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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Installation

Use uv to install the libraries used in this skill. Pin versions in production; unpinned installs are fine for exploration.

# 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; ArviZ 0.23+ requires Python 3.12+)
uv pip install "pymc>=5.0" "arviz>=0.17"

Compatibility notes (2025–2026):

• Pingouin 0.5+ renamed output columns (p_val, cohen_d, CI95, p_unc) — examples below use the current names.
• statsmodels + SciPy: use statsmodels>=0.14.6 with scipy>=1.11 to avoid _lazywhere import errors on SciPy 1.16+.
• Pingouin Bayes Factors: one-sided BF for t-tests was removed in 0.5+; use dedicated packages (e.g. JASP, BayesFactor via R) or PyMC for hypothesis testing.

For model-specific APIs (OLS, GLM, ARIMA), see the statsmodels skill. For PyMC workflows, see the pymc skill.

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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 bundled scripts/assumption_checks.py module for automated checking. Run Python from the skill directory (skills/statistical-analysis/) or add scripts/ to sys.path:

from 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:

from 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

import pingouin as pg
import numpy as np

# Run independent t-test
result = pg.ttest(group_a, group_b, correction='auto')

# Extract results (Pingouin 0.5+ 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 = result['CI95'].values[0]
ci_lower, ci_upper = ci[0], ci[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

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

import