statistical-analysis

# Statistical Analysis ## Overview Conduct hypothesis tests (t-tests, ANOVA, chi-square), regression, correlation, and Bayesian analyses with systematic assumption checking, effect sizes, and APA-style reporting. The goal is an analysis a reviewer could not tear apart: the right test, verified assumptions, honest effect sizes, and a complete write-up. ## When to Use This Skill Use this skill when: - Conducting statistical hypothesis tests (t-tests, ANOVA, chi-square, non-parametric) - 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 --- ## Installation Use **uv** to install the libraries used in this skill. Pin versions in production; unpinned installs are fine for exploration. ```bash # Core frequentist stack (Python 3.10+; 3.12+ recommended for latest SciPy/ArviZ) uv pip install "pingouin>=0.6" "scipy>=1.11" "statsmodels>=0.14.6" pandas matplotlib seaborn # Bayesian modeling (PyMC 5 + ArviZ) uv pip install "pymc>=5.0" "arviz>=1.0" ``` **Compatibility notes (verified against pingouin 0.6.1, statsmodels 0.14.6, arviz 1.2, 2026):** - **Pingouin 0.6.0** renamed output columns to remove special characters: `p_val`, `cohen_d`, `CI95`, `p_unc` (previously `p-val`, `cohen-d`, `CI95%`, `p-unc` in 0.5.x). Examples below use the current names; if stuck on 0.5.x, use the hyphenated forms. - **statsmodels + SciPy**: use `statsmodels>=0.14.6` with `scipy>=1.11` to avoid `_lazywhere` import errors on SciPy 1.16+. - **ArviZ 1.x**: `az.summary()` now defaults to **89% intervals** (`eti89` columns) and the width parameter is `ci_prob` (not `hdi_prob`). To report a conventional 95% credible interval, pass `az.summary(trace, ci_prob=0.95)`. - **One-sided Bayes Factors are gone from Pingouin**: `pg.ttest(..., alternative='greater')` silently drops the `BF10` column, and `pg.bayesfactor_ttest` raises on one-sided alternatives. For one-sided Bayesian tests, use PyMC directly (compute the posterior probability of the directional hypothesis) or JASP/R's BayesFactor. For model-specific APIs (OLS, GLM, ARIMA), see the **statsmodels** skill. For PyMC workflows, see the **pymc** skill. --- ## Analysis Workflow Every sound analysis follows the same arc. Skipping steps is how analyses end up retracted, so work through them in order and say what you did at each one. 1. **Frame the question before touching the data.** State the hypothesis, the outcome and predictor variables, and the design (independent vs. paired, number of groups). Commit to a planned test now — choosing the test after peeking at results is p-hacking, even when done innocently. 2. **Inspect the data.** Per group: n, mean, SD, median, missing values. Plot the raw data (histograms or box plots) before any test. Unequal group sizes, missingness, floor/ceiling effects, and outliers all change what test is appropriate — surface them to the user rather than silently working around them. 3. **Select the test** using the quick reference below, or `references/test_selection_guide.md` for designs beyond the basics (counts, time-to-event, reliability, factorial). 4. **Check assumptions** with `scripts/assumption_checks.py`. If an assumption fails, switch to the remedial test (table below) and report both the plan and the change. 5. **Run the test** and always compute the effect size alongside it — a p-value says an effect exists; the effect size says whether anyone should care. 6. **Report** using the APA templates below, including descriptives, exact statistics, effect sizes with CIs, and the assumption checks performed. If the user only needs one step (e.g., "how many participants do I need?"), jump straight to that section — but still confirm the design assumptions the calculation rests on. --- ## Test Selection Guide ### Quick Reference: Choosing the Right Test Use `references/test_selection_guide.md` for comprehensive guidance (counts, survival, reliability, factorial designs). 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 providing direct probability statements about hypotheses, Bayes Factors quantifying evidence, and the ability to support the null. See `references/bayesian_statistics.md`. --- ## Assumption Checking **Always check assumptions before interpreting test results**, and report the checks — reviewers look for them. Use the bundled `scripts/assumption_checks.py` module. Run Python from the skill directory (`skills/statistical-analysis/`) or add `scripts/` to `sys.path`: ```python from assumption_checks import comprehensive_assumption_check # Outliers + normality (per group) + homogeneity of variance, with plots results = comprehensive_assumption_check( data=df, value_col='score', group_col='group', # Optional: for group comparisons alpha=0.05 ) ``` For targeted checks, import individual functions: ```python from assumption_checks import ( check_normality, # Shapiro-Wilk + Q-Q plot + histogram check_normality_per_group, check_homogeneity_of_variance, # Levene's test + box plots check_linearity, # scatter + residual plot for simple regression check_regression_diagnostics, # full OLS diagnostics (see Regression below) detect_outliers # IQR or z-score methods ) 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 (`pg.ttest` applies it automatically with `correction='auto'`) - For ANOVA → Use Welch's ANOVA (`pg.welch_anova`) or Brown-Forsythe - For regression → Use robust standard errors or weighted least squares **Linearity violated (regression):** - Add polynomial terms, transform variables, or use non-linear models / GAM Formal tests get oversensitive as n grows: for n ≥ 100, weigh the Q-Q plot more heavily than the Shapiro-Wilk p-value. See `references/assumptions_and_diagnostics.md` for comprehensive guidance. --- ## Running Statistical Tests Primary libraries: - **pingouin**: user-friendly tests that return effect sizes by default — prefer it for standard tests - **scipy.stats**: core statistical tests - **statsmodels**: regression, diagnostics, power analysis - **pymc** + **arviz**: Bayesian modeling and diagnostics ### T-Test with Complete Reporting ```python import pingouin as pg # correction='auto' applies Welch's correction when variances are unequal result = pg.ttest(group_a, group_b, correction='auto') # Pingouin >= 0.6 column names 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, ci_upper = result['CI95'].values[0] # CI for the mean difference print(f"t({df:.0f}) = {t_stat:.2f}, p = {p_value:.3f}, d = {cohens_d:.2f}") ``` ### ANOVA with Post-Hoc Tests ```python import pingouin as pg aov = pg.anova(dv='score', between='group', data=df, detailed=True) print(aov) # Effect size: partial eta-squared eta_p2 = aov['np2'].values[0] # If significant, conduct post-hoc tests (Tukey HSD controls family-wise error) if aov['p_unc'].values[0] < 0.05: posthoc = pg.pairwise_tukey(dv='score', between='group', data=df) print(posthoc) # includes Hedges' g per pair ``` ### Linear Regression with Diagnostics ```python import statsmodels.api as sm from assumption_checks import check_regression_diagnostics X = sm.add_constant(X_predictors) # Add intercept model = sm.OLS(y, X).fit() print(model.summary()) # 4-panel residual plot + Shapiro-Wilk, Breusch-Pagan, Durbin-Watson, VIF diag = check_regression_diagnostics(model) print(diag['interpretation']) print(diag['vif']) # If heteroscedasticity was flagged, report robust standard errors instead robust = model.get_robustcov_results('HC3') ``` ### 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) trace = pm.sample(2000, tune=1000) # ArviZ 1.x defaults to 89% intervals; request 95% explicitly for reporting print(az.summary(trace, var_names=['difference'], ci_prob=0.95)) # Direct probability statement (this is what one-sided questions become) prob_greater = np.mean(trace.posterior['difference'].values > 0) print(f"P(mu1 > mu2 | data) = {prob_greater:.3f}") # ArviZ 1.x removed az.plot_posterior; use plot_dist (on 0.x, plot_posterior still works) az.plot_dist(trace, var_names=['difference'], ci_prob=0.95) ``` Scale priors to the data (e.g., `sigma=10` suits outcomes with SD near 10; use the observed SD as a guide) and state the priors in the report. --- ## Effect Sizes **Effect sizes quantify magnitude; p-values only indicate existence.** Report one for every test. See `references/effect_sizes_and_power.md` for the full guide. ### 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 | Benchmarks are conventions, not laws — a "small" effect can matter enormously (drug side effects) and a "large" one can be trivial. Interpret in context. ### Calculating Effect Sizes Pingouin returns effect sizes with its tests (`cohen_d` from `pg.ttest`, `np2` from `pg.anova`, `hedges` from `pg.pairwise_tukey`; `r` from `pg.corr` is already an effect size). ### Confidence Intervals for Effect Sizes Report a CI for the effect size to show its precision. Use `pg.compute_esci` (note: `pg.compute_effsize_from_t` returns only the point estimate — it does **not** return a CI): ```python import pingouin as pg d = pg.compute_effsize(group_a, group_b, eftype='cohen') ci_lower, ci_upper = pg.compute_esci(stat=d, nx=len(group_a), ny=len(group_b), eftype='cohen', confidence=0.95) print(f"d = {d:.2f}, 95% CI [{ci_lower:.2f}, {ci_upper:.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 per group 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}") # One-way ANOVA: What n is needed to detect Cohen's f = 0.25? # Notes: the parameter is k_groups; effect_size is Cohen's f (f = sqrt(eta2/(1-eta2))); # and solve_power returns the TOTAL sample size, not n per group. import math anova_power = FTestAnovaPower() n_total = anova_power.solve_power( effect_size=0.25, k_groups=3, alpha=0.05, power=0.80 ) print(f"Required total N: {math.ceil(n_total)} ({math.ceil(n_total / 3)} per group)") ``` ### Sensitivity Analysis (Post-Study) Determine what effect size the study could detect: ```python # With n=50 per group, what effect could we detect at 80% power? 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 "observed power" (computing power from the observed effect) is circular and misleading — it is a deterministic function of the p-value. If a study is done and someone asks about power, run a sensitivity analysis instead. See `references/effect_sizes_and_power.md` for detailed guidance. --- ## Reporting Results Follow `references/reporting_standards.md` for APA style. Every report needs: 1. **Descriptive statistics**: M, SD, n for all groups/variables