Statistical Analysis

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
• Analysing 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 visualisations (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:

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 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 modelling
• arviz: Bayesian visualisation 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

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

1. Descriptive statistics: M, SD, n for all groups/variables
2. Test statistics: Test name, statistic, df, exact p-value
3. Effect sizes: With confidence intervals
4. Assumption checks: Which tests were done, results, actions taken
5. 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

1. Intuitive interpretation: "Given the data, there is a 95% probability the parameter is in this interval"
2. Evidence for null: Can quantify support for no effect
3. Flexible: No p-hacking concerns; can analyse data as it arrives
4. 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 visualisations
  • 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

1. Pre-register analyses when possible to distinguish confirmatory from exploratory
2. Always check assumptions before interpreting results
3. Report effect sizes with confidence intervals
4. Report all planned analyses including non-significant results
5. Distinguish statistical from practical significance
6. Visualise data before and after analysis
7. Check diagnostics for regression/ANOVA (residual plots, VIF, etc.)
8. Conduct sensitivity analyses to assess robustness
9. Share data and code for reproducibility
10. Be transparent about violations, transformations, and decisions

Common Pitfalls to Avoid

1. P-hacking: Don't test multiple ways until something is significant
2. HARKing: Don't present exploratory findings as confirmatory
3. Ignoring assumptions: Check them and report violations
4. Confusing significance with importance: p < .05 ≠ meaningful effect
5. Not reporting effect sizes: Essential for interpretation
6. Cherry-picking results: Report all planned analyses
7. Misinterpreting p-values: They're NOT probability that hypothesis is true
8. Multiple comparisons: Correct for family-wise error when appropriate
9. Ignoring missing data: Understand mechanism (MCAR, MAR, MNAR)
10. 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 visualisations
• 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 Behavioural 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)