||
|
|
|
||
|
|
Please see the Notes section below for more detailed descriptions
|
||
|
|
of the permutation types.
|
||
|
|
|
||
|
|
vectorized : bool, optional
|
||
|
|
If `vectorized` is set ``False``, `statistic` will not be passed
|
||
|
|
keyword argument `axis` and is expected to calculate the statistic
|
||
|
|
only for 1D samples. If ``True``, `statistic` will be passed keyword
|
||
|
|
argument `axis` and is expected to calculate the statistic along `axis`
|
||
|
|
when passed an ND sample array. If ``None`` (default), `vectorized`
|
||
|
|
will be set ``True`` if ``axis`` is a parameter of `statistic`. Use
|
||
|
|
of a vectorized statistic typically reduces computation time.
|
||
|
|
n_resamples : int or np.inf, default: 9999
|
||
|
|
Number of random permutations (resamples) used to approximate the null
|
||
|
|
distribution. If greater than or equal to the number of distinct
|
||
|
|
permutations, the exact null distribution will be computed.
|
||
|
|
Note that the number of distinct permutations grows very rapidly with
|
||
|
|
the sizes of samples, so exact tests are feasible only for very small
|
||
|
|
data sets.
|
||
|
|
batch : int, optional
|
||
|
|
The number of permutations to process in each call to `statistic`.
|
||
|
|
Memory usage is O( `batch` * ``n`` ), where ``n`` is the total size
|
||
|
|
of all samples, regardless of the value of `vectorized`. Default is
|
||
|
|
``None``, in which case ``batch`` is the number of permutations.
|
||
|
|
alternative : {'two-sided', 'less', 'greater'}, optional
|
||
|
|
The alternative hypothesis for which the p-value is calculated.
|
||
|
|
For each alternative, the p-value is defined for exact tests as
|
||
|
|
follows.
|
||
|
|
|
||
|
|
- ``'greater'`` : the percentage of the null distribution that is
|
||
|
|
greater than or equal to the observed value of the test statistic.
|
||
|
|
- ``'less'`` : the percentage of the null distribution that is
|
||
|
|
less than or equal to the observed value of the test statistic.
|
||
|
|
- ``'two-sided'`` (default) : twice the smaller of the p-values above.
|
||
|
|
|
||
|
|
Note that p-values for randomized tests are calculated according to the
|
||
|
|
conservative (over-estimated) approximation suggested in [2]_ and [3]_
|
||
|
|
rather than the unbiased estimator suggested in [4]_. That is, when
|
||
|
|
calculating the proportion of the randomized null distribution that is
|
||
|
|
as extreme as the observed value of the test statistic, the values in
|
||
|
|
the numerator and denominator are both increased by one. An
|
||
|
|
interpretation of this adjustment is that the observed value of the
|
||
|
|
test statistic is always included as an element of the randomized
|
||
|
|
null distribution.
|
||
|
|
The convention used for two-sided p-values is not universal;
|
||
|
|
the observed test statistic and null distribution are returned in
|
||
|
|
case a different definition is preferred.
|
||
|
|
|
||
|
|
axis : int, default: 0
|
||
|
|
The axis of the (broadcasted) samples over which to calculate the
|
||
|
|
statistic. If samples have a different number of dimensions,
|
||
|
|
singleton dimensions are prepended to samples with fewer dimensions
|
||
|
|
before `axis` is considered.
|
||
|
|
random_state : {None, int, `numpy.random.Generator`,
|
||
|
|
`numpy.random.RandomState`}, optional
|
||
|
|
|
||
|
|
Pseudorandom number generator state used to generate permutations.
|
||
|
|
|
||
|
|
If `random_state` is ``None`` (default), the
|
||
|
|
`numpy.random.RandomState` singleton is used.
|
||
|
|
If `random_state` is an int, a new ``RandomState`` instance is used,
|
||
|
|
seeded with `random_state`.
|
||
|
|
If `random_state` is already a ``Generator`` or ``RandomState``
|
||
|
|
instance then that instance is used.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
res : PermutationTestResult
|
||
|
|
An object with attributes:
|
||
|
|
|
||
|
|
statistic : float or ndarray
|
||
|
|
The observed test statistic of the data.
|
||
|
|
pvalue : float or ndarray
|
||
|
|
The p-value for the given alternative.
|
||
|
|
null_distribution : ndarray
|
||
|
|
The values of the test statistic generated under the null
|
||
|
|
hypothesis.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
|
||
|
|
The three types of permutation tests supported by this function are
|
||
|
|
described below.
|
||
|
|
|
||
|
|
**Unpaired statistics** (``permutation_type='independent'``):
|
||
|
|
|
||
|
|
The null hypothesis associated with this permutation type is that all
|
||
|
|
observations are sampled from the same underlying distribution and that
|
||
|
|
they have been assigned to one of the samples at random.
|
||
|
|
|
||
|
|
Suppose ``data`` contains two samples; e.g. ``a, b = data``.
|
||
|
|
When ``1 < n_resamples < binom(n, k)``, where
|
||
|
|
|
||
|
|
* ``k`` is the number of observations in ``a``,
|
||
|
|
* ``n`` is the total number of observations in ``a`` and ``b``, and
|
||
|
|
* ``binom(n, k)`` is the binomial coefficient (``n`` choose ``k``),
|
||
|
|
|
||
|
|
the data are pooled (concatenated), randomly assigned to either the first
|
||
|
|
or second sample, and the statistic is calculated. This process is
|
||
|
|
performed repeatedly, `permutation` times, generating a distribution of the
|
||
|
|
statistic under the null hypothesis. The statistic of the original
|
||
|
|
data is compared to this distribution to determine the p-value.
|
||
|
|
|
||
|
|
When ``n_resamples >= binom(n, k)``, an exact test is performed: the data
|
||
|
|
are *partitioned* between the samples in each distinct way exactly once,
|
||
|
|
and the exact null distribution is formed.
|
||
|
|
Note that for a given partitioning of the data between the samples,
|
||
|
|
only one ordering/permutation of the data *within* each sample is
|
||
|
|
considered. For statistics that do not depend on the order of the data
|
||
|
|
within samples, this dramatically reduces computational cost without
|
||
|
|
affecting the shape of the null distribution (because the frequency/count
|
||
|
|
of each value is affected by the same factor).
|
||
|
|
|
||
|
|
For ``a = [a1, a2, a3, a4]`` and ``b = [b1, b2, b3]``, an example of this
|
||
|
|
permutation type is ``x = [b3, a1, a2, b2]`` and ``y = [a4, b1, a3]``.
|
||
|
|
Because only one ordering/permutation of the data *within* each sample
|
||
|
|
is considered in an exact test, a resampling like ``x = [b3, a1, b2, a2]``
|
||
|
|
and ``y = [a4, a3, b1]`` would *not* be considered distinct from the
|
||
|
|
example above.
|
||
|
|
|
||
|
|
``permutation_type='independent'`` does not support one-sample statistics,
|
||
|
|
but it can be applied to statistics with more than two samples. In this
|
||
|
|
case, if ``n`` is an array of the number of observations within each
|
||
|
|
sample, the number of distinct partitions is::
|
||
|
|
|
||
|
|
np.prod([binom(sum(n[i:]), sum(n[i+1:])) for i in range(len(n)-1)])
|
||
|
|
|
||
|
|
**Paired statistics, permute pairings** (``permutation_type='pairings'``):
|
||
|
|
|
||
|
|
The null hypothesis associated with this permutation type is that
|
||
|
|
observations within each sample are drawn from the same underlying
|
||
|
|
distribution and that pairings with elements of other samples are
|
||
|
|
assigned at random.
|
||
|
|
|
||
|
|
Suppose ``data`` contains only one sample; e.g. ``a, = data``, and we
|
||
|
|
wish to consider all possible pairings of elements of ``a`` with elements
|
||
|
|
of a second sample, ``b``. Let ``n`` be the number of observations in
|
||
|
|
``a``, which must also equal the number of observations in ``b``.
|
||
|
|
|
||
|
|
When ``1 < n_resamples < factorial(n)``, the elements of ``a`` are
|
||
|
|
randomly permuted. The user-supplied statistic accepts one data argument,
|
||
|
|
say ``a_perm``, and calculates the statistic considering ``a_perm`` and
|
||
|
|
``b``. This process is performed repeatedly, `permutation` times,
|
||
|
|
generating a distribution of the statistic under the null hypothesis.
|
||
|
|
The statistic of the original data is compared to this distribution to
|
||
|
|
determine the p-value.
|
||
|
|
|
||
|
|
When ``n_resamples >= factorial(n)``, an exact test is performed:
|
||
|
|
``a`` is permuted in each distinct way exactly once. Therefore, the
|
||
|
|
`statistic` is computed for each unique pairing of samples between ``a``
|
||
|
|
and ``b`` exactly once.
|
||
|
|
|
||
|
|
For ``a = [a1, a2, a3]`` and ``b = [b1, b2, b3]``, an example of this
|
||
|
|
permutation type is ``a_perm = [a3, a1, a2]`` while ``b`` is left
|
||
|
|
in its original order.
|
||
|
|
|
||
|
|
``permutation_type='pairings'`` supports ``data`` containing any number
|
||
|
|
of samples, each of which must contain the same number of observations.
|
||
|
|
All samples provided in ``data`` are permuted *independently*. Therefore,
|
||
|
|
if ``m`` is the number of samples and ``n`` is the number of observations
|
||
|
|
within each sample, then the number of permutations in an exact test is::
|
||
|
|
|
||
|
|
factorial(n)**m
|
||
|
|
|
||
|
|
Note that if a two-sample statistic, for example, does not inherently
|
||
|
|
depend on the order in which observations are provided - only on the
|
||
|
|
*pairings* of observations - then only one of the two samples should be
|
||
|
|
provided in ``data``. This dramatically reduces computational cost without
|
||
|
|
affecting the shape of the null distribution (because the frequency/count
|
||
|
|
of each value is affected by the same factor).
|
||
|
|
|
||
|
|
**Paired statistics, permute samples** (``permutation_type='samples'``):
|
||
|
|
|
||
|
|
The null hypothesis associated with this permutation type is that
|
||
|
|
observations within each pair are drawn from the same underlying
|
||
|
|
distribution and that the sample to which they are assigned is random.
|
||
|
|
|
||
|
|
Suppose ``data`` contains two samples; e.g. ``a, b = data``.
|
||
|
|
Let ``n`` be the number of observations in ``a``, which must also equal
|
||
|
|
the number of observations in ``b``.
|
||
|
|
|
||
|
|
When ``1 < n_resamples < 2**n``, the elements of ``a`` are ``b`` are
|
||
|
|
randomly swapped between samples (maintaining their pairings) and the
|
||
|
|
statistic is calculated. This process is performed repeatedly,
|
||
|
|
`permutation` times, generating a distribution of the statistic under the
|
||
|
|
null hypothesis. The statistic of the original data is compared to this
|
||
|
|
distribution to determine the p-value.
|
||
|
|
|
||
|
|
When ``n_resamples >= 2**n``, an exact test is performed: the observations
|
||
|
|
are assigned to the two samples in each distinct way (while maintaining
|
||
|
|
pairings) exactly once.
|
||
|
|
|
||
|
|
For ``a = [a1, a2, a3]`` and ``b = [b1, b2, b3]``, an example of this
|
||
|
|
permutation type is ``x = [b1, a2, b3]`` and ``y = [a1, b2, a3]``.
|
||
|
|
|
||
|
|
``permutation_type='samples'`` supports ``data`` containing any number
|
||
|
|
of samples, each of which must contain the same number of observations.
|
||
|
|
If ``data`` contains more than one sample, paired observations within
|
||
|
|
``data`` are exchanged between samples *independently*. Therefore, if ``m``
|
||
|
|
is the number of samples and ``n`` is the number of observations within
|
||
|
|
each sample, then the number of permutations in an exact test is::
|
||
|
|
|
||
|
|
factorial(m)**n
|
||
|
|
|
||
|
|
Several paired-sample statistical tests, such as the Wilcoxon signed rank
|
||
|
|
test and paired-sample t-test, can be performed considering only the
|
||
|
|
*difference* between two paired elements. Accordingly, if ``data`` contains
|
||
|
|
only one sample, then the null distribution is formed by independently
|
||
|
|
changing the *sign* of each observation.
|
||
|
|
|
||
|
|
.. warning::
|
||
|
|
The p-value is calculated by counting the elements of the null
|
||
|
|
distribution that are as extreme or more extreme than the observed
|
||
|
|
value of the statistic. Due to the use of finite precision arithmetic,
|
||
|
|
some statistic functions return numerically distinct values when the
|
||
|
|
theoretical values would be exactly equal. In some cases, this could
|
||
|
|
lead to a large error in the calculated p-value. `permutation_test`
|
||
|
|
guards against this by considering elements in the null distribution
|
||
|
|
that are "close" (within a relative tolerance of 100 times the
|
||
|
|
floating point epsilon of inexact dtypes) to the observed
|
||
|
|
value of the test statistic as equal to the observed value of the
|
||
|
|
test statistic. However, the user is advised to inspect the null
|
||
|
|
distribution to assess whether this method of comparison is
|
||
|
|
appropriate, and if not, calculate the p-value manually. See example
|
||
|
|
below.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
|
||
|
|
.. [1] R. A. Fisher. The Design of Experiments, 6th Ed (1951).
|
||
|
|
.. [2] B. Phipson and G. K. Smyth. "Permutation P-values Should Never Be
|
||
|
|
Zero: Calculating Exact P-values When Permutations Are Randomly Drawn."
|
||
|
|
Statistical Applications in Genetics and Molecular Biology 9.1 (2010).
|
||
|
|
.. [3] M. D. Ernst. "Permutation Methods: A Basis for Exact Inference".
|
||
|
|
Statistical Science (2004).
|
||
|
|
.. [4] B. Efron and R. J. Tibshirani. An Introduction to the Bootstrap
|
||
|
|
(1993).
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
|
||
|
|
Suppose we wish to test whether two samples are drawn from the same
|
||
|
|
distribution. Assume that the underlying distributions are unknown to us,
|
||
|
|
and that before observing the data, we hypothesized that the mean of the
|
||
|
|
first sample would be less than that of the second sample. We decide that
|
||
|
|
we will use the difference between the sample means as a test statistic,
|
||
|
|
and we will consider a p-value of 0.05 to be statistically significant.
|
||
|
|
|
||
|
|
For efficiency, we write the function defining the test statistic in a
|
||
|
|
vectorized fashion: the samples ``x`` and ``y`` can be ND arrays, and the
|
||
|
|
statistic will be calculated for each axis-slice along `axis`.
|
||
|
|
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> def statistic(x, y, axis):
|
||
|
|
... return np.mean(x, axis=axis) - np.mean(y, axis=axis)
|
||
|
|
|
||
|
|
After collecting our data, we calculate the observed value of the test
|
||
|
|
statistic.
|
||
|
|
|
||
|
|
>>> from scipy.stats import norm
|
||
|
|
>>> rng = np.random.default_rng()
|
||
|
|
>>> x = norm.rvs(size=5, random_state=rng)
|
||
|
|
>>> y = norm.rvs(size=6, loc = 3, random_state=rng)
|
||
|
|
>>> statistic(x, y, 0)
|
||
|
|
-3.5411688580987266
|
||
|
|
|
||
|
|
Indeed, the test statistic is negative, suggesting that the true mean of
|
||
|
|
the distribution underlying ``x`` is less than that of the distribution
|
||
|
|
underlying ``y``. To determine the probability of this occurring by chance
|
||
|
|
if the two samples were drawn from the same distribution, we perform
|
||
|
|
a permutation test.
|
||
|
|
|
||
|
|
>>> from scipy.stats import permutation_test
|
||
|
|
>>> # because our statistic is vectorized, we pass `vectorized=True`
|
||
|
|
>>> # `n_resamples=np.inf` indicates that an exact test is to be performed
|
||
|
|
>>> res = permutation_test((x, y), statistic, vectorized=True,
|
||
|
|
... n_resamples=np.inf, alternative='less')
|
||
|
|
>>> print(res.statistic)
|
||
|
|
-3.5411688580987266
|
||
|
|
>>> print(res.pvalue)
|
||
|
|
0.004329004329004329
|
||
|
|
|
||
|
|
The probability of obtaining a test statistic less than or equal to the
|
||
|
|
observed value under the null hypothesis is 0.4329%. This is less than our
|
||
|
|
chosen threshold of 5%, so we consider this to be significant evidence
|
||
|
|
against the null hypothesis in favor of the alternative.
|
||
|
|
|
||
|
|
Because the size of the samples above was small, `permutation_test` could
|
||
|
|
perform an exact test. For larger samples, we resort to a randomized
|
||
|
|
permutation test.
|
||
|
|
|
||
|
|
>>> x = norm.rvs(size=100, random_state=rng)
|
||
|
|
>>> y = norm.rvs(size=120, loc=0.2, random_state=rng)
|
||
|
|
>>> res = permutation_test((x, y), statistic, n_resamples=9999,
|
||
|
|
... vectorized=True, alternative='less',
|
||
|
|
... random_state=rng)
|
||
|
|
>>> print(res.statistic)
|
||
|
|
-0.4230459671240913
|
||
|
|
>>> print(res.pvalue)
|
||
|
|
0.0015
|
||
|
|
|
||
|
|
The approximate probability of obtaining a test statistic less than or
|
||
|
|
equal to the observed value under the null hypothesis is 0.0225%. This is
|
||
|
|
again less than our chosen threshold of 5%, so again we have significant
|
||
|
|
evidence to reject the null hypothesis in favor of the alternative.
|
||
|
|
|
||
|
|
For large samples and number of permutations, the result is comparable to
|
||
|
|
that of the corresponding asymptotic test, the independent sample t-test.
|
||
|
|
|
||
|
|
>>> from scipy.stats import ttest_ind
|
||
|
|
>>> res_asymptotic = ttest_ind(x, y, alternative='less')
|
||
|
|
>>> print(res_asymptotic.pvalue)
|
||
|
|
0.0014669545224902675
|
||
|
|
|
||
|
|
The permutation distribution of the test statistic is provided for
|
||
|
|
further investigation.
|
||
|
|
|
||
|
|
>>> import matplotlib.pyplot as plt
|
||
|
|
>>> plt.hist(res.null_distribution, bins=50)
|
||
|
|
>>> plt.title("Permutation distribution of test statistic")
|
||
|
|
>>> plt.xlabel("Value of Statistic")
|
||
|
|
>>> plt.ylabel("Frequency")
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
Inspection of the null distribution is essential if the statistic suffers
|
||
|
|
from inaccuracy due to limited machine precision. Consider the following
|
||
|
|
case:
|
||
|
|
|
||
|
|
>>> from scipy.stats import pearsonr
|
||
|
|
>>> x = [1, 2, 4, 3]
|
||
|
|
>>> y = [2, 4, 6, 8]
|
||
|
|
>>> def statistic(x, y, axis=-1):
|
||
|
|
... return pearsonr(x, y, axis=axis).statistic
|
||
|
|
>>> res = permutation_test((x, y), statistic, vectorized=True,
|
||
|
|
... permutation_type='pairings',
|
||
|
|
... alternative='greater')
|
||
|
|
>>> r, pvalue, null = res.statistic, res.pvalue, res.null_distribution
|
||
|
|
|
||
|
|
In this case, some elements of the null distribution differ from the
|
||
|
|
observed value of the correlation coefficient ``r`` due to numerical noise.
|
||
|
|
We manually inspect the elements of the null distribution that are nearly
|
||
|
|
the same as the observed value of the test statistic.
|
||
|
|
|
||
|
|
>>> r
|
||
|
|
0.7999999999999999
|
||
|
|
>>> unique = np.unique(null)
|
||
|
|
>>> unique
|
||
|
|
array([-1. , -1. , -0.8, -0.8, -0.8, -0.6, -0.4, -0.4, -0.2, -0.2, -0.2,
|
||
|
|
0. , 0.2, 0.2, 0.2, 0.4, 0.4, 0.6, 0.8, 0.8, 0.8, 1. ,
|
||
|
|
1. ]) # may vary
|
||
|
|
>>> unique[np.isclose(r, unique)].tolist()
|
||
|
|
[0.7999999999999998, 0.7999999999999999, 0.8] # may vary
|
||
|
|
|
||
|
|
If `permutation_test` were to perform the comparison naively, the
|
||
|
|
elements of the null distribution with value ``0.7999999999999998`` would
|
||
|
|
not be considered as extreme or more extreme as the observed value of the
|
||
|
|
statistic, so the calculated p-value would be too small.
|
||
|
|
|
||
|
|
>>> incorrect_pvalue = np.count_nonzero(null >= r) / len(null)
|
||
|
|
>>> incorrect_pvalue
|
||
|
|
0.14583333333333334 # may vary
|
||
|
|
|
||
|
|
Instead, `permutation_test` treats elements of the null distribution that
|
||
|
|
are within ``max(1e-14, abs(r)*1e-14)`` of the observed value of the
|
||
|
|
statistic ``r`` to be equal to ``r``.
|
||
|
|
|
||
|
|
>>> correct_pvalue = np.count_nonzero(null >= r - 1e-14) / len(null)
|
||
|
|
>>> correct_pvalue
|
||
|
|
0.16666666666666666
|
||
|
|
>>> res.pvalue == correct_pvalue
|
||
|
|
True
|
||
|
|
|
||
|
|
This method of comparison is expected to be accurate in most practical
|
||
|
|
situations, but the user is advised to assess this by inspecting the
|
||
|
|
elements of the null distribution that are close to the observed value
|
||
|
|
of the statistic. Also, consider the use of statistics that can be
|
||
|
|
calculated using exact arithmetic (e.g. integer statistics).
|
||
|
|
|
||
|
|
"""
|
||
|
|
args = _permutation_test_iv(data, statistic, permutation_type, vectorized,
|
||
|
|
n_resamples, batch, alternative, axis,
|
||
|
|
random_state)
|
||
|
|
(data, statistic, permutation_type, vectorized, n_resamples, batch,
|
||
|
|
alternative, axis, random_state) = args
|
||
|
|
|
||
|
|
observed = statistic(*data, axis=-1)
|
||
|
|
|
||
|
|
null_calculators = {"pairings": _calculate_null_pairings,
|
||
|
|
"samples": _calculate_null_samples,
|
||
|
|
"independent": _calculate_null_both}
|
||
|
|
null_calculator_args = (data, statistic, n_resamples,
|
||
|
|
batch, random_state)
|
||
|
|
calculate_null = null_calculators[permutation_type]
|
||
|
|
null_distribution, n_resamples, exact_test = (
|
||
|
|
calculate_null(*null_calculator_args))
|
||
|
|
|
||
|
|
# See References [2] and [3]
|
||
|
|
adjustment = 0 if exact_test else 1
|
||
|
|
|
||
|
|
# relative tolerance for detecting numerically distinct but
|
||
|
|
# theoretically equal values in the null distribution
|
||
|
|
eps = (0 if not np.issubdtype(observed.dtype, np.inexact)
|
||
|
|
else np.finfo(observed.dtype).eps*100)
|
||
|
|
gamma = np.abs(eps * observed)
|
||
|
|
|
||
|
|
def less(null_distribution, observed):
|
||
|
|
cmps = null_distribution <= observed + gamma
|
||
|
|
pvalues = (cmps.sum(axis=0) + adjustment) / (n_resamples + adjustment)
|
||
|
|
return pvalues
|
||
|
|
|
||
|
|
def greater(null_distribution, observed):
|
||
|
|
cmps = null_distribution >= observed - gamma
|
||
|
|
pvalues = (cmps.sum(axis=0) + adjustment) / (n_resamples + adjustment)
|
||
|
|
return pvalues
|
||
|
|
|
||
|
|
def two_sided(null_distribution, observed):
|
||
|
|
pvalues_less = less(null_distribution, observed)
|
||
|
|
pvalues_greater = greater(null_distribution, observed)
|
||
|
|
pvalues = np.minimum(pvalues_less, pvalues_greater) * 2
|
||
|
|
return pvalues
|
||
|
|
|
||
|
|
compare = {"less": less,
|
||
|
|
"greater": greater,
|
||
|
|
"two-sided": two_sided}
|
||
|
|
|
||
|
|
pvalues = compare[alternative](null_distribution, observed)
|
||
|
|
pvalues = np.clip(pvalues, 0, 1)
|
||
|
|
|
||
|
|
return PermutationTestResult(observed, pvalues, null_distribution)
|
||
|
|
|
||
|
|
|
||
|
|
@dataclass
|
||
|
|
class ResamplingMethod:
|
||
|
|
"""Configuration information for a statistical resampling method.
|
||
|
|
|
||
|
|
Instances of this class can be passed into the `method` parameter of some
|
||
|
|
hypothesis test functions to perform a resampling or Monte Carlo version
|
||
|
|
of the hypothesis test.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
----------
|
||
|
|
n_resamples : int
|
||
|
|
The number of resamples to perform or Monte Carlo samples to draw.
|
||
|
|
batch : int, optional
|
||
|
|
The number of resamples to process in each vectorized call to
|
||
|
|
the statistic. Batch sizes >>1 tend to be faster when the statistic
|
||
|
|
is vectorized, but memory usage scales linearly with the batch size.
|
||
|
|
Default is ``None``, which processes all resamples in a single batch.
|
||
|
|
"""
|
||
|
|
n_resamples: int = 9999
|
||
|
|
batch: int = None # type: ignore[assignment]
|
||
|
|
|
||
|
|
|
||
|
|
@dataclass
|
||
|
|
class MonteCarloMethod(ResamplingMethod):
|
||
|
|
"""Configuration information for a Monte Carlo hypothesis test.
|
||
|
|
|
||
|
|
Instances of this class can be passed into the `method` parameter of some
|
||
|
|
hypothesis test functions to perform a Monte Carlo version of the
|
||
|
|
hypothesis tests.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
----------
|
||
|
|
n_resamples : int, optional
|
||
|
|
The number of Monte Carlo samples to draw. Default is 9999.
|
||
|
|
batch : int, optional
|
||
|
|
The number of Monte Carlo samples to process in each vectorized call to
|
||
|
|
the statistic. Batch sizes >>1 tend to be faster when the statistic
|
||
|
|
is vectorized, but memory usage scales linearly with the batch size.
|
||
|
|
Default is ``None``, which processes all samples in a single batch.
|
||
|
|
rvs : callable or tuple of callables, optional
|
||
|
|
A callable or sequence of callables that generates random variates
|
||
|
|
under the null hypothesis. Each element of `rvs` must be a callable
|
||
|
|
that accepts keyword argument ``size`` (e.g. ``rvs(size=(m, n))``) and
|
||
|
|
returns an N-d array sample of that shape. If `rvs` is a sequence, the
|
||
|
|
number of callables in `rvs` must match the number of samples passed
|
||
|
|
to the hypothesis test in which the `MonteCarloMethod` is used. Default
|
||
|
|
is ``None``, in which case the hypothesis test function chooses values
|
||
|
|
to match the standard version of the hypothesis test. For example,
|
||
|
|
the null hypothesis of `scipy.stats.pearsonr` is typically that the
|
||
|
|
samples are drawn from the standard normal distribution, so
|
||
|
|
``rvs = (rng.normal, rng.normal)`` where
|
||
|
|
``rng = np.random.default_rng()``.
|
||
|
|
"""
|
||
|
|
rvs: object = None
|
||
|
|
|
||
|
|
def _asdict(self):
|
||
|
|
# `dataclasses.asdict` deepcopies; we don't want that.
|
||
|
|
return dict(n_resamples=self.n_resamples, batch=self.batch,
|
||
|
|
rvs=self.rvs)
|
||
|
|
|
||
|
|
|
||
|
|
@dataclass
|
||
|
|
class PermutationMethod(ResamplingMethod):
|
||
|
|
"""Configuration information for a permutation hypothesis test.
|
||
|
|
|
||
|
|
Instances of this class can be passed into the `method` parameter of some
|
||
|
|
hypothesis test functions to perform a permutation version of the
|
||
|
|
hypothesis tests.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
----------
|
||
|
|
n_resamples : int, optional
|
||
|
|
The number of resamples to perform. Default is 9999.
|
||
|
|
batch : int, optional
|
||
|
|
The number of resamples to process in each vectorized call to
|
||
|
|
the statistic. Batch sizes >>1 tend to be faster when the statistic
|
||
|
|
is vectorized, but memory usage scales linearly with the batch size.
|
||
|
|
Default is ``None``, which processes all resamples in a single batch.
|
||
|
|
random_state : {None, int, `numpy.random.Generator`,
|
||
|
|
`numpy.random.RandomState`}, optional
|
||
|
|
|
||
|
|
Pseudorandom number generator state used to generate resamples.
|
||
|
|
|
||
|
|
If `random_state` is already a ``Generator`` or ``RandomState``
|
||
|
|
instance, then that instance is used.
|
||
|
|
If `random_state` is an int, a new ``RandomState`` instance is used,
|
||
|
|
seeded with `random_state`.
|
||
|
|
If `random_state` is ``None`` (default), the
|
||
|
|
`numpy.random.RandomState` singleton is used.
|
||
|
|
"""
|
||
|
|
random_state: object = None
|
||
|
|
|
||
|
|
def _asdict(self):
|
||
|
|
# `dataclasses.asdict` deepcopies; we don't want that.
|
||
|
|
return dict(n_resamples=self.n_resamples, batch=self.batch,
|
||
|
|
random_state=self.random_state)
|
||
|
|
|
||
|
|
|
||
|
|
@dataclass
|
||
|
|
class BootstrapMethod(ResamplingMethod):
|
||
|
|
"""Configuration information for a bootstrap confidence interval.
|
||
|
|
|
||
|
|
Instances of this class can be passed into the `method` parameter of some
|
||
|
|
confidence interval methods to generate a bootstrap confidence interval.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
----------
|
||
|
|
n_resamples : int, optional
|
||
|
|
The number of resamples to perform. Default is 9999.
|
||
|
|
batch : int, optional
|
||
|
|
The number of resamples to process in each vectorized call to
|
||
|
|
the statistic. Batch sizes >>1 tend to be faster when the statistic
|
||
|
|
is vectorized, but memory usage scales linearly with the batch size.
|
||
|
|
Default is ``None``, which processes all resamples in a single batch.
|
||
|
|
random_state : {None, int, `numpy.random.Generator`,
|
||
|
|
`numpy.random.RandomState`}, optional
|
||
|
|
|
||
|
|
Pseudorandom number generator state used to generate resamples.
|
||
|
|
|
||
|
|
If `random_state` is already a ``Generator`` or ``RandomState``
|
||
|
|
instance, then that instance is used.
|
||
|
|
If `random_state` is an int, a new ``RandomState`` instance is used,
|
||
|
|
seeded with `random_state`.
|
||
|
|
If `random_state` is ``None`` (default), the
|
||
|
|
`numpy.random.RandomState` singleton is used.
|
||
|
|
|
||
|
|
method : {'bca', 'percentile', 'basic'}
|
||
|
|
Whether to use the 'percentile' bootstrap ('percentile'), the 'basic'
|
||
|
|
(AKA 'reverse') bootstrap ('basic'), or the bias-corrected and
|
||
|
|
accelerated bootstrap ('BCa', default).
|
||
|
|
"""
|
||
|
|
random_state: object = None
|
||
|
|
method: str = 'BCa'
|
||
|
|
|
||
|
|
def _asdict(self):
|
||
|
|
# `dataclasses.asdict` deepcopies; we don't want that.
|
||
|
|
return dict(n_resamples=self.n_resamples, batch=self.batch,
|
||
|
|
random_state=self.random_state, method=self.method)
|