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- 1. Quantitative Methods for Lawyers Power Laws, Hypothesis Testing & Statistical Significance Class #14 professor daniel martin katz computationallegalstudies.com @ computational
- 2. Power Law Distribution (Scale Free) This is a Classic and Very Important Distribution A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event (e.g. its size), the frequency is said to follow a power law.
- 3. Power Law Distribution (Scale Free) Pareto distribution ( Wealth Distribution ) Zipf's law ( Natural Language Frequency ) Links on the Internet Citations Richardson's Law for the severity of violent conflicts (wars and terrorism) Population of cities Etc. Examples:
- 4. Power Laws Appear to be a Common Feature of Legal Systems Katz, et al (2011) American Legal Academy Katz & Stafford (2010) American Federal Judges Geist (2009) Austrian Supreme Court Smith (2007) U.S. Supreme Court Smith (2007) U.S. Law Reviews Post & Eisen (2000) NY Ct of Appeals
- 5. Rare Events, Criticality Power Laws Rare Events Criticality Disorder Induction
- 6. “ [T]here are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – there are things we do not know we don't know. ” United States Secretary of Defense Donald Rumsfeld
- 7. Unknown, Unknowns and Inductivist Reasoning Philosophy of Science = How do we Know What We Know? Black Swan Problem Even If We Observe White Swan after White Swan cannot induce that all swans are white
- 8. Learning by Falsification Popperian Perspective Karl Popper Rejected Inductivist Reasoning Science Advances Incrementally as Hypotheses are Falsified
- 9. Learning by Falsification Of Course, Certain Hypothesis cannot likely be falsified on a Reasonable Time Scale The problem of induction: the sun has risen every day for as long as anyone can remember. what is the rational proof that it will rise tomorrow? How can one rationally prove that past events will continue to repeat in the future, just because they have repeated in the past?
- 10. Learning by Falsification Popper Solution to the Question: No Need to Reject the Hypothesis of Sun Rising Cannot Really Formulate a Theory that Can Prove that the Sun Will Always Rise Can Develop a Theory that It Rise which will be falsified if the sun fails to rise
- 11. Hypothesis Testing & Statistical Significance
- 12. The Null and Alternative Hypothesis Example from Criminal Law: Criminal Trial Burden of Proof Presumption of Innocence Not Possible to Conclusively Prove a Lack of Innocence (with zero doubt) Must Be Overruled Beyond a Reasonable Doubt
- 13. The Null and Alternative Hypothesis Switch Now To a Scientific Inquiry: Study is Typically Designed to Determine Whether a Particular Hypothesis is Supported Start with Presumption that Hypothesis is Not True (Null Hypothesis) Researcher Must Demonstrate That The Presumption is Unlikely to Be True given the Population
- 14. Example: Coin Flip Nostradamus Predicting Coin Flips - Does you Friend Have the General Ability to Actually Predict Coin Flips? How Would You Evaluate This Proposition? How Many Predictions Would Your Friend Have to Get Right For You To Believe They Actually Have Real Ability?
- 15. Example: Coin Flip Nostradamus Ho: Cannot Actually Predict Coin Flips Ho is the Null Hypothesis H1: Can Actually Predict Coin Flip (i.e. do so at a rate greater than chance) H1 is the Alternative Hypothesis
- 16. Reject the Null versus Failing to Reject the Null In the Coin Flip Example, We might have enough evidence to reject the null Remember the default (null) is that there is no relationship If We Fail to Reject the Null, we are left with the assumption of no relationship Although a Relationship might actually exist
- 17. Coin Flip Nostradamus: Binomial Distribution Here is the Formula for a binomial experiment consisting of n trials and results in x successes. If the probability of success on an individual trial is P, then the binomial probability is: b(x; n, P) = nCx * Px * (1 - P)n - x What is the Probability Coin Flip Nostradamus Predicts at least 3 of 4 Coin Tosses ?
- 18. Coin Flip Nostradamus: Binomial Distribution ( 4! ) 3! (4-3)! (.53) (.54-3) (.125) (.5) ( 24 Here is the Prob of Getting Exactly 3 of 4 correct 6(1) ) = .25
- 19. Coin Flip Nostradamus: Binomial Distribution ( 4! ) 3! (4-3)! (.53) (.54-3) (.125) (.5) ( 24 Here is the Prob of Getting Exactly 3 of 4 correct 6(1) ) = .25 We Want “At Least” Which Implies BOTH 3 and 4 = .3125 .25 + .0625 Exactly 3 Exactly 4 at least 3 of 4 Coin Tosses
- 20. Coin Flip Nostradamus: Binomial Distribution If Our Would Be Coin Flip Nostradamus were able to get 3 out 4 Correct - we would not generally be prepared to give him/her credit just yet Namely, there is a 31.25% Probability that by Chance he/she would be able to predict at least 3 out of 4
- 21. Coin Flip Nostradamus: Binomial Distribution Now We Can Calculate Probability Associated of Prediction across some arbitrary number of trials How Much Do We Need to Be Convinced that Our Friend is Actually Coin Flip Nostradamus? This is a Question of Type I and Type II Error
- 22. Type I v. Type II Error
- 23. Type I v. Type II Error
- 24. Type I v. Type II Error Typical Convention is that a 5% Chance of Error is Acceptable for Purposes of Statistical Significance It is Depends Upon the Application Social Science = 5% Medicine with Serious Side Effects might Require Greater Level of Significance 1% or even less
- 25. Back To Coin Flip Nostradamus Okay let say Our Coin Flip Nostradamus agrees to run 75 coins flips in order to demonstrate his/her true powers Predicts 43 out of 75 Correct Is this Sufficient to Label Our Friend the Coin Flip Nostradamus?
- 26. Binomial Probability Calculator http://stattrek.com/tables/binomial.aspx
- 27. Binomial Probability Calculator http://stattrek.com/tables/binomial.aspx Enter These Three Values + Hit Calculate
- 28. Binomial Probability Calculator http://stattrek.com/tables/binomial.aspx And These are the Results Our P value Here is 12.4%
- 29. Coin Flip Nostradamus Our P Value is the Probability of Observing this Data Given the Null (i.e. that our friend does not have psychic powers) In this Case, the P Value is Our Pvalue > 5% Statistical Significance Threshold “Fail to Reject” Our Null of No Psychic Powers (We Do not Say Accept -- see the induction problem)
- 30. One Tailed -or- Two Tailed Tests There is a Difference Between a Directional and a Non- Directional Hypothesis In the Coin Flip Nostradamus Example it would be amazing if our friend could actually fail to predict 75 consecutive events Note: These are Symmetric
- 31. One Tailed -or- Two Tailed Tests We are Often Interested in a Non- Directional Hypothesis Stricter Crime Law and the Crime Rate We are Interested in Whether there is Deterrence and if there were to be higher crime rates New Drug and Health We Want to Both if It Makes the Patient Better and if the Patient’s condition get worse
- 32. One Tailed -or- Two Tailed Tests Two Tailed Test One Tailed Test (Positive direction) One Tailed Test (negative direction)
- 33. An Example of a Hypothesis Test Note: π is Prob α is the Significance Level https://onlinecourses.science.psu.edu/stat500/book/export/html/43
- 34. An Example of a Hypothesis Test Note: π is Prob α is the Significance Level Want to Make Sure Sample is Large Enough https://onlinecourses.science.psu.edu/stat500/book/export/html/43
- 35. An Example of a Hypothesis Test Note: π is Prob α is the Significance Level Want to Make Sure Sample is Large Enough If you Do Equal vs. Does Not Equal -- Two Tail https://onlinecourses.science.psu.edu/stat500/book/export/html/43
- 36. An Example of a Hypothesis Test z = (p - P) / σ where p is our sample prov P is theorized population prob σ is our Standard Deviation https://onlinecourses.science.psu.edu/stat500/book/export/html/43
- 37. An Example of a Hypothesis Test https://onlinecourses.science.psu.edu/stat500/book/export/html/43
- 38. Another Example Question I roll a single die 1,000 times and obtain a "6" on 204 rolls. Is there significant evidence to suggest that the die is not fair?
- 39. Another Example Question

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