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Medical Biometry I

Medical Biometry I. ( Biostatistics 511) Week 5 Discussion Section Mike Garcia. Outline. Brief review of HW #4 Introduction to probability Types of Probability Definition of useful terms Useful rules for probability Probability examples “revisited” (two-way tables) More examples.

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Medical Biometry I

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  1. Medical Biometry I (Biostatistics 511) Week 5 Discussion Section Mike Garcia Biostat 511

  2. Outline • Brief review of HW #4 • Introduction to probability • Types of Probability • Definition of useful terms • Useful rules for probability • Probability examples “revisited” (two-way tables) • More examples Biostat 511

  3. Homework #4 • A couple common mistakes: • If the researchers influence the study conditions, the study design is experimental – even if no control group! • Confusion between case-control and cohort designs. Both are observational, but case-control samples disease/non-diseased individuals and compares exposure. Cohort is the other way around: sample exposed and non-exposed, then compare disease rate/frequency etc. Biostat 511

  4. Probability • Probability - meanings • 1) classical • 2) frequentist • 3) subjective (personal) • Sample space, events • Mutually exclusive, independence • and, or, complement • Joint, marginal, conditional probability • Probability - rules • 1) Addition • 2) Multiplication • 3) Total probability • 4) Bayes • Screening • sensitivity • specificity • predictive values Biostat 511

  5. Probability Probabilities allow us to quantify the uncertainty associated with the occurrence of some eventof interest Example events: • E1 = Rolling a total of 7 on two dice • E2 = Observing 2 heads in two flips of a fair coin • A = You pass Biost 511 (without laughing at Dr. Yanez’sterrible jokes) • D = Observing more than 12 cases of ovarian cancer in a random sample of n=100,000 US women. • We label events with capital letters (e.g., E, A, B,…) instead of writing the name of the event repeatedly. • Sometimes we write E1, E2, etc. for events so we don’t run out of letters to label our events. Biostat 511

  6. Probability The Sample Space (W) defines the list of all the possible outcomes that can take place when an experiment is performed Sample spaces for our examples: E1 : W ={(1,1), (1,2), (1,3),…(1,6), (2,1), (2,2),…(6,5), (6,6) }where the first number corresponds to the number shown on one die and the second number corresponds to the number on the other die. There are 36 possible outcomes. E2: W = {(T,T), (T,H), (H,T), (H,H) }. There are 4 outcomes. A : W = ? (Not easily quantified) D : W = Something very big. There are 2100000 possibilities! Biostat 511

  7. Probability Definitions: 1. Classical: P(E) = m/N m = the number of ways the event E can happen N = n(W) = size of the sample space = the total number of possible outcomes For our coin flipping example, E2, there are 4 possible equally likely things that can happen: (T, T), (T, H), (H, T) and (H, H). We have, N=4 and m=1. (H, H) is the only outcome that satisfies our event, E2. So… P(E2) = P(2 heads) = ¼ = 0.25. Sometimes it is difficult or impossible to know all of the possible things that can happen (to get N). We may choose to look at probability in a slightly different way… Biostat 511

  8. Relative Frequency 2. Relative frequency: If a process or an experiment is repeated a large number of times, n, and if the event, E, occurs m times, then the relative frequency, m/n, of E will be approximately equal to the probability of E. • Experiment: • Please everyone take a coin from your pocket. Flip it 10 times. • Do you think the coins are “balanced”, i.e., Prob(H) = .5 ? Biostat 511

  9. Personal Probability 3. Personal probability (subjective) • Your personal degree of uncertainty. You incorporate subjective information. • Examples • What is the probability you pass Biost 511? • The sample space is not easily defined (to use the classical definition); we really can’t do repeated experiments (upon you) to get a relative frequency probability, either. • What is the probability of life on Mars? • Same issues as with the preceding example Biostat 511

  10. Basic Terminology and Properties • Mutually exclusive (or disjoint) events: • Two events, A1and A2, are said to be mutually exclusive (disjoint) if only one or the other, but not both, can occur in a particular experiment. • Examples • Rolling a six-sided die Flipping a coin • Some possible events: Possible events: • A1 = rolling an even number B1 = H • A2 = rolling an odd number B2 = T • A3 = rolling a 1, 2 or 3 • A1 and A2 mutually exclusive? B1 and B2 mutually exclusive? • A1 and A3 mutually exclusive? Biostat 511

  11. Basic Terminology and Properties • Possible probability values for any event: • The probability of an event A, denoted P(A), must be between 0 and 1 0 < P(A) < 1 • Probabilities of any exhaustive collection of possible events (i.e. at least one must occur) that are mutually exclusive must be equal to 1 • Examples • Rolling a six-sided die Flipping a coin • Some possible events: Possible events • A1 = rolling an even number B1 = H • A2 = rolling an odd number B2 = T • A3 = rolling a 1, 2 or 3 • P(A1) + P(A2) = 1 P(B1) + P(B2) = 1 Biostat 511

  12. Basic Terminology and Properties • More on mutually exclusive and exhaustive events: • Mutually exclusive means one and only one of the events can occur (of the events being considered) • e.g., you can’t observe a tail on the flip of a coin if you observe a heads • Exhaustive means that your collection of events encompasses all the possible things that could occur (i.e., the sample space) • e.g., for the roll of a die, events A1={1,2,3}, A2={4}, A3={5,6} together encompass the sample space. They are exhaustive. They are also mutually exclusive events. • The events B1={H} and B2={T} are also exhaustive events. Biostat 511

  13. Basic Terminology and Properties • Complement events: • Event Ac is a complement of event A if Ac includes all possible outcomes in the sample space other than the outcomes in A. We denote the complement event as Ac. • Examples: • In our coin flipping experiment, the events B1={H} and B2={T} are complements of the other. • In our die rolling experiment the complement of the event A1={1,2,3} is A1c={4,5,6}. The complement of event A2={4} is A2c={1,2,3,5,6} • The probability of Ac is equal to 1 minus P(A). That is, • P(Ac) = 1 - P(A) Biostat 511

  14. Combining events • If A and B are any two events then we write • P(A and B) = P(A , B) = P(AB) • to indicate the probability that event Aandevent B (both) occurred. • It is often written as the probability of the intersection of the two events • P(A B) • Examples • For the education (A) versus willingness to participate (B) in the study of HIV prevention, we define the following events: • A1 = < HS B1 = definitely not participate • A2 = HS degree B2 = probably not participate • A3 = some college B3 = probably participate • A4 = post grad B4 = definitely participate Biostat 511

  15. Combining events • We have the following table for the population of individuals surveyed. • Example: • What is the probability of selecting an individual who has some college and will probably not participate? • What are the events in question? • A3 = some college, B2 = probably not participate • How do we combine them? • A3 and B2 (aka A3 B2) Biostat 511

  16. Combining events • We can find the probability of the combined event, P(A3 B2), using the classical definition of probability • P(A3 B2) = n(A3 B2)/n(W) • = 213 / 4850. • Note that 213 is the number of participants that satisfy the intersection of the two events. Biostat 511

  17. Combining events • If A and B are any two events then we write • P(A or B) • to indicate the probability that event A or event B (or both) occurred. • It is often written as the probability of the union of the two events • P(A U B) • Example: • What is the probability of selecting an individual who has some college or will probably not participate? • What are the events in question? • A3 = some college, B2 = probably not participate • How do we combine them? • A3or B2 (aka A3U B2) Biostat 511

  18. Combining Events • Addition (“or”) rule • If two events A and B are not mutually exclusive, then the probability that event A or event B occurs is: • P(A U B) = P(A) + P(B) – P(A B) • We subtract P(A B) so we do not double count it! Biostat 511

  19. Combining events • We find the probability of P(A3 U B2), using the classical definition of probability • P(A3U B2) = P(A3) + P(B2) - P(A3 B2) • = n(A3)/n(W) + n(B2)/n(W) - n(A3 B2)/n(W) • = 1270/4850 + 861/4850 - 213/4850 • = 1918/4850 • Note that 213 participants would have been counted twice had we not subtracted them with the intersection of the two events. Biostat 511

  20. Combining events • If A and B are any two events then we write • P(A given B) or P(A|B) • to indicate the probability of A among the subset of cases in which B is known to have occurred. • We have • P(A | B) = P(A B)/P(B) • Example: • What is the probability of selecting an individual who has some college given that they will probably not participate? • What are the events in question? • A3 = some college, B2 = probably not participate • How do we combine them? • A3given B2 (aka A3 | B2) Biostat 511

  21. Combining events • We find the probability of P(A3 | B2), using the classical definition of probability • P(A3| B2) = P(A3 B2)/P(B2) • = [n(A3 B2)/n(W)] / [n(B2)/n(W)] • = [213/4850] / [861/4850] • = 213/861 • The answer is the same as “conditioning out” the parts of the table that can no longer happen (i.e., events B1, B3, B4). Biostat 511

  22. General Rules Multiplication (“and”) rule (special case – independence) If two events, A and B, are “independent” (probability of one does not depend on the outcome of the other) then we can simply multiply their marginal probabilities together to get the joint probability P(A B) = P(A) P(B) Easy to extend for more than two independent events, e.g., A,B,C P(A B C) = P(A) P(B) P(C) Biostat 511

  23. Independence • To check for independence, you can check any of the following … • P(A | B) = P(A) or • P(B | A) = P(B) or • P(A B) = P(A) P(B). • If one holds, then all three hold; if one is violated, then all are violated • Question: • Are events A (getting mumps) and B (getting measles) • independent of each other? The notion of independent events is pervasive throughout statistics … Biostat 511

  24. Independence 2) Multiplication (“and”) rule – general case The general formula for the probability that both A B will occur is Our example: P(A) = 40/100 = 0.40 P(A | B) = 32/70 = 0.46 P(B) = 70/100 = 0.70 P(B | A) = 32/40 = 0.80 P(A B) = 32/100 = 0.32 Question: Is Mumps (A) independent of Measles (B)? Biostat 511

  25. Summary • Probability - meanings • 1) classical • 2) frequentist • 3) subjective (personal) • Sample space, events • Mutually exclusive, independence • and, or, complement • Joint, marginal, conditional probability • Probability - rules • 1) Addition • 2) Multiplication • 3) Independence • 4) Total Probability • 5) Bayes • Screening • sensitivity • specificity • predictive values Biostat 511

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