Medical False Positives and False Negatives
Copyright © 2001–2022 by Stan Brown, BrownMath.com
(frequency table idea adapted from pages 136–137 of
John Allen Paulos,
A Mathematician Reads the Newspaper)
If a test for a disease is 98% accurate, and you test positive, the
probability you actually have the disease is not 98%. In fact,
the more rare the disease, the lower the probability that a positive result means you actually have it,
despite that 98% accuracy. The
difference lies in the rules of conditional or contingent probability.
You’ve taken a test for a deadly disease D, and the doctor
tells you that you’ve tested positive. How bad is the news? You need
to know P(D | pos), the probability that your positive test
result actually means you have D.
Don’t let the vertical-bar notation
you. P(A | B)
just means “the probability of A given that B
occurs” or “if B, then the probability of A”.
Suppose I draw a card
from the deck, and you guess that it’s the king of clubs. There are 52
cards, so you have 1 chance in 52 of being right:
P(♣K) = 1/52.
But if I tell you I drew a black
card and then you guess the king of clubs, with 26
black cards you have 1 chance in 26 of being right:
P(♣K | black) = 1/26.
To interpret your test result correctly,
you need to know three numbers:
- The false positive rate (FPR), the likelihood that you’ll
get a wrong positive test result when you actually don’t have D. This
is P(pos | no D), the probability of a type I error.
Sometimes instead of a false positive rate you’ll know
the specificity, which is P(neg | no D), the
probability of getting a correct negative result when you don’t
have D. If you don’t have D, you must get either a positive or
negative result, so P(pos | no D) +
P(neg | no D) = 100%. Therefore,
specificity = 100% − FPR, or FPR = 100% − specificity.
- The false negative rate (FNR), the likelihood that you’ll
get a wrong negative test result when you actually have D. This
is P(neg | D), the probability of a type II error.
Sometimes instead of a false negative rate you’ll know
the sensitivity, which is P(pos | D), the
probability of getting a correct positive result when you
have D. If you have D, you must get either a positive or
negative result, so P(pos | D) +
P(neg | D) = 100%. Therefore,
sensitivity = 100% − FNR, or FNR = 100% − sensitivity.
- The prevalence of D, the percentage of people that have
D, which can be written as P(D). Surprise! you need to know this to
interpret your test result.
Whether you consider sensitivity and specificity, or the false
negative and false positive rates, the two numbers can be the same but
typically they are different. But notice that none of them tells you
directly what you want to know: does my positive result mean I have D?
Suppose you’re told the test for D is “98% accurate” in the
following sense: If you have D, the test will be positive 99% of the
time, and if you don’t have it, the test will be negative
97% of the
time. In other words , the sensitivity is 99% and so the false
negative rate is 1%; the specificity is 97% and therefore the false
positive rate is 3%.
Suppose further that 0.1% — one out of every
thousand people — have D.
You might think that a positive result means you’re
99% likely to
have the disease. But 99% is the probability that if you have the
disease then you test positive, not the probability that if you test
positive then you have the disease. In symbols,
P(pos | D) = 99%, but you want to know
P(D | pos).
This kind of thing is easier to understand if you
work with numbers of people rather than percentages, and lay the numbers out in a chart
like the one below.
Suppose 100,000 people are tested for disease D. Consider the
people who actually have the disease (column 1).
Since the disease prevalence is 0.1% or 1
in 1000, about 100,000 × 0.1% =
100 actually have the disease.
And since 99% of people with the disease test positive, those
100 people will get about 99 positive
tests and 1 negative test.
Now look at the healthy people (column 2). Out of the
100,000 who took the test,
100,000 − 100 =
99,900 don’t have the disease.
Of those healthy people,
2,997 will test positive,
and the other 99,900 × 97% =
96,903 will test negative.
Finally, add across to find the row totals (column 3). There are
99 + 2,997 =
3,096 positive test results and
1 + 96,903 =
negative test results.
|Test result positive
|Test result negative
|(For a disease that
in 1000 actually has, and|
test with false positive rate 3% and false
negative rate 1%.)
From this chart, you can easily answer the questions,
“What’s the probability that testing positive means I have
the disease? What’s the probability that testing negative means
Out of the 3,096 tests
that report positive results,
2,997 (97%) are false positives, and only
The probability that you actually have D, when you’re given a
positive test result, is just
3%, so we can say that the
positive predictive value (PPV) of this test is
for a test that is 98% accurate!
you can write this as P(D | pos) =
(Remember that P(A|B) is the probability of “if B then A” or
“A given that B is true”.)
Let’s recap. The conditional probability that
you test positive, given that you have the disease, is
P(pos | D) = 99 ÷
100 = 99%
and this is what people sometimes call the “accuracy”
of the test.
(It’s actually the definition of the sensitivity of the
But the conditional probability that you have the disease if you
test positive, the positive predictive value, is
P(D | pos) = 99 ÷
3,096 = about
The number of sick people with positive results is on top of
both fractions, but the first fraction has the total sick people on
the bottom and the second fraction has the total positive results.
Can you believe a negative result? Well, if you test
negative, then the probability that you are actually negative is
P(no D | neg) =
96,903 ÷ 96,904, close to
so a negative result is almost certainly correct. You can say that the
negative predictive value (NPV) is close to
The exact probabilities will vary depending on the
accuracy of the test and the actual incidence of the disease, but
always you have to look at the conditional probability. This is one
reason why, for a disease like AIDS, patients are never told they test
positive until the blood has been retested with a different test, to
minimize the chance of a false positive.
See Repeating a test, later.
Doctors should be familiar with the probabilities when they give
test results to patients, but if you get a positive result from a test
for an uncommon disease, make sure your doctor
“Houston, we have a problem!”
Gordon MacGregor points out (email dated 27 Jan 2013) one
giant unstated assumption here: that people who have the disease and
people who don’t have the disease are equally likely to be
tested for it. That’s probably true or nearly true for diseases like HIV or
Huntington’s, where people with no symptoms are encouraged to
get tested and do.
But it’s emphatically not true for diseases
where people are typically not tested unless they have symptoms. So
really what we need to know is not the prevalence of the disease among
the general population — the 0.1% in the example
above — but the proportion of people who take the test
that actually have the disease.
Let’s think about biopsy results in testing for breast cancer.
(Please understand that what follows is not medical advice, and your
own personal family history and risk factors mean that these figures
may not apply to you.)
There are different types of biopsies, ordered by doctors for
different reasons including the particular patient’s
but Figure E in
Comparative Effectiveness of Core-Needle and Open Surgical Biopsy for the Diagnosis of Breast Lesions: Executive Summary
from the US Agency for Healthcare Research and Quality indicates that
26%–35% of women biopsied actually have breast cancer.
Fine Needle Aspiration Cytology (breast)
from the General Practice Notebook in the UK indicates a
false-positive rate of 1% to 3% and a false-negative rate of
10% to 18%. (The sensitivity is therefore 82%–90%, and the
specificity is 97%–99%.)
You could do the analysis using the above ranges. But to
keep things simple I’m just going to use the approximate
midpoint of each range: say that 30% of women biopsied actually have breast
cancer, and FNA biopsies yield 2% false positives and 14% false
negatives. Using those figures, here’s the table:
|Test result positive
|Test result negative
30% of women biopsied actually have breast cancer, and
the biopsy has a false positive rate of 2% and false
negative rate of 14%.)
Now you can compute probabilities.
First, the false-positive rate, the likelihood
of a positive result where there’s actually no cancer, was given
P(pos | no cancer) = 1% to 3% (I used 2%)
But you’re interested in the probability that a positive result
has actually detected cancer, and this is not 100% minus 2%.
P(cancer | pos) = 25,800 ÷
27,200 = 95%
The given false-negative rate, the probability that a woman
who has breast cancer gets a negative biopsy result, was given as
P(neg | cancer) = 10% to 18% (I used 14%)
But what’s the probability that a woman with a negative
result actually has breast cancer?
P(cancer | neg) = 4,200 ÷
72,800 = 6%
These discrepancies come from the difference between P(A|B)
and P(B|A), such as the difference between “getting a positive
result if cancer is present” and “having cancer if the test
result was positive”. The differences are less than they were in
the original example, because the incidence is greater (30% versus
Caution: Again, don’t use this page to make
medical decisions. You should work with your doctor, in light of your
unique medical situation.
Repeating a test
Reader Jarno Makkonen writes in to ask, “if you repeat
the test and get a confirming result, then what does that do to the
probability that a positive result is accurate?”
We can say that two positive results give us greater
confidence than one, but how much greater? This depends on the exact
mechanism that causes a false positive or false negative result, and
this will be different for different tests.
One important question is whether a false result is
essentially a random occurrence, or is tied in some way to
characteristics of an individual. For example, suppose that using
alcohol or other recreational drugs makes you more likely to get a
false positive result, or suppose having diabetes makes it more
likely, or a recent broken bone. The body is so fantastically
complicated that I would imagine each of those could affect
But I don’t have medical training, so let’s stick
with pure probability. In other words, let’s make an assumption
that any person is as likely to get a false positive (or false
negative) as any other person, so that nothing in an
individual’s biology has a significant effect on the chance of a
false positive (or negative). If we make that simplifying assumption,
then my reader’s question can be answered.
We want to know, “If I tested positive twice, how likely
is it that I have (or don’t have) the disease?” Well,
let’s expand the breast-cancer table to show the results of a
second test for people who tested positive the first time, or people
who tested negative the first time.
To help you read the table a little more easily, I’ve
italicized the results of the second tests. For example, in column 1,
we see that of the 25,800 women who actually had breast cancer and got
a correct positive result the first time, 22,188 got a positive
second result and 3,612 got a negative second result: that’s our
false negative rate of 14%, and 14% of 25,800 is 3,612.
|1st result positive
| 2nd result positive
| 2nd result negative
|1st result negative
| 2nd result positive
| 2nd result negative
|(Assuming that 30% of women
biopsied actually have breast cancer, and that|
the biopsy has a
false positive rate of 2% and false negative rate of 14%, and
a false positive is equally likely for everyone, and the same
for a false negative.)
What about the 70,000 women in column 2 who don’t have
breast cancer? 1,400 will nonetheless get a positive result:
that’s our 2% false positive rate.
(Remember, we’re assuming that false positives and false
negatives don’t depend significantly on any characteristics of
the individual, but only on the test itself.)
Of course we don’t know which 1,400 women got the wrong result.
But we do know that 2% of any women without breast cancer get
a false positive, and 2% of 1,400 is 28.
The other 1,372 get a correct negative result.
Once the table numbers are filled in, we can answer my reader’s
questions. If you have one test, and the result is positive,
there’s a 95% chance you have breast cancer (row 2,
25,800/27,200 = 95%). If you have two tests, both positive, the
probability rises to nearly 100% (row 3, 22,188/22,216 = 99.87%). On the
other hand, if you have one negative result, you have a 94% chance of
not having breast cancer (row 5, 68,600/72,800 = 94%); a
second negative test pushes that to 99% (row 7, 67,228/67,816 =
Filling in all those numbers is a fair amount of work, and
mistakes are easy to make. You might want to create formulas in terms
of these four variables:
- p = proportion of people tested who actually have the
disease. (This is not the proportion that test
- FPR = false positive rate
- FNR = false negative rate
- N = total number of people tested (Changing this
won’t change the final probabilities.)
It’s a nice intellectual exercise to develop formulas,
but if you just want to know the probabilities, take a look at the
accompanying Excel workbook.
It’s already set up with the breast-cancer example using
the midpoint figures for the variables. You might want to see how the
probabilities change when you vary the proportion of women with breast
cancer between 26% and 35%, the false positive rate between 1% and 3%,
and the false negative rate between 10% and 18%. But of course you can
enter numbers for any problem of your own.
As I write this, on 9 May 2020, it’s unfortunately true
that a positive or negative result on a COVID-19 test can’t be
interpreted using probability. To answer the questions “I tested
positive; what’s the chance I actually have the virus?” or
“I tested negative; what’s the chance I have the virus
anyway?” you need to know three things:
- The false positive rate for the test.
- The false negative rate for the test.
- The percentage of people taking the test w/ho actually have the disease.
We didn’t know any of those when I first wrote this
section, but three and a half weeks later (28 May 2020) we’re getting some ideas.
The FDA has published estimated sensitivity and specificity in
EUA Authorized Serology Test Performance.
(Remember that the false positive rate is 100% minus the specificity,
and the false negative rate is 100% minus the sensitivity.) And the
CDC is telling us, in the Test Performance section of
Interim Guidelines for COVID-19 Antibody Testing:
In most of the country, including areas that have been
heavily impacted, the prevalence of SARS-CoV-2 antibody is expected to
be low, ranging from <5% to 25%, so that testing at this point might
result in relatively more false positive results and fewer
In some settings, such as COVID-19 outbreaks in food
processing plants and congregate living facilities, the prevalence of
infection in the population may be significantly higher. In such
settings, serologic testing at appropriate intervals following
outbreaks might result in relatively fewer false positive results and
more false-negative results.
(But remember that what matters is not
the prevalence of a disease in the population, but the proportion
among tested people who actually have the disease.)
- 16–18 Nov 2021: Updated links here and
here; replaced links to withdrawn resources
with links to copies in web.archive.org.
- 27 May 2021: Explained vertical-bar
notation at the beginning, with an example from a deck of cards.
- 29 July 2020: Corrected an occurrence of “specificity” to
“sensitivity”; thanks to reader
Charlie Kufs for reporting this!
- 28 May 2020:
- 9 May 2020:
- 3 Feb 2013: Added Houston, we have a problem!
and the breast-cancer example.
- (intervening changes suppressed)
- 2 June 2002: New article.