[NEW ALGORITHM] Kruskal-Wallis Test Algorithm #1362

Miscellaneous Algorithms/Kruskal-Wallis Algorithm/Kruskal-Wallis.c

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+//Kruskal-Wallis Algorithm
+
+#include <stdio.h>
+#include <stdlib.h>
+
+typedef struct {
+    double value;
+    int group;
+} DataPoint;
+
+int compare(const void *a, const void *b) {
+    return ((DataPoint *)a)->value > ((DataPoint *)b)->value ? 1 : -1;
+}
+
+void kruskal_wallis(int *groups, double *data, int n, int k) {
+    DataPoint *points = (DataPoint *)malloc(n * sizeof(DataPoint));
+
+    for (int i = 0; i < n; i++) {
+        points[i].value = data[i];
+        points[i].group = groups[i];
+    }
+
+    // Rank data
+    qsort(points, n, sizeof(DataPoint), compare);
+    double *rank = (double *)malloc(n * sizeof(double));
+    for (int i = 0; i < n; i++) {
+        rank[i] = i + 1;  // Ranks start at 1
+    }
+
+    // Sum of ranks for each group
+    double *rank_sum = (double *)calloc(k, sizeof(double));
+    int *group_count = (int *)calloc(k, sizeof(int));
+    for (int i = 0; i < n; i++) {
+        rank_sum[points[i].group - 1] += rank[i];
+        group_count[points[i].group - 1]++;
+    }
+
+    // Calculate H
+    double H = 0.0;
+    for (int j = 0; j < k; j++) {
+        H += (rank_sum[j] * rank_sum[j]) / group_count[j];
+    }
+    H = (12.0 / (n * (n + 1))) * H - 3 * (n + 1);
+
+    // Free allocated memory
+    free(points);
+    free(rank);
+    free(rank_sum);
+    free(group_count);
+
+    printf("Kruskal-Wallis H statistic: %lf\n", H);
+}
+
+int main() {
+    int groups[] = {1, 1, 2, 2, 3, 3}; // Group identifiers
+    double data[] = {10, 12, 20, 22, 30, 32}; // Observations
+    int n = sizeof(data) / sizeof(data[0]);
+    int k = 3; // Number of groups
+
+    kruskal_wallis(groups, data, n, k);
+    return 0;
+}

Miscellaneous Algorithms/Kruskal-Wallis Algorithm/readme.md

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+### Kruskal-Wallis Test Algorithm in C
+
+The Kruskal-Wallis test is a non-parametric statistical test used to determine whether there are statistically significant differences between the medians of three or more independent groups. It is an extension of the Mann-Whitney U test and is particularly useful when the assumptions of ANOVA (normality and homogeneity of variance) are not met.
+
+#### Steps of the Kruskal-Wallis Test
+
+1. **Rank all the data:** Combine all the groups' data into one dataset and rank the values. If there are ties, assign the average rank to the tied values.
+
+2. **Calculate the rank sums:** For each group, sum the ranks of the observations.
+
+3. **Compute the test statistic (H):** The Kruskal-Wallis test statistic is calculated using the formula:
+
+      H=N(N+1)12​∑nj​Rj2​​−3(N+1)
+
+   where:
+   - \( N \) = total number of observations across all groups
+   - \( R_j \) = sum of ranks for group \( j \)
+   - \( n_j \) = number of observations in group \( j \)
+
+4. **Determine the p-value:** Compare the test statistic \( H \) to the chi-squared distribution with \( k - 1 \) degrees of freedom (where \( k \) is the number of groups).
+
+5. **Decision:** If the p-value is less than the significance level (e.g., 0.05), reject the null hypothesis, indicating that at least one group median is different from the others.
+
+### Importance of the Kruskal-Wallis Test
+
+1. **Non-Parametric Nature:** It does not assume a normal distribution of the data, making it applicable in various real-world scenarios where data may not meet ANOVA assumptions.
+
+2. **Robustness:** The Kruskal-Wallis test is robust against outliers and non-homogeneity of variance, which are common issues in practical data analysis.
+
+3. **Multiple Groups Comparison:** It allows for the comparison of three or more groups simultaneously, saving time and resources compared to multiple pairwise tests.
+
+4. **Broad Applications:** It is widely used in various fields, including biology, psychology, and social sciences, for analyzing experimental and observational data.
+
+5. **Foundation for Further Analysis:** If the Kruskal-Wallis test indicates significant differences, post-hoc tests (like Dunn’s test) can be conducted to identify specific group differences.