statistical-analysis

# 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 --- ## 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 --- ## 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" ``` --- ## 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` --- ## 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. --- ## 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) ``` --- ## 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}]") ``` --- ## 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. --- ## Reporting Results ### APA Style Statistical Reporting Follow guidelines in `references/reporting_standards.md`. ### Essential Reporting Elements