Question 1191803: A certain disease has an incidence rate of 0.9%. If the false negative rate is 4% and the false positive rate is 5%, compute the probability that a person who tests positive actually has the disease.
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Edit: ikleyn is using the wrong numbers in her solution. The 1-0.05 = 0.95 is NOT the true positive rate. The true positive rate is 1 - 0.04 = 0.96 = 96%
The false negative rate is indeed relevant.
Edit2: Her solution has been fixed. Thank you ikleyn.
D = person has the disease
~D = person does not have the disease
P(D) = 0.009
P(~D) = 1-P(D) = 1 - 0.009 = 0.991
Terms to know:- Positive test result = the test claims the person has the disease (the claim may be true or false)
- Negative test result = the test claims the person does not have the disease (the claim may be true or false)
- False negative = when the test says "negative", but the reality is that the person actually has the disease
- False positive = when the test says "positive", but the reality is that the person does not actually have the disease
T = person tests positive
~T = person tests negative
False negative rate = 4% = 0.04
P(~T given D) = 0.04
Its complementary event is
P(T given D) = 1 - P(~T given D) = 1 - 0.04 = 0.96
False positive rate = 5% = 0.05
P(T given ~D) = 0.05
Its complementary event is
P(~T given ~D) = 1 - P(T given ~D) = 1 - 0.05 = 0.95
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Summary so far:
P(D) = 0.009 = incidence rate
P(~D) = 0.991
P(~T given D) = 0.04 = false negative rate
P(T given D) = 0.96 = true positive rate
P(T given ~D) = 0.05 = false positive rate
P(~T given ~D) = 0.95 = true negative rate
Let's compute the probability of testing positive
Use the law of total probability
P(T) = P(T and D) + P(T and ~D)
P(T) = P(T given D)*P(D) + P(T given ~D)*P(~D)
P(T) = 0.96*0.009 + 0.05*0.991
P(T) = 0.05819
Now we can use Bayes Theorem to get the following.
P(D given T) = probability person has disease given they tested positive
P(D given T) = P(T given D)*P(D)/P(T)
P(D given T) = 0.96*0.009/0.05819
P(D given T) = 0.14847912012373
P(D given T) = 0.1485
You may be asking yourself "why is this value so low?"
It's not a typo or a mistake that we got something this small.
The reason why is because the disease incidence rate (0.9% = 0.009) is very small.
The more rare the disease, the lower the P(D given T) value will get.
This makes sense because the person is more likely to be clear of the disease even if the test says "positive" because the incidence rate is so low.
In cases like this, it might be recommended to do a second medical test or perhaps an alternative test all together.
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We can create a table like this to help get a better understanding of everything
| Tests Positive | Tests Negative | Total | | Has Disease | 864 | 36 | 900 | | Does Not Have Disease | 4,955 | 94,145 | 99,100 | | Total | 5,819 | 94,181 | 100,000 |
I'm considering a hypothetical city of 100,000 people
0.9% has the disease so 0.009*100,000 = 900 people have the disease and 100,000-900 = 99,100 do not.
Of the 900 people who have the disease, 4% of the tests say "negative" and claims the person doesn't have the disease when it should say "positive".
4% of 900 = 0.04*900 = 36 people get false negative results. 900-36 = 864 people get true positive results, i.e correct positive results.
That fills up the first row of the table.
The second row has 99,100 people who don't have the disease.
5% of this sub-population will get a false positive unfortunately
5% of 99,100 = 0.05*99100 = 4955
Thankfully, the remaining 99,100 - 4,955 = 94,145 people get true negative tests.
To fill out the rest of the table, add up the items in each separate column
column1: 864 + 4,955 = 5,819
column2: 36 + 94,145 = 94,181
These sums are the totals for those who tested positive and negative in that order.
The table shows that 5,819 people tested positive. Of those who tested positive, 864 actually have the disease.
864/5819 = 0.1485 approximately
So this is one way to use an example to create a hypothetical population and help back up the results of the previous section.
You can change the "100,000" to any large number of your choice.
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Answer: Approximately 0.1485
Answer by ikleyn(52817)  (Show Source):
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