Ch 6, Principle of Biostatistics 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 Ac or , “not A” A ∩ B “both A and B” A ∪ B “A or B”

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

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)

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

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(A2B1) P(A1)

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

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,

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

‘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.

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

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

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

Examples of marginal probabilities Find the marginal probabilities

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

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

Examples of conditional probabilities Looking at each employment status subgroup separately

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

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

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

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)

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)

Example D+ D- Total T+ 20 180 200 T- 10 1820 1830 30 2000 2030 Frequency Table of response by age response age 35 55 75 Total 0 3 7 23 33 1 31 3 6 40 Total 34 10 29 73

Example D+ D- Total T+ 67 9 76 T- 33 91 124 100 200 Frequency Table of response by age response age 35 55 75 Total 0 3 7 23 33 1 31 3 6 40 Total 34 10 29 73