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Probability Ch 6, Principle of Biostatistics Pagano & Gauvreau Prepared by Yu-Fen Li 1

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Operations on Events a Venn diagram is a useful device for depicting the relationships among events 2 A B both A and B A B A or B A c or, not A

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

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For any two events A and B – If A and B are disjoint (mutually exclusive) 4

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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 5 A2 A4 A5 A3 A8 A6 A7

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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. 6 A1A2 B1aba+b B2cdc+d a+cb+dn P(A1) P(A2 B1)

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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 two events 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 7

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

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

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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. 10

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

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The Law of Total Probability P(A)=P(A1 A2 A3 A4) =P(A1) + P(A2 ) + P(A3 ) + P(A4) = 1 P(B)=P(BA1) + P(BA2) + P(BA3) + P(BA4) =P(A1)P(B|A1) + P(A2)P(B|A2) + P(A3)P(BA3) + P(A4)P(B|A4) 12 B

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

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Examples of marginal probabilities Find the marginal probabilities 14

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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 = E 1 E 2 E the additive rule of probability

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

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

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

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

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

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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 TestGold Standard ResultsD+D- T+ a (TP) b (FP) T- c (FN) d (TN) 21 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) ??

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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) 22

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Example 23 D+D-Total T T Total

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Example 24 D+D-Total T T Total

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