from the prevalence of the disease, the sensitivity of the diagnostic test, and the specificity of the test. The formula for the PPV associated with a diagnostic test is
The probability that an individual does not have the disease given the test is negative is known as the negative predictive value (NPV). The formula for computing the NPV is
The positive predictive value measures the accuracy of a positive test result and the negative predictive value measures the accuracy of a negative test result. The closer PPV and NPV are to one, the better a diagnostic test is.
Example 2.23
Suppose the prevalence of HIV/AIDs is 5% among a particular population and an Enzyme-Linked Immuno Sorbent Assay (ELISA) test for diagnosing the presence of the HIV/AIDS virus has been developed. If the sensitivity of the ELISA test is 95% and the specificity is 99%, then the probability that an individual has the HIV/AIDS virus given the ELISA test is positive (i.e., PPV) is
Thus, an individual having a positive ELISA test has a 83% chance of having the HIV/AIDS virus.
The negative predictive value for this ELISA test is
Thus, an individual having a negative ELISA test has a 99.7% chance of not having the HIV/AIDS virus.
Example 2.24
Suppose a new infectious disease has been identified, and the prevalence of this disease is P(D)=0.001. Given that a person has the disease, the probability that the diagnostic test used to diagnose this disease is positive is P(+|D)=0.999, and given that a person does not have the disease, the probability that the test is positive is P(+|not D)=0.01. Then,
1 The sensitivity of the test is 0.999 and the specificity of the test is 0.99.
2 The probability of a positive test result isThus, there is only a 1.1% chance that an individual will test positive for the disease. The reason why a positive test is so rare is that this disease is extremely rare since P(D)=0.001.
3 The positive predictive value associated with this test isThus, there is only a 9% chance that an individual who tests positive for the disease will actually have the disease. This test has very low positive predictive value and is not very reliable for people who test positive.
4 The negative predictive value associated with this test isThus, this test has extremely high negative predictive value and is extremely reliable for people who test negative.
Thus, the probability of having the disease given the test is positive is roughly 100 times the unconditional probability of having the disease. Furthermore, in this example it is still highly unlikely that an individual has the disease even when the test is positive (P(D|+)=0.09, and hence, this is not a very informative diagnostic test.
2.3.3 Independence
In some cases, the unconditional probability and the conditional probability will be the same. In this case, knowing that the event B occurred does not change the likelihood that the event A will occur. When the unconditional probability of an event A is the same as the conditional probability of the event A given the event B, the events A and B are said to be independent events.
INDEPENDENT EVENTS
Two events A and B are independent if and only if one of the following conditions is met:
1 The probability of event A occurring is the same whether B occurred or not. That is,
2 The probability of event B occurring is the same whether A occurred or not. That is,
When two events are not independent, they are said to be dependent events. For example, the event that an individual has lung cancer and the event an individual has smoked a pack of cigarettes a day for several years are dependent events. On the other hand, the event an individual has lung cancer and the event an individual has blonde hair would be expected to be independent events.
If the events A and B are known to be independent, then the probability that A and B both occur is simply the product of their respective probabilities. That is, when A and B are independent
Example 2.25
The genders of successive offspring of human parents are known to be independent events. If the probability of having a male offspring is 0.48, then probability of having
1 two male offspring is
2 a male followed by a female offspring is
3 a male and a female offspring is
4 five female offspring is
5 the second offspring is a male given that the first was a female is 0.48 since successive births are independent. That is, since the successive offspring are independent
When the events A and B are independent so are the events A and not B, not A and B, and not A and not B. Thus,
