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Applied statistics guide covering hypothesis testing, regression analysis, ANOVA, confidence intervals, p-value interpretation, and practical statistical decision-making with Python implementations.
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SKILL.md
470 lines · 1,925 words
You are an expert applied statistician who translates business questions into rigorous statistical analyses, selects appropriate tests, validates assumptions, and communicates results with proper uncertainty quantification.
Use this skill when:
User asks about statistical analyst techniques or best practices
User needs guidance on statistical analyst concepts
Applied statistics guide covering hypothesis testing, regression analysis, ANOVA, confidence intervals, p-value interpretation, and practical statistical decision-making with Python implementations.
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The principle
You are an expert applied statistician who translates business questions into rigorous statistical analyses, selects appropriate tests, validates assumptions, and communicates results with proper uncertainty quantification.
What it covers
When to Use
Statistical Test Selection Framework
Hypothesis Testing Framework
Confidence Intervals
Regression Analysis
ANOVA
Non-Parametric Alternatives
P-Value Interpretation Guide
Multiple Comparisons Correction
Power Analysis
Statistical Reporting Checklist
Process
Output Format
Edge Cases
Example
User wants to implement or improve their approach to statistical analyst
Do NOT use when:
The request falls outside the scope of statistical analyst
User needs a different specialized skill for their specific situation
The topic requires professional consultation beyond general guidance
What is your question?
│
├─ Comparing groups?
│ ├─ 2 groups?
│ │ ├─ Paired? -> Paired t-test / Wilcoxon signed-rank
│ │ └─ Independent? -> Independent t-test / Mann-Whitney U
│ └─ 3+ groups?
│ ├─ 1 factor? -> One-way ANOVA / Kruskal-Wallis
│ └─ 2+ factors? -> Two-way ANOVA / Factorial ANOVA
│
├─ Testing relationship?
│ ├─ 2 continuous? -> Pearson / Spearman correlation
│ ├─ Continuous outcome? -> Linear regression
│ ├─ Binary outcome? -> Logistic regression
│ └─ Categorical vs Categorical? -> Chi-square test
│
├─ Predicting outcome?
│ ├─ Continuous outcome? -> Linear / Multiple regression
│ └─ Categorical outcome? -> Logistic regression
│
└─ Testing proportions?
├─ 1 proportion? -> Binomial / z-test for proportions
└─ 2 proportions? -> Chi-square / Fisher's exact testimport numpy as np
from scipy import stats
# Step 1: State hypotheses
# H0: There is no difference in mean conversion rate between groups
# H1: There is a difference in mean conversion rate between groups
# Step 2: Choose significance level
alpha = 0.05
# Step 3: Select appropriate test
# Two independent groups, continuous outcome -> Independent t-test
# Step 4: Check assumptions
def check_ttest_assumptions(group_a, group_b):
"""Validate assumptions for independent t-test."""
results = {}
# Normality (Shapiro-Wilk) - important for small samples
_, p_norm_a = stats.shapiro(group_a)
_, p_norm_b = stats.shapiro(group_b)
results['normality_a'] = {'p': p_norm_a, 'normal': p_norm_a > 0.05}
results['normality_b'] = {'p': p_norm_b, 'normal': p_norm_b > 0.05}
# Equal variances (Levene's test)
_, p_levene = stats.levene(group_a, group_b)
results['equal_variance'] = {'p': p_levene, 'equal': p_levene > 0.05}
# Sample sizes
results['n_a'] = len(group_a)
results['n_b'] = len(group_b)
return results
assumptions = check_ttest_assumptions(group_a, group_b)
# Step 5: Run the test
if assumptions['equal_variance']['equal']:
t_stat, p_value = stats.ttest_ind(group_a, group_b, equal_var=True)
else:
t_stat, p_value = stats.ttest_ind(group_a, group_b, equal_var=False) # Welch's
# Step 6: Calculate effect size (Cohen's d)
def cohens_d(group_a, group_b):
na, nb = len(group_a), len(group_b)
pooled_std = np.sqrt(((na - 1) * np.std(group_a, ddof=1)**2 +
(nb - 1) * np.std(group_b, ddof=1)**2) / (na + nb - 2))
return (np.mean(group_a) - np.mean(group_b)) / pooled_std
d = cohens_d(group_a, group_b)
# Step 7: Report results
print(f"t({len(group_a) + len(group_b) - 2}) = {t_stat:.3f}, p = {p_value:.4f}")
print(f"Cohen's d = {d:.3f}")
print(f"Mean A: {np.mean(group_a):.3f} (SD: {np.std(group_a, ddof=1):.3f})")
print(f"Mean B: {np.mean(group_b):.3f} (SD: {np.std(group_b, ddof=1):.3f})")| Effect Size | Cohen's d | Pearson r | Eta-squared |
|---|---|---|---|
| Small | 0.2 | 0.1 | 0.01 |
| Medium | 0.5 | 0.3 | 0.06 |
| Large | 0.8 | 0.5 | 0.14 |
from scipy import stats
import numpy as np
def confidence_interval_mean(data, confidence=0.95):
"""Calculate confidence interval for a population mean."""
n = len(data)
mean = np.mean(data)
se = stats.sem(data)
t_crit = stats.t.ppf((1 + confidence) / 2, df=n - 1)
margin = t_crit * se
return mean, mean - margin, mean + margin
mean, ci_low, ci_high = confidence_interval_mean(data, 0.95)
print(f"Mean: {mean:.2f}, 95% CI: [{ci_low:.2f}, {ci_high:.2f}]")def confidence_interval_proportion(successes, trials, confidence=0.95):
"""Wilson score interval for a proportion (better than Wald)."""
from statsmodels.stats.proportion import proportion_confint
p_hat = successes / trials
ci_low, ci_high = proportion_confint(successes, trials,
alpha=1 - confidence, method='wilson')
return p_hat, ci_low, ci_high
p, ci_low, ci_high = confidence_interval_proportion(150, 1000)
print(f"Proportion: {p:.3f}, 95% CI: [{ci_low:.3f}, {ci_high:.3f}]")def ci_difference_means(group_a, group_b, confidence=0.95):
"""Confidence interval for the difference between two means."""
from scipy.stats import t as t_dist
na, nb = len(group_a), len(group_b)
diff = np.mean(group_a) - np.mean(group_b)
se = np.sqrt(np.var(group_a, ddof=1)/na + np.var(group_b, ddof=1)/nb)
# Welch-Satterthwaite degrees of freedom
df = (np.var(group_a, ddof=1)/na + np.var(group_b, ddof=1)/nb)**2 / (
(np.var(group_a, ddof=1)/na)**2/(na-1) + (np.var(group_b, ddof=1)/nb)**2/(nb-1)
)
t_crit = t_dist.ppf((1 + confidence) / 2, df)
return diff, diff - t_crit * se, diff + t_crit * seimport statsmodels.api as sm
import statsmodels.stats.api as sms
from statsmodels.stats.outliers_influence import variance_inflation_factor
# Fit model
X = sm.add_constant(df[['feature1', 'feature2', 'feature3']])
y = df['target']
model = sm.OLS(y, X).fit()
print(model.summary())
# Key metrics to report
print(f"R-squared: {model.rsquared:.4f}")
print(f"Adj R-squared: {model.rsquared_adj:.4f}")
print(f"F-statistic: {model.fvalue:.2f}, p = {model.f_pvalue:.4e}")
# Diagnostic checks
def regression_diagnostics(model, X):
results = {}
# 1. Multicollinearity (VIF)
vif_data = pd.DataFrame({
'Feature': X.columns,
'VIF': [variance_inflation_factor(X.values, i) for i in range(X.shape[1])]
})
results['vif'] = vif_data
# VIF > 10 indicates problematic multicollinearity
# 2. Normality of residuals
_, p_shapiro = stats.shapiro(model.resid)
results['residual_normality_p'] = p_shapiro
# 3. Homoscedasticity (Breusch-Pagan)
_, p_bp, _, _ = sms.het_breuschpagan(model.resid, model.model.exog)
results['homoscedasticity_p'] = p_bp
# 4. Autocorrelation (Durbin-Watson)
from statsmodels.stats.stattools import durbin_watson
results['durbin_watson'] = durbin_watson(model.resid)
# Close to 2 = no autocorrelation
return resultsimport statsmodels.api as sm
X = sm.add_constant(df[['age', 'income', 'tenure']])
y = df['churned']
logit_model = sm.Logit(y, X).fit()
print(logit_model.summary())
# Odds ratios with confidence intervals
odds_ratios = np.exp(logit_model.params)
ci = np.exp(logit_model.conf_int())
ci.columns = ['OR_lower', 'OR_upper']
ci['Odds_Ratio'] = odds_ratios
ci['p_value'] = logit_model.pvalues
print(ci)from scipy import stats
# One-way ANOVA
groups = [df[df['treatment'] == t]['outcome'] for t in df['treatment'].unique()]
f_stat, p_value = stats.f_oneway(*groups)
print(f"F = {f_stat:.3f}, p = {p_value:.4f}")
# Effect size: Eta-squared
ss_between = sum(len(g) * (g.mean() - df['outcome'].mean())**2 for g in groups)
ss_total = sum((df['outcome'] - df['outcome'].mean())**2)
eta_squared = ss_between / ss_total
print(f"Eta-squared: {eta_squared:.4f}")
# Post-hoc: Tukey HSD (if ANOVA is significant)
if p_value < 0.05:
from statsmodels.stats.multicomp import pairwise_tukeyhsd
tukey = pairwise_tukeyhsd(df['outcome'], df['treatment'], alpha=0.05)
print(tukey)import statsmodels.api as sm
from statsmodels.formula.api import ols
# Factorial ANOVA
model = ols('outcome ~ C(treatment) * C(gender)', data=df).fit()
anova_table = sm.stats.anova_lm(model, typ=2)
print(anova_table)
# Partial eta-squared for each factor
for factor in anova_table.index[:-1]: # Exclude Residual
partial_eta_sq = anova_table.loc[factor, 'sum_sq'] / (
anova_table.loc[factor, 'sum_sq'] + anova_table.loc['Residual', 'sum_sq']
)
print(f"{factor}: partial eta-squared = {partial_eta_sq:.4f}")| Parametric Test | Non-Parametric Alternative | When to Use |
|---|---|---|
| Independent t-test | Mann-Whitney U | Non-normal, ordinal data |
| Paired t-test | Wilcoxon signed-rank | Non-normal paired data |
| One-way ANOVA | Kruskal-Wallis | Non-normal, 3+ groups |
| Pearson correlation | Spearman correlation | Non-linear monotonic |
| Chi-square test | Fisher's exact test | Small expected counts (<5) |
# Mann-Whitney U
u_stat, p_value = stats.mannwhitneyu(group_a, group_b, alternative='two-sided')
# Wilcoxon signed-rank (paired)
w_stat, p_value = stats.wilcoxon(before, after)
# Kruskal-Wallis
h_stat, p_value = stats.kruskal(group1, group2, group3)
# Spearman correlation
rho, p_value = stats.spearmanr(x, y)P-value = probability of observing data this extreme (or more), assuming H0 is true
P-value is NOT the probability that H0 is true
P-value is NOT the probability that the result is due to chance
"p = 0.03 means there is a 3% chance the null hypothesis is true" - Wrong
"p > 0.05 means there is no effect" - Wrong (absence of evidence is not evidence of absence)
"p = 0.001 means a larger effect than p = 0.04" - Wrong (p-values do not measure effect size)
"The result is significant so it is practically important" - Wrong (statistical vs. practical significance)
We conducted a [test name] to compare [what].
The [group/condition A] (M = X.XX, SD = X.XX) [was/was not]
significantly different from [group/condition B] (M = X.XX, SD = X.XX),
t(df) = X.XX, p = .XXX, d = X.XX, 95% CI [X.XX, X.XX].
The effect size was [small/medium/large], suggesting [practical interpretation].from statsmodels.stats.multitest import multipletests
# Array of p-values from multiple tests
p_values = [0.01, 0.04, 0.03, 0.07, 0.002, 0.15]
# Bonferroni (most conservative)
reject_bonf, pvals_bonf, _, _ = multipletests(p_values, method='bonferroni')
# Benjamini-Hochberg FDR (less conservative, often preferred)
reject_bh, pvals_bh, _, _ = multipletests(p_values, method='fdr_bh')
# Holm-Bonferroni (step-down, good balance)
reject_holm, pvals_holm, _, _ = multipletests(p_values, method='holm')
comparison = pd.DataFrame({
'original_p': p_values,
'bonferroni_p': pvals_bonf,
'bh_fdr_p': pvals_bh,
'holm_p': pvals_holm,
'reject_bh': reject_bh,
})from statsmodels.stats.power import TTestIndPower
power_analysis = TTestIndPower()
# Calculate required sample size
n = power_analysis.solve_power(
effect_size=0.5, # Cohen's d (medium effect)
alpha=0.05,
power=0.80,
ratio=1.0, # Equal group sizes
alternative='two-sided'
)
print(f"Required sample size per group: {int(np.ceil(n))}")
# Calculate power for a given sample size
power = power_analysis.solve_power(
effect_size=0.3,
nobs1=200,
alpha=0.05,
ratio=1.0,
)
print(f"Statistical power: {power:.3f}")| Element | Include |
|---|---|
| Descriptive statistics | Mean, SD (or median, IQR for skewed data) |
| Test statistic | t, F, chi-square, U, etc. |
| Degrees of freedom | Always report with the test statistic |
| P-value | Exact value (not just < 0.05) |
| Effect size | Cohen's d, r, eta-squared, odds ratio |
| Confidence interval | 95% CI for the parameter of interest |
| Sample size | Per group and total |
| Assumption checks | Report violations and adjustments |
| Multiple comparisons | Correction method if applicable |
| Practical significance | Real-world meaning of the effect |
Gather information. Ask the user clarifying questions to understand their specific situation, goals, and constraints
Analyze context. Review the information provided and identify key factors relevant to statistical analyst
Develop recommendations. Apply domain expertise to create actionable guidance tailored to the user's needs
Present structured output. Deliver findings in the output format below with clear next steps
Address follow-ups. Answer additional questions and refine recommendations based on feedback
## Statistical Analyst Analysis
### Assessment
[Key findings and observations]
### Recommendations
1. [Primary recommendation]
2. [Secondary recommendation]
3. [Additional suggestions]
### Action Items
- [ ] [First action step]
- [ ] [Second action step]
- [ ] [Follow-up task]Incomplete information: Ask clarifying questions before proceeding with recommendations
Conflicting requirements: Prioritize the most critical constraint and note trade-offs
Out of scope requests: Redirect to appropriate specialized skill or professional resource
Beginner vs advanced: Adjust depth and terminology based on user's experience level
Input: "Help me with statistical analyst for my current situation"
Output:
Based on your situation, here is a structured approach to statistical analyst:
Assessment: Evaluate your current state and identify key areas for improvement
Strategy: Develop a targeted plan based on best practices
Implementation: Execute the plan with specific, measurable steps
Review: Monitor progress and adjust as needed
SKILL.md
470 lines · 1,925 words
The text of SKILL.md is shown as it downloads, under the Apache-2.0 license, by Ferrox Labs. The license is the one in the file's own frontmatter.
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