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- 1. Crash Course in A/B testing A statistical perspective Wayne Tai Lee
- 2. Roadmap • What is A/B testing? • • • • • • Good experiments and the role of statistics Similar to proof by contradiction “Tests” Big data meets classic asymptotics Complaints with classical hypothesis testing Alternatives?
- 3. What is A/B Testing • An industry term for controlled and randomized experiment between treatment/control groups. • Age old problem….especially with humans
- 4. What most people know: Gather samples Apply treatments Compare Measure Outcome Assign treatments A ? B
- 5. What most people know: Only difference is in the treatment! A ? B
- 6. Reality: Variability from Samples/Inputs Variability from Treatment/function Variability from Measurement A ?????? B How do we account for all that?
- 7. Confounding: • If there are variabilities in addition to the treatment effect, how can we identify/isolate the effect from the treatment?
- 8. 3 Types of Variability: • Controlled variability • Systematic and desired • i.e. our treatment • Bias • Systematic but not desired • Anything that can confound our study • Noise • Random error but not desired • Won’t confound the study but makes it hard to make a decision.
- 9. How do we categorize each? Variability from Samples/Inputs Variability from Treatment/function Variability from Measurement A ?????? B
- 10. Reality: Good instrumentation! A ?????? B
- 11. Reality: Randomize assignment! Convert bias to noise A ?????? B
- 12. Reality: Randomize assignment! Convert bias to noise A ?????? B Your population can be skewed or biased….but that only restricts the generalizability of the results
- 13. Reality: Think about what you want to measure and how! Minimize the noise level/variability in the metric. A ? B
- 14. A good experiment in general: - Good design and implementation should be used to avoid bias. - For unavoidable biases, use randomization to turn it into noise. - Good planning to minimize noise in data.
- 15. How do we deal with noise? - Bread and butter of statisticians! - Quantify the magnitude of the treatment - Quantify the magnitude of the noise - Just compare…..most of the time
- 16. Formalizing the Comparison Similar to proof by contradiction - You assume the difference is by chance (noise)
- 17. Formalizing the Comparison Similar to proof by contradiction - You assume the difference is by chance (noise) - See how the data contradicts the assumption
- 18. Formalizing the Comparison Similar to proof by contradiction - You assume the difference is by chance (noise) - See how the data contradicts the assumption - If the surprise surpasses a threshold, we reject the assumption. - ….nothing is “100%”
- 19. Difference due to chance? Red -> treatment; Black -> control ID Person 1 Person 2 Person 3 Person 4 Person 5 Person 6 PV 39 209 31 98 9 151
- 20. Difference due to chance? Red -> treatment; Black -> control ID Person 1 Person 2 Person 3 Person 4 Person 5 Person 6 PV 39 209 31 98 9 151 | | | | | | | Let’s measure the difference in means! mean 72 mean 124.5 Diff = -52.5 ….so what?
- 21. Difference due to chance? Red -> treatment; Black -> control ID PV ID PV Person 1 Person 2 Person 3 Person 4 Person 5 Person 6 39 209 31 98 9 151 1 2 3 4 5 6 39 209 31 98 9 151 If there was no difference from the treatment, shuffling the treatment status can emulate the randomization of the samples.
- 22. Difference due to chance? Red -> treatment; Black -> control ID PV ID PV Person 1 Person 2 Person 3 Person 4 Person 5 Person 6 39 209 31 98 9 151 1 2 3 4 5 6 39 209 31 98 9 151 Diff = 122.25 – 24 = 98.25
- 23. Difference due to chance? Red -> treatment; Black -> control ID PV ID PV Person 1 Person 2 Person 3 Person 4 Person 5 Person 6 39 209 31 98 9 151 1 2 3 4 5 6 39 209 31 98 9 151 Diff = 107. 5 – 53.5 = 54
- 24. Difference due to chance? 50000 repeats later….. Our original -52.5
- 25. Difference due to chance? Our original -52.5 46.5% of the permutations yielded a larger if not the same difference as our original sample (in magnitude). Are you surprised by the initial results?
- 26. “Tests” Congratulations! - You just learned the permutation test! - The 46.5% is the p-value under the permutation test.
- 27. “Tests” Congratulations! - You just learned the permutation test! - The 46.5% is the p-value under the permutation test. Problems: - Permuting the labels can be computationally costly. - Not possible before computers! - Statistical theory says there are many tests out there.
- 28. Standard t-test: 1) Calculate delta: = mean_treatment – mean_control 2) Assumes follows a Normal distribution then calculate the p-value. p-value = sum of red areas - 0 3) If p-value < 0.05 then we reject the assumption that there is no difference between treatment and control. 28 “Tests”
- 29. Big data meets classic Stats 29 Wait, our metrics may not be Normal!
- 30. Big Data meets Classic Stat We care about the “mean of the metric” and not the actual metric distribution. 30 Wait, our metrics may not be Normal!
- 31. Big Data meets Classic Stat We care about the “mean of the metric” and not the actual metric distribution. 31 Wait, our metrics may not be Normal! Central Limit Theorem: The “mean of the metric” will be Normal if the sample size is LARGE!
- 32. Assumptions with t-test - Normality of %delta - Guaranteed with large sample sizes - Independent Samples - Not too many 0’s That’s IT!!! - Easy to automate. - Simple and general. 32 Big Data meets Classic Stat
- 33. What are “Tests”? 33 • Statistical tests are just procedures that depend on data to make a decision. • Engineerify: Statistical tests are functions that take in data, treatments, and return a boolean.
- 34. • Statistical tests are just procedures that depend on data to make a decision. • Engineerify: Statistical tests are functions that take in data, treatments, and return a boolean. Guarantees: • By setting the p-value to compare to a 5% threshold, we control P( Test says difference exists | In reality NO difference) <= 5% 34 What are “Tests”?
- 35. • Statistical tests are just procedures that depend on data to make a decision. • Engineerify: Statistical tests are functions that take in data, treatments, and return a boolean. Guarantees: • By setting the p-value to compare to a 5% threshold, we control P( Test says difference exists | In reality NO difference) <= 5% • By setting the power of the test to be 80%, we control P( Test says difference exists | In reality difference exists) >= 80% 35 What are “Tests”?
- 36. • Statistical tests are just procedures that depend on data to make a decision. • Engineerify: Statistical tests are functions that take in data, treatments, and return a boolean. Guarantees: • By setting the p-value to compare to a 5% threshold, we control P( Test says difference exists | In reality NO difference) <= 5% • By setting the power of the test to be 80%, we control P( Test says difference exists | In reality difference exists) >= 80% • Increasing this often requires more data 36 What are “Tests”?
- 37. Meaning: All treatments No difference Difference exist 37 Reality Useless treatments Impactful treatments
- 38. Meaning: All treatments No difference Difference exist 38 Reality Useless treatments Test Decision No difference Difference Exists Impactful treatments No difference Difference Exists
- 39. Meaning: All treatments No difference Difference exist 39 Reality Useless treatments Test Decision Guarantees through conventional thresholds No difference >95% Difference Exists <=5% Impactful treatments No difference Difference Exists <20% >=80%
- 40. Meaning: All treatments No difference Difference exist 40 Reality Useless treatments Test Decision Guarantees through conventional thresholds Jargon No difference >95% Difference Exists <=5% Significance level Impactful treatments No difference Difference Exists <20% >=80% Power
- 41. Meaning: 41 - Most appropriate over repeated decision making - E.g. spammer or not
- 42. - Most appropriate over repeated decision making - E.g. spammer or not - Not seeing a difference could mean - There is no difference - Not enough power 42 Meaning:
- 43. - Most appropriate over repeated decision making - E.g. spammer or not - Not seeing a difference could mean - There is no difference - Not enough power - Seeing a difference could mean - There is a difference - Got unlucky/lucky 43 Meaning:
- 44. - Most appropriate over repeated decision making - E.g. spammer or not - Not seeing a difference could mean - There is no difference - Not enough power - Seeing a difference could mean - There is a difference - Got unlucky/lucky - Your specific test is either impactful or not. (100% or 0%) Not what most people want to hear…. 44 Meaning:
- 45. Complaints with Hypth Testing 45 • People get really stuck on p-values and tests. • Confusing, boring, and formulaic.
- 46. Complaints with Hypth Testing 46 • People get really stuck on p-values and tests. • Confusing, boring, and formulaic. • Statistical significance != Scientific significance • You could detect a .000001 difference, so what?
- 47. • People get really stuck on p-values and tests. • Confusing, boring, and formulaic. • Statistical significance != Scientific significance • You could detect a .000001 difference, so what? • Multiple Hypothesis testing • 5% false positive is 1 out of 20. Quite high! • http://xkcd.com/882/ • Most published results are false still (Ioannidis 2005) 47 Complaints with Hypth Testing
- 48. • People get really stuck on p-values and tests. • Confusing, boring, and formulaic. • Statistical significance != Scientific significance • You could detect a .000001 difference, so what? • Multiple Hypothesis testing • 5% false positive is 1 out of 20. Quite high! • http://xkcd.com/882/ • Most published results are false still (Ioannidis 2005) • What is it answering? • Nothing specific about your test…. probabilities are over repeated trials. 48 Complaints with Hypth Testing
- 49. Both children of a British mother died within a short period of time. Mother was convicted of murder because p-value was low. If she was innocent, the chance of both children dying is low p-value = P( two deaths | innocent ) 49 Abuse: Prosecutor Fallacy
- 50. Both children of a British mother died within a short period of time. Mother was convicted of murder because p-value was low. If she was innocent, the chance of both children dying is low p-value = P( two deaths | innocent ) In fact, we should be looking at P( innocent | two deaths ) This is the prosecutor’s fallacy. 50 Abuse: Prosecutor Fallacy
- 51. Example: 51 All Mothers Guilty Mothers Two deaths Innocent Mothers Two deaths
- 52. Example: base line matters! 52 All Mothers Guilty Mothers Two deaths Innocent Mothers Two deaths P-value can be small. But base line can be huge.
- 53. Any Alternatives? 53 P( innocent | two deaths ) is what we want…… but does it make sense? Bayesian methodology: P( difference exists | data ) This requires knowing P(difference exists), i.e. the prior - Philosophical debate, “What is a probability?” - Easy to cheat the numbers
- 54. - How to deal with multiple hypothesis testing? - What are we doing in the company? - Rumor has it that “Multi-armed bandit > A/B testing”? 54 Questions?

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