Add biomedical statistical tests for non-parametric analysis

- Implement Wilcoxon Signed-Rank Test for paired samples and one-sample tests
- Implement Mann-Whitney U Test for independent group comparisons
- Include comprehensive biomedical examples with dummy clinical data
- Add detailed documentation explaining importance for biomedical students
- Cover common use cases: treatment effects, biomarker analysis, clinical trials
- Provide robust error handling and statistical interpretations

Features:
- Blood pressure, pain score, weight loss, and cholesterol analysis examples
- Drug efficacy, gender differences, disease staging, and immune response examples
- Clinical interpretation guidelines and statistical theory explanations
- Ready-to-use functions with medical data applications

Author: Brijesh03032001

Biomedical/README.md

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+# R for Biomedical Data Analysis
+
+This directory contains R implementations of essential statistical tests commonly used in biomedical research and clinical studies. These non-parametric tests are fundamental tools for biomedical students and researchers working with medical data.
+
+## 📋 Contents
+
+- [`wilcoxon_signed_rank_test.r`](wilcoxon_signed_rank_test.r) - Wilcoxon Signed-Rank Test for paired samples
+- [`mann_whitney_u_test.r`](mann_whitney_u_test.r) - Mann-Whitney U Test for independent samples
+- [`README.md`](README.md) - This documentation file
+
+## 🏥 Why These Tests Are Essential for Biomedical Students
+
+### 1. **Real-World Medical Data Challenges**
+Medical data rarely follows normal distributions due to:
+- **Skewed distributions**: Lab values, reaction times, survival data
+- **Outliers**: Extreme measurements that are medically significant
+- **Small sample sizes**: Pilot studies, rare diseases, expensive procedures
+- **Ordinal scales**: Pain scores, quality of life indices, severity ratings
+
+### 2. **Robust and Assumption-Free**
+Non-parametric tests like Wilcoxon and Mann-Whitney:
+- **No normality assumption**: Work with any distribution shape
+- **Outlier resistant**: Not affected by extreme values
+- **Rank-based**: Focus on relative ordering rather than exact values
+- **Clinically meaningful**: Often more relevant for medical decisions
+
+### 3. **Common Applications in Medicine**
+- **Clinical trials**: Comparing treatment effectiveness
+- **Before/after studies**: Evaluating intervention outcomes
+- **Biomarker research**: Comparing levels between groups
+- **Quality of life studies**: Analyzing patient-reported outcomes
+- **Diagnostic accuracy**: Comparing test performance
+
+## 📊 Statistical Tests Overview
+
+### Wilcoxon Signed-Rank Test
+**Purpose**: Compare paired samples or test single sample against a median
+
+**When to use**:
+- Before/after treatment comparisons
+- Matched pairs (e.g., twins, matched controls)
+- Repeated measurements on same subjects
+- Testing if median differs from a specific value
+
+**Medical Examples**:
+- Blood pressure before vs after medication
+- Pain scores before vs after treatment
+- Weight loss in diet studies
+- Biomarker changes over time
+
+**Assumptions**:
+- Paired observations or single sample
+- Data can be ranked
+- Differences are symmetrically distributed around the median
+
+### Mann-Whitney U Test (Wilcoxon Rank-Sum Test)
+**Purpose**: Compare two independent groups
+
+**When to use**:
+- Comparing treatment vs control groups
+- Comparing different populations
+- When groups are independent (not paired)
+- Alternative to two-sample t-test
+
+**Medical Examples**:
+- Drug efficacy: treatment vs placebo
+- Gender differences in biomarkers
+- Disease severity between stages
+- Age-related immune responses
+- Comparing diagnostic methods
+
+**Assumptions**:
+- Two independent samples
+- Data can be ranked
+- Similar distribution shapes (for location comparison)
+
+## 🚀 Quick Start Guide
+
+### Installation
+No additional packages required! Just R base installation.
+
+```r
+# Clone or download the files to your R working directory
+# Source the functions
+source("wilcoxon_signed_rank_test.r")
+source("mann_whitney_u_test.r")
+```
+
+### Basic Usage
+
+#### Wilcoxon Signed-Rank Test
+```r
+# Example: Blood pressure before and after treatment
+before <- c(145, 150, 138, 155, 142, 148, 152, 140)
+after <- c(138, 142, 135, 148, 136, 140, 145, 133)
+
+# Perform the test
+result <- wilcoxon_signed_rank_test(before, after, alternative = "greater")
+print(result)
+```
+
+#### Mann-Whitney U Test
+```r
+# Example: Comparing treatment vs control groups
+treatment <- c(78, 85, 92, 73, 88, 91, 76, 83)
+control <- c(65, 58, 71, 62, 69, 54, 67, 60)
+
+# Perform the test
+result <- mann_whitney_u_test(treatment, control, alternative = "greater")
+print(result)
+```
+
+## 🔬 Detailed Examples with Biomedical Data
+
+### Running the Examples
+Each R file contains comprehensive examples with dummy biomedical data:
+
+```r
+# Run Wilcoxon examples
+source("wilcoxon_signed_rank_test.r")
+run_biomedical_examples()
+
+# Run Mann-Whitney examples
+source("mann_whitney_u_test.r")
+run_biomedical_examples()
+```
+
+## 📈 Understanding the Results
+
+### Key Output Elements
+
+#### For Wilcoxon Signed-Rank Test:
+- **W+**: Sum of ranks for positive differences
+- **W-**: Sum of ranks for negative differences  
+- **Test statistic W**: Usually min(W+, W-)
+- **P-value**: Probability of observing the result by chance
+- **Effect size**: Magnitude of the difference
+
+#### For Mann-Whitney U Test:
+- **U1, U2**: U statistics for each group
+- **W1, W2**: Sum of ranks for each group
+- **Test statistic U**: Depends on alternative hypothesis
+- **P-value**: Probability of observing the result by chance
+- **Effect size estimate**: Magnitude of group difference
+
+### Interpreting P-values in Medical Context
+- **p < 0.001**: Highly significant - very strong evidence
+- **p < 0.01**: Very significant - strong evidence  
+- **p < 0.05**: Significant - moderate evidence
+- **p ≥ 0.05**: Not significant - insufficient evidence
+
+**Important**: Always consider clinical significance alongside statistical significance!
+
+## 🎯 Choosing the Right Test
+
+| Scenario | Test | Example |
+|----------|------|---------|
+| Same subjects, before/after | Wilcoxon Signed-Rank | Pre/post treatment blood pressure |
+| Paired subjects | Wilcoxon Signed-Rank | Twins, matched controls |
+| Single sample vs reference | Wilcoxon Signed-Rank | Patient cholesterol vs normal (200) |
+| Two independent groups | Mann-Whitney U | Treatment vs control groups |
+| Gender/age comparisons | Mann-Whitney U | Male vs female biomarker levels |
+| Disease stage comparison | Mann-Whitney U | Stage I vs Stage II severity |
+
+## 🧮 Statistical Theory (Simplified)
+
+### Wilcoxon Signed-Rank Test
+1. **Calculate differences** between paired observations
+2. **Rank the absolute differences** (ignore zeros)
+3. **Sum ranks** for positive and negative differences separately
+4. **Test statistic** is typically the smaller sum
+5. **Compare** to expected distribution under null hypothesis
+
+### Mann-Whitney U Test
+1. **Combine all observations** from both groups
+2. **Rank all values** from smallest to largest
+3. **Sum ranks** for each group separately
+4. **Calculate U statistics** using rank sums
+5. **Compare** to expected distribution under null hypothesis
+
+## 🔧 Advanced Features
+
+### Alternative Hypotheses
+- **"two.sided"**: Groups/conditions are different (default)
+- **"greater"**: First group/condition is greater than second
+- **"less"**: First group/condition is less than second
+
+### Handling Ties
+Both implementations use average ranks for tied values, which is the standard approach.
+
+### Effect Size
+- **Wilcoxon**: Based on Z-score and sample size
+- **Mann-Whitney**: Based on U statistic relative to maximum possible
+
+## 📚 Further Reading
+
+### Recommended Resources for Biomedical Students
+1. **"Biostatistics: A Foundation for Analysis in the Health Sciences"** - Wayne Daniel
+2. **"Statistical Methods in Medical Research"** - Armitage, Berry & Matthews  
+3. **"Nonparametric Statistical Methods"** - Hollander, Wolfe & Chicken
+4. **"Medical Statistics from Scratch"** - David Bowers
+
+### Online Resources
+- [Laerd Statistics](https://statistics.laerd.com/) - Excellent step-by-step guides
+- [StatsDirect](https://www.statsdirect.com/) - Comprehensive statistical reference
+- [BMJ Statistics Notes](https://www.bmj.com/about-bmj/resources-readers/publications/statistics-square-one) - Medical statistics primer
+
+## ⚠️ Important Considerations
+
+### When NOT to Use These Tests
+- **Large samples with normal data**: t-tests might be more powerful
+- **Survival data**: Use specialized survival analysis methods
+- **Repeated measures**: Consider mixed-effects models
+- **Multiple comparisons**: Adjust p-values appropriately
+
+### Common Pitfalls
+1. **Multiple testing**: Correct for multiple comparisons
+2. **Effect size**: Don't ignore practical significance
+3. **Sample size**: Very small samples need exact methods
+4. **Assumptions**: Ensure data can be meaningfully ranked
+
+### Data Quality Checks
+- Check for outliers and data entry errors
+- Verify assumptions are met
+- Consider the clinical context
+- Validate results with domain experts