Topic 6 Probability

1. Topic 6Probability Modified from the notes of Professor A. Kuk P&G pp. 125-134

2. Events: • passing an exam • getting a disease • surviving beyond a certain age • treatment effective An event may occuror may not occur. What is the probability of occurrence of an event? Use letters A, B, C, … to denote events

3. Operations on events 1º Intersection A = “A woman has cervical cancer” B = “Positive Pap smear test” “A woman has cervical cancer and is tested positive”

4. Venn Diagram S A B

5. 2° Union • • • • e.g. 6 sided die • • • • • • • • • • • • • • • • • A=“Roll a 3” B=“Roll a 5”

6. Venn Diagram S A B

7. 3° Complement “A complement,” denoted by Ac, is the event “not A.” A = “live to be 25” Ac= “do not live to be 25” = “dead by 25”

8. Venn Diagram S Ac A

9. Definitions: Null event Cannot happen --- contradiction

10. Mutually exclusive events: Cannot happen together: A = “live to be 25” B =“die before 10th birthday”

11. Venn Diagram S B A

12. Meaning of probability What do we mean when we say P(Head turns up in a coin toss) ? Frequency interpretation of probability Number of tosses 10 100 1000 10000 Proportion of heads .200 .410 .494 .5017

13. More generally, If an experiment is repeated n times under essentially identical conditions and the event A occurs m times, then as n gets large the ratio approaches the probability of A. as n gets large

14. For any event A Complement

15. Venn Diagram Repeat experiment n times Ac=n-m A=m

16. Mutually exclusive events If A and B are mutually exclusive i.e.cannot occur together

17. Venn Diagram when A and B are mutually exclusive Conduct experiment n times B=k A=m

18. Additive Law If the events A, B, C, …. are mutually exclusive – so at most one of them may occur at any one time – then :

19. In general, B A

24. Multiplicative rule Note:

25. Diagnostic tests D = “have disease” Dc =“do not have disease” T+=“positive screening result P(T+|D)=sensitivity P(T-| Dc)=specificity Note: sensitivity & specificity are properties of the test

26. PRIOR TO TEST P(D)= prevalence AFTER TEST: For someone tested positive, consider P(D|T+)=positive predictive value. For someone tested negative, consider P(Dc |T-)=negative predictive value. Update probability in presence ofadditional information

27. D T+ Dc

28. Using multiplicative rule prevalencex sensitivity = prev x sens + (1-prev)x(1-specifity) = positive predictive value = PPV This is called Bayes’ theorem

29. X-ray Tuberculosis Yes Positive 22 Negative 8 Total 30 Example: X-ray screening for tuberculosis

30. X-ray Tuberculosis Yes No Positive 22 51 Negative 8 1739 Total 30 1790 Example: X-ray screening for tuberculosis

31. X-ray Tuberculosis Yes No Positive 22 51 Negative 8 1739 Total 30 1790 Example: X-ray screening for tuberculosis

32. Screening for TB Population: 1,000,000

33. Population: 1,000,000 Prevalence = 9.3 per 100,000 No TB: 999,907 TB: 93

34. Population: 1,000,000 TB: 93No TB: 999,907 = 0.7333 Sensitivity T+ 68 T- 25

35. Population: 1,000,000 TB: 93No TB: 999,907 Specificity 0.9715 = T+ 68 T- 25 T+ 28,497 T- 971,410

36. Population: 1,000,000 TB: 93No TB: 999,907 T+ 68 T- 25 T+ 28,497 T- 971,410 T+ 28,565 T- 971,435

37. Population: 1,000,000 TB: 93No TB: 999,907 T+ 28,497 T+ 68 T+ 28,565 compared with prevalence of 0.00093

38. Population: 1,000,000 TB: 93No TB: 999,907 T- 971,410 T- 25 T- 971,445

39. Ingelfinger et.al (1983) Biostatistics in Clinical Medicine

40. Ingelfinger et.al (1983) Biostatistics in Clinical Medicine