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
7Note: This module was copied from the Performance Dashboard code, and changed
8to 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
15There are several conditions that the data under test should meet in order
16for 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
20References:
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
26import math
27
28import math_utils
29
30
31def 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
57def _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
80def _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.
117TWO_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.
121TABLE = {
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
180def _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]
210