What are the chances…

1. What are the chances… Conditional Probability&Introduction to Bayes’ Theorem

2. Agenda • Introduction • Definitions and equations • Odds and probability • Likelihood ratios • Bayes’ Theorem

3. Examples: • If you flipped a coin 10 times, what is the probability that the first 5 come up heads? • What is the probability that the 6th toss comes up heads? • Given a positive dobutamine stress echo, what is the probability that the patient does NOT have CAD?

4. The probability of an event is the proportion of times the event is expected to occur in repeated experiments • The probability of an event, say event A, is denoted P(A). • All probabilities are between 0 and 1. (i.e. 0 < P(A) < 1) • The sum of the probabilities of all possible outcomes must be 1.

5. Assigning Probabilities • Guess based on prior knowledge alone • Guess based on knowledge of probability distribution (to be discussed later) • Assume equally likely outcomes • Use relative frequencies

6. Conditional Probability • The probability of event A occurring, given that event B has occurred, is called the conditional probability of event A given event B, denoted P(A|B) Example • Among women with a (+) mammogram, how often does a patient have breast cancer • P(breast CA +|+ mammogram)

7. Mutually Exclusive Events • Two events are mutually exclusive if their intersection is empty. • Two events, A and B, are mutually exclusive if and only if P(AB) = 0 • a child is a red head and a brunette. • P(A U B) = P(A) + P(B) “And”

8. Odds • The concept of "odds" is familiar from gambling • For instance, one might say the odds of a particular horse winning a race are "3 to 1"; • This means the probability of the horse winning is 3 times the probability of not winning. • Odds of 1 to 1 means a 50% chance of something happening (as in tossing a coin and getting a head), and odds of 99 to 1 means it will happen 99 times out of 100 (as in bad weather on a public holiday).

9. Odds and Probability • Both are ways to express chance or likelihood of an event • Example: • What is the chance that a coin flip will result in “heads”? • Probability: expected number of “heads” 1 total number of options 2 • Odds: expected number of “heads” 1 expected number of non “heads” 1

10. Odds and Probability • Example: • What is the chance that you will roll a 7 at the craps table and “crap out”? • Probability: number of ways to roll a 7 6 16.7% total number of options 36 • Odds: number of ways to roll a 7 6 20% number of ways to not roll a 7 30

11. Odds and Probability • Odds = probability / (1-probability) • Probability = odds / (1+odds) • Use the craps example: if the probability of rolling a 7 is 16.77777%, what are the odds of rolling a seven

12. - + + - Likelihood Ratio Likelihood of a given test result in a patient with the target disorder compared to the likelihood of the same result in a patient without that disorder LR+ = sensitivity / (1-specificity) = (a/(a+c)) / (b/(b+d)) LR- = (1-sensitivity) / specificity = (c/(a+c)) / (d/(b+d)) Gold Standard Test

13. Result in probability theory Relates the conditional and marginal probability distributions of random variables In some interpretations of probability, tells how to update or revise beliefs in light of new evidence Bayes’ Theorem: Definition Thomas Bayes (1702-1761) British mathematician and minister http://en.wikipedia.org/wiki/Bayes'_theorem

14. Bayes’ Theorem: Definition • Bayes’ Rule underlies reasoning systems in artificial intelligence, decision analysis, and everyday medical decision making • we often know the probabilities on the right hand side of Bayes’ Rule and wish to estimate the probability on the left.

15. Example from Wikipedia… From which bowl is the cookie? • To illustrate, suppose there are two full bowls of cookies. • Bowl #1 has 10 chocolate chip and 30 plain cookies, • Bowl #2 has 20 of each • Fred picks a bowl at random, and then picks a cookie at random. • (Assume there is no reason to believe Fred treats one bowl differently from another, likewise for the cookies) • The cookie turns out to be a plain one…

16. Example from Wikipedia… • How probable is it that Fred picked it out of bowl #1? • Intuitively, it seems clear that the answer should be more than a half, since there are more plain cookies in bowl #1. • The precise answer is given by Bayes' theorem.

17. Example from Wikipedia… • Let B1 correspond to Bowl #1 and B2 to bowl #2 • Since the bowls are identical to Fred, P(B1) = P(B2) and there is a 50:50 shot of picking either bowl so the P(B1)=P(B2)=0.5 • P(C)=probability of a plain cookie P(B1) * P(C│B1) P(B1│C) = P(B1) * P(C│B1) + P(B2) * P(C│B2) 0.5 * 0.75 = 0.6 = 0.5 * 0.75 + 0.5 * 0.5

18. Posterior Probability Prior Probability Evidence Bayesian Analysis New Information Background Information Updated Information x =

19. Activity Background Activity Background Borrow money Credit history Buy a stock Market trends Bet a horse Past performance Sentence a criminal Previous convictions Treat a patient Past medical history Interpret a test Pre-test probability Borrow money Credit history Buy a stock Market trends Bet a horse Past performance Sentence a criminal Previous convictions Treat a patient Past medical history Interpret a test Pre-test probability Bayesian Analysis Prior Clinical trial analysis NONE!

20. Women Typical Angina Atypical Angina Nonanginal No Pain Prior Information in Diagnostic Testing Bayesian Analysis Prior N Engl J Med 1979;300:1350

21. Women Atypical Angina 0.2 Prior 0.17 0.17 Odds = = 0.2 1 – 0.17 Bayesian Analysis 0.1 1 10 Prior Odds N Engl J Med 1979;300:1350

22. Men Atypical Angina 0.8 0.44 0.44 Odds = = 0.8 1 – 0.44 Bayesian Analysis Prior 0.1 1 10 Prior Odds N Engl J Med 1979;300:1350

23. Disease + - b a + - Test c d Quantifying the Evidence Bayesian Analysis x 0.8 Evidence 0.1 1 10 Prior Odds LR+ = sensitivity / (1-specificity) = (a/(a+c)) / (b/(b+d))

24. Quantifying the Evidence Bayesian Analysis Disease + - + - 80 40 x 0.8 4.0 Test 20 160 100 200 0.1 1 10 0.1 1 10 Prior Odds Likelihood Ratio LR+ = sensitivity / (1-specificity) = (a/(a+c)) / (b/(b+d)) = 80/100 / 40/200 = 4.0

25. Computing the Post-test Odds Bayesian Analysis x 0.8 4.0 = 3.2 0.1 1 10 0.1 1 10 0.1 1 10 Posterior Odds Prior Odds Likelihood Ratio 45 year old man with atypical angina and 2.0 mm ST depression CAD probability = 3.2/4.2 = 76% 45 year old man with atypical angina CAD probability = 0.8/1.8 = 44% 2.0 mm horizontal ST depression

26. Computing the Post-test Odds Bayesian Analysis x 0.2 4.0 = 0.8 0.1 1 10 0.1 1 10 0.1 1 10 Posterior Odds Prior Odds Likelihood Ratio 45 year old woman with atypical angina and 2.0 mm ST depression CAD probability = 0.8/1.8 = 44% 45 year old woman with atypical angina CAD probability = 0.2/1.2 = 17% 2.0 mm horizontal ST depression

27. Review Bayesian Analysis Posterior Odds Ratio Prior Odds Ratio Evidential Odds Ratio x =

28. A Sample Problem Bayesian Analysis • Here's a story problem about a situation that doctors often encounter: • 1% of women at age forty who participate in routine screening have breast cancer. • 80% of women with breast cancer will get positive mammographies. • 9.6% of women without breast cancer will also get positive mammographies.  • A woman in this age group had a positive mammography in a routine screening.  • What is the probability that she actually has breast cancer? http://www.sysopmind.com/bayes

29. Evidence Posterior Prior Bayesian Analysis New Information Background Information Updated Information x =

30. Bayesian Analysis • Pre-test probability = .01 • Pre-test odds: • Odds = probability / (1-probability) • = .01/(1-.01) = 0.01

31. Evidence Posterior Prior Odds Bayesian Analysis New Information Background Information Updated Information x = x 0.01

32. - + + - • Evidence = Likelihood Ratio LR+ = sensitivity / (1-specificity) = (a/(a+c)) / (b/(b+d)) Gold Standard Test

33. - + + - A Sample Problem Bayesian Analysis • Here's a story problem about a situation that doctors often encounter: • 1% of women at age forty who participate in routine screening have breast cancer. • 80% of women with breast cancer will get positive mammographies. • 9.6% of women without breast cancer will also get positive mammographies.  Gold Standard 80 9.6 Test 20 90.4 100 100 http://www.sysopmind.com/bayes

34. - + + - • Evidence = Likelihood Ratio LR+ = sensitivity / (1-specificity) = (a/(a+c)) / (b/(b+d)) Gold Standard = (80/100) / (9.6/100) = 8.33 Test

35. Evidence Posterior Odds Prior Odds Bayesian Analysis New Information Background Information Updated Information x = x 0.01 8.33

36. Evidence Posterior Odds Prior Odds Bayesian Analysis New Information Background Information Updated Information x = x 0.01 8.33 = 0.0833

37. Evidence Posterior Odds Prior Odds Bayesian Analysis • Given the low pre-test probability, even a + test did not dramatically effect the post-test probability New Information Background Information Updated Information x = x = 0.01 8.33 0.0833 7.7% probability

39. 7.7%

40. Conclusions • Probability and odds are different ways to express chance • Conditional probability allows us to calculate the probability of an event given another event has or has not occurred (allows us to incorporate more information) • Bayes’ theorem incorporates results of trials/research to update our baseline assumptions

41. Bayesian Analysis Events + - b a A B Posterior Odds Ratio Prior Risk Ratio Evidential Odds Ratio x = Treatment c d Odds Ratio = ad/bc

42. Bayesian Analysis of Clinical Trials Quantifying the Prior

43. Bayesian Analysis of Clinical Trials Quantifying the Prior Events + - A B 1925 174 Posterior Odds Ratio Prior Risk Ratio Evidential Odds Ratio x = Treatment 198 1865 PROVE-IT Odds Ratio = 0.85 N Engl J Med 2004;350:1495

44. Bayesian Analysis of Clinical Trials Quantifying the Prior Posterior Odds Ratio Evidential Odds Ratio x 0.85 = 0.8 1 1.25 Prior Odds Ratio

45. Bayesian Analysis of Clinical Trials Quantifying the Evidence Events + - A B 309 1956 Posterior Odds Ratio 0.85 = Treatment 343 1889 0.8 1 1.25 Prior Odds Ratio A to Z Odds Ratio = 0.87 JAMA 2004;292:1307

46. Posterior Odds Ratio Bayesian Analysis of Clinical Trials Quantifying the Evidence x = 0.85 0.87 0.8 1 1.25 0.8 1 1.25 Prior Odds Ratio Evidential Odds Ratio

47. Posterior Risk Ratio I figure a 40% chance of rain, and a 10% chance we know what we’re talking about. Bayesian Analysis of Clinical Trials Considering the Uncertainties x = 0.85 0.87 0.8 1 1.25 0.8 1 1.25 Posterior Risk Ratio Prior Odds Ratio Evidential Odds Ratio

48. Bayesian Analysis of Clinical Trials Computing the Posterior x = 0.8 1 1.25 0.8 1 1.25 0.8 1 1.25 Posterior Odds Ratio Prior Odds Ratio Evidential Odds Ratio

49. Posterior Risk Ratio p = 0.10 CI Bayesian Analysis of Clinical Trials Interpreting the Posterior Risk Reduction > 10% Area = 0.8 x = 0.8 1 1.25 0.8 1 1.25 0.8 1 1.25 Posterior Odds Ratio Prior Odds Ratio Evidential Odds Ratio

50. Bayesian Analysis of Clinical Trials Interpreting the Posterior 1 0 Area = 0.8 Posterior Probability 10 0.8 1 1.25 0.8 1 1.25 0 50 100 Risk Reduction Threshold Prior Odds Ratio Evidential Odds Ratio