tools/auto_bisect/ttest.py - chromium/src.git - Git at Google

 # Copyright 2014 The Chromium Authors. All rights reserved.
 # Use of this source code is governed by a BSD-style license that can be
 # found in the LICENSE file.

 """Functions for doing independent two-sample t-tests and looking up p-values.

 Note: This module was copied from the Performance Dashboard code, and changed
 to use definitions of mean and variance from math_utils instead of numpy.

 > A t-test is any statistical hypothesis test in which the test statistic
 > follows a Student's t distribution if the null hypothesis is supported.
 > It can be used to determine if two sets of data are significantly different
 > from each other.

 There are several conditions that the data under test should meet in order
 for a t-test to be completely applicable:
  - The data should be roughly normal in distribution.
  - The two samples that are compared should be roughly similar in size.

 References:
   http://en.wikipedia.org/wiki/Student%27s_t-test
   http://en.wikipedia.org/wiki/Welch%27s_t-test
   https://github.com/scipy/scipy/blob/master/scipy/stats/stats.py#L3244
 """

 import math

 import math_utils


 def WelchsTTest(sample1, sample2):
   """Performs Welch's t-test on the two samples.

   Welch's t-test is an adaptation of Student's t-test which is used when the
   two samples may have unequal variances. It is also an independent two-sample
   t-test.

   Args:
     sample1: A collection of numbers.
     sample2: Another collection of numbers.

   Returns:
     A 3-tuple (t-statistic, degrees of freedom, p-value).
   """
   mean1 = math_utils.Mean(sample1)
   mean2 = math_utils.Mean(sample2)
   v1 = math_utils.Variance(sample1)
   v2 = math_utils.Variance(sample2)
   n1 = len(sample1)
   n2 = len(sample2)
   t = _TValue(mean1, mean2, v1, v2, n1, n2)
   df = _DegreesOfFreedom(v1, v2, n1, n2)
   p = _LookupPValue(t, df)
   return t, df, p


 def _TValue(mean1, mean2, v1, v2, n1, n2):
   """Calculates a t-statistic value using the formula for Welch's t-test.

   The t value can be thought of as a signal-to-noise ratio; a higher t-value
   tells you that the groups are more different.

   Args:
     mean1: Mean of sample 1.
     mean2: Mean of sample 2.
     v1: Variance of sample 1.
     v2: Variance of sample 2.
     n1: Sample size of sample 1.
     n2: Sample size of sample 2.

   Returns:
     A t value, which may be negative or positive.
   """
   # If variance of both segments is zero, return some large t-value.
   if v1 == 0 and v2 == 0:
     return 1000.0
   return (mean1 - mean2) / (math.sqrt(v1 / n1 + v2 / n2))


 def _DegreesOfFreedom(v1, v2, n1, n2):
   """Calculates degrees of freedom using the Welch-Satterthwaite formula.

   Degrees of freedom is a measure of sample size. For other types of tests,
   degrees of freedom is sometimes N - 1, where N is the sample size. However,

   Args:
     v1: Variance of sample 1.
     v2: Variance of sample 2.
     n1: Size of sample 2.
     n2: Size of sample 2.

   Returns:
     An estimate of degrees of freedom. Must be at least 1.0.
   """
   # When there's no variance in either sample, return 1.
   if v1 == 0 and v2 == 0:
     return 1
   # If the sample size is too small, also return the minimum (1).
   if n1 <= 1 or n2 <= 2:
     return 1
   df = (((v1 / n1 + v2 / n2) ** 2) /
         ((v1 ** 2) / ((n1 ** 2) * (n1 - 1)) +
          (v2 ** 2) / ((n2 ** 2) * (n2 - 1))))
   return max(1, df)


 # Below is a hard-coded table for looking up p-values.
 #
 # Normally, p-values are calculated based on the t-distribution formula.
 # Looking up pre-calculated values is a less accurate but less complicated
 # alternative.
 #
 # Reference: http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf

 # A list of p-values for a two-tailed test. The entries correspond to to
 # entries in the rows of the table below.
 TWO_TAIL = [1, 0.20, 0.10, 0.05, 0.02, 0.01, 0.005, 0.002, 0.001]

 # A map of degrees of freedom to lists of t-values. The index of the t-value
 # can be used to look up the corresponding p-value.
 TABLE = {
     1: [0, 3.078, 6.314, 12.706, 31.820, 63.657, 127.321, 318.309, 636.619],
     2: [0, 1.886, 2.920, 4.303, 6.965, 9.925, 14.089, 22.327, 31.599],
     3: [0, 1.638, 2.353, 3.182, 4.541, 5.841, 7.453, 10.215, 12.924],
     4: [0, 1.533, 2.132, 2.776, 3.747, 4.604, 5.598, 7.173, 8.610],
     5: [0, 1.476, 2.015, 2.571, 3.365, 4.032, 4.773, 5.893, 6.869],
     6: [0, 1.440, 1.943, 2.447, 3.143, 3.707, 4.317, 5.208, 5.959],
     7: [0, 1.415, 1.895, 2.365, 2.998, 3.499, 4.029, 4.785, 5.408],
     8: [0, 1.397, 1.860, 2.306, 2.897, 3.355, 3.833, 4.501, 5.041],
     9: [0, 1.383, 1.833, 2.262, 2.821, 3.250, 3.690, 4.297, 4.781],
     10: [0, 1.372, 1.812, 2.228, 2.764, 3.169, 3.581, 4.144, 4.587],
     11: [0, 1.363, 1.796, 2.201, 2.718, 3.106, 3.497, 4.025, 4.437],
     12: [0, 1.356, 1.782, 2.179, 2.681, 3.055, 3.428, 3.930, 4.318],
     13: [0, 1.350, 1.771, 2.160, 2.650, 3.012, 3.372, 3.852, 4.221],
     14: [0, 1.345, 1.761, 2.145, 2.625, 2.977, 3.326, 3.787, 4.140],
     15: [0, 1.341, 1.753, 2.131, 2.602, 2.947, 3.286, 3.733, 4.073],
     16: [0, 1.337, 1.746, 2.120, 2.584, 2.921, 3.252, 3.686, 4.015],
     17: [0, 1.333, 1.740, 2.110, 2.567, 2.898, 3.222, 3.646, 3.965],
     18: [0, 1.330, 1.734, 2.101, 2.552, 2.878, 3.197, 3.610, 3.922],
     19: [0, 1.328, 1.729, 2.093, 2.539, 2.861, 3.174, 3.579, 3.883],
     20: [0, 1.325, 1.725, 2.086, 2.528, 2.845, 3.153, 3.552, 3.850],
     21: [0, 1.323, 1.721, 2.080, 2.518, 2.831, 3.135, 3.527, 3.819],
     22: [0, 1.321, 1.717, 2.074, 2.508, 2.819, 3.119, 3.505, 3.792],
     23: [0, 1.319, 1.714, 2.069, 2.500, 2.807, 3.104, 3.485, 3.768],
     24: [0, 1.318, 1.711, 2.064, 2.492, 2.797, 3.090, 3.467, 3.745],
     25: [0, 1.316, 1.708, 2.060, 2.485, 2.787, 3.078, 3.450, 3.725],
     26: [0, 1.315, 1.706, 2.056, 2.479, 2.779, 3.067, 3.435, 3.707],
     27: [0, 1.314, 1.703, 2.052, 2.473, 2.771, 3.057, 3.421, 3.690],
     28: [0, 1.313, 1.701, 2.048, 2.467, 2.763, 3.047, 3.408, 3.674],
     29: [0, 1.311, 1.699, 2.045, 2.462, 2.756, 3.038, 3.396, 3.659],
     30: [0, 1.310, 1.697, 2.042, 2.457, 2.750, 3.030, 3.385, 3.646],
     31: [0, 1.309, 1.695, 2.040, 2.453, 2.744, 3.022, 3.375, 3.633],
     32: [0, 1.309, 1.694, 2.037, 2.449, 2.738, 3.015, 3.365, 3.622],
     33: [0, 1.308, 1.692, 2.035, 2.445, 2.733, 3.008, 3.356, 3.611],
     34: [0, 1.307, 1.691, 2.032, 2.441, 2.728, 3.002, 3.348, 3.601],
     35: [0, 1.306, 1.690, 2.030, 2.438, 2.724, 2.996, 3.340, 3.591],
     36: [0, 1.306, 1.688, 2.028, 2.434, 2.719, 2.991, 3.333, 3.582],
     37: [0, 1.305, 1.687, 2.026, 2.431, 2.715, 2.985, 3.326, 3.574],
     38: [0, 1.304, 1.686, 2.024, 2.429, 2.712, 2.980, 3.319, 3.566],
     39: [0, 1.304, 1.685, 2.023, 2.426, 2.708, 2.976, 3.313, 3.558],
     40: [0, 1.303, 1.684, 2.021, 2.423, 2.704, 2.971, 3.307, 3.551],
     42: [0, 1.302, 1.682, 2.018, 2.418, 2.698, 2.963, 3.296, 3.538],
     44: [0, 1.301, 1.680, 2.015, 2.414, 2.692, 2.956, 3.286, 3.526],
     46: [0, 1.300, 1.679, 2.013, 2.410, 2.687, 2.949, 3.277, 3.515],
     48: [0, 1.299, 1.677, 2.011, 2.407, 2.682, 2.943, 3.269, 3.505],
     50: [0, 1.299, 1.676, 2.009, 2.403, 2.678, 2.937, 3.261, 3.496],
     60: [0, 1.296, 1.671, 2.000, 2.390, 2.660, 2.915, 3.232, 3.460],
     70: [0, 1.294, 1.667, 1.994, 2.381, 2.648, 2.899, 3.211, 3.435],
     80: [0, 1.292, 1.664, 1.990, 2.374, 2.639, 2.887, 3.195, 3.416],
     90: [0, 1.291, 1.662, 1.987, 2.369, 2.632, 2.878, 3.183, 3.402],
     100: [0, 1.290, 1.660, 1.984, 2.364, 2.626, 2.871, 3.174, 3.391],
     120: [0, 1.289, 1.658, 1.980, 2.358, 2.617, 2.860, 3.160, 3.373],
     150: [0, 1.287, 1.655, 1.976, 2.351, 2.609, 2.849, 3.145, 3.357],
     200: [0, 1.286, 1.652, 1.972, 2.345, 2.601, 2.839, 3.131, 3.340],
     300: [0, 1.284, 1.650, 1.968, 2.339, 2.592, 2.828, 3.118, 3.323],
     500: [0, 1.283, 1.648, 1.965, 2.334, 2.586, 2.820, 3.107, 3.310],
 }


 def _LookupPValue(t, df):
   """Looks up a p-value in a t-distribution table.

   Args:
     t: A t statistic value; the result of a t-test.
     df: Number of degrees of freedom.

   Returns:
     A p-value, which represents the likelihood of obtaining a result at least
     as extreme as the one observed just by chance (the null hypothesis).
   """
   assert df >= 1, 'Degrees of freedom must be positive'

   # We ignore the negative sign on the t-value because our null hypothesis
   # is that the two samples are the same; our alternative hypothesis is that
   # the second sample is lesser OR greater than the first.
   t = abs(t)

   def GreatestSmaller(nums, target):
     """Returns the largest number that is <= the target number."""
     lesser_equal = [n for n in nums if n <= target]
     assert lesser_equal, 'No number in number list <= target.'
     return max(lesser_equal)

   df_key = GreatestSmaller(TABLE.keys(), df)
   t_table_row = TABLE[df_key]
   approximate_t_value = GreatestSmaller(t_table_row, t)
   t_value_index = t_table_row.index(approximate_t_value)

   return TWO_TAIL[t_value_index]
	# Copyright 2014 The Chromium Authors. All rights reserved.
	# Use of this source code is governed by a BSD-style license that can be
	# found in the LICENSE file.

	"""Functions for doing independent two-sample t-tests and looking up p-values.

	Note: This module was copied from the Performance Dashboard code, and changed
	to use definitions of mean and variance from math_utils instead of numpy.

	> A t-test is any statistical hypothesis test in which the test statistic
	> follows a Student's t distribution if the null hypothesis is supported.
	> It can be used to determine if two sets of data are significantly different
	> from each other.

	There are several conditions that the data under test should meet in order
	for a t-test to be completely applicable:
	- The data should be roughly normal in distribution.
	- The two samples that are compared should be roughly similar in size.

	References:
	http://en.wikipedia.org/wiki/Student%27s_t-test
	http://en.wikipedia.org/wiki/Welch%27s_t-test
	https://github.com/scipy/scipy/blob/master/scipy/stats/stats.py#L3244
	"""

	import math

	import math_utils


	def WelchsTTest(sample1, sample2):
	"""Performs Welch's t-test on the two samples.

	Welch's t-test is an adaptation of Student's t-test which is used when the
	two samples may have unequal variances. It is also an independent two-sample
	t-test.

	Args:
	sample1: A collection of numbers.
	sample2: Another collection of numbers.

	Returns:
	A 3-tuple (t-statistic, degrees of freedom, p-value).
	"""
	mean1 = math_utils.Mean(sample1)
	mean2 = math_utils.Mean(sample2)
	v1 = math_utils.Variance(sample1)
	v2 = math_utils.Variance(sample2)
	n1 = len(sample1)
	n2 = len(sample2)
	t = _TValue(mean1, mean2, v1, v2, n1, n2)
	df = _DegreesOfFreedom(v1, v2, n1, n2)
	p = _LookupPValue(t, df)
	return t, df, p


	def _TValue(mean1, mean2, v1, v2, n1, n2):
	"""Calculates a t-statistic value using the formula for Welch's t-test.

	The t value can be thought of as a signal-to-noise ratio; a higher t-value
	tells you that the groups are more different.

	Args:
	mean1: Mean of sample 1.
	mean2: Mean of sample 2.
	v1: Variance of sample 1.
	v2: Variance of sample 2.
	n1: Sample size of sample 1.
	n2: Sample size of sample 2.

	Returns:
	A t value, which may be negative or positive.
	"""
	# If variance of both segments is zero, return some large t-value.
	if v1 == 0 and v2 == 0:
	return 1000.0
	return (mean1 - mean2) / (math.sqrt(v1 / n1 + v2 / n2))


	def _DegreesOfFreedom(v1, v2, n1, n2):
	"""Calculates degrees of freedom using the Welch-Satterthwaite formula.

	Degrees of freedom is a measure of sample size. For other types of tests,
	degrees of freedom is sometimes N - 1, where N is the sample size. However,

	Args:
	v1: Variance of sample 1.
	v2: Variance of sample 2.
	n1: Size of sample 2.
	n2: Size of sample 2.

	Returns:
	An estimate of degrees of freedom. Must be at least 1.0.
	"""
	# When there's no variance in either sample, return 1.
	if v1 == 0 and v2 == 0:
	return 1
	# If the sample size is too small, also return the minimum (1).
	if n1 <= 1 or n2 <= 2:
	return 1
	df = (((v1 / n1 + v2 / n2) ** 2) /
	((v1 2) / ((n1 2) * (n1 - 1)) +
	(v2 2) / ((n2 2) * (n2 - 1))))
	return max(1, df)


	# Below is a hard-coded table for looking up p-values.
	#
	# Normally, p-values are calculated based on the t-distribution formula.
	# Looking up pre-calculated values is a less accurate but less complicated
	# alternative.
	#
	# Reference: http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf

	# A list of p-values for a two-tailed test. The entries correspond to to
	# entries in the rows of the table below.
	TWO_TAIL = [1, 0.20, 0.10, 0.05, 0.02, 0.01, 0.005, 0.002, 0.001]

	# A map of degrees of freedom to lists of t-values. The index of the t-value
	# can be used to look up the corresponding p-value.
	TABLE = {
	1: [0, 3.078, 6.314, 12.706, 31.820, 63.657, 127.321, 318.309, 636.619],
	2: [0, 1.886, 2.920, 4.303, 6.965, 9.925, 14.089, 22.327, 31.599],
	3: [0, 1.638, 2.353, 3.182, 4.541, 5.841, 7.453, 10.215, 12.924],
	4: [0, 1.533, 2.132, 2.776, 3.747, 4.604, 5.598, 7.173, 8.610],
	5: [0, 1.476, 2.015, 2.571, 3.365, 4.032, 4.773, 5.893, 6.869],
	6: [0, 1.440, 1.943, 2.447, 3.143, 3.707, 4.317, 5.208, 5.959],
	7: [0, 1.415, 1.895, 2.365, 2.998, 3.499, 4.029, 4.785, 5.408],
	8: [0, 1.397, 1.860, 2.306, 2.897, 3.355, 3.833, 4.501, 5.041],
	9: [0, 1.383, 1.833, 2.262, 2.821, 3.250, 3.690, 4.297, 4.781],
	10: [0, 1.372, 1.812, 2.228, 2.764, 3.169, 3.581, 4.144, 4.587],
	11: [0, 1.363, 1.796, 2.201, 2.718, 3.106, 3.497, 4.025, 4.437],
	12: [0, 1.356, 1.782, 2.179, 2.681, 3.055, 3.428, 3.930, 4.318],
	13: [0, 1.350, 1.771, 2.160, 2.650, 3.012, 3.372, 3.852, 4.221],
	14: [0, 1.345, 1.761, 2.145, 2.625, 2.977, 3.326, 3.787, 4.140],
	15: [0, 1.341, 1.753, 2.131, 2.602, 2.947, 3.286, 3.733, 4.073],
	16: [0, 1.337, 1.746, 2.120, 2.584, 2.921, 3.252, 3.686, 4.015],
	17: [0, 1.333, 1.740, 2.110, 2.567, 2.898, 3.222, 3.646, 3.965],
	18: [0, 1.330, 1.734, 2.101, 2.552, 2.878, 3.197, 3.610, 3.922],
	19: [0, 1.328, 1.729, 2.093, 2.539, 2.861, 3.174, 3.579, 3.883],
	20: [0, 1.325, 1.725, 2.086, 2.528, 2.845, 3.153, 3.552, 3.850],
	21: [0, 1.323, 1.721, 2.080, 2.518, 2.831, 3.135, 3.527, 3.819],
	22: [0, 1.321, 1.717, 2.074, 2.508, 2.819, 3.119, 3.505, 3.792],
	23: [0, 1.319, 1.714, 2.069, 2.500, 2.807, 3.104, 3.485, 3.768],
	24: [0, 1.318, 1.711, 2.064, 2.492, 2.797, 3.090, 3.467, 3.745],
	25: [0, 1.316, 1.708, 2.060, 2.485, 2.787, 3.078, 3.450, 3.725],
	26: [0, 1.315, 1.706, 2.056, 2.479, 2.779, 3.067, 3.435, 3.707],
	27: [0, 1.314, 1.703, 2.052, 2.473, 2.771, 3.057, 3.421, 3.690],
	28: [0, 1.313, 1.701, 2.048, 2.467, 2.763, 3.047, 3.408, 3.674],
	29: [0, 1.311, 1.699, 2.045, 2.462, 2.756, 3.038, 3.396, 3.659],
	30: [0, 1.310, 1.697, 2.042, 2.457, 2.750, 3.030, 3.385, 3.646],
	31: [0, 1.309, 1.695, 2.040, 2.453, 2.744, 3.022, 3.375, 3.633],
	32: [0, 1.309, 1.694, 2.037, 2.449, 2.738, 3.015, 3.365, 3.622],
	33: [0, 1.308, 1.692, 2.035, 2.445, 2.733, 3.008, 3.356, 3.611],
	34: [0, 1.307, 1.691, 2.032, 2.441, 2.728, 3.002, 3.348, 3.601],
	35: [0, 1.306, 1.690, 2.030, 2.438, 2.724, 2.996, 3.340, 3.591],
	36: [0, 1.306, 1.688, 2.028, 2.434, 2.719, 2.991, 3.333, 3.582],
	37: [0, 1.305, 1.687, 2.026, 2.431, 2.715, 2.985, 3.326, 3.574],
	38: [0, 1.304, 1.686, 2.024, 2.429, 2.712, 2.980, 3.319, 3.566],
	39: [0, 1.304, 1.685, 2.023, 2.426, 2.708, 2.976, 3.313, 3.558],
	40: [0, 1.303, 1.684, 2.021, 2.423, 2.704, 2.971, 3.307, 3.551],
	42: [0, 1.302, 1.682, 2.018, 2.418, 2.698, 2.963, 3.296, 3.538],
	44: [0, 1.301, 1.680, 2.015, 2.414, 2.692, 2.956, 3.286, 3.526],
	46: [0, 1.300, 1.679, 2.013, 2.410, 2.687, 2.949, 3.277, 3.515],
	48: [0, 1.299, 1.677, 2.011, 2.407, 2.682, 2.943, 3.269, 3.505],
	50: [0, 1.299, 1.676, 2.009, 2.403, 2.678, 2.937, 3.261, 3.496],
	60: [0, 1.296, 1.671, 2.000, 2.390, 2.660, 2.915, 3.232, 3.460],
	70: [0, 1.294, 1.667, 1.994, 2.381, 2.648, 2.899, 3.211, 3.435],
	80: [0, 1.292, 1.664, 1.990, 2.374, 2.639, 2.887, 3.195, 3.416],
	90: [0, 1.291, 1.662, 1.987, 2.369, 2.632, 2.878, 3.183, 3.402],
	100: [0, 1.290, 1.660, 1.984, 2.364, 2.626, 2.871, 3.174, 3.391],
	120: [0, 1.289, 1.658, 1.980, 2.358, 2.617, 2.860, 3.160, 3.373],
	150: [0, 1.287, 1.655, 1.976, 2.351, 2.609, 2.849, 3.145, 3.357],
	200: [0, 1.286, 1.652, 1.972, 2.345, 2.601, 2.839, 3.131, 3.340],
	300: [0, 1.284, 1.650, 1.968, 2.339, 2.592, 2.828, 3.118, 3.323],
	500: [0, 1.283, 1.648, 1.965, 2.334, 2.586, 2.820, 3.107, 3.310],
	}


	def _LookupPValue(t, df):
	"""Looks up a p-value in a t-distribution table.

	Args:
	t: A t statistic value; the result of a t-test.
	df: Number of degrees of freedom.

	Returns:
	A p-value, which represents the likelihood of obtaining a result at least
	as extreme as the one observed just by chance (the null hypothesis).
	"""
	assert df >= 1, 'Degrees of freedom must be positive'

	# We ignore the negative sign on the t-value because our null hypothesis
	# is that the two samples are the same; our alternative hypothesis is that
	# the second sample is lesser OR greater than the first.
	t = abs(t)

	def GreatestSmaller(nums, target):
	"""Returns the largest number that is <= the target number."""
	lesser_equal = [n for n in nums if n <= target]
	assert lesser_equal, 'No number in number list <= target.'
	return max(lesser_equal)

	df_key = GreatestSmaller(TABLE.keys(), df)
	t_table_row = TABLE[df_key]
	approximate_t_value = GreatestSmaller(t_table_row, t)
	t_value_index = t_table_row.index(approximate_t_value)

	return TWO_TAIL[t_value_index]