tools/auto_bisect/ttest.py

   1 # Copyright 2014 The Chromium Authors. All rights reserved.
   2 # Use of this source code is governed by a BSD-style license that can be
   3 # found in the LICENSE file.
   4
   5 """Functions for doing independent two-sample t-tests and looking up p-values.
   6
   7 Note: This module was copied from the Performance Dashboard code, and changed
   8 to use definitions of mean and variance from math_utils instead of numpy.
   9
  10 > A t-test is any statistical hypothesis test in which the test statistic
  11 > follows a Student's t distribution if the null hypothesis is supported.
  12 > It can be used to determine if two sets of data are significantly different
  13 > from each other.
  14
  15 There are several conditions that the data under test should meet in order
  16 for a t-test to be completely applicable:
  17  - The data should be roughly normal in distribution.
  18  - The two samples that are compared should be roughly similar in size.
  19
  20 References:
  21   http://en.wikipedia.org/wiki/Student%27s_t-test
  22   http://en.wikipedia.org/wiki/Welch%27s_t-test
  23   https://github.com/scipy/scipy/blob/master/scipy/stats/stats.py#L3244
  24 """
  25
  26 import math
  27
  28 import math_utils
  29
  30
  31 def WelchsTTest(sample1, sample2):
  32   """Performs Welch's t-test on the two samples.
  33
  34   Welch's t-test is an adaptation of Student's t-test which is used when the
  35   two samples may have unequal variances. It is also an independent two-sample
  36   t-test.
  37
  38   Args:
  39     sample1: A collection of numbers.
  40     sample2: Another collection of numbers.
  41
  42   Returns:
  43     A 3-tuple (t-statistic, degrees of freedom, p-value).
  44   """
  45   mean1 = math_utils.Mean(sample1)
  46   mean2 = math_utils.Mean(sample2)
  47   v1 = math_utils.Variance(sample1)
  48   v2 = math_utils.Variance(sample2)
  49   n1 = len(sample1)
  50   n2 = len(sample2)
  51   t = _TValue(mean1, mean2, v1, v2, n1, n2)
  52   df = _DegreesOfFreedom(v1, v2, n1, n2)
  53   p = _LookupPValue(t, df)
  54   return t, df, p
  55
  56
  57 def _TValue(mean1, mean2, v1, v2, n1, n2):
  58   """Calculates a t-statistic value using the formula for Welch's t-test.
  59
  60   The t value can be thought of as a signal-to-noise ratio; a higher t-value
  61   tells you that the groups are more different.
  62
  63   Args:
  64     mean1: Mean of sample 1.
  65     mean2: Mean of sample 2.
  66     v1: Variance of sample 1.
  67     v2: Variance of sample 2.
  68     n1: Sample size of sample 1.
  69     n2: Sample size of sample 2.
  70
  71   Returns:
  72     A t value, which may be negative or positive.
  73   """
  74   # If variance of both segments is zero, return some large t-value.
  75   if v1 == 0 and v2 == 0:
  76     return 1000.0
  77   return (mean1 - mean2) / (math.sqrt(v1 / n1 + v2 / n2))
  78
  79
  80 def _DegreesOfFreedom(v1, v2, n1, n2):
  81   """Calculates degrees of freedom using the Welch-Satterthwaite formula.
  82
  83   Degrees of freedom is a measure of sample size. For other types of tests,
  84   degrees of freedom is sometimes N - 1, where N is the sample size. However,
  85
  86   Args:
  87     v1: Variance of sample 1.
  88     v2: Variance of sample 2.
  89     n1: Size of sample 2.
  90     n2: Size of sample 2.
  91
  92   Returns:
  93     An estimate of degrees of freedom. Must be at least 1.0.
  94   """
  95   # When there's no variance in either sample, return 1.
  96   if v1 == 0 and v2 == 0:
  97     return 1
  98   # If the sample size is too small, also return the minimum (1).
  99   if n1 <= 1 or n2 <= 2:
 100     return 1
 101   df = (((v1 / n1 + v2 / n2) ** 2) /
 102         ((v1 ** 2) / ((n1 ** 2) * (n1 - 1)) +
 103          (v2 ** 2) / ((n2 ** 2) * (n2 - 1))))
 104   return max(1, df)
 105
 106
 107 # Below is a hard-coded table for looking up p-values.
 108 #
 109 # Normally, p-values are calculated based on the t-distribution formula.
 110 # Looking up pre-calculated values is a less accurate but less complicated
 111 # alternative.
 112 #
 113 # Reference: http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf
 114
 115 # A list of p-values for a two-tailed test. The entries correspond to to
 116 # entries in the rows of the table below.
 117 TWO_TAIL = [1, 0.20, 0.10, 0.05, 0.02, 0.01, 0.005, 0.002, 0.001]
 118
 119 # A map of degrees of freedom to lists of t-values. The index of the t-value
 120 # can be used to look up the corresponding p-value.
 121 TABLE = {
 122     1: [0, 3.078, 6.314, 12.706, 31.820, 63.657, 127.321, 318.309, 636.619],
 123     2: [0, 1.886, 2.920, 4.303, 6.965, 9.925, 14.089, 22.327, 31.599],
 124     3: [0, 1.638, 2.353, 3.182, 4.541, 5.841, 7.453, 10.215, 12.924],
 125     4: [0, 1.533, 2.132, 2.776, 3.747, 4.604, 5.598, 7.173, 8.610],
 126     5: [0, 1.476, 2.015, 2.571, 3.365, 4.032, 4.773, 5.893, 6.869],
 127     6: [0, 1.440, 1.943, 2.447, 3.143, 3.707, 4.317, 5.208, 5.959],
 128     7: [0, 1.415, 1.895, 2.365, 2.998, 3.499, 4.029, 4.785, 5.408],
 129     8: [0, 1.397, 1.860, 2.306, 2.897, 3.355, 3.833, 4.501, 5.041],
 130     9: [0, 1.383, 1.833, 2.262, 2.821, 3.250, 3.690, 4.297, 4.781],
 131     10: [0, 1.372, 1.812, 2.228, 2.764, 3.169, 3.581, 4.144, 4.587],
 132     11: [0, 1.363, 1.796, 2.201, 2.718, 3.106, 3.497, 4.025, 4.437],
 133     12: [0, 1.356, 1.782, 2.179, 2.681, 3.055, 3.428, 3.930, 4.318],
 134     13: [0, 1.350, 1.771, 2.160, 2.650, 3.012, 3.372, 3.852, 4.221],
 135     14: [0, 1.345, 1.761, 2.145, 2.625, 2.977, 3.326, 3.787, 4.140],
 136     15: [0, 1.341, 1.753, 2.131, 2.602, 2.947, 3.286, 3.733, 4.073],
 137     16: [0, 1.337, 1.746, 2.120, 2.584, 2.921, 3.252, 3.686, 4.015],
 138     17: [0, 1.333, 1.740, 2.110, 2.567, 2.898, 3.222, 3.646, 3.965],
 139     18: [0, 1.330, 1.734, 2.101, 2.552, 2.878, 3.197, 3.610, 3.922],
 140     19: [0, 1.328, 1.729, 2.093, 2.539, 2.861, 3.174, 3.579, 3.883],
 141     20: [0, 1.325, 1.725, 2.086, 2.528, 2.845, 3.153, 3.552, 3.850],
 142     21: [0, 1.323, 1.721, 2.080, 2.518, 2.831, 3.135, 3.527, 3.819],
 143     22: [0, 1.321, 1.717, 2.074, 2.508, 2.819, 3.119, 3.505, 3.792],
 144     23: [0, 1.319, 1.714, 2.069, 2.500, 2.807, 3.104, 3.485, 3.768],
 145     24: [0, 1.318, 1.711, 2.064, 2.492, 2.797, 3.090, 3.467, 3.745],
 146     25: [0, 1.316, 1.708, 2.060, 2.485, 2.787, 3.078, 3.450, 3.725],
 147     26: [0, 1.315, 1.706, 2.056, 2.479, 2.779, 3.067, 3.435, 3.707],
 148     27: [0, 1.314, 1.703, 2.052, 2.473, 2.771, 3.057, 3.421, 3.690],
 149     28: [0, 1.313, 1.701, 2.048, 2.467, 2.763, 3.047, 3.408, 3.674],
 150     29: [0, 1.311, 1.699, 2.045, 2.462, 2.756, 3.038, 3.396, 3.659],
 151     30: [0, 1.310, 1.697, 2.042, 2.457, 2.750, 3.030, 3.385, 3.646],
 152     31: [0, 1.309, 1.695, 2.040, 2.453, 2.744, 3.022, 3.375, 3.633],
 153     32: [0, 1.309, 1.694, 2.037, 2.449, 2.738, 3.015, 3.365, 3.622],
 154     33: [0, 1.308, 1.692, 2.035, 2.445, 2.733, 3.008, 3.356, 3.611],
 155     34: [0, 1.307, 1.691, 2.032, 2.441, 2.728, 3.002, 3.348, 3.601],
 156     35: [0, 1.306, 1.690, 2.030, 2.438, 2.724, 2.996, 3.340, 3.591],
 157     36: [0, 1.306, 1.688, 2.028, 2.434, 2.719, 2.991, 3.333, 3.582],
 158     37: [0, 1.305, 1.687, 2.026, 2.431, 2.715, 2.985, 3.326, 3.574],
 159     38: [0, 1.304, 1.686, 2.024, 2.429, 2.712, 2.980, 3.319, 3.566],
 160     39: [0, 1.304, 1.685, 2.023, 2.426, 2.708, 2.976, 3.313, 3.558],
 161     40: [0, 1.303, 1.684, 2.021, 2.423, 2.704, 2.971, 3.307, 3.551],
 162     42: [0, 1.302, 1.682, 2.018, 2.418, 2.698, 2.963, 3.296, 3.538],
 163     44: [0, 1.301, 1.680, 2.015, 2.414, 2.692, 2.956, 3.286, 3.526],
 164     46: [0, 1.300, 1.679, 2.013, 2.410, 2.687, 2.949, 3.277, 3.515],
 165     48: [0, 1.299, 1.677, 2.011, 2.407, 2.682, 2.943, 3.269, 3.505],
 166     50: [0, 1.299, 1.676, 2.009, 2.403, 2.678, 2.937, 3.261, 3.496],
 167     60: [0, 1.296, 1.671, 2.000, 2.390, 2.660, 2.915, 3.232, 3.460],
 168     70: [0, 1.294, 1.667, 1.994, 2.381, 2.648, 2.899, 3.211, 3.435],
 169     80: [0, 1.292, 1.664, 1.990, 2.374, 2.639, 2.887, 3.195, 3.416],
 170     90: [0, 1.291, 1.662, 1.987, 2.369, 2.632, 2.878, 3.183, 3.402],
 171     100: [0, 1.290, 1.660, 1.984, 2.364, 2.626, 2.871, 3.174, 3.391],
 172     120: [0, 1.289, 1.658, 1.980, 2.358, 2.617, 2.860, 3.160, 3.373],
 173     150: [0, 1.287, 1.655, 1.976, 2.351, 2.609, 2.849, 3.145, 3.357],
 174     200: [0, 1.286, 1.652, 1.972, 2.345, 2.601, 2.839, 3.131, 3.340],
 175     300: [0, 1.284, 1.650, 1.968, 2.339, 2.592, 2.828, 3.118, 3.323],
 176     500: [0, 1.283, 1.648, 1.965, 2.334, 2.586, 2.820, 3.107, 3.310],
 177 }
 178
 179
 180 def _LookupPValue(t, df):
 181   """Looks up a p-value in a t-distribution table.
 182
 183   Args:
 184     t: A t statistic value; the result of a t-test.
 185     df: Number of degrees of freedom.
 186
 187   Returns:
 188     A p-value, which represents the likelihood of obtaining a result at least
 189     as extreme as the one observed just by chance (the null hypothesis).
 190   """
 191   assert df >= 1, 'Degrees of freedom must be positive'
 192
 193   # We ignore the negative sign on the t-value because our null hypothesis
 194   # is that the two samples are the same; our alternative hypothesis is that
 195   # the second sample is lesser OR greater than the first.
 196   t = abs(t)
 197
 198   def GreatestSmaller(nums, target):
 199     """Returns the largest number that is <= the target number."""
 200     lesser_equal = [n for n in nums if n <= target]
 201     assert lesser_equal, 'No number in number list <= target.'
 202     return max(lesser_equal)
 203
 204   df_key = GreatestSmaller(TABLE.keys(), df)
 205   t_table_row = TABLE[df_key]
 206   approximate_t_value = GreatestSmaller(t_table_row, t)
 207   t_value_index = t_table_row.index(approximate_t_value)
 208
 209   return TWO_TAIL[t_value_index]