Applied Biostatistics for the Health Sciences. Richard J. Rossi

Applied Biostatistics for the Health Sciences - Richard J. Rossi


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for the n=242 Volunteers in a University Study of Rapid Eye Movement (REM)

Gender Age
<18 18–20 21–25 >25
Female 3 58 42 25
Male 1 61 43 9

      1 a female volunteer is selected,

      2 a male volunteer younger than 21 is selected,

      3 a male volunteer or a volunteer older than 25 is selected.

      Solutions Let F be the event a female volunteer is selected, M the event a male volunteer is selected, 21 the event a volunteer younger than 21 is selected, and 25 the event that a volunteer older than 25 is selected. Since a volunteer will be selected at random, each volunteer is equally likely to be selected, and thus,

      1 the probability that a female volunteer is selected is

      2 the probability that a male volunteer younger than 21 is selected is

      3 the probability that a male volunteer or a volunteer older than 25 is selected is

       Example 2.21

Blood Type Rh Factor
+
O 38% 7%
A 34% 6%
B 9% 2%
AB 3% 1%

      1 has Rh positive blood,

      2 has type A or type B blood,

      3 has type A blood or Rh positive blood.

      Solutions Based on the percentages given in Table 2.9

      1 Rh positive blood is O+, A+, B+, and AB+, and thus, the probability that a randomly selected individual has Rh positive blood is

      2 type A or type B blood is A+, A−, B+, B−. Thus,

      3 the probability an individual has type A blood or Rh positive blood is

      In many biomedical studies, the probabilities associated with a qualitative variable will be modeled. A good probability model will take into account all of the factors believed to cause or explain the occurrence of the event. For example, the probability of survival for a melanoma patient depends on many factors including tumor thickness, tumor ulceration, gender, and age. Probabilities that are functions of a particular set of conditions are called conditional probabilities.

      Conditional probabilities are simply probabilities based on well-defined subpopulations defined by a particular set of conditions (i.e., explanatory variables). The conditional probability of the event A given that the event B has occurred is denoted by P(A|B) and is defined as

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      provided that P(B)≠0. Specifying that the event B has occurred places restrictions on the outcomes of the chance experiment that are possible. Thus, the outcomes in the event B become the subpopulation upon which the probability of the event A is based.

       Example 2.22

      Suppose that in a population of 100 units 35 units are in event A, 48 of the units are in event B, and 22 units are in both events A and B. The unconditional probability of event A is P(A)=35100=0.35. The conditional probability of event A given that event B has occurred is

dollar-sign upper P left-parenthesis upper A Math bar pipe bar symblom upper B right-parenthesis equals StartFraction 0.22 Over 0.48 EndFraction equals 0.46

      In this example, knowing that event B has occurred increases the probability that event A will occur from 0.35 to 0.46. That is, event A is more likely to occur if it is known that event B has occurred.

      Since conditional probabilities are probabilities, the rules associated with conditional probabilities are similar to the rules of probability. In particular,

      1 conditional probabilities are always between 0 and 1. That is, 0≤P(A|B)≤1.

      2 P(A does not occur|B)=1−P(A|B).

      3 P(A or B|C)=P(A|C)+P(B|C)−P(A and B|C).

      The probability that an individual will test positive in a diagnostic test is a weighted average of the proportion of correct diagnoses (i.e., true positives) and the false diagnoses (i.e., false positives). The formula for computing the probability that an individual tests positive is

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      A good diagnostic test for detecting a disease must have high sensitivity and specificity


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