Applied statistics/Tutorials: Difference between revisions
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==The false positive question== | ==The false positive question== | ||
The question:<br> | The question:<br> | ||
If a test of a disease that has a prevalence rate of 1 in 1000 has a false positive rate of 5%, what is the chance that a person who has positive result actually has the disease.<br> | If a test of a disease that has a prevalence rate of 1 in 1000 has a false positive rate of 5%, what is the chance that a person who has been given a positive result actually has the disease.<br> | ||
The answer:<br> | The answer:<br> | ||
2%<br> | 2%<br> | ||
Proof:<br><small> | Proof:<br><small> | ||
Let A denote the event of having the disease and, B the event of having been tested positive (for the purpose of applying Bayes'theorem) | Let A denote the event of having the disease and, B the event of having been tested positive (for the purpose of applying Bayes'theorem),<br> | ||
Then P(B/A) which is the probability of having been tested positive when having the disease, can be taken | Then P(B/A) which is the probability of having been tested positive when having the disease, can be taken as equal to 1;<br> | ||
And P(A) is the probability of having the disease, which with a prevalence of 1 in 1000 must be equal to 1/1000<<br> | And P(A) is the probability of having the disease, which with a prevalence of 1 in 1000 must be equal to 1/1000<<br> | ||
And P(B) is the probability of being tested positive, which can be arrived at by 3 steps:<br> | And P(B) is the probability of being tested positive, which can be arrived at by 3 steps:<br> | ||
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::P(B) = 999/1000 multiplied by 1/20 or, near enough 1/20.<br> | ::P(B) = 999/1000 multiplied by 1/20 or, near enough 1/20.<br> | ||
So applying Bayes' theorem, the probability of having the disease, given that you have been tested positive is given by | So applying Bayes' theorem, the probability of having the disease, given that you have been tested positive is given by | ||
::: P(A/B) = P(B/A) x P(A)/P(B), or: | ::: P(A/B) = P(B/A) x P(A)/P(B), or: | ||
:::: = 1 x (1/1000)/(1/20) - which is 0.02, or 2%. | :::: = 1 x (1/1000)/(1/20) - which is 0.02, or 2%. | ||
</small> | </small> | ||
Revision as of 11:21, 29 June 2009
Rules of chance
The addition rule
For two mutually exclusive events, A and B,
the probability that either A or B will occur is equal to the probability that A will occur plus the probability that B will occur,
- P(A or B) = P(A) + P(B).
The multiplication rule
For two independent (unrelated) events, A and B,
the probability that A and B will both occur is equal to the probability that A will occur multiplied by the probability that B will occur,
- P(A and B) = P(A) x P(B)
Bayes' theorem
The probability that event A will occur, given that event B has occurred is equal to the probability that B will occur, given that A has occurred, mutiplied by the probability that A will occur divided by the probability that B will occur,
- P(A/B) = P(B/A) x P(A)/P(B).
The false positive question
The question:
If a test of a disease that has a prevalence rate of 1 in 1000 has a false positive rate of 5%, what is the chance that a person who has been given a positive result actually has the disease.
The answer:
2%
Proof:
Let A denote the event of having the disease and, B the event of having been tested positive (for the purpose of applying Bayes'theorem),
Then P(B/A) which is the probability of having been tested positive when having the disease, can be taken as equal to 1;
And P(A) is the probability of having the disease, which with a prevalence of 1 in 1000 must be equal to 1/1000<
And P(B) is the probability of being tested positive, which can be arrived at by 3 steps:
Step 1 is to observe that since the prevalence of the disease is 1 in 1000, 999 persons out of every 1000 are healthy.
Step 2 is to recall that for each healthy person the probability of being tested positive is 5% or 1 in 20.
Step 3 is to apply the multiplication rule and get the answer:
- P(B) = 999/1000 multiplied by 1/20 or, near enough 1/20.
- P(B) = 999/1000 multiplied by 1/20 or, near enough 1/20.
So applying Bayes' theorem, the probability of having the disease, given that you have been tested positive is given by
- P(A/B) = P(B/A) x P(A)/P(B), or:
- = 1 x (1/1000)/(1/20) - which is 0.02, or 2%.
- P(A/B) = P(B/A) x P(A)/P(B), or: