statistical-analysis

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Statistical analysis toolkit. Hypothesis tests (t-test, ANOVA, chi-square), regression, correlation, Bayesian stats, power analysis, assumption checks, APA reporting, for academic research.

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npxskills add davila7/claude-code-templates--skill statistical-analysis

Instructions

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:

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:

Outlier detection (IQR and z-score methods)
Normality testing (Shapiro-Wilk test + Q-Q plots)
Homogeneity of variance (Levene's test + box plots)
Interpretation and recommendations

Individual Assumption Checks

For targeted checks, use individual functions:

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

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

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 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

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:

# 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:

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:

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:

# 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

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).

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

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

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

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

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

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)

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