Hypothesis Testing

Framework

IRON LAW: Statistical Significance ≠ Practical Significance

A p-value < 0.05 means the result is unlikely under the null hypothesis.
It does NOT mean the result is important, large, or practically meaningful.
With a large enough sample, a 0.1% conversion rate difference becomes
"statistically significant" but is practically worthless.

ALWAYS report effect size alongside p-value.

IRON LAW: State Hypotheses BEFORE Looking at Data

H₀ (null) and H₁ (alternative) must be defined before data analysis.
Choosing hypotheses after seeing the data = p-hacking = scientific fraud.
"We found an interesting pattern, let's test it on the same data" is invalid.

Core Concepts

Concept	Definition
H₀ (Null)	Default assumption: no effect, no difference
H₁ (Alternative)	What you want to show: there IS an effect/difference
p-value	Probability of seeing this result (or more extreme) IF H₀ is true
α (significance level)	Threshold for rejecting H₀ (typically 0.05)
Type I error (α)	Rejecting H₀ when it's actually true (false positive)
Type II error (β)	Failing to reject H₀ when H₁ is true (false negative)
Power (1-β)	Probability of detecting a real effect (target: ≥ 0.8)
Effect size	Magnitude of the difference (Cohen's d, odds ratio, R²)

Test Selection Guide

Data Type	Groups	Test
Continuous, normal, 2 groups	Independent	Independent t-test
Continuous, normal, 2 groups	Paired/before-after	Paired t-test
Continuous, normal, 3+ groups	Independent	One-way ANOVA
Continuous, non-normal	2 groups	Mann-Whitney U
Categorical	2+ groups	Chi-square test
Continuous, relationship	2 variables	Pearson correlation (normal) / Spearman (non-normal)
Binary outcome	Predictors	Logistic regression

Testing Process

1. State hypotheses: H₀ and H₁ with specific parameters
2. Choose test: Based on data type, distribution, and groups (use guide above)
3. Set α: Usually 0.05 (justify if different)
4. Calculate: Run the test, get test statistic and p-value
5. Decide: p < α → reject H₀; p ≥ α → fail to reject H₀
6. Report: Effect size + confidence interval + p-value (not just "significant")

Output Format

# Hypothesis Test: {Research Question}

## Hypotheses
- H₀: {null — no effect/difference}
- H₁: {alternative — there IS an effect/difference}
- α = {0.05 or other}

## Test Selection
- Test: {name}
- Rationale: {why this test fits the data}
- Assumptions checked: {normality, independence, equal variance}

## Results
- Test statistic: {value}
- p-value: {value}
- Effect size: {value and interpretation}
- 95% CI: [{lower}, {upper}]

## Decision
{Reject / Fail to reject H₀}

## Interpretation
{What this means in practical terms, with effect size context}

Gotchas

• "Fail to reject H₀" ≠ "H₀ is true": Absence of evidence is not evidence of absence. You may lack power to detect a real effect.
• Multiple comparisons inflate Type I error: Testing 20 hypotheses at α=0.05 → expect 1 false positive by chance. Apply Bonferroni or FDR correction.
• Check assumptions before testing: t-test assumes normality and equal variance. Violating assumptions invalidates results. Use non-parametric alternatives when assumptions fail.
• Sample size determines power: Small samples miss real effects (Type II error). Calculate required sample size BEFORE collecting data.
• p-value is NOT the probability that H₀ is true: It's the probability of the data given H₀. These are fundamentally different things (base rate fallacy).

References

• For sample size calculation, see references/sample-size.md
• For non-parametric test alternatives, see references/nonparametric-tests.md