In this article, I want to explain why we decided to completely change the way we perform A/B tests in Pixel Federation. I will be comparing the method we used before to the new method in context of A/B testing new features in F2P mobile games.
But first, a preface. We usually test two versions of a game. Version A is the current version of the game, and Version B is different with a new feature, rebalance or some other change. About a year ago, we were using the null hypothesis significance testing (NHST), also known as frequentist testing. Specifically, we used Chi-squared test for testing conversion metrics and the t-test for revenue metrics like ARPU (average revenue per user) or ARPPU (average revenue per paying user). Now, we are using Bayesian statistics.
I will mention these main reasons why we switched from NHST to Bayes:
- Interpretability of results;
- Sample size and setup of the test.
1. How to interpret the results?
The frequentist and Bayesian statistics have a very different approach to probability and hypothesis testing. So, the results we get from the same data are also quite different.
Interpretation of frequentist tests
The frequentist tests give us p-value and confidence interval as results. By comparing the p-value to predetermined alpha level (usually 5%) we conclude whether the difference is significant or not. This approach has been used for over 100 years and it works fine. But there is one big problem with p-values: many people think they understand them, but only a few actually do. And this can lead to a lot of problems, like unintentional p-hacking.
The reason is that p-values are very unintuitive. If we want to test that there is a difference between A and B, we must form a null hypothesis assuming the opposite (A is equal to B). Then we collect the data and observe how far are they from this assumption. The p-value then tells us the probability of observing data that are even further from the null hypothesis.
In other words, the p-value tells us the probability of getting an outcome as extreme as (or more extreme than) the actual observed outcome if we would have run an A/A test. This is the simplest description of p-value I know but it still seems too confusing. I am not saying that complicated things are not useful. But in our business, we need also managers and game designers to understand our results.
So how do we interpret the result? If we get a result of p-value less than alpha, everything is fine. We make a conclusion that A and B are different which has clear business value. Everyone will be happy that we made a successful A/B test, and no one cares that they do not understand what the p-value actually means.
But what if we are not so lucky? It is hard to make a conclusion that has some value when we get p-value higher than alpha. We can see that on following example. This is the result from a Chi-squared test (calculated by Evan’s Awesome A/B Tools).
Not only we cannot say that one version is better, but we also cannot say that they are the same. Now, our only result is a p-value equal to 7.2%. We also have confidence intervals but that is not much help here since the confidence interval of difference would include zero. Now, the chances are that people from the production team will not understand what p-value equal to 7.2% means and if you somehow manage to explain it to them it can be even worse. Because who cares what is the probability of getting an outcome more extreme than the observed outcome with an assumption that is the opposite of what we hope for?
In this situation, it can be tempting to say “Well, 7% is quite close to 5% and it looks like there is an effect. Let’s say that B is better”. But you cannot do that, because p-values are uniformly distributed between 0 and 1 if there is no real effect. In other words: if there really is no difference between A and B, there is an equal chance of p-value being equal to 7% or 80% or any other number. So, the only thing you can declare and keep some statistician’s integrity is that we do not have enough data to make any decision in the current setup of the test.
This is very disappointing because A/B tests are generally expensive. From the manager’s point of view, when we run a test for a few weeks and then analysts conclude that we know basically nothing, it can look like a waste of time and money. I saw how this leads to frustration and unwillingness to do A/B tests in the future. And this is not what we want.
Interpretation of Bayesian tests
Bayesian approach solves this very well, because it gives us meaningful and understandable information in any situation. It does not operate with any null hypothesis assumptions and this simplifies a lot in terms of interpretability.
Instead of p-value, it gives us these three numbers:
- P(B > A) – probability that version B is better than A
- E(loss|B) – expected loss with version B if B is actually worse
- E(uplift|B) – expected uplift with version B if B is actually better
E(loss|B) and E(uplift|B) are measured in the same units as the tested measure. In case of conversion, it is a percentage. Let’s see the result using the same data as before:
This result tells us that if we select version B, but it is actually worse (probability 3.6%) we can expect to lose 0.026% on conversion. On the other hand, if our choice is correct (probability 96.4%), we can expect to gain 3.3% resulting in 23% conversion. This is a simple result that can be well understood by both game designers and managers.
To decide whether to use A or B, we should determine a decision rule before calculating the results of the test. You can decide based on the P(B > A) but it is much smarter to use E(loss|B) and declare B as the winner only if E(loss|B) is smaller than some predefined threshold.
This decision rule is not as clear as with the p-value, but I would argue that it is a good thing. The “p-value < 5%” has become so wired in the brain of every statistician that we often do not even think about whether it makes sense in our context. With Bayesian testing we should discuss the threshold for E(loss|B) with game designers before running the test. Then we should set it to a number so low that we do not care if we make an error smaller than this threshold.
But even if the predefined decision rule does not distinguish between variations (because we do not have enough data, or we set the threshold too low), we can always do at least an informed decision. We can compare how much we can gain with the probability of P(B > A) and how much we can lose with the probability of 1 – P(B > A). This is the key advantage of Bayesian testing in business applications: no matter the results, you always get meaningful information that is easy to understand.
2. Sample size and setup of the test
So far, I have only mentioned issues about the understanding of test results. But there is another very practical reason to use Bayesian testing, which has to do with sample size. The most often mentioned drawback of NHST is the requirement to fix the sample size before running the test. The Bayesian tests do not have this requirement. Also, they usually work better with smaller sample sizes.
Sample size in frequentist tests
The reason that fixing the sample size is required in NHST is the fact that the p-value is dependent on how we collect the data. We would calculate the p-value differently if we set up the test with the intention to gather observations from 10,000 players versus a test where we intend to gather data until we observe 100 conversions. This sounds counterintuitive and it is often disregarded; one would expect that the data itself is sufficient, but that is not the case if you want to calculate p-values correctly. I am not going to try to explain it here since it is very well analyzed in a paper (see Figure 10) by John K. Kruschke and explained in even more detail in chapter 11 of his book, Doing Bayesian Data Analysis. Another reason to fix the sample size is that we want some estimate of the power of the test.
This means that we must determine how many players will be included in the test before collecting the data and then evaluate the results only after the predefined sample size has been reached. But since we usually have no idea about the real size of the effect, it is very hard to set up a test correctly and efficiently. And because A/B tests are expensive, we want to set up them as efficiently as possible. This topic is very well covered in an article by Evan Miller. He clearly shows how “peeking” at the results before reaching the fixed sample size can considerably increase the type I error rate, and it completely invalidates the result. He also suggests two possible approaches to fix the “peeking problem”: Sequential testing or Bayesian testing.
Sample size in Bayesian tests
Bayesian tests do not require fixed sample sizes to provide valid results. You can even evaluate the results repeatedly (which is considered almost a blasphemy with NHST) and the results will still hold. Note that I write this with a big warning: some articles claim that Bayesian approach is “immune to peeking”, but this can be very misleading, because it suggests the type I error rate is kept even if we regularly check results and stop the test prematurely when significant. This is just not true. No statistical method can do that if you have the stopping rule based on observing the extreme values. The fact is that Bayesian tests never keep type I error rate bounded in the first place — they work differently. Instead, they keep the expected loss bounded. It is shown in this article by David Robinson that the expected loss in Bayesian tests is bounded even when peeking. So, the claim is partly true but just be aware that being “immune to peeking” means one thing for NHST and something different for Bayesian tests.
But I am convinced that type I error rate is not very important in our context. Sure, it is super important in scientific research like medicine. But the situation is different in A/B testing. In the end we always choose A or B. And if there really is no difference or the difference is very small we do not care that we made a type I error; we only care if the error is too big. So, in the context of A/B testing, it makes much more sense to bound the expected loss instead of type I error rate.
On top of this, there is another practical advantage in Bayesian tests: they usually require considerably smaller sample sizes to achieve the same level of statistical power compared to NHST. This is valuable because it allows us to make decisions faster.
The following plots show simulations of conversion tests. We simulated 1,000 Chi-squared tests and 1,000 Bayesian tests for each sample size and calculated the percentage of tests that correctly declared B as a winner. That tells us the approximate power of the test with a specific sample size.
We can see that the sample size required to get 80% power is significantly lower for Bayesian tests. It is less than half. On the other hand, if we plotted the situation where the conversion of A and B is the same, the NHST would have better results (95% for any sample size).
For Chi-squared test, we used alpha equal to 5%. The Bayesian test declared B as a winner when the expected loss was lower than 0.1%.
How this works in practice
After talking about the reasons for this transition away from NHST, I should also mention how we use it in practice. For now, we are using two separate Bayesian models for two different types of tests. Without detailed explanations (which would need a separate article), I will just list what we use right now.
Instead of Chi-squared test, we use a model that assumes that the conversion rate has Bernoulli distribution. All the formulas are from following sources:
- Formulas for Bayesian A/B Testing by Evan Miller
- Easy Evaluation of Decision Rules in Bayesian A/B testing by Chris Stucchio
For revenue data, we have an extension of the conversion model. It separately estimates the conversion rate with the Bernoulli distribution and then the paid amount using the Exponential distribution. I implemented it based on this paper:
- Bayesian A/B Testing at VWO by Chris Stucchio
Code available with Kruschke’s book was also very useful in understanding the Bayesian statistics. I used parts of it to calculate the highest density intervals:
Using these sources, I implemented A/B test calculators in R. You can give them a try here:
Using Bayesian A/B testing, we can now carry out tests faster with more actionable results. Let me summarize the advantages below:
But what are the disadvantages and why is the Bayesian approach not used more commonly? In my opinion, the main reason is that it is far more complicated to implement compared to traditional NHST tests which can be calculated using one function in Excel. Another reason some are hesitant to use the Bayesian approach is the prior distribution. Prior and posterior distributions are key elements of Bayesian statistics. You have to determine the prior distribution before doing any calculations. At Pixel Federation, we are currently using non-informative priors for A/B testing purposes.
In conclusion, I want to emphasize that I am not claiming that frequentist statistics is generally inferior to Bayesian. They are both useful for different purposes. In this article, I wanted to argue that Bayesian approach is more appropriate for testing features and configurations in games.
If you happened to face similar problems and solved them differently or you used Bayesian statistics and have some experience you wanted to share, please let me know.
- Bayesian estimation supersedes the t test
- Doing Bayesian Data Analysis, A Tutorial with R, JAGS and Stan
- How Not To Run an A/B Test
- Is Bayesian A/B Testing Immune to Peeking? Not Exactly
- Bayesian A/B Testing at VWO
- Formulas for Bayesian A/B Testing
- Easy Evaluation of Decision Rules in Bayesian A/B testing