power-analysis

# Statistical Power Analysis Skill

## Purpose

Calculate statistical power and determine required sample sizes for research studies. Essential for experimental design, grant writing, and meeting NIH rigor and reproducibility standards.

## Core Concepts

### Statistical Power
**Definition:** Probability of detecting a true effect when it exists (1 - β)

**Standard:** Power ≥ 0.80 (80%) is typically required for NIH grants and pre-registration

### Key Parameters
1. **Effect Size (d, r, η²)** - Magnitude of the phenomenon
2. **Alpha (α)** - Type I error rate (typically 0.05)
3. **Power (1-β)** - Probability of detecting effect (typically 0.80)
4. **Sample Size (N)** - Number of participants/observations needed

### The Relationship
```
Power = f(Effect Size, Sample Size, Alpha, Test Type)

For given effect size and alpha:
↑ Sample Size → ↑ Power
↑ Effect Size → ↓ Sample Size needed
```

## When to Use This Skill

### Pre-Study (Prospective Power Analysis)
1. **Grant Proposals** - Justify requested sample size
2. **Study Design** - Determine recruitment needs
3. **Pre-Registration** - Document planned sample size with justification
4. **Resource Planning** - Estimate time and cost requirements
5. **Ethical Review** - Minimize participants while maintaining power

### Post-Study (Retrospective/Sensitivity Analysis)
1. **Null Results** - Was study adequately powered?
2. **Publication** - Report achieved power
3. **Meta-Analysis** - Assess individual study adequacy
4. **Study Critique** - Evaluate power of published work

## Common Study Designs

### 1. Independent Samples T-Test
**Use:** Compare two independent groups

**Formula:**
```
N per group = 2 * (z_α/2 + z_β)² * σ² / d²

Where:
- d = effect size (Cohen's d)
- α = significance level (typ. 0.05)
- β = Type II error (1 - power)
- σ² = pooled variance
```

**Example:**
```
Research Question: Does intervention improve test scores vs. control?
Effect Size: d = 0.5 (medium effect)
Alpha: 0.05
Power: 0.80

Result: N =