Statistical Analysis

Overview

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. For academic research.

When to Use This Skill

Use when:

• Conducting 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

Core Capabilities

1. Test selection & planning — choose tests by research question and data type; a priori power analysis; multiple-comparison strategy.
2. Assumption checking — verify normality, homogeneity, linearity; diagnostic plots; remediation when violated.
3. Statistical testing — parametric and non-parametric tests; regression; correlation; Bayesian alternatives with Bayes Factors.
4. Effect sizes & interpretation — appropriate effect sizes with CIs; statistical vs. practical significance.
5. Professional reporting — APA-style reports, publication-ready figures and tables.

Workflow

SELECT a test?      → Test Selection Guide
CHECK assumptions?  → Assumption Checking
RUN analysis?       → Running Statistical Tests + references/code_examples.md
REPORT results?     → Reporting Results + references/apa_report_templates.md

Worked code for every step is in references/code_examples.md.

Test Selection Guide

Quick reference (full decision tree: references/test_selection_guide.md):

Two groups — independent + normal → independent t-test; independent + non-normal → Mann-Whitney U; paired + normal → paired t-test; paired + non-normal → Wilcoxon signed-rank; binary outcome → chi-square or Fisher's exact.

3+ groups — independent + normal → one-way ANOVA; independent + non-normal → Kruskal-Wallis; paired + normal → repeated-measures ANOVA; paired + non-normal → Friedman.

Relationships — two continuous → Pearson (normal) or Spearman (non-normal); continuous outcome + predictors → linear regression; binary outcome + predictors → logistic regression.

Bayesian alternatives exist for all of the above, giving direct probability statements, Bayes Factors, and the ability to support the null. See references/bayesian_statistics.md.

Assumption Checking

ALWAYS check assumptions before interpreting results. Use the bundled scripts/assumption_checks.py (comprehensive_assumption_check() and targeted functions — code in references/code_examples.md). It performs outlier detection (IQR + z-score), normality testing (Shapiro-Wilk + Q-Q), homogeneity of variance (Levene + box plots), and interpretation.

When violated:

• Normality — mild + n>30/group → proceed (robust); moderate → non-parametric; severe → transform or non-parametric.
• Homogeneity — t-test → Welch's; ANOVA → Welch's or Brown-Forsythe; regression → robust SE or WLS.
• Linearity (regression) — polynomial terms, transform variables, or GAM.

Detail: references/assumptions_and_diagnostics.md.

Running Statistical Tests

Primary libraries: scipy.stats (core tests), statsmodels (regression/diagnostics), pingouin (tests + effect sizes), pymc + arviz (Bayesian).

Copy-paste examples for t-test, ANOVA + post-hoc, linear regression with full diagnostics, and a Bayesian t-test: references/code_examples.md.

Effect Sizes

Effect sizes quantify magnitude; p-values only indicate existence. Always report them, with CIs. Benchmarks (guidelines only — context matters):

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

Calculation code and CI computation: references/code_examples.md. Full guidance: references/effect_sizes_and_power.md.

Power Analysis

Run a priori power analysis to size a study before collecting data, and sensitivity analysis afterward to report the smallest detectable effect. Avoid post-hoc power. Code: references/code_examples.md. Guidance: references/effect_sizes_and_power.md.

Reporting Results

Follow APA style (references/reporting_standards.md). Report descriptive statistics (M, SD, n), test statistics (name, statistic, df, exact p), effect sizes with CIs, assumption checks, and all planned analyses.

Ready-to-adapt report paragraphs for t-test, ANOVA, multiple regression, and Bayesian analysis: references/apa_report_templates.md.

Bayesian Statistics

Consider Bayesian methods when you have prior information, want direct probability statements, have small samples or sequential data, need to quantify evidence for the null, or have a complex (hierarchical, missing-data) model. Key advantages: intuitive interpretation, evidence for the null, flexibility, full uncertainty quantification. Comprehensive guide (priors, Bayes Factors, credible intervals, convergence checks): references/bayesian_statistics.md.

Best Practices

Pre-register analyses; always check assumptions; report effect sizes with CIs; report all planned analyses including non-significant ones; distinguish statistical from practical significance; visualize before and after; check regression/ANOVA diagnostics; run sensitivity analyses; share data and code; be transparent about violations and decisions.

Common Pitfalls

P-hacking; HARKing; ignoring assumptions; confusing significance with importance; not reporting effect sizes; cherry-picking results; misinterpreting p-values; uncorrected multiple comparisons; ignoring missing-data mechanism (MCAR/MAR/MNAR); overinterpreting non-significant results.

Getting Started Checklist

• Define research question and hypotheses
• Determine the test (references/test_selection_guide.md)
• Power analysis for sample size
• Load and inspect data; check missing data and outliers
• Verify assumptions (scripts/assumption_checks.py)
• Run primary analysis; compute effect sizes with CIs
• Post-hoc tests if needed (with corrections)
• Create visualizations
• Write results (references/reporting_standards.md)
• Sensitivity analyses; share data and code

Index of Bundled Resources

References (references/)

• test_selection_guide.md — decision tree for choosing a test
• assumptions_and_diagnostics.md — checking and handling violations
• effect_sizes_and_power.md — effect sizes and power analysis
• bayesian_statistics.md — complete Bayesian guide
• reporting_standards.md — APA reporting rules
• code_examples.md — worked Python for tests, diagnostics, Bayesian, power, effect sizes
• apa_report_templates.md — ready-to-adapt APA results paragraphs

Scripts (scripts/)

• assumption_checks.py — automated assumption checking with visualizations (comprehensive_assumption_check, check_normality, check_homogeneity_of_variance, check_linearity, detect_outliers)

Further Reading

Key textbooks: Cohen (1988), Field (2013), Gelman & Hill (2006), Kruschke (2014). Online: APA Style Guide (https://apastyle.apa.org/), Cross Validated (stats.stackexchange.com).