# Statistical Analysis
Overview
Statistical analysis is a systematic process for testing hypotheses and quantifying relationships. Conduct hypothesis tests (t-test, ANOVA, chi-square), regression, correlation, and Bayesian analyses with assumption checks and APA reporting. Apply this skill for academic research.
When to Use This Skill
This skill should be used when: - Conducting statistical hypothesis tests (t-tests, ANOVA, chi-square) - Performing regression or correlation analyses - Running Bayesian statistical analyses - Checking statistical assumptions and diagnostics - Calculating effect sizes and conducting power analyses - Reporting statistical results in APA format - Analyzing experimental or observational data for research
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Core Capabilities
### 1. Test Selection and Planning - Choose appropriate statistical tests based on research questions and data characteristics - Conduct a priori power analyses to determine required sample sizes - Plan analysis strategies including multiple comparison corrections
### 2. Assumption Checking - Automatically verify all relevant assumptions before running tests - Provide diagnostic visualizations (Q-Q plots, residual plots, box plots) - Recommend remedial actions when assumptions are violated
### 3. Statistical Testing - Hypothesis testing: t-tests, ANOVA, chi-square, non-parametric alternatives - Regression: linear, multiple, logistic, with diagnostics - Correlations: Pearson, Spearman, with confidence intervals - Bayesian alternatives: Bayesian t-tests, ANOVA, regression with Bayes Factors
### 4. Effect Sizes and Interpretation - Calculate and interpret appropriate effect sizes for all analyses - Provide confidence intervals for effect estimates - Distinguish statistical from practical significance
### 5. Professional Reporting - Generate APA-style statistical reports - Create publication-ready figures and tables - Provide complete interpretation with all required statistics
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Workflow Decision Tree
Use this decision tree to determine your analysis path:
``` START │ ├─ Need to SELECT a statistical test? │ └─ YES → See "Test Selection Guide" │ └─ NO → Continue │ ├─ Ready to check ASSUMPTIONS? │ └─ YES → See "Assumption Checking" │ └─ NO → Continue │ ├─ Ready to run ANALYSIS? │ └─ YES → See "Running Statistical Tests" │ └─ NO → Continue │ └─ Need to REPORT results? └─ YES → See "Reporting Results" ```
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Test Selection Guide
Quick Reference: Choosing the Right Test
Use `references/test_selection_guide.md` for comprehensive guidance. Quick reference:
**Comparing Two Groups:** - Independent, continuous, normal → Independent t-test - Independent, continuous, non-normal → Mann-Whitney U test - Paired, continuous, normal → Paired t-test - Paired, continuous, non-normal → Wilcoxon signed-rank test - Binary outcome → Chi-square or Fisher's exact test
**Comparing 3+ Groups:** - Independent, continuous, normal → One-way ANOVA - Independent, continuous, non-normal → Kruskal-Wallis test - Paired, continuous, normal → Repeated measures ANOVA - Paired, continuous, non-normal → Friedman test
**Relationships:** - Two continuous variables → Pearson (normal) or Spearman correlation (non-normal) - Continuous outcome with predictor(s) → Linear regression - Binary outcome with predictor(s) → Logistic regression
**Bayesian Alternatives:** All tests have Bayesian versions that provide: - Direct probability statements about hypotheses - Bayes Factors quantifying evidence - Ability to support null hypothesis - See `references/bayesian_statistics.md`
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Assumption Checking
Systematic Assumption Verification
**ALWAYS check assumptions before interpreting test results.**
Use the provided `scripts/assumption_checks.py` module for automated checking:
```python from scripts.assumption_checks import comprehensive_assumption_check
# Comprehensive check with visualizations results = comprehensive_assumption_check( data=df, value_col='score', group_col='group', # Optional: for group comparisons alpha=0.05 ) ```
This performs: 1. **Outlier detection** (IQR and z-score methods) 2. **Normality testing** (Shapiro-Wilk test + Q-Q plots) 3. **Homogeneity of variance** (Levene's test + box plots) 4. **Interpretation and recommendations**
Individual Assumption Checks
For targeted checks, use individual functions:
```python from scripts.assumption_checks import ( check_normality, check_normality_per_group, check_homogeneity_of_variance, check_linearity, detect_outliers )
# Example: Check normality with visualization result = check_normality( data=df['score'], name='Test Score', alpha=0.05, plot=True ) print(result['interpretation']) print(result['recommendation']) ```
What to Do When Assumptions Are Violated
**Normality violated:** - Mild violation + n > 30 per group → Proceed with parametric test (robust) - Moderate violation → Use non-parametric alternative - Severe violation → Transform data or use non-parametric test
**Homogeneity of variance violated:** - For t-test → Use Welch's t-test - For ANOVA → Use Welch's ANOVA or Brown-Forsythe ANOVA - For regression → Use robust standard errors or weighted least squares
**Linearity violated (regression):** - Add polynomial terms - Transform variables - Use non-linear models or GAM
See `references/assumptions_and_diagnostics.md` for comprehensive guidance.
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Running Statistical Tests
Python Libraries
Primary libraries for statistical analysis: - **scipy.stats**: Core statistical tests - **statsmodels**: Advanced regression and diagnostics - **pingouin**: User-friendly statistical testing with effect sizes - **pymc**: Bayesian statistical modeling - **arviz**: Bayesian visualization and diagnostics
Example Analyses
T-Test with Complete Reporting
```python import pingouin as pg import numpy as np
# Run independent t-test result = pg.ttest(group_a, group_b, correction='auto')
# Extract results t_stat = result['T'].values[0] df = result['dof'].values[0] p_value = result['p-val'].values[0] cohens_d = result['cohen-d'].values[0] ci_lower = result['CI95%'].values[0][0] ci_upper = result['CI95%'].values[0][1]
# Report print(f"t({df:.0f}) = {t_stat:.2f}, p = {p_value:.3f}") print(f"Cohen's d = {cohens_d:.2f}, 95% CI [{ci_lower:.2f}, {ci_upper:.2f}]") ```
ANOVA with Post-Hoc Tests
```python import pingouin as pg
# One-way ANOVA aov = pg.anova(dv='score', between='group', data=df, detailed=True) print(aov)
# If significant, conduct post-hoc tests if aov['p-unc'].values[0] < 0.05: posthoc = pg.pairwise_tukey(dv='score', between='group', data=df) print(posthoc)
# Effect size eta_squared = aov['np2'].values[0] # Partial eta-squared print(f"Partial η² = {eta_squared:.3f}") ```
Linear Regression with Diagnostics
```python import statsmodels.api as sm from statsmodels.stats.outliers_influence import variance_inflation_factor
# Fit model X = sm.add_constant(X_predictors) # Add intercept model = sm.OLS(y, X).fit()
# Summary print(model.summary())
# Check multicollinearity (VIF) vif_data = pd.DataFrame() vif_data["Variable"] = X.columns vif_data["VIF"] = [variance_inflation_factor(X.values, i) for i in range(X.shape[1])] print(vif_data)
# Check assumptions residuals = model.resid fitted = model.fittedvalues
# Residual plots import matplotlib.pyplot as plt fig, axes = plt.subplots(2, 2, figsize=(12, 10))
# Residuals vs fitted axes[0, 0].scatter(fitted, residuals, alpha=0.6) axes[0, 0].axhline(y=0, color='r', linestyle='--') axes[0, 0].set_xlabel('Fitted values') axes[0, 0].set_ylabel('Residuals') axes[0, 0].set_title('Residuals vs Fitted')
# Q-Q plot from scipy import stats stats.probplot(residuals, dist="norm", plot=axes[0, 1]) axes[0, 1].set_title('Normal Q-Q')
# Scale-Location axes[1, 0].scatter(fitted, np.sqrt(np.abs(residuals / residuals.std())), alpha=0.6) axes[1, 0].set_xlabel('Fitted values') axes[1, 0].set_ylabel('√|Standardized residuals|') axes[1, 0].set_title('Scale-Location')
# Residuals histogram axes[1, 1].hist(residuals, bins=20, edgecolor='black', alpha=0.7) axes[1, 1].set_xlabel('Residuals') axes[1, 1].set_ylabel('Frequency') axes[1, 1].set_title('Histogram of Residuals')
plt.tight_layout() plt.show() ```
Bayesian T-Test
```python import pymc as pm import arviz as az import numpy as np
with pm.Model() as model: # Priors mu1 = pm.Normal('mu_group1', mu=0, sigma=10) mu2 = pm.Normal('mu_group2', mu=0, sigma=10) sigma = pm.HalfNormal('sigma', sigma=10)
# Likelihood y1 = pm.Normal('y1', mu=mu1, sigma=sigma, observed=group_a) y2 = pm.Normal('y2', mu=mu2, sigma=sigma, observed=group_b)
# Derived quantity diff = pm.Deterministic('difference', mu1 - mu2)
# Sample trace = pm.sample(2000, tune=1000, return_inferencedata=True)
# Summarize print(az.summary(trace, var_names=['difference']))
# Probability that group1 > group2 prob_greater = np.mean(trace.posterior['difference'].values > 0) print(f"P(μ₁ > μ₂ | data) = {prob_greater:.3f}")
# Plot posterior az.plot_posterior(trace, var_names=['difference'], ref_val=0) ```
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Effect Sizes
Always Calculate Effect Sizes
**Effect sizes quantify magnitude, while p-values only indicate existence of an effect.**
See `references/effect_sizes_and_power.md` for comprehensive guidance.
Quick Reference: Common Effect Sizes
| Test | Effect Size | Small | Medium | Large | |------|-------------|-------|--------|-------| | T-test | Cohen's d | 0.20 | 0.50 | 0.80 | | ANOVA | η²_p | 0.01 | 0.06 | 0.14 | | Correlation | r | 0.10 | 0.30 | 0.50 | | Regression | R² | 0.02 | 0.13 | 0.26 | | Chi-square | Cramér's V | 0.07 | 0.21 | 0.35 |
**Important**: Benchmarks are guidelines. Context matters!
Calculating Effect Sizes
Most effect sizes are automatically calculated by pingouin:
```python # T-test returns Cohen's d result = pg.ttest(x, y) d = result['cohen-d'].values[0]
# ANOVA returns partial eta-squared aov = pg.anova(dv='score', between='group', data=df) eta_p2 = aov['np2'].values[0]
# Correlation: r is already an effect size corr = pg.corr(x, y) r = corr['r'].values[0] ```
Confidence Intervals for Effect Sizes
Always report CIs to show precision:
```python from pingouin import compute_effsize_from_t
# For t-test d, ci = compute_effsize_from_t( t_statistic, nx=len(group1), ny=len(group2), eftype='cohen' ) print(f"d = {d:.2f}, 95% CI [{ci[0]:.2f}, {ci[1]:.2f}]") ```
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Power Analysis
A Priori Power Analysis (Study Planning)
Determine required sample size before data collection:
```python from statsmodels.stats.power import ( tt_ind_solve_power, FTestAnovaPower )
# T-test: What n is needed to detect d = 0.5? n_required = tt_ind_solve_power( effect_size=0.5, alpha=0.05, power=0.80, ratio=1.0, alternative='two-sided' ) print(f"Required n per group: {n_required:.0f}")
# ANOVA: What n is needed to detect f = 0.25? anova_power = FTestAnovaPower() n_per_group = anova_power.solve_power( effect_size=0.25, ngroups=3, alpha=0.05, power=0.80 ) print(f"Required n per group: {n_per_group:.0f}") ```
Sensitivity Analysis (Post-Study)
Determine what effect size you could detect:
```python # With n=50 per group, what effect could we detect? detectable_d = 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"Study could detect d ≥ {detectable_d:.2f}") ```
**Note**: Post-hoc power analysis (calculating power after study) is generally not recommended. Use sensitivity analysis instead.
See `references/effect_sizes_and_power.md` for detailed guidance.
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Reporting Results
APA Style Statistical Reporting
Follow guidelines in `references/reporting_standards.md`.
Essential Reporting Elements
- **Descriptive statistics**: M, SD, n for all groups/variables
- **Test statistics**: Test name, statistic, df, exact p-value
- **Effect sizes**: With confidence intervals
- **Assumption checks**: Which tests were done, results, actions taken
- **All planned analyses**: Including non-significant findings
Example Report Templates
Independent T-Test
``` Group A (n = 48, M = 75.2, SD = 8.5) scored significantly higher than Group B (n = 52, M = 68.3, SD = 9.2), t(98) = 3.82, p < .001, d = 0.77, 95% CI [0.36, 1.18], two-tailed. Assumptions of normality (Shapiro-Wilk: Group A W = 0.97, p = .18; Group B W = 0.96, p = .12) and homogeneity of variance (Levene's F(1, 98) = 1.23, p = .27) were satisfied. ```
One-Way ANOVA
``` A one-way ANOVA revealed a significant main effect of treatment condition on test scores, F(2, 147) = 8.45, p < .001, η²_p = .10. Post hoc comparisons using Tukey's HSD indicated that Condition A (M = 78.2, SD = 7.3) scored significantly higher than Condition B (M = 71.5, SD = 8.1, p = .002, d = 0.87) and Condition C (M = 70.1, SD = 7.9, p < .001, d = 1.07). Conditions B and C did not differ significantly (p = .52, d = 0.18). ```
Multiple Regression
``` Multiple linear regression was conducted to predict exam scores from study hours, prior GPA, and attendance. The overall model was significant, F(3, 146) = 45.2, p < .001, R² = .48, adjusted R² = .47. Study hours (B = 1.80, SE = 0.31, β = .35, t = 5.78, p < .001, 95% CI [1.18, 2.42]) and prior GPA (B = 8.52, SE = 1.95, β = .28, t = 4.37, p < .001, 95% CI [4.66, 12.38]) were significant predictors, while attendance was not (B = 0.15, SE = 0.12, β = .08, t = 1.25, p = .21, 95% CI [-0.09, 0.39]). Multicollinearity was not a concern (all VIF < 1.5). ```
Bayesian Analysis
``` A Bayesian independent samples t-test was conducted using weakly informative priors (Normal(0, 1) for mean difference). The posterior distribution indicated that Group A scored higher than Group B (M_diff = 6.8, 95% credible interval [3.2, 10.4]). The Bayes Factor BF₁₀ = 45.3 provided very strong evidence for a difference between groups, with a 99.8% posterior probability that Group A's mean exceeded Group B's mean. Convergence diagnostics were satisfactory (all R̂ < 1.01, ESS > 1000). ```
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Bayesian Statistics
When to Use Bayesian Methods
Consider Bayesian approaches when: - You have prior information to incorporate - You want direct probability statements about hypotheses - Sample size is small or planning sequential data collection - You need to quantify evidence for the null hypothesis - The model is complex (hierarchical, missing data)
See `references/bayesian_statistics.md` for comprehensive guidance on: - Bayes' theorem and interpretation - Prior specification (informative, weakly informative, non-informative) - Bayesian hypothesis testing with Bayes Factors - Credible intervals vs. confidence intervals - Bayesian t-tests, ANOVA, regression, and hierarchical models - Model convergence checking and posterior predictive checks
Key Advantages
- **Intuitive interpretation**: "Given the data, there is a 95% probability the parameter is in this interval"
- **Evidence for null**: Can quantify support for no effect
- **Flexible**: No p-hacking concerns; can analyze data as it arrives
- **Uncertainty quantification**: Full posterior distribution
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Resources
This skill includes comprehensive reference materials:
References Directory
- **test_selection_guide.md**: Decision tree for choosing appropriate statistical tests
- **assumptions_and_diagnostics.md**: Detailed guidance on checking and handling assumption violations
- **effect_sizes_and_power.md**: Calculating, interpreting, and reporting effect sizes; conducting power analyses
- **bayesian_statistics.md**: Complete guide to Bayesian analysis methods
- **reporting_standards.md**: APA-style reporting guidelines with examples
Scripts Directory
- **assumption_checks.py**: Automated assumption checking with visualizations - `comprehensive_assumption_check()`: Complete workflow - `check_normality()`: Normality testing with Q-Q plots - `check_homogeneity_of_variance()`: Levene's test with box plots - `check_linearity()`: Regression linearity checks - `detect_outliers()`: IQR and z-score outlier detection
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Best Practices
- **Pre-register analyses** when possible to distinguish confirmatory from exploratory
- **Always check assumptions** before interpreting results
- **Report effect sizes** with confidence intervals
- **Report all planned analyses** including non-significant results
- **Distinguish statistical from practical significance**
- **Visualize data** before and after analysis
- **Check diagnostics** for regression/ANOVA (residual plots, VIF, etc.)
- **Conduct sensitivity analyses** to assess robustness
- **Share data and code** for reproducibility
- **Be transparent** about violations, transformations, and decisions
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Common Pitfalls to Avoid
- **P-hacking**: Don't test multiple ways until something is significant
- **HARKing**: Don't present exploratory findings as confirmatory
- **Ignoring assumptions**: Check them and report violations
- **Confusing significance with importance**: p < .05 ≠ meaningful effect
- **Not reporting effect sizes**: Essential for interpretation
- **Cherry-picking results**: Report all planned analyses
- **Misinterpreting p-values**: They're NOT probability that hypothesis is true
- **Multiple comparisons**: Correct for family-wise error when appropriate
- **Ignoring missing data**: Understand mechanism (MCAR, MAR, MNAR)
- **Overinterpreting non-significant results**: Absence of evidence ≠ evidence of absence
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Getting Started Checklist
When beginning a statistical analysis:
- [ ] Define research question and hypotheses
- [ ] Determine appropriate statistical test (use test_selection_guide.md)
- [ ] Conduct power analysis to determine sample size
- [ ] Load and inspect data
- [ ] Check for missing data and outliers
- [ ] Verify assumptions using assumption_checks.py
- [ ] Run primary analysis
- [ ] Calculate effect sizes with confidence intervals
- [ ] Conduct post-hoc tests if needed (with corrections)
- [ ] Create visualizations
- [ ] Write results following reporting_standards.md
- [ ] Conduct sensitivity analyses
- [ ] Share data and code
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Support and Further Reading
For questions about: - **Test selection**: See references/test_selection_guide.md - **Assumptions**: See references/assumptions_and_diagnostics.md - **Effect sizes**: See references/effect_sizes_and_power.md - **Bayesian methods**: See references/bayesian_statistics.md - **Reporting**: See references/reporting_standards.md
**Key textbooks**: - Cohen, J. (1988). *Statistical Power Analysis for the Behavioral Sciences* - Field, A. (2013). *Discovering Statistics Using IBM SPSS Statistics* - Gelman, A., & Hill, J. (2006). *Data Analysis Using Regression and Multilevel/Hierarchical Models* - Kruschke, J. K. (2014). *Doing Bayesian Data Analysis*
**Online resources**: - APA Style Guide: https://apastyle.apa.org/ - Statistical Consulting: Cross Validated (stats.stackexchange.com)