nonparametric_tests.hpp

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#pragma once
 
#include "statcpp/basic_statistics.hpp"
#include "statcpp/order_statistics.hpp"
#include "statcpp/continuous_distributions.hpp"
#include "statcpp/discrete_distributions.hpp"
#include "statcpp/parametric_tests.hpp"
 
#include <algorithm>
#include <cmath>
#include <cstddef>
#include <cstdint>
#include <limits>
#include <stdexcept>
#include <utility>
#include <vector>
 
namespace statcpp {
 
// ============================================================================
// Helper: Compute Ranks with Tie Handling
// ============================================================================
 
template <typename Iterator>
std::vector<double> compute_ranks_with_ties(Iterator first, Iterator last)
{
    auto n = statcpp::count(first, last);
    if (n == 0) return {};
 
    // Create index-value pairs
    std::vector<std::pair<double, std::size_t>> indexed(n);
    std::size_t i = 0;
    for (auto it = first; it != last; ++it, ++i) {
        indexed[i] = {static_cast<double>(*it), i};
    }
 
    // Sort by value
    std::sort(indexed.begin(), indexed.end(),
              [](const auto& a, const auto& b) { return a.first < b.first; });
 
    // Assign ranks with tie handling (average rank)
    std::vector<double> ranks(n);
    std::size_t j = 0;
    while (j < n) {
        std::size_t k = j;
        // Find all elements with same value
        while (k < n && indexed[k].first == indexed[j].first) {
            ++k;
        }
        // Average rank for tied elements
        double avg_rank = (static_cast<double>(j + 1) + static_cast<double>(k)) / 2.0;
        for (std::size_t m = j; m < k; ++m) {
            ranks[indexed[m].second] = avg_rank;
        }
        j = k;
    }
 
    return ranks;
}
 
inline std::vector<std::size_t> compute_tie_groups(const std::vector<double>& sorted_values)
{
    std::vector<std::size_t> tie_groups;
    std::size_t n = sorted_values.size();
    std::size_t i = 0;
    while (i < n) {
        std::size_t j = i;
        while (j < n && sorted_values[j] == sorted_values[i]) {
            ++j;
        }
        std::size_t t = j - i;
        if (t > 1) {
            tie_groups.push_back(t);
        }
        i = j;
    }
    return tie_groups;
}
 
// ============================================================================
// Shapiro-Wilk Test for Normality
// ============================================================================
 
template <typename Iterator>
test_result shapiro_wilk_test(Iterator first, Iterator last)
{
    auto n = statcpp::count(first, last);
    if (n < 3) {
        throw std::invalid_argument("statcpp::shapiro_wilk_test: need at least 3 elements");
    }
    if (n > 5000) {
        throw std::invalid_argument("statcpp::shapiro_wilk_test: n > 5000 not supported");
    }
 
    // Copy and sort data
    std::vector<double> sorted_data;
    sorted_data.reserve(n);
    for (auto it = first; it != last; ++it) {
        sorted_data.push_back(static_cast<double>(*it));
    }
    std::sort(sorted_data.begin(), sorted_data.end());
 
    double mean_val = statcpp::mean(sorted_data.begin(), sorted_data.end());
 
    // Compute SS (sum of squared deviations)
    double ss = 0.0;
    for (double x : sorted_data) {
        double d = x - mean_val;
        ss += d * d;
    }
 
    if (ss == 0.0) {
        throw std::invalid_argument("statcpp::shapiro_wilk_test: zero variance");
    }
 
    // Compute W statistic using Royston's approximation
    // First compute all m_i values (expected order statistics of standard normal)
    std::vector<double> m_vals(n);
    for (std::size_t i = 0; i < n; ++i) {
        double p = (static_cast<double>(i + 1) - 0.375) / (static_cast<double>(n) + 0.25);
        m_vals[i] = norm_quantile(p);
    }
 
    // Compute sum of squared m values
    double sum_m2 = 0.0;
    for (std::size_t i = 0; i < n; ++i) {
        sum_m2 += m_vals[i] * m_vals[i];
    }
 
    // Compute a coefficients using Royston's algorithm
    std::vector<double> a(n, 0.0);
 
    if (n <= 5) {
        // For very small n, use simple approximation
        double sqrt_sum_m2 = std::sqrt(sum_m2);
        for (std::size_t i = 0; i < n; ++i) {
            a[i] = m_vals[i] / sqrt_sum_m2;
        }
    } else {
        // Royston's polynomial approximation for a_n
        double sqrt_n = std::sqrt(static_cast<double>(n));
        double u = 1.0 / sqrt_n;
 
        // a_n coefficient (for largest order statistic)
        double a_n = -2.706056 * std::pow(u, 5) + 4.434685 * std::pow(u, 4)
                     - 2.071190 * std::pow(u, 3) - 0.147981 * std::pow(u, 2)
                     + 0.221157 * u + m_vals[n - 1] / std::sqrt(sum_m2);
 
        // a_{n-1} coefficient
        double a_n1 = -3.582633 * std::pow(u, 5) + 5.682633 * std::pow(u, 4)
                      - 1.752461 * std::pow(u, 3) - 0.293762 * std::pow(u, 2)
                      + 0.042981 * u + m_vals[n - 2] / std::sqrt(sum_m2);
 
        // Set the extreme coefficients
        a[n - 1] = a_n;
        a[0] = -a_n;
        if (n > 3) {
            a[n - 2] = a_n1;
            a[1] = -a_n1;
        }
 
        // Compute phi for intermediate coefficients
        double phi = (sum_m2 - 2.0 * m_vals[n - 1] * m_vals[n - 1]
                      - 2.0 * m_vals[n - 2] * m_vals[n - 2])
                     / (1.0 - 2.0 * a_n * a_n - 2.0 * a_n1 * a_n1);
 
        if (phi > 0) {
            double sqrt_phi = std::sqrt(phi);
            for (std::size_t i = 2; i < n - 2; ++i) {
                a[i] = m_vals[i] / sqrt_phi;
            }
        }
    }
 
    // Compute W = (sum(a_i * x_(i)))^2 / SS
    double b = 0.0;
    for (std::size_t i = 0; i < n; ++i) {
        b += a[i] * sorted_data[i];
    }
 
    double w = (b * b) / ss;
 
    // Clamp W to valid range [0, 1]
    w = std::max(0.0, std::min(1.0, w));
 
    // Approximation for p-value using transformation to normal
    double ln_n = std::log(static_cast<double>(n));
    double mu, sigma, gamma;
 
    if (n <= 11) {
        gamma = 0.459 * static_cast<double>(n) - 2.273;
        mu = -0.0006714 * std::pow(static_cast<double>(n), 3)
             + 0.025054 * std::pow(static_cast<double>(n), 2)
             - 0.39978 * static_cast<double>(n) + 0.5440;
        sigma = std::exp(-0.0020322 * std::pow(static_cast<double>(n), 3)
                         + 0.062767 * std::pow(static_cast<double>(n), 2)
                         - 0.77857 * static_cast<double>(n) + 1.3822);
    } else {
        gamma = 0.0;
        mu = 0.0038915 * std::pow(ln_n, 3) - 0.083751 * std::pow(ln_n, 2)
             - 0.31082 * ln_n - 1.5861;
        sigma = std::exp(0.0030302 * std::pow(ln_n, 2) - 0.082676 * ln_n - 0.4803);
    }
 
    double z;
    if (gamma != 0.0 && w < 1.0) {
        double arg = gamma - std::log(1.0 - w);
        if (arg > 0) {
            z = (-std::log(arg) - mu) / sigma;
        } else {
            z = -8.0;  // Very high W (very normal): p -> 1
        }
    } else if (w < 1.0) {
        z = (std::log(1.0 - w) - mu) / sigma;
    } else {
        z = -8.0;  // Perfect W = 1 (maximally normal): p -> 1
    }
 
    double p_value = norm_sf(z);
    p_value = std::max(0.0, std::min(1.0, p_value));
 
    return {w, p_value, static_cast<double>(n), alternative_hypothesis::less};
}
 
// ============================================================================
// Lilliefors Test for Normality
// ============================================================================
 
template <typename Iterator>
test_result lilliefors_test(Iterator first, Iterator last)
{
    auto n = statcpp::count(first, last);
    if (n < 2) {
        throw std::invalid_argument("statcpp::lilliefors_test: need at least 2 elements");
    }
 
    // Standardize data
    double mean_val = statcpp::mean(first, last);
    double sd = statcpp::sample_stddev(first, last);
 
    if (sd == 0.0) {
        throw std::invalid_argument("statcpp::lilliefors_test: zero variance");
    }
 
    std::vector<double> standardized;
    standardized.reserve(n);
    for (auto it = first; it != last; ++it) {
        standardized.push_back((static_cast<double>(*it) - mean_val) / sd);
    }
    std::sort(standardized.begin(), standardized.end());
 
    // Compute D statistic
    double d_plus = 0.0;
    double d_minus = 0.0;
 
    for (std::size_t i = 0; i < n; ++i) {
        double f_x = norm_cdf(standardized[i]);
        double f_n_upper = static_cast<double>(i + 1) / static_cast<double>(n);
        double f_n_lower = static_cast<double>(i) / static_cast<double>(n);
 
        d_plus = std::max(d_plus, f_n_upper - f_x);
        d_minus = std::max(d_minus, f_x - f_n_lower);
    }
 
    double d = std::max(d_plus, d_minus);
 
    // p-value from the Dallal and Wilkinson (1986) analytic approximation to the
    // null distribution of D when the mean and variance are estimated from the
    // sample. See "An analytic approximation to the distribution of Lilliefors's
    // test statistic for normality", The American Statistician 40:294-296.
    //
    // The approximation is published for p <= 0.10 and is calibrated for n >= 5.
    // Every conventional significance level (0.10, 0.05, 0.01) falls inside that
    // range; above 0.10 the returned value only indicates that the sample is
    // consistent with normality and should not be read as an accurate p-value.
    const double n_d = static_cast<double>(n);
    const double a = n_d + 2.78019;
 
    // As a function of d the exponent is a downward parabola, so to the left of
    // its vertex the approximation would increase with d. Clamping at the vertex
    // keeps the p-value monotonically non-increasing in d over the whole domain.
    const double d_vertex = 2.99587 / (2.0 * 7.01256 * std::sqrt(a));
    const double d_eff = std::max(d, d_vertex);
 
    double p_value = std::exp(-7.01256 * d_eff * d_eff * a + 2.99587 * d_eff * std::sqrt(a) -
                              0.122119 + 0.974598 / std::sqrt(n_d) + 1.67997 / n_d);
    p_value = std::max(0.0, std::min(1.0, p_value));
 
    return {d, p_value, static_cast<double>(n), alternative_hypothesis::greater};
}
 
template <typename Iterator>
[[deprecated("Use lilliefors_test() instead. ks_test_normal() will be removed in a future version.")]]
test_result ks_test_normal(Iterator first, Iterator last)
{
    return lilliefors_test(first, last);
}
 
// ============================================================================
// Levene's Test for Homogeneity of Variance
// ============================================================================
 
inline test_result levene_test(const std::vector<std::vector<double>>& groups)
{
    std::size_t k = groups.size();
    if (k < 2) {
        throw std::invalid_argument("statcpp::levene_test: need at least 2 groups");
    }
 
    // Compute group medians and deviations from median
    std::vector<std::vector<double>> z_values(k);
    std::size_t total_n = 0;
 
    for (std::size_t i = 0; i < k; ++i) {
        if (groups[i].size() < 2) {
            throw std::invalid_argument("statcpp::levene_test: each group needs at least 2 elements");
        }
 
        std::vector<double> sorted = groups[i];
        std::sort(sorted.begin(), sorted.end());
        double med = statcpp::median(sorted.begin(), sorted.end());
 
        for (double x : groups[i]) {
            z_values[i].push_back(std::abs(x - med));
        }
        total_n += groups[i].size();
    }
 
    // Compute group means of z values
    std::vector<double> z_means(k);
    double z_grand_mean = 0.0;
 
    for (std::size_t i = 0; i < k; ++i) {
        z_means[i] = statcpp::mean(z_values[i].begin(), z_values[i].end());
        z_grand_mean += z_means[i] * static_cast<double>(z_values[i].size());
    }
    z_grand_mean /= static_cast<double>(total_n);
 
    // Compute test statistic
    double ss_between = 0.0;
    double ss_within = 0.0;
 
    for (std::size_t i = 0; i < k; ++i) {
        double ni = static_cast<double>(z_values[i].size());
        ss_between += ni * (z_means[i] - z_grand_mean) * (z_means[i] - z_grand_mean);
 
        for (double z : z_values[i]) {
            ss_within += (z - z_means[i]) * (z - z_means[i]);
        }
    }
 
    double df1 = static_cast<double>(k - 1);
    double df2 = static_cast<double>(total_n - k);
 
    // Guard: if all deviations are zero (all values identical within each group),
    // variances are trivially equal → F = 0, p = 1
    if (ss_within == 0.0) {
        return {0.0, 1.0, df1, alternative_hypothesis::greater};
    }
 
    double f = (ss_between / df1) / (ss_within / df2);
    double p_value = 1.0 - f_cdf(f, df1, df2);
 
    return {f, p_value, df1, alternative_hypothesis::greater};
}
 
// ============================================================================
// Bartlett's Test for Homogeneity of Variance
// ============================================================================
 
inline test_result bartlett_test(const std::vector<std::vector<double>>& groups)
{
    std::size_t k = groups.size();
    if (k < 2) {
        throw std::invalid_argument("statcpp::bartlett_test: need at least 2 groups");
    }
 
    std::vector<double> vars(k);
    std::vector<std::size_t> ns(k);
    std::size_t total_n = 0;
    double pooled_var_num = 0.0;
 
    for (std::size_t i = 0; i < k; ++i) {
        if (groups[i].size() < 2) {
            throw std::invalid_argument("statcpp::bartlett_test: each group needs at least 2 elements");
        }
        ns[i] = groups[i].size();
        total_n += ns[i];
        vars[i] = statcpp::sample_variance(groups[i].begin(), groups[i].end());
 
        if (vars[i] <= 0.0) {
            throw std::invalid_argument("statcpp::bartlett_test: zero or negative variance in group");
        }
 
        pooled_var_num += (ns[i] - 1) * vars[i];
    }
 
    double pooled_var = pooled_var_num / static_cast<double>(total_n - k);
 
    // Compute Bartlett's statistic
    double sum_log = 0.0;
    double sum_inv = 0.0;
 
    for (std::size_t i = 0; i < k; ++i) {
        double df_i = static_cast<double>(ns[i] - 1);
        sum_log += df_i * std::log(vars[i]);
        sum_inv += 1.0 / df_i;
    }
 
    double df_total = static_cast<double>(total_n - k);
    double chi2 = df_total * std::log(pooled_var) - sum_log;
 
    // Correction factor
    double c = 1.0 + (sum_inv - 1.0 / df_total) / (3.0 * (k - 1));
    chi2 /= c;
 
    double df = static_cast<double>(k - 1);
    double p_value = 1.0 - chisq_cdf(chi2, df);
 
    return {chi2, p_value, df, alternative_hypothesis::greater};
}
 
// ============================================================================
// Wilcoxon Signed-Rank Test
// ============================================================================
 
template <typename Iterator>
test_result wilcoxon_signed_rank_test(Iterator first, Iterator last, double mu0 = 0.0,
                                       alternative_hypothesis alt = alternative_hypothesis::two_sided)
{
    auto n = statcpp::count(first, last);
    if (n < 2) {
        throw std::invalid_argument("statcpp::wilcoxon_signed_rank_test: need at least 2 elements");
    }
 
    // Compute differences from mu0, excluding zeros
    std::vector<double> diffs;
    for (auto it = first; it != last; ++it) {
        double d = static_cast<double>(*it) - mu0;
        if (d != 0.0) {
            diffs.push_back(d);
        }
    }
 
    std::size_t n_nonzero = diffs.size();
    if (n_nonzero < 2) {
        throw std::invalid_argument("statcpp::wilcoxon_signed_rank_test: need at least 2 non-zero differences");
    }
 
    // Compute ranks of absolute differences
    std::vector<double> abs_diffs(n_nonzero);
    for (std::size_t i = 0; i < n_nonzero; ++i) {
        abs_diffs[i] = std::abs(diffs[i]);
    }
 
    auto ranks = compute_ranks_with_ties(abs_diffs.begin(), abs_diffs.end());
 
    // Compute W+ (sum of ranks of positive differences)
    double w = 0.0;
 
    for (std::size_t i = 0; i < n_nonzero; ++i) {
        if (diffs[i] > 0.0) {
            w += ranks[i];
        }
    }
 
    // Normal approximation for p-value (with continuity correction)
    double nn = static_cast<double>(n_nonzero);
    double mean_w = nn * (nn + 1.0) / 4.0;
    double var_w = nn * (nn + 1.0) * (2.0 * nn + 1.0) / 24.0;
 
    // Tie correction for variance
    // Sort absolute differences to compute tie groups
    std::vector<double> sorted_abs(abs_diffs.begin(), abs_diffs.end());
    std::sort(sorted_abs.begin(), sorted_abs.end());
    auto tie_groups = compute_tie_groups(sorted_abs);
    // Subtract tie correction: sum(t^3 - t) / 48
    double tie_correction = 0.0;
    for (auto t : tie_groups) {
        double td = static_cast<double>(t);
        tie_correction += td * td * td - td;
    }
    var_w -= tie_correction / 48.0;
 
    double se = std::sqrt(std::max(0.0, var_w));
 
    if (se == 0.0) {
        return {w, 1.0, static_cast<double>(n), alt};
    }
 
    double z;
    double p_value;
 
    switch (alt) {
        case alternative_hypothesis::less:
            z = (w - mean_w + 0.5) / se;
            p_value = norm_cdf(z);
            break;
        case alternative_hypothesis::greater:
            z = (w - mean_w - 0.5) / se;
            p_value = norm_sf(z);
            break;
        case alternative_hypothesis::two_sided:
        default:
            z = (w - mean_w) / se;
            if (z < 0) z = (w - mean_w + 0.5) / se;
            else z = (w - mean_w - 0.5) / se;
            p_value = 2.0 * std::min(norm_cdf(z), norm_sf(z));
            break;
    }
 
    return {w, p_value, static_cast<double>(n_nonzero), alt};
}
 
// ============================================================================
// Mann-Whitney U Test
// ============================================================================
 
template <typename Iterator1, typename Iterator2>
test_result mann_whitney_u_test(Iterator1 first1, Iterator1 last1,
                                 Iterator2 first2, Iterator2 last2,
                                 alternative_hypothesis alt = alternative_hypothesis::two_sided,
                                 bool correct = true)
{
    auto n1 = statcpp::count(first1, last1);
    auto n2 = statcpp::count(first2, last2);
 
    if (n1 < 2 || n2 < 2) {
        throw std::invalid_argument("statcpp::mann_whitney_u_test: need at least 2 elements in each sample");
    }
 
    // Combine and rank all observations
    std::vector<std::pair<double, int>> combined;
    combined.reserve(n1 + n2);
 
    for (auto it = first1; it != last1; ++it) {
        combined.push_back({static_cast<double>(*it), 1}); // group 1
    }
    for (auto it = first2; it != last2; ++it) {
        combined.push_back({static_cast<double>(*it), 2}); // group 2
    }
 
    // Sort by value
    std::sort(combined.begin(), combined.end(),
              [](const auto& a, const auto& b) { return a.first < b.first; });
 
    // Assign ranks with tie handling
    std::size_t total_n = n1 + n2;
    std::vector<double> ranks(total_n);
    std::size_t i = 0;
    while (i < total_n) {
        std::size_t j = i;
        while (j < total_n && combined[j].first == combined[i].first) {
            ++j;
        }
        double avg_rank = (static_cast<double>(i + 1) + static_cast<double>(j)) / 2.0;
        for (std::size_t k = i; k < j; ++k) {
            ranks[k] = avg_rank;
        }
        i = j;
    }
 
    // Compute R1 (sum of ranks in group 1)
    double r1 = 0.0;
    for (std::size_t k = 0; k < total_n; ++k) {
        if (combined[k].second == 1) {
            r1 += ranks[k];
        }
    }
 
    // Compute U1
    double u1 = r1 - n1 * (n1 + 1.0) / 2.0;
    (void)(static_cast<double>(n1 * n2) - u1);  // u2 not used but computed for reference
 
    // Normal approximation with tie correction
    double N = static_cast<double>(total_n);
    double mean_u = static_cast<double>(n1 * n2) / 2.0;
    double var_u = static_cast<double>(n1 * n2) / 12.0 * (N + 1.0);
 
    // Tie correction: compute tie groups from sorted combined values
    {
        std::vector<double> sorted_vals;
        sorted_vals.reserve(total_n);
        for (const auto& p : combined) {
            sorted_vals.push_back(p.first);
        }
        // combined is already sorted by value
        auto tie_groups = compute_tie_groups(sorted_vals);
        double tie_sum = 0.0;
        for (auto t : tie_groups) {
            double td = static_cast<double>(t);
            tie_sum += td * td * td - td;
        }
        // Corrected variance: n1*n2/(12*N*(N-1)) * (N^3 - N - tie_sum)
        // which equals n1*n2/12 * (N+1) - n1*n2/(12*N*(N-1)) * tie_sum
        if (N > 1.0 && tie_sum > 0.0) {
            var_u = static_cast<double>(n1 * n2) / (12.0 * N * (N - 1.0))
                    * (N * N * N - N - tie_sum);
        }
    }
 
    double se = std::sqrt(std::max(0.0, var_u));
 
    if (se == 0.0) {
        return {u1, 1.0, static_cast<double>(n1 + n2), alt};
    }
 
    // Continuity correction (Yates)
    double diff = u1 - mean_u;
    if (correct) {
        if (diff > 0) diff -= 0.5;
        else if (diff < 0) diff += 0.5;
    }
    double z = diff / se;
 
    double p_value;
    switch (alt) {
        case alternative_hypothesis::less:
            p_value = norm_cdf(z);
            break;
        case alternative_hypothesis::greater:
            p_value = norm_sf(z);
            break;
        case alternative_hypothesis::two_sided:
        default:
            p_value = 2.0 * std::min(norm_cdf(z), norm_sf(z));
            break;
    }
 
    return {u1, p_value, static_cast<double>(n1 + n2), alt};
}
 
// ============================================================================
// Kruskal-Wallis Test
// ============================================================================
 
inline test_result kruskal_wallis_test(const std::vector<std::vector<double>>& groups)
{
    std::size_t k = groups.size();
    if (k < 2) {
        throw std::invalid_argument("statcpp::kruskal_wallis_test: need at least 2 groups");
    }
 
    // Combine all observations with group labels
    std::vector<std::pair<double, std::size_t>> combined;
    std::vector<std::size_t> ns(k);
    std::size_t total_n = 0;
 
    for (std::size_t i = 0; i < k; ++i) {
        if (groups[i].empty()) {
            throw std::invalid_argument("statcpp::kruskal_wallis_test: empty group");
        }
        ns[i] = groups[i].size();
        total_n += ns[i];
        for (double x : groups[i]) {
            combined.push_back({x, i});
        }
    }
 
    // Sort by value
    std::sort(combined.begin(), combined.end(),
              [](const auto& a, const auto& b) { return a.first < b.first; });
 
    // Assign ranks with tie handling
    std::vector<double> ranks(total_n);
    std::size_t i = 0;
    while (i < total_n) {
        std::size_t j = i;
        while (j < total_n && combined[j].first == combined[i].first) {
            ++j;
        }
        double avg_rank = (static_cast<double>(i + 1) + static_cast<double>(j)) / 2.0;
        for (std::size_t m = i; m < j; ++m) {
            ranks[m] = avg_rank;
        }
        i = j;
    }
 
    // Compute sum of ranks for each group
    std::vector<double> rank_sums(k, 0.0);
    for (std::size_t m = 0; m < total_n; ++m) {
        rank_sums[combined[m].second] += ranks[m];
    }
 
    // Compute H statistic
    double n_d = static_cast<double>(total_n);
    double sum_term = 0.0;
    for (std::size_t g = 0; g < k; ++g) {
        double r_bar = rank_sums[g] / static_cast<double>(ns[g]);
        sum_term += static_cast<double>(ns[g]) * r_bar * r_bar;
    }
 
    double h = (12.0 / (n_d * (n_d + 1.0))) * sum_term - 3.0 * (n_d + 1.0);
 
    // Tie correction: H_corrected = H / (1 - sum(t^3 - t) / (N^3 - N))
    {
        std::vector<double> sorted_vals;
        sorted_vals.reserve(total_n);
        for (const auto& p : combined) {
            sorted_vals.push_back(p.first);
        }
        auto tie_groups = compute_tie_groups(sorted_vals);
        double tie_sum = 0.0;
        for (auto t : tie_groups) {
            double td = static_cast<double>(t);
            tie_sum += td * td * td - td;
        }
        double denom = n_d * n_d * n_d - n_d;
        if (tie_sum > 0.0 && denom > 0.0) {
            double correction = 1.0 - tie_sum / denom;
            if (correction > 0.0) {
                h /= correction;
            } else {
                // All observations are tied: the corrected statistic is 0 (avoid /0).
                h = 0.0;
            }
        }
    }
 
    // Approximate p-value using chi-square distribution
    double df = static_cast<double>(k - 1);
    double p_value = 1.0 - chisq_cdf(h, df);
 
    return {h, p_value, df, alternative_hypothesis::greater};
}
 
// ============================================================================
// Fisher's Exact Test (2x2 table)
// ============================================================================
 
inline test_result fisher_exact_test(std::uint64_t a, std::uint64_t b,
                                      std::uint64_t c, std::uint64_t d,
                                      alternative_hypothesis alt = alternative_hypothesis::two_sided)
{
    // 2x2 contingency table:
    //       | Col1 | Col2 | Row Total
    // Row1  |  a   |  b   |   a+b
    // Row2  |  c   |  d   |   c+d
    // Col   | a+c  | b+d  |   n
 
    std::uint64_t n = a + b + c + d;
    std::uint64_t row1 = a + b;
    (void)(c + d);  // row2 not used
    std::uint64_t col1 = a + c;
    std::uint64_t col2 = b + d;
 
    // Compute p-value under hypergeometric distribution
    // P(X = a) where X ~ Hypergeometric(n, col1, row1)
 
    // Probability of observed table
    double log_p_obs = log_binomial_coef(col1, a) + log_binomial_coef(col2, b) - log_binomial_coef(n, row1);
    double p_obs = std::exp(log_p_obs);
 
    double p_value = 0.0;
 
    // Compute range of possible values for a
    std::uint64_t a_min = (row1 > col2) ? row1 - col2 : 0;
    std::uint64_t a_max = std::min(row1, col1);
 
    switch (alt) {
        case alternative_hypothesis::less:
            for (std::uint64_t x = a_min; x <= a; ++x) {
                double log_p = log_binomial_coef(col1, x) + log_binomial_coef(col2, row1 - x) - log_binomial_coef(n, row1);
                p_value += std::exp(log_p);
            }
            break;
        case alternative_hypothesis::greater:
            for (std::uint64_t x = a; x <= a_max; ++x) {
                double log_p = log_binomial_coef(col1, x) + log_binomial_coef(col2, row1 - x) - log_binomial_coef(n, row1);
                p_value += std::exp(log_p);
            }
            break;
        case alternative_hypothesis::two_sided:
        default:
            // Sum probabilities of tables as extreme or more extreme
            for (std::uint64_t x = a_min; x <= a_max; ++x) {
                double log_p = log_binomial_coef(col1, x) + log_binomial_coef(col2, row1 - x) - log_binomial_coef(n, row1);
                double p = std::exp(log_p);
                if (p <= p_obs + 1e-10) {
                    p_value += p;
                }
            }
            break;
    }
 
    p_value = std::min(1.0, p_value);
 
    // Odds ratio as test statistic
    double odds_ratio = (b == 0 || c == 0) ? std::numeric_limits<double>::infinity()
                                           : (static_cast<double>(a) * d) / (static_cast<double>(b) * c);
 
    return {odds_ratio, p_value, std::numeric_limits<double>::quiet_NaN(), alt};
}
 
} // namespace statcpp