When do I use negative binomial vs binomial distribution for an A/B test/experiment Why?

Motivation for A/B Testing and Statistical Models

A/B testing, also known as split testing, is a widely used methodology in various industries, especially in digital marketing, to determine which version of a variable performs better. Often, the success of an A/B test is gauged by the difference in outcomes between two groups: the control group and the test group. Understanding the appropriate statistical models to apply, such as the binomial and negative binomial distributions, is crucial for accurate hypothesis testing and reliable results.

Binomial Distribution

The binomial distribution is perhaps the most well-known distribution in probability theory. It models the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes: success or failure. This makes it particularly suitable for scenarios where the outcome of interest is a count of successes, such as the number of users clicking on an advertisement.

Key Features of Binomial Distribution

Fixed number of trials. Each trial has only two possible outcomes: success or failure. Trials are independent. The probability of success is constant across all trials.

Negative Binomial Distribution

The negative binomial distribution, on the other hand, generalizes the binomial distribution by allowing for overdispersion. It models the number of failures before a fixed number of successes in a sequence of independent and identically distributed Bernoulli trials.

Key Features of Negative Binomial Distribution

Flexible for modeling count data with overdispersion. Non-constant variance, allowing for greater variability than the binomial distribution. Can accommodate a broader range of data, particularly when the data shows a lot of variability or clusters.

Applications in A/B Testing

When conducting A/B tests, the choice of distribution can significantly impact the validity and reliability of the results. Let’s consider an example where we want to test the effectiveness of a new antibiotic. We might set up an experiment where each petri dish, representing an experimental unit, either is treated with the antibiotic (treatment group) or remains untreated (control group). The outcome is the number of bacterial colonies formed in each dish.

For a simple model where the number of bacterial colonies is expected to maintain a consistent average across all dishes, the Poisson distribution might be appropriate. However, if there is variation in the number of colonies due to factors such as different contamination rates or random biological variations, the negative binomial distribution could provide a better fit.

Choosing the Right Distribution

The selection between a binomial and a negative binomial distribution for A/B testing depends on the specific characteristics of the data:

Binomial Distribution

Use when you have a fixed number of trials and the outcome is either success or failure. Appropriate for binary outcomes, such as whether a user converts (success) or not (failure).

Negative Binomial Distribution

Use when the data shows overdispersion, meaning there is more variability in the outcomes than expected under the binomial distribution. Use when the outcome involves counts and the data show higher variance compared to the Poisson distribution.

Conclusion

Understanding the theoretical differences between the binomial and negative binomial distributions is essential for accurately interpreting A/B test results. The appropriate use of these distributions can lead to more reliable conclusions and more effective decisions in data-driven marketing and experimentation.

References

Kindly refer to statistical textbooks and research articles for a deeper understanding of the concepts discussed here. You can also consult with a statistician or data scientist for more tailored advice on the application of these models in your specific A/B testing scenarios.