statistical-analysis
Guided statistical analysis with test selection and reporting. Use when you need help choosing appropriate tests for your data, assumption checking, power analysis, and APA-formatted results. Best for academic research reporting, test selection guidance. For implementing specific models programmatically use statsmodels.
name:statistical-analysisdescription:Guided statistical analysis with test selection and reporting. Use when you need help choosing appropriate tests for your data, assumption checking, power analysis, and APA-formatted results. Best for academic research reporting, test selection guidance. For implementing specific models programmatically use statsmodels.license:MIT licensemetadata:skill-author:K-Dense Inc.
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:
Core Capabilities
1. Test Selection and Planning
2. Assumption Checking
3. Statistical Testing
4. Effect Sizes and Interpretation
5. Professional Reporting
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:
Comparing 3+ Groups:
Relationships:
Bayesian Alternatives:
All tests have Bayesian versions that provide:
references/bayesian_statistics.mdAssumption 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_checkComprehensive check with visualizations
results = comprehensive_assumption_check(
data=df,
value_col='score',
group_col='group', # Optional: for group comparisons
alpha=0.05
)This performs:
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:
Homogeneity of variance violated:
Linearity violated (regression):
See references/assumptions_and_diagnostics.md for comprehensive guidance.
Running Statistical Tests
Python Libraries
Primary libraries for statistical analysis:
Example Analyses
T-Test with Complete Reporting
import pingouin as pg
import numpy as npRun 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 pgOne-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_factorFit 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.fittedvaluesResidual 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 npwith 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_tFor 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
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:
See references/bayesian_statistics.md for comprehensive guidance on:
Key Advantages
Resources
This skill includes comprehensive reference materials:
References Directory
Scripts Directory
-
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 detectionBest Practices
Common Pitfalls to Avoid
Getting Started Checklist
When beginning a statistical analysis:
Support and Further Reading
For questions about:
Key textbooks:
Online resources:
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