# How to Calculate Power Statistics for A/B Testing?

A/B testing is a powerful technique for optimizing conversions and revenue. But without proper statistical power analysis, your tests may fail to detect real differences between variants. Low statistical power leads to more Type II errors, where you incorrectly accept the null hypothesis and miss out on lifts from improved variants.

This article will provide a comprehensive overview of how to calculate and interpret statistical power for planning robust A/B tests.

Whether you're new to A/B testing or looking to optimize your methodology, this guide will give you the knowledge to accurately compute power and ensure your tests can reliably detect true variant differences.

## What is Statistical Power?

Statistical power indicates the likelihood that a test will detect a real effect or difference, if one truly exists. It's defined as the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true.

The null hypothesis (H0) assumes there is no difference between your control and variant. The alternative hypothesis (H1) is what you want to prove - that a difference exists.

Power is represented as a value between 0 and 1. A power of 0.8 means there is an 80% chance of accurately rejecting the null and detecting the effect size specified by your alternative hypothesis. This is why you'll often see 0.8 or 80% stated as the recommended power level.

Low power means you are more likely to miss real lifts from positive variants. High power gives you more confidence in ruling out false positives and only detecting true effects.

## Type I and Type II Errors in A/B Testing

There are two types of errors that are possible in hypothesis testing:

### Type I error

The probability of falsely rejecting the null hypothesis when it is actually true. Also known as a "false positive" - detecting a difference when there is none.

### Type II error

The probability of failing to reject the null hypothesis when the alternative hypothesis is true. Also known as a "false negative" - missing a real difference between variants.

The rate of Type I errors is set by your statistical significance threshold (usually 5% at p=0.05). But Type II error rates are directly related to the power of your test. The lower the power, the higher the chance of missing a real lift with a false negative.

Optimizing power minimizes Type II errors. You want to design tests that maximize the probability of detecting true differences.

## Factors That Influence Statistical Power

There are four key factors that determine the statistical power of an A/B test:

**Sample Size** - The number of data points or users assigned to each variant. More samples mean higher power. This is the most important factor for improving power.

**Minimum Detectable Effect (MDE)** - The smallest difference or lift between variants you want to detect. A higher minimum detectable effect requires less power to identify.

**Significance Level** - The p-value threshold for statistical significance. 0.05 or 5% is standard. More stringent thresholds like 0.01 require more power.

**Base Conversion Rate** - The baseline conversion rate for your control variant. Higher rates mean more conversions per user, increasing power.

Understanding how each of these factors impacts power will allow you to optimize your test design and analysis.

## How To Calculate Sample Size for Statistical Power

Here is the process for determining the required sample size per variant to achieve your desired level of statistical power:

- Define your minimum detectable effect (MDE)
- Set your statistical significance level (often 5%)
- Define your target power level (80% or 90% typically)
- Estimate your base conversion rate
- Use a sample size calculator to determine the necessary sample size

Sample size calculators allow you to plug in these variables and will output the minimum number of users needed per variant. Some popular calculators include:

For example, say you want to detect a 5% lift in conversion rate, with 90% power and 5% significance. Your current conversion rate is 10%. Using Mida's calculator, you would need 1597 users per variant. Running the test until you reach this sample provides a 90% chance of detecting a 5% increase in conversions.

## Statistical Power Analysis Tools

In addition to sample size calculators, there are tools that provide full power analysis reports for your A/B tests. These let you calculate power before running your test and re-analyze power after you have collected all your data. Popular options include:

**G*Power**- Free power analysis tool**Power and Sample Size**- Web-based power calculator**R PWR Package**- Power analysis in R**PostHoc Power Calculator**- Retrospective power calculator

These tools make it easy to compute achieved power based on the actual sample size and effect size from your completed A/B test. This helps you interpret whether a non-significant result was due to low power or truly no difference between the variants.

## Minimum Detectable Effect (MDE)

The minimum detectable effect (MDE) is the magnitude of difference between your variants you want to be able to identify. This is a key input that determines how much power you need.

How do you set an appropriate MDE? Here are some tips:

- Start with a target increase based on business goals like 5% lift in conversion rate.
- Consider typical impact sizes seen from past tests and changes.
- Factor in feasibility - how easy is it to move the metric by X%?
- Balance desired power and sample size requirements.

A smaller MDE requires more power and larger sample sizes. If getting sufficient sample is difficult, you may need to test larger differences initially.

It's also important to think about the dollar value of different effect sizes. A 2% increase may not be worth detecting if a 5% lift would be far more meaningful business-wise.

## Optimizing Power When Sample Size is Limited

In some cases, you may not be able to achieve the sample size needed for your desired power. When sample size is capped, you have a few options to optimize power:

**Increase the MDE** - Relaxing your minimum detectable effect reduces the required power and sample size. Test bigger differences.

**Extend the test duration** - Run the test longer to accumulate more users over time until you reach the needed sample.

**Reduce variants** - Comparing just two variants means more users per branch, increasing power over tests with multiple variants.

**Leverage historical data** - Use conversion rates from past experiments as the control variant to gain more baseline data and power.

**Simplify metrics** - Testing aggregate-level metrics like overall conversion rate requires smaller samples than specific micro-conversions.

**Statistical techniques** - Methods like Bayesian analysis can potentially improve power in smaller samples, but introduce additional complexity.

While no perfect solutions, these tactics can help you maximize the power possible when sample size is low or availability is limited. Any power gains are beneficial.

## Interpreting Results and Power Analysis

Power analysis isn't just useful for calculating the required sample size prior to testing. It's also crucial for properly interpreting your results after completing an experiment.

Once you’ve run the test and collected all your data, recalculate the achieved power using the actual sample size and observed effect size.

If the measured power is low, a non-significant result may simply indicate insufficient power rather than no real difference between variants. In this case, you cannot confidently accept the null hypothesis. Running the test longer or repeating it with a larger sample may reveal a significant effect.

Power analysis provides context around whether negative test results stem from low power or truly no impact. This helps you avoid incorrectly rejecting variants due to underpowered tests.

## Conclusion

Computing and optimizing statistical power is vital for reliable, impactful A/B testing. Low power leads to frequent Type II errors and inability to detect real lifts. Factors like sample size, minimum detectable effect size, statistical significance, and base rates all impact power calculations.

Tools make it easy to determine the sample size needed to achieve your target power and sensitivity. When sample is constrained, adjusting the MDE, durations, historical data use, and simplifying metrics can help maximize power.

Analyzing power after your test provides context on non-significant results. With proper power analysis, you can optimize your testing methodology, avoid false negatives, and ensure you are accurately identifying the best-performing variants. This ultimately translates into more revenue gains and better customer experiences driven by your tests.