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 # 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]