False positives

False positives result when a test falsely (incorrectly) reports a positive result. For example, a medical test for a disease may return a positive result indicating that the patient has the disease even if the patient does not have the disease. False positive is commonly denoted as the top right (Condition negative X test outcome positive) unit in a Confusion matrix.

False negatives

On the other hand, false negatives result when a test falsely or incorrectly reports a negative result. For example, a medical test for a disease may return a negative result indicating that patient does not have a disease even though the patient actually has the disease. False negative is commonly denoted as the bottom left (Condition positive X test outcome negative) unit in a Confusion matrix.

True positives

True positives result when a test correctly reports a positive result. As an example, a medical test for a disease may return a positive result indicating that the patient has the disease. This is shown to be true when the patient test confirms the existence of the disease. True positive is commonly denoted as the top left (Condition positive X test outcome positive) unit in a Confusion matrix.

True negatives

True negative result when a test correctly reports a negative result. As an example, a medical test for a disease may return a positive result indicating that the patient does not have the disease. This is shown to be true when the patients test also reports not having the disease. True negative is commonly denoted as the bottom right (Condition negative X test outcome negative) unit in a Confusion matrix.

False positives

We can use Bayes' theorem to determine the probability that a positive result is in fact a false positive. We find that if a disease is rare, then the majority of positive results may be false positives, even if the test is relatively accurate.

Naively, one might think that only 5% of positive test results are false, but that is quite wrong, as we shall see.

Suppose that only 0.1% of the population has that disease, so that a randomly selected patient has a 0.001 prior probability of having the disease.

We can use Bayes' theorem to calculate the probability that a positive test result is a false positive.

${\displaystyle {\begin{aligned}P(A|B)&={\frac {P(B|A)P(A)}{P(B|A)P(A)+P(B|\neg A)P(\neg A)}}\\\\&={\frac {0.99\times 0.001}{0.99\times 0.001+0.05\times 0.999}}\\~\\&\approx 0.019.\end{aligned}}}$

and hence the probability that a positive result is a false positive is about 1 − 0.019 = 0.98, or 98%.

Despite the apparent high accuracy of the test, the incidence of the disease is so low that the vast majority of patients who test positive do not have the disease. Nonetheless, the fraction of patients who test positive who do have the disease (0.019) is 19 times the fraction of people who have not yet taken the test who have the disease (0.001). Thus the test is not useless, and re-testing may improve the reliability of the result.

In order to reduce the problem of false positives, a test should be very accurate in reporting a negative result when the patient does not have the disease. If the test reported a negative result in patients without the disease with probability 0.999, then

${\displaystyle P(A|B)={\frac {0.99\times 0.001}{0.99\times 0.001+0.001\times 0.999}}\approx 0.5,}$

so that 1 − 0.5 = 0.5 now is the probability of a false positive.

False negatives

We can use Bayes' theorem to determine the probability that the negative result is in fact a false negative using the example from above:

${\displaystyle {\begin{aligned}P(A|\neg B)&={\frac {P(\neg B|A)P(A)}{P(\neg B|A)P(A)+P(\neg B|\neg A)P(\neg A)}}\\\\&={\frac {0.01\times 0.001}{0.01\times 0.001+0.95\times 0.999}}\\~\\&\approx 0.0000105.\end{aligned}}}$

The probability that a negative result is a false negative is about 0.0000105 or 0.00105%. When a disease is rare, false negatives will not be a major problem with the test.

But if 60% of the population had the disease, then the probability of a false negative would be greater. With the above test, the probability of a false negative would be

${\displaystyle {\begin{aligned}P(A|\neg B)&={\frac {P(\neg B|A)P(A)}{P(\neg B|A)P(A)+P(\neg B|\neg A)P(\neg A)}}\\\\&={\frac {0.01\times 0.6}{0.01\times 0.6+0.95\times 0.4}}\\~\\&\approx 0.0155.\end{aligned}}}$

The probability that a negative result is a false negative rises to 0.0155 or 1.55%.

True positives

We can use Bayes' theorem to determine the probability that the positive result is in fact a true positive using the example from above:

• If a tested patient has the disease, the test returns a positive result 99% of the time, or with a probability of 0.99.
• If a tested patient does not have the disease, the test returns a positive result 5% of the time, or with a probability of 0.05.
• Suppose that only 0.1% of the population has that disease, so that a randomly selected patient has a 0.001 prior probability of having the disease.

Let A represent the condition in which the patient has the disease, and B represent the evidence of a positive test result. Then, the probability that the patient actually has the disease given a positive test result is:

${\displaystyle {\begin{aligned}P(A|B)&={\frac {P(B|A)P(A)}{P(B|A)P(A)+P(B|\neg A)P(\neg A)}}\\\\&={\frac {0.99\times 0.001}{0.99\times 0.001+0.05\times 0.999}}\\~\\&\approx 0.019.\end{aligned}}}$

The probability that a positive result is a true positive is about 0.019%

True negatives

We can also use Bayes' theorem to calculate the probability of true negative. Using the examples above:

• If a tested patient has the disease, the test returns a positive result 99% of the time, or with a probability of 0.99.

${\displaystyle {\begin{aligned}P(\neg A|\neg B)&={\frac {P(\neg B|\neg A)P(A)}{P(\neg B|\neg A)P(\neg A)+P(\neg B|A)P(A)}}\\\\&={\frac {0.99\times 0.999}{0.99\times 0.999+0.05\times 0.001}}\\~\\&\approx 0.0000105.\end{aligned}}}$

The probability that a negative result is a true negative is 0.9999494 or 99.99%. Since the disease is rare and the positive to positive rate is high and the negative to negative rate is also high, this will produce a large True Negative rate.