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+# Effect Sizes and Power Analysis
+
+This document provides guidance on calculating, interpreting, and reporting effect sizes, as well as conducting power analyses for study planning.
+
+## Why Effect Sizes Matter
+
+1. **Statistical significance ≠ practical significance**: p-values only tell if an effect exists, not how large it is
+2. **Sample size dependent**: With large samples, trivial effects become "significant"
+3. **Interpretation**: Effect sizes provide magnitude and practical importance
+4. **Meta-analysis**: Effect sizes enable combining results across studies
+5. **Power analysis**: Required for sample size determination
+
+**Golden rule**: ALWAYS report effect sizes alongside p-values.
+
+---
+
+## Effect Sizes by Analysis Type
+
+### T-Tests and Mean Differences
+
+#### Cohen's d (Standardized Mean Difference)
+
+**Formula**:
+- Independent groups: d = (M₁ - M₂) / SD_pooled
+- Paired groups: d = M_diff / SD_diff
+
+**Interpretation** (Cohen, 1988):
+- Small: |d| = 0.20
+- Medium: |d| = 0.50
+- Large: |d| = 0.80
+
+**Context-dependent interpretation**:
+- In education: d = 0.40 is typical for successful interventions
+- In psychology: d = 0.40 is considered meaningful
+- In medicine: Small effect sizes can be clinically important
+
+**Python calculation**:
+```python
+import pingouin as pg
+import numpy as np
+
+# Independent t-test with effect size
+result = pg.ttest(group1, group2, correction=False)
+cohens_d = result['cohen-d'].values[0]
+
+# Manual calculation
+mean_diff = np.mean(group1) - np.mean(group2)
+pooled_std = np.sqrt((np.var(group1, ddof=1) + np.var(group2, ddof=1)) / 2)
+cohens_d = mean_diff / pooled_std
+
+# Paired t-test
+result = pg.ttest(pre, post, paired=True)
+cohens_d = result['cohen-d'].values[0]
+```
+
+**Confidence intervals for d**:
+```python
+from pingouin import compute_effsize_from_t
+
+d, ci = compute_effsize_from_t(t_statistic, nx=n1, ny=n2, eftype='cohen')
+```
+
+---
+
+#### Hedges' g (Bias-Corrected d)
+
+**Why use it**: Cohen's d has slight upward bias with small samples (n < 20)
+
+**Formula**: g = d × correction_factor, where correction_factor = 1 - 3/(4df - 1)
+
+**Python calculation**:
+```python
+result = pg.ttest(group1, group2, correction=False)
+hedges_g = result['hedges'].values[0]
+```
+
+**Use Hedges' g when**:
+- Sample sizes are small (n < 20 per group)
+- Conducting meta-analyses (standard in meta-analysis)
+
+---
+
+#### Glass's Δ (Delta)
+
+**When to use**: When one group is a control with known variability
+
+**Formula**: Δ = (M₁ - M₂) / SD_control
+
+**Use cases**:
+- Clinical trials (use control group SD)
+- When treatment affects variability
+
+---
+
+### ANOVA
+
+#### Eta-squared (η²)
+
+**What it measures**: Proportion of total variance explained by factor
+
+**Formula**: η² = SS_effect / SS_total
+
+**Interpretation**:
+- Small: η² = 0.01 (1% of variance)
+- Medium: η² = 0.06 (6% of variance)
+- Large: η² = 0.14 (14% of variance)
+
+**Limitation**: Biased with multiple factors (sums to > 1.0)
+
+**Python calculation**:
+```python
+import pingouin as pg
+
+# One-way ANOVA
+aov = pg.anova(dv='value', between='group', data=df)
+eta_squared = aov['SS'][0] / aov['SS'].sum()
+
+# Or use pingouin directly
+aov = pg.anova(dv='value', between='group', data=df, detailed=True)
+eta_squared = aov['np2'][0]  # Note: pingouin reports partial eta-squared
+```
+
+---
+
+#### Partial Eta-squared (η²_p)
+
+**What it measures**: Proportion of variance explained by factor, excluding other factors
+
+**Formula**: η²_p = SS_effect / (SS_effect + SS_error)
+
+**Interpretation**: Same benchmarks as η²
+
+**When to use**: Multi-factor ANOVA (standard in factorial designs)
+
+**Python calculation**:
+```python
+aov = pg.anova(dv='value', between=['factor1', 'factor2'], data=df)
+# pingouin reports partial eta-squared by default
+partial_eta_sq = aov['np2']
+```
+
+---
+
+#### Omega-squared (ω²)
+
+**What it measures**: Less biased estimate of population variance explained
+
+**Why use it**: η² overestimates effect size; ω² provides better population estimate
+
+**Formula**: ω² = (SS_effect - df_effect × MS_error) / (SS_total + MS_error)
+
+**Interpretation**: Same benchmarks as η², but typically smaller values
+
+**Python calculation**:
+```python
+def omega_squared(aov_table):
+    ss_effect = aov_table.loc[0, 'SS']
+    ss_total = aov_table['SS'].sum()
+    ms_error = aov_table.loc[aov_table.index[-1], 'MS']  # Residual MS
+    df_effect = aov_table.loc[0, 'DF']
+
+    omega_sq = (ss_effect - df_effect * ms_error) / (ss_total + ms_error)
+    return omega_sq
+```
+
+---
+
+#### Cohen's f
+
+**What it measures**: Effect size for ANOVA (analogous to Cohen's d)
+
+**Formula**: f = √(η² / (1 - η²))
+
+**Interpretation**:
+- Small: f = 0.10
+- Medium: f = 0.25
+- Large: f = 0.40
+
+**Python calculation**:
+```python
+eta_squared = 0.06  # From ANOVA
+cohens_f = np.sqrt(eta_squared / (1 - eta_squared))
+```
+
+**Use in power analysis**: Required for ANOVA power calculations
+
+---
+
+### Correlation
+
+#### Pearson's r / Spearman's ρ
+
+**Interpretation**:
+- Small: |r| = 0.10
+- Medium: |r| = 0.30
+- Large: |r| = 0.50
+
+**Important notes**:
+- r² = coefficient of determination (proportion of variance explained)
+- r = 0.30 means 9% shared variance (0.30² = 0.09)
+- Consider direction (positive/negative) and context
+
+**Python calculation**:
+```python
+import pingouin as pg
+
+# Pearson correlation with CI
+result = pg.corr(x, y, method='pearson')
+r = result['r'].values[0]
+ci = [result['CI95%'][0][0], result['CI95%'][0][1]]
+
+# Spearman correlation
+result = pg.corr(x, y, method='spearman')
+rho = result['r'].values[0]
+```
+
+---
+
+### Regression
+
+#### R² (Coefficient of Determination)
+
+**What it measures**: Proportion of variance in Y explained by model
+
+**Interpretation**:
+- Small: R² = 0.02
+- Medium: R² = 0.13
+- Large: R² = 0.26
+
+**Context-dependent**:
+- Physical sciences: R² > 0.90 expected
+- Social sciences: R² > 0.30 considered good
+- Behavior prediction: R² > 0.10 may be meaningful
+
+**Python calculation**:
+```python
+from sklearn.metrics import r2_score
+from statsmodels.api import OLS
+
+# Using statsmodels
+model = OLS(y, X).fit()
+r_squared = model.rsquared
+adjusted_r_squared = model.rsquared_adj
+
+# Manual
+r_squared = 1 - (SS_residual / SS_total)
+```
+
+---
+
+#### Adjusted R²
+
+**Why use it**: R² artificially increases when adding predictors; adjusted R² penalizes model complexity
+
+**Formula**: R²_adj = 1 - (1 - R²) × (n - 1) / (n - k - 1)
+
+**When to use**: Always report alongside R² for multiple regression
+
+---
+
+#### Standardized Regression Coefficients (β)
+
+**What it measures**: Effect of one-SD change in predictor on outcome (in SD units)
+
+**Interpretation**: Similar to Cohen's d
+- Small: |β| = 0.10
+- Medium: |β| = 0.30
+- Large: |β| = 0.50
+
+**Python calculation**:
+```python
+from scipy import stats
+
+# Standardize variables first
+X_std = (X - X.mean()) / X.std()
+y_std = (y - y.mean()) / y.std()
+
+model = OLS(y_std, X_std).fit()
+beta = model.params
+```
+
+---
+
+#### f² (Cohen's f-squared for Regression)
+
+**What it measures**: Effect size for individual predictors or model comparison
+
+**Formula**: f² = R²_AB - R²_A / (1 - R²_AB)
+
+Where:
+- R²_AB = R² for full model with predictor
+- R²_A = R² for reduced model without predictor
+
+**Interpretation**:
+- Small: f² = 0.02
+- Medium: f² = 0.15
+- Large: f² = 0.35
+
+**Python calculation**:
+```python
+# Compare two nested models
+model_full = OLS(y, X_full).fit()
+model_reduced = OLS(y, X_reduced).fit()
+
+r2_full = model_full.rsquared
+r2_reduced = model_reduced.rsquared
+
+f_squared = (r2_full - r2_reduced) / (1 - r2_full)
+```
+
+---
+
+### Categorical Data Analysis
+
+#### Cramér's V
+
+**What it measures**: Association strength for χ² test (works for any table size)
+
+**Formula**: V = √(χ² / (n × (k - 1)))
+
+Where k = min(rows, columns)
+
+**Interpretation** (for k > 2):
+- Small: V = 0.07
+- Medium: V = 0.21
+- Large: V = 0.35
+
+**For 2×2 tables**: Use phi coefficient (φ)
+
+**Python calculation**:
+```python
+from scipy.stats.contingency import association
+
+# Cramér's V
+cramers_v = association(contingency_table, method='cramer')
+
+# Phi coefficient (for 2x2)
+phi = association(contingency_table, method='pearson')
+```
+
+---
+
+#### Odds Ratio (OR) and Risk Ratio (RR)
+
+**For 2×2 contingency tables**:
+
+|           | Outcome + | Outcome - |
+|-----------|-----------|-----------|
+| Exposed   | a         | b         |
+| Unexposed | c         | d         |
+
+**Odds Ratio**: OR = (a/b) / (c/d) = ad / bc
+
+**Interpretation**:
+- OR = 1: No association
+- OR > 1: Positive association (increased odds)
+- OR < 1: Negative association (decreased odds)
+- OR = 2: Twice the odds
+- OR = 0.5: Half the odds
+
+**Risk Ratio**: RR = (a/(a+b)) / (c/(c+d))
+
+**When to use**:
+- Cohort studies: Use RR (more interpretable)
+- Case-control studies: Use OR (RR not available)
+- Logistic regression: OR is natural output
+
+**Python calculation**:
+```python
+import statsmodels.api as sm
+
+# From contingency table
+odds_ratio = (a * d) / (b * c)
+
+# Confidence interval
+table = np.array([[a, b], [c, d]])
+oddsratio, pvalue = stats.fisher_exact(table)
+
+# From logistic regression
+model = sm.Logit(y, X).fit()
+odds_ratios = np.exp(model.params)  # Exponentiate coefficients
+ci = np.exp(model.conf_int())  # Exponentiate CIs
+```
+
+---
+
+### Bayesian Effect Sizes
+
+#### Bayes Factor (BF)
+
+**What it measures**: Ratio of evidence for alternative vs. null hypothesis
+
+**Interpretation**:
+- BF₁₀ = 1: Equal evidence for H₁ and H₀
+- BF₁₀ = 3: H₁ is 3× more likely than H₀ (moderate evidence)
+- BF₁₀ = 10: H₁ is 10× more likely than H₀ (strong evidence)
+- BF₁₀ = 100: H₁ is 100× more likely than H₀ (decisive evidence)
+- BF₁₀ = 0.33: H₀ is 3× more likely than H₁
+- BF₁₀ = 0.10: H₀ is 10× more likely than H₁
+
+**Classification** (Jeffreys, 1961):
+- 1-3: Anecdotal evidence
+- 3-10: Moderate evidence
+- 10-30: Strong evidence
+- 30-100: Very strong evidence
+- >100: Decisive evidence
+
+**Python calculation**:
+```python
+import pingouin as pg
+
+# Bayesian t-test
+result = pg.ttest(group1, group2, correction=False)
+# Note: pingouin doesn't include BF; use other packages
+
+# Using JASP or BayesFactor (R) via rpy2
+# Or implement using numerical integration
+```
+
+---
+
+## Power Analysis
+
+### Concepts
+
+**Statistical power**: Probability of detecting an effect if it exists (1 - β)
+
+**Conventional standards**:
+- Power = 0.80 (80% chance of detecting effect)
+- α = 0.05 (5% Type I error rate)
+
+**Four interconnected parameters** (given 3, can solve for 4th):
+1. Sample size (n)
+2. Effect size (d, f, etc.)
+3. Significance level (α)
+4. Power (1 - β)
+
+---
+
+### A Priori Power Analysis (Planning)
+
+**Purpose**: Determine required sample size before study
+
+**Steps**:
+1. Specify expected effect size (from literature, pilot data, or minimum meaningful effect)
+2. Set α level (typically 0.05)
+3. Set desired power (typically 0.80)
+4. Calculate required n
+
+**Python implementation**:
+```python
+from statsmodels.stats.power import (
+    tt_ind_solve_power,
+    zt_ind_solve_power,
+    FTestAnovaPower,
+    NormalIndPower
+)
+
+# T-test power analysis
+n_required = tt_ind_solve_power(
+    effect_size=0.5,  # Cohen's d
+    alpha=0.05,
+    power=0.80,
+    ratio=1.0,  # Equal group sizes
+    alternative='two-sided'
+)
+
+# ANOVA power analysis
+anova_power = FTestAnovaPower()
+n_per_group = anova_power.solve_power(
+    effect_size=0.25,  # Cohen's f
+    ngroups=3,
+    alpha=0.05,
+    power=0.80
+)
+
+# Correlation power analysis
+from pingouin import power_corr
+n_required = power_corr(r=0.30, power=0.80, alpha=0.05)
+```
+
+---
+
+### Post Hoc Power Analysis (After Study)
+
+**⚠️ CAUTION**: Post hoc power is controversial and often not recommended
+
+**Why it's problematic**:
+- Observed power is a direct function of p-value
+- If p > 0.05, power is always low
+- Provides no additional information beyond p-value
+- Can be misleading
+
+**When it might be acceptable**:
+- Study planning for future research
+- Using effect size from multiple studies (not just your own)
+- Explicit goal is sample size for replication
+
+**Better alternatives**:
+- Report confidence intervals for effect sizes
+- Conduct sensitivity analysis
+- Report minimum detectable effect size
+
+---
+
+### Sensitivity Analysis
+
+**Purpose**: Determine minimum detectable effect size given study parameters
+
+**When to use**: After study is complete, to understand study's capability
+
+**Python implementation**:
+```python
+# What effect size could we detect with n=50 per group?
+detectable_effect = tt_ind_solve_power(
+    effect_size=None,  # Solve for this
+    nobs1=50,
+    alpha=0.05,
+    power=0.80,
+    ratio=1.0,
+    alternative='two-sided'
+)
+
+print(f"With n=50 per group, we could detect d ≥ {detectable_effect:.2f}")
+```
+
+---
+
+## Reporting Effect Sizes
+
+### APA Style Guidelines
+
+**T-test example**:
+> "Group A (M = 75.2, SD = 8.5) scored significantly higher than Group B (M = 68.3, SD = 9.2), t(98) = 3.82, p < .001, d = 0.77, 95% CI [0.36, 1.18]."
+
+**ANOVA example**:
+> "There was a significant main effect of treatment condition on test scores, F(2, 87) = 8.45, p < .001, η²p = .16. Post hoc comparisons using Tukey's HSD revealed..."
+
+**Correlation example**:
+> "There was a moderate positive correlation between study time and exam scores, r(148) = .42, p < .001, 95% CI [.27, .55]."
+
+**Regression example**:
+> "The regression model significantly predicted exam scores, F(3, 146) = 45.2, p < .001, R² = .48. Study hours (β = .52, p < .001) and prior GPA (β = .31, p < .001) were significant predictors."
+
+**Bayesian example**:
+> "A Bayesian independent samples t-test provided strong evidence for a difference between groups, BF₁₀ = 23.5, indicating the data are 23.5 times more likely under H₁ than H₀."
+
+---
+
+## Effect Size Pitfalls
+
+1. **Don't only rely on benchmarks**: Context matters; small effects can be meaningful
+2. **Report confidence intervals**: CIs show precision of effect size estimate
+3. **Distinguish statistical vs. practical significance**: Large n can make trivial effects "significant"
+4. **Consider cost-benefit**: Even small effects may be valuable if intervention is low-cost
+5. **Multiple outcomes**: Effect sizes vary across outcomes; report all
+6. **Don't cherry-pick**: Report effects for all planned analyses
+7. **Publication bias**: Published effects are often overestimated
+
+---
+
+## Quick Reference Table
+
+| Analysis | Effect Size | Small | Medium | Large |
+|----------|-------------|-------|--------|-------|
+| T-test | Cohen's d | 0.20 | 0.50 | 0.80 |
+| ANOVA | η², ω² | 0.01 | 0.06 | 0.14 |
+| ANOVA | Cohen's f | 0.10 | 0.25 | 0.40 |
+| Correlation | r, ρ | 0.10 | 0.30 | 0.50 |
+| Regression | R² | 0.02 | 0.13 | 0.26 |
+| Regression | f² | 0.02 | 0.15 | 0.35 |
+| Chi-square | Cramér's V | 0.07 | 0.21 | 0.35 |
+| Chi-square (2×2) | φ | 0.10 | 0.30 | 0.50 |
+
+---
+
+## Resources
+
+- Cohen, J. (1988). *Statistical Power Analysis for the Behavioral Sciences* (2nd ed.)
+- Lakens, D. (2013). Calculating and reporting effect sizes
+- Ellis, P. D. (2010). *The Essential Guide to Effect Sizes*