1 / 24

Probability

Probability. Ch 6, Principle of Biostatistics Pagano & Gauvreau Prepared by Yu-Fen Li. Operations on Events. a Venn diagram is a useful device for depicting the relationships among events. A ∪ B “A or B”. A ∩ B “both A and B”. A c or , “not A”. Probability.

zarek
Download Presentation

Probability

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Probability Ch6, Principle of Biostatistics Pagano & Gauvreau Prepared by Yu-Fen Li

  2. Operations on Events • a Venn diagram is a useful device for depicting the relationships among events A ∪ B “A or B” A ∩ B “both A and B” Ac or , “not A”

  3. Probability • The numerical value of a probability lies between 0 and 1. • We have The additive rule of probability

  4. The additive rule of probability • For any two events A and B • If A and B are disjoint (mutually exclusive)

  5. The additive rule of probability • The additive rule can be extended to the cases of three or more mutually exclusive events • If A1, A2, · · · , and An are n mutually exclusive events, then A7 A5 A3 A6 A2 A4 A8

  6. Joint and Marginal Probabilities • Joint probability is the probability that two events will occur simultaneously. • Marginal probability is the probability of the occurrence of the single event. P(A2B1) P(A1)

  7. Conditional Probability • We are often interested in determining the probability that an event B will occur given that we already know the outcome of another event A • The multiplicative rule of probability states that the probability that twoevents A and B will both occur is equal to the probability of A multiplied by the probability of B given that A has already occurred

  8. Independence • Two events are said to be independent, if the outcome of oneevent has no effect on the occurrence of the other. • If A and B are independent events,

  9. Multiplicative rule of probability • For any events A and B • If A and B are independent

  10. ‘independent’ vs ‘mutually exclusive’ • the terms ‘independent’ and ‘mutually exclusive’ do not mean the same thing. • If A and B are independent and event A occurs, the outcome of B is not affected, i.e. P(B|A) = P(B). • If A and B are mutually exclusive and event A occurs, then event B cannot occur, i.e. P(B|A) = 0.

  11. Bayes’ Theorem • If A1, A2, · · · , and An are n mutually exclusive and exhaustive events • Bayes’ theorem states mutually exclusive exhaustive

  12. The Law of Total Probability • P(A)=P(A1∪A2∪A3∪A4) =P(A1) + P(A2 ) + P(A3 ) + P(A4) = 1 • P(B)=P(B∩A1) + P(B∩A2) + P(B∩A3) + P(B∩A4) =P(A1)P(B|A1) + P(A2)P(B|A2) + P(A3)P(B∩A3) + P(A4)P(B|A4) B ∩ ∩B B ∩ ∩B

  13. Examples • For example, the 163157 persons in the National Health Interview Survey of 1980-1981 (S) were subdivided into three mutually exclusive categories:

  14. Examples of marginal probabilities • Find the marginal probabilities

  15. Example of the additive rule of probability • If S is the event that an individual in the study is currently employed or currently unemployed or not in the labor force, i.e. S = E1∪ E2 ∪ E3. the additive rule of probability

  16. Example of the law of total probability • H may be expressed as the union of three exclusive events: the law of total probability

  17. Examples of conditional probabilities • Looking at each employment status subgroup separately

  18. Example of Bayes’ theorem • What is the probability of being current employed given on having hearing impairment?

  19. Diagnostic Tests • Bayes’ theorem is often employed in issues of diagnostic testing or screening • Sensitivity and Specificity

  20. Positive and Negative Predictive Values (PPV and NPV) • PPV • NPV Sensitivity (SE) 1-Specificity (1-SP)

  21. A 2 x 2 table • The diagnostic test is compared against a reference ('gold') standard, and results are tabulated in a 2 x 2 table Sensitivity = a / a+c Specificity = d / b+d Positive Predictive Value (PPV) = a / a+b ?? Negative Predictive Value (NPV) = d / c +d ?? Prevalence = a+c / (a+b+c+d) ??

  22. Relationship of Disease Prevalence to Predictive Values • The probability that he or she has the disease depends on the prevalence of the disease in the population tested and the validity of the test (sensitivity and specificity)

  23. Example

  24. Example

More Related