4 ms·

It's been years since I last cracked a statistics book. Would a t-test be more accurate than a z-test or chi-squared for small samples? I've love to see a write

by nhebb 16y ago

It's been years since I last cracked a statistics book. Would a t-test be more accurate than a z-test or chi-squared for small samples? I've love to see a write-up on the statistics equations for A/B testing and guidelines on when to use which.

btilly 16y ago
Would a t-test be more accurate than a z-test or chi-squared for small samples? No. Here is a brief overview of the different tests. The z-test depends on the assumption that the sum of random events is well approximated by a normal distribution. This happens with a large enough sample size. But is wrong for small sample sizes. A further weakness with the z-test is that the same sample data is being used to estimate variance and averages. Depending on the specifics of your actual sample, both estimates can be off in a way that confuses the test. The t-test corrects for this possibility but assumes that the individual measurements you are making are normally distributed. It is quite sensitive to this assumption. Unfortunately in A/B testing the measurements are binary yes/no, 1/0 decisions, which are decidedly not normally distributed. The g-test is designed to be a very good approximation to the exact problem that A/B testing tries to solve. Unfortunately it requires you to compute logs, which was hard before computers. The chi-square test is designed as an easily computed good approximation to the g-test. It only needs you to square numbers and add them together. It therefore became the most widely used algorithm, and is the one that people have heard of. Now that we have computers, the g-test is the test we should use. But nobody has heard of it. (However you're still free to use it! And should.) If you get a large enough body of measurements then all tests converge to the z-test. I've love to see a write-up on the statistics equations for A/B testing and guidelines on when to use which. http://elem.com/~btilly/effective-ab-testing/ http://elem.com/~btilly/effective-ab-testing/ goes through this in some detail.
- equark 16y ago
  A/B testing is a Bayesian decision problem. Hypothesis testing is really the wrong way to frame this problem. Dynamically it is a bandit problem and statically it's just about maximizing expected utility, where the expectation is taken with respect to the posterior distribution. This becomes especially crucial when doing lots of tests or experimenting sequentially. All the frequentist proposals don't have their advertised coverage rates in these situations. Unfortunately most write-ups seem to be completely ignorant of Bayesian statistics.
  - btilly 16y ago
    I mostly agree with you. However if you've ever tried to explain those concepts to a bunch of PMs, you'll quickly discover that it isn't worth the complication of trying to describe things the "right" way. There are more important problems to deal with. My one point of disagreement is that dynamically it is a bandit problem. The reason that I disagree is that the things we are tracking and trying to optimize have very strong time dependencies that aren't obvious. (Behavior varies a lot by time of day, day of week, week of month, and month of year.) Therefore attempting to dynamically optimize can very, very easily go astray. That's not to say you can't do it. For instance I am quite supportive of using dynamic techniques to explore a multi-variate space. But in the end I wouldn't be comfortable unless you did a classic A/B test to validate the results.
- srgseg 16y ago
  Thanks, can't believe you only got 2 points for this.

by nhebb 16y ago

btilly 16y ago

Would a t-test be more accurate than a z-test or chi-squared for small samples? No. Here is a brief overview of the different tests. The z-test depends on the assumption that the sum of random events is well approximated by a normal distribution. This happens with a large enough sample size. But is wrong for small sample sizes. A further weakness with the z-test is that the same sample data is being used to estimate variance and averages. Depending on the specifics of your actual sample, both estimates can be off in a way that confuses the test. The t-test corrects for this possibility but assumes that the individual measurements you are making are normally distributed. It is quite sensitive to this assumption. Unfortunately in A/B testing the measurements are binary yes/no, 1/0 decisions, which are decidedly not normally distributed. The g-test is designed to be a very good approximation to the exact problem that A/B testing tries to solve. Unfortunately it requires you to compute logs, which was hard before computers. The chi-square test is designed as an easily computed good approximation to the g-test. It only needs you to square numbers and add them together. It therefore became the most widely used algorithm, and is the one that people have heard of. Now that we have computers, the g-test is the test we should use. But nobody has heard of it. (However you're still free to use it! And should.) If you get a large enough body of measurements then all tests converge to the z-test. I've love to see a write-up on the statistics equations for A/B testing and guidelines on when to use which. http://elem.com/~btilly/effective-ab-testing/ http://elem.com/~btilly/effective-ab-testing/ goes through this in some detail.

equark 16y ago
A/B testing is a Bayesian decision problem. Hypothesis testing is really the wrong way to frame this problem. Dynamically it is a bandit problem and statically it's just about maximizing expected utility, where the expectation is taken with respect to the posterior distribution. This becomes especially crucial when doing lots of tests or experimenting sequentially. All the frequentist proposals don't have their advertised coverage rates in these situations. Unfortunately most write-ups seem to be completely ignorant of Bayesian statistics.
- btilly 16y ago
  I mostly agree with you. However if you've ever tried to explain those concepts to a bunch of PMs, you'll quickly discover that it isn't worth the complication of trying to describe things the "right" way. There are more important problems to deal with. My one point of disagreement is that dynamically it is a bandit problem. The reason that I disagree is that the things we are tracking and trying to optimize have very strong time dependencies that aren't obvious. (Behavior varies a lot by time of day, day of week, week of month, and month of year.) Therefore attempting to dynamically optimize can very, very easily go astray. That's not to say you can't do it. For instance I am quite supportive of using dynamic techniques to explore a multi-variate space. But in the end I wouldn't be comfortable unless you did a classic A/B test to validate the results.
srgseg 16y ago
Thanks, can't believe you only got 2 points for this.