statistical-analysis
用于研究数据的引导式统计分析——包括检验选择、假设检验、效应量、功效分析、贝叶斯替代方案以及符合 APA 格式的报告撰写。只要用户想要比较组别、检验假设、分析实验或调查数据、检查统计假设、计算所需样本量或撰写结果（即使他们从未点名某种特定检验），都适用。涵盖 t 检验、方差分析（ANOVA）、卡方检验、相关分析、回归分析、非参数方法以及贝叶斯方法。关于低层级模型 API，请参见 statsmodels 和 pymc 的相关技能。
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Statistical Analysis
Overview
Conduct hypothesis tests (t-tests, ANOVA, chi-square), regression, correlation, and Bayesian analyses with systematic assumption checking, effect sizes, and APA-style reporting. The goal is an analysis a reviewer could not tear apart: the right test, verified assumptions, honest effect sizes, and a complete write-up.
When to Use This Skill
Use this skill when:
Installation
Use uv to install the libraries used in this skill. Pin versions in production; unpinned installs are fine for exploration.
# Core frequentist stack (Python 3.10+; 3.12+ recommended for latest SciPy/ArviZ)
uv pip install "pingouin>=0.6" "scipy>=1.11" "statsmodels>=0.14.6" pandas matplotlib seaborn
# Bayesian modeling (PyMC 5 + ArviZ)
uv pip install "pymc>=5.0" "arviz>=1.0"Compatibility notes (verified against pingouin 0.6.1, statsmodels 0.14.6, arviz 1.2, 2026):
p_val, cohen_d, CI95, p_unc (previously p-val, cohen-d, CI95%, p-unc in 0.5.x). Examples below use the current names; if stuck on 0.5.x, use the hyphenated forms.statsmodels>=0.14.6 with scipy>=1.11 to avoid _lazywhere import errors on SciPy 1.16+.az.summary() now defaults to 89% intervals (eti89 columns) and the width parameter is ci_prob (not hdi_prob). To report a conventional 95% credible interval, pass az.summary(trace, ci_prob=0.95).pg.ttest(..., alternative='greater') silently drops the BF10 column, and pg.bayesfactor_ttest raises on one-sided alternatives. For one-sided Bayesian tests, use PyMC directly (compute the posterior probability of the directional hypothesis) or JASP/R's BayesFactor.For model-specific APIs (OLS, GLM, ARIMA), see the statsmodels skill. For PyMC workflows, see the pymc skill.
Analysis Workflow
Every sound analysis follows the same arc. Skipping steps is how analyses end up retracted, so work through them in order and say what you did at each one.
references/test_selection_guide.md for designs beyond the basics (counts, time-to-event, reliability, factorial).scripts/assumption_checks.py. If an assumption fails, switch to the remedial test (table below) and report both the plan and the change.If the user only needs one step (e.g., "how many participants do I need?"), jump straight to that section — but still confirm the design assumptions the calculation rests on.
Test Selection Guide
Quick Reference: Choosing the Right Test
Use references/test_selection_guide.md for comprehensive guidance (counts, survival, reliability, factorial designs). Quick reference:
Comparing Two Groups:
Comparing 3+ Groups:
Relationships:
Bayesian Alternatives:
All tests have Bayesian versions providing direct probability statements about hypotheses, Bayes Factors quantifying evidence, and the ability to support the null. See references/bayesian_statistics.md.
Assumption Checking
Always check assumptions before interpreting test results, and report the checks — reviewers look for them.
Use the bundled scripts/assumption_checks.py module. Run Python from the skill directory (skills/statistical-analysis/) or add scripts/ to sys.path:
from assumption_checks import comprehensive_assumption_check
# Outliers + normality (per group) + homogeneity of variance, with plots
results = comprehensive_assumption_check(
data=df,
value_col='score',
group_col='group', # Optional: for group comparisons
alpha=0.05
)For targeted checks, import individual functions:
from assumption_checks import (
check_normality, # Shapiro-Wilk + Q-Q plot + histogram
check_normality_per_group,
check_homogeneity_of_variance, # Levene's test + box plots
check_linearity, # scatter + residual plot for simple regression
check_regression_diagnostics, # full OLS diagnostics (see Regression below)
detect_outliers # IQR or z-score methods
)
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:
pg.ttest applies it automatically with correction='auto')pg.welch_anova) or Brown-ForsytheLinearity violated (regression):
Formal tests get oversensitive as n grows: for n ≥ 100, weigh the Q-Q plot more heavily than the Shapiro-Wilk p-value. See references/assumptions_and_diagnostics.md for comprehensive guidance.
Running Statistical Tests
Primary libraries:
T-Test with Complete Reporting
import pingouin as pg
# correction='auto' applies Welch's correction when variances are unequal
result = pg.ttest(group_a, group_b, correction='auto')
# Pingouin >= 0.6 column names
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, ci_upper = result['CI95'].values[0] # CI for the mean difference
print(f"t({df:.0f}) = {t_stat:.2f}, p = {p_value:.3f}, d = {cohens_d:.2f}")ANOVA with Post-Hoc Tests
import pingouin as pg
aov = pg.anova(dv='score', between='group', data=df, detailed=True)
print(aov)
# Effect size: partial eta-squared
eta_p2 = aov['np2'].values[0]
# If significant, conduct post-hoc tests (Tukey HSD controls family-wise error)
if aov['p_unc'].values[0] < 0.05:
posthoc = pg.pairwise_tukey(dv='score', between='group', data=df)
print(posthoc) # includes Hedges' g per pairLinear Regression with Diagnostics
import statsmodels.api as sm
from assumption_checks import check_regression_diagnostics
X = sm.add_constant(X_predictors) # Add intercept
model = sm.OLS(y, X).fit()
print(model.summary())
# 4-panel residual plot + Shapiro-Wilk, Breusch-Pagan, Durbin-Watson, VIF
diag = check_regression_diagnostics(model)
print(diag['interpretation'])
print(diag['vif'])
# If heteroscedasticity was flagged, report robust standard errors instead
robust = model.get_robustcov_results('HC3')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)
trace = pm.sample(2000, tune=1000)
# ArviZ 1.x defaults to 89% intervals; request 95% explicitly for reporting
print(az.summary(trace, var_names=['difference'], ci_prob=0.95))
# Direct probability statement (this is what one-sided questions become)
prob_greater = np.mean(trace.posterior['difference'].values > 0)
print(f"P(mu1 > mu2 | data) = {prob_greater:.3f}")
# ArviZ 1.x removed az.plot_posterior; use plot_dist (on 0.x, plot_posterior still works)
az.plot_dist(trace, var_names=['difference'], ci_prob=0.95)Scale priors to the data (e.g., sigma=10 suits outcomes with SD near 10; use the observed SD as a guide) and state the priors in the report.
Effect Sizes
Effect sizes quantify magnitude; p-values only indicate existence. Report one for every test. See references/effect_sizes_and_power.md for the full guide.
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 |
Benchmarks are conventions, not laws — a "small" effect can matter enormously (drug side effects) and a "large" one can be trivial. Interpret in context.
Calculating Effect Sizes
Pingouin returns effect sizes with its tests (cohen_d from pg.ttest, np2 from pg.anova, hedges from pg.pairwise_tukey; r from pg.corr is already an effect size).
Confidence Intervals for Effect Sizes
Report a CI for the effect size to show its precision. Use pg.compute_esci (note: pg.compute_effsize_from_t returns only the point estimate — it does not return a CI):
import pingouin as pg
d = pg.compute_effsize(group_a, group_b, eftype='cohen')
ci_lower, ci_upper = pg.compute_esci(stat=d, nx=len(group_a), ny=len(group_b),
eftype='cohen', confidence=0.95)
print(f"d = {d:.2f}, 95% CI [{ci_lower:.2f}, {ci_upper:.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 per group 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}")
# One-way ANOVA: What n is needed to detect Cohen's f = 0.25?
# Notes: the parameter is k_groups; effect_size is Cohen's f (f = sqrt(eta2/(1-eta2)));
# and solve_power returns the TOTAL sample size, not n per group.
import math
anova_power = FTestAnovaPower()
n_total = anova_power.solve_power(
effect_size=0.25,
k_groups=3,
alpha=0.05,
power=0.80
)
print(f"Required total N: {math.ceil(n_total)} ({math.ceil(n_total / 3)} per group)")Sensitivity Analysis (Post-Study)
Determine what effect size the study could detect:
# With n=50 per group, what effect could we detect at 80% power?
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 "observed power" (computing power from the observed effect) is circular and misleading — it is a deterministic function of the p-value. If a study is done and someone asks about power, run a sensitivity analysis instead.
See references/effect_sizes_and_power.md for detailed guidance.
Reporting Results
Follow references/reporting_standards.md for APA style. Every report needs:
p = .034, not p < .05; use p < .001 only below .001)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, 10) for group means). The posterior
distribution indicated that Group A scored higher than Group B
(M_diff = 6.8, 95% credible interval [3.2, 10.4]), with a 99.8%
posterior probability that Group A's mean exceeded Group B's mean.
Convergence diagnostics were satisfactory (all R-hat < 1.01, ESS > 1000).If a non-parametric test was used, report medians rather than means, the U/W/H statistic, and a rank-based effect size (e.g., rank-biserial correlation, returned by pg.mwu as RBC).
Bayesian Statistics
Consider Bayesian approaches when:
See references/bayesian_statistics.md for prior specification, Bayes Factors, credible intervals, hierarchical models, and convergence checking (R-hat < 1.01, sufficient ESS, posterior predictive checks).
Bundled Resources
References (references/)
Scripts (scripts/)
-
comprehensive_assumption_check(): outliers + normality + variance homogeneity in one call-
check_normality(), check_normality_per_group(): Shapiro-Wilk with Q-Q plots-
check_homogeneity_of_variance(): Levene's test with box plots-
check_regression_diagnostics(): 4-panel residual plots + Shapiro-Wilk, Breusch-Pagan, Durbin-Watson, VIF for fitted OLS models-
check_linearity(), detect_outliers()Statistical Integrity
These are the practices that keep an analysis defensible. They matter because the most common statistical failures are not computational errors — they are silent flexibility (testing until something works) and selective reporting.