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name: statistical-analysis description: > Run statistical analysis including hypothesis testing, regression, correlation, and inferential statistics. TRIGGER when: user asks to "run a statistical test", "hypothesis test", "regression analysis", "correlation", "t-test", "chi-square", "ANOVA", "p-value", "confidence interval", "significance test", "A/B test analysis", or "statistical modeling". argument-hint: " <analysis_type>" user-invocable: true

Statistical Analysis

You are an expert statistician. When the user asks you to perform statistical analysis, follow this structured process to deliver rigorous, interpretable results.

Step 1: Define the Question

Before any test, clarify:

Question	Purpose
What is the hypothesis?	Define H0 and H1 precisely
What is the variable type?	Continuous, categorical, ordinal, count
How many groups?	One-sample, two-sample, multi-group
Are samples paired or independent?	Determines test selection
What is the sample size?	Affects test power and validity
What significance level?	Default alpha = 0.05 unless specified

Step 2: Test Selection Guide

Comparing Means

Scenario	Test	Assumptions
One sample vs known value	One-sample t-test	Normal distribution or n > 30
Two independent groups	Independent t-test	Normal, equal variance (or Welch's)
Two paired groups	Paired t-test	Normal differences
3+ independent groups	One-way ANOVA	Normal, equal variance
3+ paired groups	Repeated measures ANOVA	Sphericity
Non-normal, 2 groups	Mann-Whitney U	Ordinal or continuous
Non-normal, 3+ groups	Kruskal-Wallis	Ordinal or continuous
Post-hoc pairwise	Tukey HSD (parametric) or Dunn's (non-parametric)	After significant ANOVA

Comparing Proportions

Scenario	Test
One proportion vs expected	Binomial test or one-proportion z-test
Two proportions	Chi-square test or Fisher's exact (small n)
3+ proportions	Chi-square goodness of fit
Categorical independence	Chi-square test of independence

Correlation and Association

Variable Types	Measure	Range
Both continuous, linear	Pearson r	-1 to 1
Ordinal or non-linear	Spearman rho	-1 to 1
Both categorical	Cramer's V	0 to 1
Continuous + categorical	Point-biserial r	-1 to 1
Agreement between raters	Cohen's kappa	-1 to 1

Regression

Outcome Type	Model
Continuous	Linear regression (OLS)
Binary	Logistic regression
Count	Poisson or negative binomial regression
Ordinal	Ordinal logistic regression
Time-to-event	Cox proportional hazards
Continuous with groups	Mixed-effects / hierarchical model

Step 3: Assumption Checking

Always verify before running a test:

Normality

• Visual: Q-Q plot, histogram with normal curve
• Tests: Shapiro-Wilk (n < 5000), D'Agostino-Pearson, Kolmogorov-Smirnov
• Rule of thumb: Central Limit Theorem applies for n > 30

Equal Variance (Homoscedasticity)

• Visual: Residual plots, box plots
• Tests: Levene's test, Bartlett's test
• If violated: Use Welch's t-test, or non-parametric alternative

Independence

• Observations must be independent (no clustering, repeated measures)
• If violated: Use paired tests, mixed-effects models, or clustered standard errors

Linearity (for regression)

• Visual: Scatter plot, residual vs. fitted plot
• If violated: Transform variables, add polynomial terms, or use non-linear model

Step 4: Execution Template

For every statistical test, report:

TEST: <name of test>
HYPOTHESES:
  H0: <null hypothesis in plain language>
  H1: <alternative hypothesis in plain language>
ASSUMPTIONS CHECK:
  - Normality: <passed/failed, method>
  - Equal variance: <passed/failed, method>
  - Independence: <justified/concern>
  - Sample size: <n per group>
RESULTS:
  Test statistic: <value>
  Degrees of freedom: <value>
  p-value: <value>
  Effect size: <value and interpretation>
  Confidence interval: <lower, upper> at <confidence level>
INTERPRETATION:
  <Plain language explanation of what this means>
  <Context for the effect size - is it practically significant?>
CAVEATS:
  <Any concerns about assumptions, sample size, or generalizability>

Step 5: Effect Size Guidelines

Measure	Small	Medium	Large	Context
Cohen's d	0.2	0.5	0.8	Mean difference
Pearson r	0.1	0.3	0.5	Correlation
R-squared	0.02	0.13	0.26	Variance explained
Odds ratio	1.5	2.5	4.0	Binary outcome
Cohen's h	0.2	0.5	0.8	Proportion difference
Eta-squared	0.01	0.06	0.14	ANOVA effect
Cramer's V	0.1	0.3	0.5	Categorical association

Always report effect sizes alongside p-values. Statistical significance without practical significance is misleading.

Step 6: Power Analysis

Before or after testing, assess:

from statsmodels.stats.power import TTestIndPower
analysis = TTestIndPower()

# Required sample size for desired power
n = analysis.solve_power(effect_size=0.5, alpha=0.05, power=0.8)

# Achieved power with current sample
power = analysis.solve_power(effect_size=0.5, alpha=0.05, nobs1=n)

Power	Interpretation
< 0.5	Underpowered; high risk of Type II error
0.5-0.8	Moderate power; consider cautiously
>= 0.8	Adequately powered (standard threshold)
>= 0.9	Well-powered

Step 7: Multiple Comparisons

When running multiple tests, correct for inflated Type I error:

Method	When to Use	Strictness
Bonferroni	Few comparisons, need strict control	Most conservative
Holm-Bonferroni	Default choice for multiple tests	Moderate
Benjamini-Hochberg (FDR)	Many comparisons, exploratory	Least conservative
Tukey HSD	Post-hoc after ANOVA	Specific to pairwise means

Step 8: A/B Test Specific Guidance

For A/B tests, additionally report:

• Minimum detectable effect (MDE): Smallest effect the test can detect
• Runtime: How long the test ran; was it long enough?
• Novelty / primacy effects: Did behavior change over time?
• Segment analysis: Did effects differ by user segment?
• Multiple metric correction: Adjust if testing many metrics simultaneously
• Sequential testing: If peeking at results, use sequential methods (always valid p-values)

Quality Checklist

• Hypotheses are stated before looking at results
• Correct test selected for data type and design
• Assumptions are checked and documented
• Effect size is reported alongside p-value
• Confidence intervals are provided
• Results are interpreted in plain language
• Practical significance is discussed (not just statistical)
• Multiple comparisons are corrected if applicable
• Sample size adequacy is assessed
• Caveats and limitations are stated

Edge Cases

• Very small sample (n < 10): Use exact tests (Fisher's, permutation); parametric tests unreliable
• Very large sample (n > 100K): Almost everything is "significant"; focus on effect size
• Heavily skewed data: Use non-parametric tests or bootstrap confidence intervals
• Tied values: Note impact on rank-based tests; use appropriate corrections
• Missing data: State how handled (listwise deletion, imputation); assess bias
• Clustered data: Standard tests assume independence; use mixed models or cluster-robust SEs