The Chance to Beat All (CTBA) represents the probability that a variation will perform better than the control version of your website or mobile app. For example, a chance of 60% means that the variation is likely to perform better than the control 60% of the time. This value also means a 40% chance that your results are completely due to a random chance.
Currently, this method of calculating the CTBA is only used for Multivariate tests and Personalization because they still follow the Frequentist method of calculating a winner. To determine the winning variation in A/B and Split URL tests, we follow the SmartStats (Bayesian) method of calculating the chance to beat all versions.
The success or failure of a variation is calculated by comparing the results against a base version (by default, the control page). You can change the base for your test at any time.
Please note that VWO calculates the chance to beat based on the test threshold settings you have defined in Settings > Campaign Settings.
To determine the chance to beat a variation, VWO calculates the:
- Conversion rates of the control and variation
- The standard error for the control and variation
- Chance to beat for a variation
Calculating Conversion Rates of the Control and Variations
Mathematically, the conversion rate is a binomial random variable, i.e., it can have two possible values. It is represented as the variable p. In A/B testing, you can calculate the value of p by observing n visitors on the webpage that is being tested. From those n visitors, you can calculate the number of conversions. The percentage value from the result is the conversion rate of your website or app.
Conversion rate (p) = Actual conversions/Visits to the website
The percentage value is the conversion rate of your website.
For example, consider that you are running a test to review the new design of your landing page. The control receives 894 visits, of which 423 are converted. On the other hand, the variation receives 863 visits, of which 458 are converted. These values imply that the conversion percentage value of the control is 47%, and that of the variation is 53%.
Calculating the Standard Error for the Control and Variations
If you repeat the same test multiple times, you will likely get a different value of p (or conversion rate of a variation) every time. This happens because, in statistical terms, we’re sampling, and like every sample, there’s a range of errors associated with it (because it does not cover the entire population). To avoid repeated tests, we can use the already established statistical formulas to calculate the standard error to determine how much deviation from the average conversion rate (p) can be expected from the test results. The deviation is the range within which the conversion rate is usually found. Smaller the deviation, the more confident you can be about estimating the true conversion rate. For a given conversion rate (p) and the number of trials (n), the standard error is calculated as:
Standard Error (SE) = Square root of (p * (1-p) / n)
In the example, the SE for control = Square root of (0.4732*(1-0.473/894) = 1.67 x 10-2. And, the SE for variation = Square root of (0.5307*(1-0.5307/863) =1.70 x 10-2.
(Please note that the percentage values are converted to decimal in the above calculations).
To get a 95% range for the conversion rate, multiply the standard error value by 2 (or 1.96 to be precise). You can be sure with 95% confidence that your true conversion rate lies within this range: p% ± 2 * SE.
In VWO, we calculate the conversion rate range for 80%, not 95%. Hence, we multiply the SE value by 1.28.
In the example discussed above, the standard error for the control is 1.67, while that of the variation is 1.70. Next, multiply the values by 1.28. Thus, the conversion rate range is ± 2.14% for control and ± 2.17% for the variation.
Calculating the CTBA for a Variation
After the Z-score is calculated, VWO calculates standard normal distribution to derive the chance percentage for the variation. You can refer to the following normal distribution table to understand how CTBA is derived for a variation:
To determine the chance to beat for a variation, we calculate the Z-score for control and variation. A Z-score is used in statistics to model any normal distribution as a standard normal distribution.
- Pv = Conversion rate of variation
- Pc = Conversion rate of control
- SEv = Standard error of variation
- SEc = Standard error of control
For our example discussed above, the Z-score can be derived as follows:
Z-score = (0.53 – 0.47)/Square root of 0.01698882 + 0.0166982 = 2.41605
As we see, for our z-score of 2.41, the normal distribution value is approximately 99%. To make the calculation easier for you, VWO provides a test significance calculator. Click here to download an Excel spreadsheet to help you determine the statistical significance of your test.
VWO does not calculate the CTBA when:
- The conversion rate of control is 0 or 1.
- The conversion rate of a variation is 0 or 1.
- The number of visitors in the variation is less than the set threshold value (sample size).
- The standard Error of either control or variation is 0.
- You can edit the CTBA threshold values and sample size using the VWO dashboard under Settings > Campaign Settings.