statistical-analysis - Agent Skills

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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