T_test.h

#pragma once

#include <vector>
#include <cmath>
#include <iostream>
#include <iomanip>
#include <sstream>

/**
 * About:
 * This is implementation of Independent two-sample t-test
 * Read about it on https://en.wikipedia.org/wiki/Student%27s_t-test (Equal or unequal sample sizes, equal variance)
 *
 * Usage:
 * It's it used to assume with some level of confidence that two distributions don't differ.
 * Values can be added with T_test.add(0/1, value) and after compared and reported with compareAndReport().
 */
struct T_test
{
    struct DistributionData
    {
        size_t size = 0;
        double sum = 0;
        double squares_sum = 0;

        void add(double value)
        {
            ++size;
            sum += value;
            squares_sum += value * value;
        }

        double avg() const
        {
            return sum / size;
        }

        double var() const
        {
            return (squares_sum - (sum * sum / size)) / static_cast<double>(size - 1);
        }

        void clear()
        {
            size = 0;
            sum = 0;
            squares_sum = 0;
        }
    };

    std::vector<DistributionData> data;

    /// First row corresponds to infinity size of distributions case
    const double students_table[101][6] =
    {
        { 1.282, 1.645, 1.960, 2.326, 2.576, 3.090 },
        { 3.078, 6.314, 12.706, 31.821, 63.657, 318.313 },
        { 1.886, 2.920, 4.303, 6.965, 9.925, 22.327 },
        { 1.638, 2.353, 3.182, 4.541, 5.841, 10.215 },
        { 1.533, 2.132, 2.776, 3.747, 4.604, 7.173 },
        { 1.476, 2.015, 2.571, 3.365, 4.032, 5.893 },
        { 1.440, 1.943, 2.447, 3.143, 3.707, 5.208 },
        { 1.415, 1.895, 2.365, 2.998, 3.499, 4.782 },
        { 1.397, 1.860, 2.306, 2.896, 3.355, 4.499 },
        { 1.383, 1.833, 2.262, 2.821, 3.250, 4.296 },
        { 1.372, 1.812, 2.228, 2.764, 3.169, 4.143 },
        { 1.363, 1.796, 2.201, 2.718, 3.106, 4.024 },
        { 1.356, 1.782, 2.179, 2.681, 3.055, 3.929 },
        { 1.350, 1.771, 2.160, 2.650, 3.012, 3.852 },
        { 1.345, 1.761, 2.145, 2.624, 2.977, 3.787 },
        { 1.341, 1.753, 2.131, 2.602, 2.947, 3.733 },
        { 1.337, 1.746, 2.120, 2.583, 2.921, 3.686 },
        { 1.333, 1.740, 2.110, 2.567, 2.898, 3.646 },
        { 1.330, 1.734, 2.101, 2.552, 2.878, 3.610 },
        { 1.328, 1.729, 2.093, 2.539, 2.861, 3.579 },
        { 1.325, 1.725, 2.086, 2.528, 2.845, 3.552 },
        { 1.323, 1.721, 2.080, 2.518, 2.831, 3.527 },
        { 1.321, 1.717, 2.074, 2.508, 2.819, 3.505 },
        { 1.319, 1.714, 2.069, 2.500, 2.807, 3.485 },
        { 1.318, 1.711, 2.064, 2.492, 2.797, 3.467 },
        { 1.316, 1.708, 2.060, 2.485, 2.787, 3.450 },
        { 1.315, 1.706, 2.056, 2.479, 2.779, 3.435 },
        { 1.314, 1.703, 2.052, 2.473, 2.771, 3.421 },
        { 1.313, 1.701, 2.048, 2.467, 2.763, 3.408 },
        { 1.311, 1.699, 2.045, 2.462, 2.756, 3.396 },
        { 1.310, 1.697, 2.042, 2.457, 2.750, 3.385 },
        { 1.309, 1.696, 2.040, 2.453, 2.744, 3.375 },
        { 1.309, 1.694, 2.037, 2.449, 2.738, 3.365 },
        { 1.308, 1.692, 2.035, 2.445, 2.733, 3.356 },
        { 1.307, 1.691, 2.032, 2.441, 2.728, 3.348 },
        { 1.306, 1.690, 2.030, 2.438, 2.724, 3.340 },
        { 1.306, 1.688, 2.028, 2.434, 2.719, 3.333 },
        { 1.305, 1.687, 2.026, 2.431, 2.715, 3.326 },
        { 1.304, 1.686, 2.024, 2.429, 2.712, 3.319 },
        { 1.304, 1.685, 2.023, 2.426, 2.708, 3.313 },
        { 1.303, 1.684, 2.021, 2.423, 2.704, 3.307 },
        { 1.303, 1.683, 2.020, 2.421, 2.701, 3.301 },
        { 1.302, 1.682, 2.018, 2.418, 2.698, 3.296 },
        { 1.302, 1.681, 2.017, 2.416, 2.695, 3.291 },
        { 1.301, 1.680, 2.015, 2.414, 2.692, 3.286 },
        { 1.301, 1.679, 2.014, 2.412, 2.690, 3.281 },
        { 1.300, 1.679, 2.013, 2.410, 2.687, 3.277 },
        { 1.300, 1.678, 2.012, 2.408, 2.685, 3.273 },
        { 1.299, 1.677, 2.011, 2.407, 2.682, 3.269 },
        { 1.299, 1.677, 2.010, 2.405, 2.680, 3.265 },
        { 1.299, 1.676, 2.009, 2.403, 2.678, 3.261 },
        { 1.298, 1.675, 2.008, 2.402, 2.676, 3.258 },
        { 1.298, 1.675, 2.007, 2.400, 2.674, 3.255 },
        { 1.298, 1.674, 2.006, 2.399, 2.672, 3.251 },
        { 1.297, 1.674, 2.005, 2.397, 2.670, 3.248 },
        { 1.297, 1.673, 2.004, 2.396, 2.668, 3.245 },
        { 1.297, 1.673, 2.003, 2.395, 2.667, 3.242 },
        { 1.297, 1.672, 2.002, 2.394, 2.665, 3.239 },
        { 1.296, 1.672, 2.002, 2.392, 2.663, 3.237 },
        { 1.296, 1.671, 2.001, 2.391, 2.662, 3.234 },
        { 1.296, 1.671, 2.000, 2.390, 2.660, 3.232 },
        { 1.296, 1.670, 2.000, 2.389, 2.659, 3.229 },
        { 1.295, 1.670, 1.999, 2.388, 2.657, 3.227 },
        { 1.295, 1.669, 1.998, 2.387, 2.656, 3.225 },
        { 1.295, 1.669, 1.998, 2.386, 2.655, 3.223 },
        { 1.295, 1.669, 1.997, 2.385, 2.654, 3.220 },
        { 1.295, 1.668, 1.997, 2.384, 2.652, 3.218 },
        { 1.294, 1.668, 1.996, 2.383, 2.651, 3.216 },
        { 1.294, 1.668, 1.995, 2.382, 2.650, 3.214 },
        { 1.294, 1.667, 1.995, 2.382, 2.649, 3.213 },
        { 1.294, 1.667, 1.994, 2.381, 2.648, 3.211 },
        { 1.294, 1.667, 1.994, 2.380, 2.647, 3.209 },
        { 1.293, 1.666, 1.993, 2.379, 2.646, 3.207 },
        { 1.293, 1.666, 1.993, 2.379, 2.645, 3.206 },
        { 1.293, 1.666, 1.993, 2.378, 2.644, 3.204 },
        { 1.293, 1.665, 1.992, 2.377, 2.643, 3.202 },
        { 1.293, 1.665, 1.992, 2.376, 2.642, 3.201 },
        { 1.293, 1.665, 1.991, 2.376, 2.641, 3.199 },
        { 1.292, 1.665, 1.991, 2.375, 2.640, 3.198 },
        { 1.292, 1.664, 1.990, 2.374, 2.640, 3.197 },
        { 1.292, 1.664, 1.990, 2.374, 2.639, 3.195 },
        { 1.292, 1.664, 1.990, 2.373, 2.638, 3.194 },
        { 1.292, 1.664, 1.989, 2.373, 2.637, 3.193 },
        { 1.292, 1.663, 1.989, 2.372, 2.636, 3.191 },
        { 1.292, 1.663, 1.989, 2.372, 2.636, 3.190 },
        { 1.292, 1.663, 1.988, 2.371, 2.635, 3.189 },
        { 1.291, 1.663, 1.988, 2.370, 2.634, 3.188 },
        { 1.291, 1.663, 1.988, 2.370, 2.634, 3.187 },
        { 1.291, 1.662, 1.987, 2.369, 2.633, 3.185 },
        { 1.291, 1.662, 1.987, 2.369, 2.632, 3.184 },
        { 1.291, 1.662, 1.987, 2.368, 2.632, 3.183 },
        { 1.291, 1.662, 1.986, 2.368, 2.631, 3.182 },
        { 1.291, 1.662, 1.986, 2.368, 2.630, 3.181 },
        { 1.291, 1.661, 1.986, 2.367, 2.630, 3.180 },
        { 1.291, 1.661, 1.986, 2.367, 2.629, 3.179 },
        { 1.291, 1.661, 1.985, 2.366, 2.629, 3.178 },
        { 1.290, 1.661, 1.985, 2.366, 2.628, 3.177 },
        { 1.290, 1.661, 1.985, 2.365, 2.627, 3.176 },
        { 1.290, 1.661, 1.984, 2.365, 2.627, 3.175 },
        { 1.290, 1.660, 1.984, 2.365, 2.626, 3.175 },
        { 1.290, 1.660, 1.984, 2.364, 2.626, 3.174 },
    };

    const std::vector<double> confidence_level = { 80, 90, 95, 98, 99, 99.5 };

    T_test()
    {
        data.resize(2);
    }

    void clear()
    {
        data[0].clear();
        data[1].clear();
    }

    void add(size_t distribution, double value)
    {
        if (distribution > 1)
            return;
        data[distribution].add(value);
    }

    /// Confidence_level_index can be set in range [0, 5]. Corresponding values can be found above.
    std::pair<bool, std::string> compareAndReport(size_t confidence_level_index = 5) const
    {
        if (confidence_level_index > 5)
            confidence_level_index = 5;

        if (data[0].size == 0 || data[1].size == 0)
            return {true, ""};

        size_t degrees_of_freedom = (data[0].size - 1) + (data[1].size - 1);

        double table_value = students_table[degrees_of_freedom > 100 ? 0 : degrees_of_freedom][confidence_level_index];

        double pooled_standard_deviation = sqrt(((data[0].size - 1) * data[0].var() + (data[1].size - 1) * data[1].var()) / degrees_of_freedom);

        double t_statistic = pooled_standard_deviation * sqrt(1.0 / data[0].size + 1.0 / data[1].size);

        double mean_difference = fabs(data[0].avg() - data[1].avg());

        double mean_confidence_interval = table_value * t_statistic;

        std::stringstream ss;
        if (mean_difference > mean_confidence_interval && (mean_difference - mean_confidence_interval > 0.0001)) /// difference must be more than 0.0001, to take into account connection latency.
        {
            ss << "Difference at " << confidence_level[confidence_level_index] <<  "% confidence : ";
            ss << std::fixed << std::setprecision(8) << "mean difference is " << mean_difference << ", but confidence interval is " << mean_confidence_interval;
            return {false, ss.str()};
        }
        else
        {
            ss << "No difference proven at " << confidence_level[confidence_level_index] <<  "% confidence";
            return {true, ss.str()};
        }
    }

};