Explain power analysis with an example

Question

Explain power analysis in simple terms and provide a concrete example.

Codemia · Accepted Answer

To conduct a power analysis for an A/B test on Airbnb, we need to determine how many users we should involve in the experiment to detect a meaningful effect of a new feature (e.g., a revised booking interface) on booking conversions.

**Data Needed:**  
1. **Current Conversion Rate:** The baseline conversion rate, say it's 5%.  
2. **Minimum Detectable Effect (MDE):** The smallest effect size we want to detect, for instance, a 1% increase in conversion rate to 6%.  
3. **Significance Level (alpha):** Typically set at 0.05 for a 95% confidence level.  
4. **Power (1-beta):** Commonly set at 0.8, indicating an 80% probability of detecting the effect if it exists.

The power analysis will use the following formula to estimate the required sample size for both the control and treatment groups:

$$ n = \frac{(Z_{\alpha/2} + Z_{\beta})^2 \cdot \left(p_1(1-p_1) + p_2(1-p_2)\right)}{(p_2 - p_1)^2} $$

Where:  
- $ p_1 $ = baseline conversion rate (0.05)  
- $ p_2 $ = expected conversion rate (0.06)  
- $ Z_{\alpha/2} $ = Z-score for alpha level (1.96 for 0.05)  
- $ Z_{\beta} $ = Z-score for power (0.84 for 80% power)

Let’s calculate the required sample size using the values provided:  
- $ p_1 = 0.05 $  
- $ p_2 = 0.06 $  
- $ Z_{\alpha/2} = 1.96 $  
- $ Z_{\beta} = 0.84 $

Plugging these into the formula:  
$$ n = \frac{(1.96 + 0.84)^2 \cdot \left(0.05(1-0.05) + 0.06(1-0.06)\right)}{(0.06 - 0.05)^2} $$  
$$ n = \frac{(2.8)^2 \cdot (0.05 \cdot 0.95 + 0.06 \cdot 0.94)}{(0.01)^2} $$  
$$ n = \frac{7.84 \cdot (0.0475 + 0.0564)}{0.0001} $$  
$$ n = \frac{7.84 \cdot 0.1039}{0.0001} $$  
$$ n = 81354 $$  
Thus, approximately 81,354 users would be needed in each group to detect a 1% increase in conversion rate.

The result indicates that to confidently detect a 1% increase in the conversion rate due to the new booking interface, we would need around 81,354 users in both the control and treatment groups. This large sample size ensures that we minimize the risk of Type II errors (failing to detect an effect that is there) while maintaining a manageable Type I error rate (false positives). The findings can guide decision-making about the feature's rollout based on statistical evidence.

Common pitfalls in power analysis include:  
1. **Ignoring Variability:** Not accounting for variability in the conversion rates across different user segments might lead to an underestimation of the required sample size.  
2. **Misestimating MDE:** Choosing an MDE that is too small can inflate the sample size unnecessarily, while a large MDE might overlook meaningful changes.  
3. **Single Testing Assumption:** Assuming that each test has independent outcomes without considering potential confounding variables can lead to misleading results.

Explain power analysis with an example

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