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name: statistical-analysis description: Comprehensive statistical analysis toolkit for research. Conduct hypothesis tests (t-test, ANOVA, chi-square), regression, correlation, Bayesian stats, power analysis, assumption checks, and APA reporting. Use when the user asks about statistics, needs help analyzing data, or when writing methods sections that include statistical approaches.

Statistical Analysis

Comprehensive statistical testing, power analysis, and experimental design for reproducible research.

When to Use

• 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
• Planning experiments with proper power calculations
• Helping with the ANALYSIS phase of a research project

Workflow Decision Tree

START
│
├─ Need to SELECT a statistical test?
│  └─ See "Test Selection Guide"
│
├─ Ready to check ASSUMPTIONS?
│  └─ See "Assumption Checking"
│
├─ Ready to run ANALYSIS?
│  └─ See "Running Statistical Tests"
│
└─ Need to REPORT results?
   └─ See "Reporting Results (APA)"

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Test Selection Guide

Quick Reference: Choosing the Right Test

Comparing Two Groups:

Data Type	Distribution	Design	Test
Continuous	Normal	Independent	Independent t-test
Continuous	Non-normal	Independent	Mann-Whitney U
Continuous	Normal	Paired	Paired t-test
Continuous	Non-normal	Paired	Wilcoxon signed-rank
Binary	-	-	Chi-square / Fisher's exact

Comparing 3+ Groups:

Data Type	Distribution	Design	Test
Continuous	Normal	Independent	One-way ANOVA
Continuous	Non-normal	Independent	Kruskal-Wallis
Continuous	Normal	Paired	Repeated measures ANOVA
Continuous	Non-normal	Paired	Friedman test

Relationships:

Analysis	Use Case	Test
Two continuous vars	Normal	Pearson correlation
Two continuous vars	Non-normal	Spearman correlation
Continuous outcome + predictor(s)	Prediction	Linear regression
Binary outcome + predictor(s)	Classification	Logistic regression

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Assumption Checking

ALWAYS check assumptions before interpreting test results.

Key Assumptions to Check

import scipy.stats as stats
import numpy as np

# 1. Normality Test (Shapiro-Wilk)
stat, p = stats.shapiro(data)
print(f"Shapiro-Wilk: W={stat:.3f}, p={p:.3f}")
if p < 0.05:
    print("⚠️ Normality assumption violated - consider non-parametric test")

# 2. Homogeneity of Variance (Levene's test)
stat, p = stats.levene(group1, group2)
print(f"Levene's: F={stat:.3f}, p={p:.3f}")
if p < 0.05:
    print("⚠️ Variance assumption violated - use Welch's t-test")

# 3. Outlier Detection (IQR method)
Q1, Q3 = np.percentile(data, [25, 75])
IQR = Q3 - Q1
outliers = data[(data < Q1 - 1.5*IQR) | (data > Q3 + 1.5*IQR)]
print(f"Outliers detected: {len(outliers)}")

What to Do When Assumptions Are Violated

Assumption	Violation	Solution
Normality (mild, n>30)	Proceed	Parametric tests are robust
Normality (severe)	Transform	Use log/sqrt or non-parametric
Homogeneity of variance	t-test	Use Welch's t-test
Homogeneity of variance	ANOVA	Use Welch's ANOVA
Linearity (regression)	Violated	Add polynomial terms or use GAM

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Running Statistical Tests

Python Libraries

import scipy.stats as stats       # Core statistical tests
import statsmodels.api as sm      # Regression, diagnostics
import pingouin as pg             # User-friendly testing
import numpy as np
import pandas as pd

Common Analyses

T-Test with Complete Reporting

import pingouin as pg

# Independent t-test with effect size
result = pg.ttest(group_a, group_b, correction='auto')
print(f"t({result['dof'].values[0]:.0f}) = {result['T'].values[0]:.2f}, "
      f"p = {result['p-val'].values[0]:.3f}, "
      f"d = {result['cohen-d'].values[0]:.2f}")

One-Way ANOVA with Post-Hoc

import pingouin as pg

# ANOVA
aov = pg.anova(dv='score', between='group', data=df, detailed=True)
print(f"F = {aov['F'].values[0]:.2f}, p = {aov['p-unc'].values[0]:.3f}, "
      f"η²_p = {aov['np2'].values[0]:.3f}")

# Post-hoc if significant
if aov['p-unc'].values[0] < 0.05:
    posthoc = pg.pairwise_tukey(dv='score', between='group', data=df)
    print(posthoc[['A', 'B', 'diff', 'p-tukey']])

Linear Regression with Diagnostics

import statsmodels.api as sm

# Fit model
X = sm.add_constant(predictors)
model = sm.OLS(outcome, X).fit()
print(model.summary())

# Key outputs
print(f"R² = {model.rsquared:.3f}, Adjusted R² = {model.rsquared_adj:.3f}")
print(f"F({model.df_model:.0f}, {model.df_resid:.0f}) = {model.fvalue:.2f}, p = {model.f_pvalue:.4f}")

Correlation with Confidence Intervals

import pingouin as pg

# Pearson correlation with CI
result = pg.corr(x, y, method='pearson')
print(f"r = {result['r'].values[0]:.3f}, "
      f"p = {result['p-val'].values[0]:.3f}, "
      f"95% CI [{result['CI95%'].values[0][0]:.3f}, {result['CI95%'].values[0][1]:.3f}]")

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Effect Sizes

Always report effect sizes alongside p-values.

Quick Reference: Effect Size Benchmarks

Test	Effect Size	Small	Medium	Large
T-test	Cohen's d	0.20	0.50	0.80
ANOVA	η²_p (partial eta²)	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: These are guidelines only. Practical significance depends on context.

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Power Analysis

A Priori Power Analysis (Before Study)

from statsmodels.stats.power import tt_ind_solve_power, FTestAnovaPower

# T-test: Required n for d=0.5, power=0.80, alpha=0.05
n = tt_ind_solve_power(effect_size=0.5, alpha=0.05, power=0.80, ratio=1.0)
print(f"Required n per group: {n:.0f}")

# ANOVA: Required n for f=0.25, 3 groups
power_anova = FTestAnovaPower()
n = power_anova.solve_power(effect_size=0.25, ngroups=3, alpha=0.05, power=0.80)
print(f"Required n per group: {n:.0f}")

Sensitivity Analysis (After Study)

# What effect could we detect with n=50 per group?
detectable_d = tt_ind_solve_power(effect_size=None, nobs1=50, alpha=0.05, 
                                   power=0.80, ratio=1.0)
print(f"Minimum detectable effect: d = {detectable_d:.2f}")

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Reporting Results (APA Format)

Templates for Common Tests

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].

One-Way ANOVA:

A one-way ANOVA revealed a significant main effect of treatment 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) 
differed significantly from Condition B (M = 71.5, SD = 8.1, p = .002).

Pearson Correlation:

There was a significant positive correlation between study hours and
exam scores, r(98) = .45, p < .001, 95% CI [.28, .59].

Multiple Regression:

Multiple regression was conducted with exam scores as the outcome.
The model was significant, F(3, 146) = 45.2, p < .001, R² = .48.
Study hours (β = .35, p < .001) and prior GPA (β = .28, p < .001)
were significant predictors.

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Integration with RA Workflow

During PLANNING Phase

• Help determine appropriate sample sizes with power analysis
• Suggest statistical approaches for research design

During ANALYSIS Phase

• Run assumption checks on collected data
• Perform planned statistical analyses
• Generate effect sizes and confidence intervals

During WRITING Phase

• Format results for methods and results sections
• Generate APA-formatted statistical reports
• Connect to /write_methods and /write_results skills

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Essential Reporting Elements

Always include:

1. Descriptive statistics: M, SD, n for all groups
2. Test statistics: Name, statistic value, df, exact p-value
3. Effect sizes: With confidence intervals when possible
4. Assumption checks: What was tested, results, any corrections
5. All planned analyses: Including non-significant findings