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**Ch 6, Principle of Biostatistics**

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

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

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**The additive rule of probability**

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

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**The additive rule of probability**

For any two events A and B If A and B are disjoint (mutually exclusive) 𝑃(𝐴∪𝐵)=𝑃(𝐴)+𝑃(𝐵)−𝑃(𝐴∩𝐵) 𝑃(𝐴∪𝐵)=𝑃(𝐴)+𝑃(𝐵)

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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 A5 A7 A3 A6 A2 A4 A8

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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. A1 A2 B1 a b a+b B2 c d c+d a+c b+d n P(A2B1) P(A1)

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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

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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,

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**Multiplicative rule of probability**

For any events A and B If A and B are independent

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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.

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

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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(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

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

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**Examples of marginal probabilities**

Find the marginal probabilities

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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 = E1 ∪ E2 ∪ E3. 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: the law of total probability

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**Examples of conditional probabilities**

Looking at each employment status subgroup separately

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**Example of Bayes’ theorem**

What is the probability of being current employed given on having hearing impairment?

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

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**Positive and Negative Predictive Values (PPV and NPV)**

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 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) ?? Test Gold Standard Results D+ D- T+ a (TP) b (FP) T- c (FN) d (TN)

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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)

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**Example D+ D- Total T+ 20 180 200 T- 10 1820 1830 30 2000 2030**

Frequency Table of response by age response age Total Total

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**Example D+ D- Total T+ 67 9 76 T- 33 91 124 100 200 Frequency**

Table of response by age response age Total Total

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