---
name: statistical-significance-checker-for-low-conversion-experiments
description: "Assesses statistical power, sample size adequacy, and significance thresholds for A/B tests running in rare-event or high-variance environments — flags underpowered tests, peeking risk, and multiple-comparison inflation, then delivers a verdict with required sample size, minimum detectable effect, and a decision recommendation. Use when validating whether a low-conversion or fintech/e-commerce experiment has enough data to declare a winner, or when a result looks significant but the base rate is tiny."
argument-hint: "[experiment-metrics] [base-rate]"
allowed-tools: ["Read"]
---

# Statistical Significance Checker for Low-Conversion Experiments

## Purpose
Determine whether a low-conversion-rate experiment has actually reached a trustworthy conclusion — or whether the observed lift is noise dressed up as significance. Produces a rigorous read on statistical power, required sample size, minimum detectable effect, and the specific failure modes (peeking, high variance, multiple comparisons) that mislead teams testing rare events at 0.5%–5% base rates.

## When to Use
Invoke this skill when the user asks to:
- Check whether an A/B test result is statistically significant with a low base conversion rate
- Determine if an experiment has enough sample size or statistical power to call
- Validate a suspiciously large lift on a rare-event metric (signup, deposit, fraud flag, checkout)
- Assess how long a test must run given current traffic and conversion volume
- Diagnose why two tests keep flipping between "winner" and "no effect"
- Sanity-check a p-value before shipping a change to production

## Inputs Required
Before starting, collect:
- **Metric under test**: The conversion event and its type (binary rate, revenue-per-user, count)
- **Base (control) conversion rate**: The historical or control-arm rate — critical for rare-event math
- **Sample sizes per arm**: Users/sessions in control and each treatment
- **Observed conversions per arm**: Raw event counts, not just percentages
- **Observed lift/effect**: The measured difference (relative % or absolute)
- **Minimum meaningful effect**: The smallest lift worth shipping (MDE the business cares about)
- **Test setup**: Number of variants, whether results were checked mid-flight, significance level and power targets (default α=0.05, power=0.80 two-sided)

If MDE or power targets are missing, default to α=0.05, power=0.80, and infer a business-relevant MDE from the base rate (typically a 5–15% relative lift for rare events) and state the assumption.

## Instructions

### Step 1: Reconstruct the Raw Data
Never trust reported percentages alone.
- Convert every arm to raw counts: successes and total N per arm
- Recompute observed rates from counts to catch rounding or reporting errors
- Flag if any arm has **fewer than ~30 conversions** — normal-approximation math becomes unreliable below this and exact/Bayesian methods are needed
- For revenue or count metrics, note the mean and variance separately (high variance is the silent killer in these tests)

### Step 2: Assess Sample Size Adequacy
Compute the sample size the test *should* have had.
- Required N per arm (two-proportion z-test approximation):
  `n = (Z_α/2 + Z_β)² · [p₁(1−p₁) + p₂(1−p₂)] / (p₂ − p₁)²`
  where p₁ = base rate, p₂ = p₁ × (1 + MDE), Z_α/2 = 1.96, Z_β = 0.84 at 80% power
- Compare required N to actual N per arm → report **% of required sample achieved**
- For rare events, note that required N scales roughly with `1/base_rate` — a 1% base rate needs ~5× the sample of a 5% base rate for the same relative MDE

### Step 3: Compute Statistical Power (Post-Hoc, With Caveats)
Estimate whether the test *could* have detected the effect it aimed for.
- Given actual N, base rate, and target MDE, back out achieved power
- Label the test:
  - **Adequately powered** (≥80%): a null result is meaningfully informative
  - **Underpowered** (50–79%): a null result is inconclusive, not evidence of no effect
  - **Severely underpowered** (<50%): the test cannot reliably detect the target effect — any "winner" here is high-risk
- Warn against interpreting post-hoc power as validation of a *significant* result; use it only to interpret null results and future planning

### Step 4: Test Significance Correctly
Run the appropriate test for the data shape.
- **Binary rate, adequate counts**: Two-proportion z-test → report z-statistic, two-sided p-value, and 95% CI on the absolute difference
- **Binary rate, low counts (<30 conversions per arm)**: Recommend Fisher's exact test or Bayesian Beta-Binomial; report a credible interval on lift
- **Revenue/count metric**: Welch's t-test (unequal variance) or Mann-Whitney; flag that outliers/whales distort means and recommend capping or log-transform
- Always report the **confidence interval on lift**, not just the p-value — a CI spanning zero or spanning both trivial and huge effects tells the real story

### Step 5: Screen for the Four Rare-Event Failure Modes
Explicitly check each:
- **Peeking / optional stopping**: If results were checked mid-flight and stopped at significance, the true false-positive rate is inflated well above 5%. Recommend a sequential correction (alpha-spending / SPRT) or discount the p-value
- **Multiple comparisons**: If ≥3 variants or ≥2 metrics were tested, apply Bonferroni (α/m) or note the family-wise error inflation and give the adjusted threshold
- **High variance / low signal**: If per-arm variance is large relative to the effect (common in revenue-per-user with heavy tails), the nominal p-value is optimistic
- **Novelty / ramp effects**: For fintech/e-commerce, flag if test duration is under one full business cycle (weekly seasonality, payday cycles, weekend patterns)

### Step 6: Render the Verdict and Decision
Deliver a clear call, not a hedge.
- One of: **Ship**, **Do Not Ship**, **Keep Running**, or **Inconclusive — Redesign**
- State required additional sample and estimated additional runtime at current traffic to reach adequate power
- Give the MDE the current sample *can* actually detect (so the team knows what effect size, if any, is trustworthy today)

## Output Format
- **Structure**: Diagnostic report with a headline verdict at top, then the six analysis sections, then a decision block
- **Length**: 700–1,000 words
- **Format**: Structured document with a summary table (per-arm N, conversions, rate, CI) plus prose reasoning
- **Sections**: Verdict headline → Raw data reconstruction → Sample size adequacy → Statistical power → Significance test → Failure-mode screen → Decision & required runtime
- **Tone**: Rigorous experimentation lead — precise, numeric, unwilling to over-claim; explains the math but always lands on an actionable call

## Quality Checklist
Before delivering output, verify:
- [ ] Raw conversion counts (not just percentages) are reconstructed and rates recomputed for every arm
- [ ] Required sample size per arm is calculated with stated α, power, base rate, and MDE
- [ ] Achieved power is reported with an explicit adequacy label (adequate / underpowered / severely underpowered)
- [ ] The significance test chosen matches the data shape (z-test vs. Fisher's exact vs. Welch's), and a confidence interval on lift is reported alongside the p-value
- [ ] All four rare-event failure modes (peeking, multiple comparisons, high variance, novelty) are explicitly checked
- [ ] A single clear verdict is given, with required additional sample and runtime when the test isn't ready

## Notes
- The single most common error in low-conversion testing is calling a winner on an underpowered test after peeking — treat any mid-flight-stopped significant result as suspect until corrected
- At base rates under 2%, favor Bayesian Beta-Binomial or Fisher's exact over normal approximations; the CI is more honest and doesn't break with low counts
- Revenue-per-user in e-commerce is heavy-tailed — a single whale can manufacture "significance"; always inspect variance and consider a capped or log-transformed metric alongside the raw one
- Runtime estimates should use realistic daily traffic and conversion volume, not idealized numbers — a test needing 400k users per arm at 5k daily eligible users is a 3-month commitment worth surfacing up front

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## How to use this skill

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