# Medical False Positives and False Negatives

(Conditional Probability)

Copyright © 2001–2021 by Stan Brown

(frequency table idea adapted from pages 136–137 of John Allen Paulos, A Mathematician Reads the Newspaper)

(Conditional Probability)

Copyright © 2001–2021 by Stan Brown

(frequency table idea adapted from pages 136–137 of John Allen Paulos, A Mathematician Reads the Newspaper)

**Summary:**
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** throw
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 = 96,904 negative test results.

Sick | Healthy | (totals) | |
---|---|---|---|

Actual status | 100 | 99,900 | 100,000 |

Test result positive | 99 | 2,997 | 3,096 |

Test result negative | 1 | 96,903 | 96,904 |

(For a disease that
1 person
in 1000 actually has, and)a 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 I don’t?”

Out of the 3,096 tests
that report positive results,
2,997 (97%) are false positives, and only
99
(3%) are
correct.
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
3% —
for a test that is 98% accurate!
Symbolically
you can write this as P(D | pos) =
3%.
(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
test.)
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 3%

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
100%,
so a negative result is almost certainly correct. You can say that the
**negative predictive value (NPV)** is close to
100%.

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
understands.

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 characteristics, 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:

Have Breast Cancer |
Don’t Have Breast Cancer |
(totals) | |
---|---|---|---|

Actual status | 30,000 | 70,000 | 100,000 |

Test result positive | 25,800 | 1,400 | 27,200 |

Test result negative | 4,200 | 68,600 | 72,800 |

(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%. |

Now you can compute probabilities. First, the false-positive rate, the likelihood of a positive result where there’s actually no cancer, was given as

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 0.1%).

**Caution**: Again, don’t use this page to make
medical decisions. You should work with your doctor, in light of your
unique medical situation.

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
*some* test.

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.

Have Breast Cancer |
Don’t Have Breast Cancer |
(totals) | |
---|---|---|---|

Actual status | 30,000 | 70,000 | 100,000 |

1st result positive | 25,800 | 1,400 | 27,200 |

2nd result positive |
22,188 |
28 |
22,216 |

2nd result negative |
3,612 |
1,372 |
4,984 |

1st result negative | 4,200 | 68,600 | 72,800 |

2nd result positive |
3,612 |
1,372 |
4,984 |

2nd result negative |
588 |
67,228 |
67,816 |

(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 that 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 =
99%).

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 positive.)*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 false-negative results.

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.)

- 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:
- Listed clearly the three numbers that go into the calculations, and added explanations of sensitivity and specificity.
- Changed the first example to give different rates for false positives and false negatives.
- Added symbols for the various conditional probabilities.
- Updated the COVID-19 section with available figures from the CDC and FDA.
- Updated the Excel workbook, to show the positive and negative predictive values.

- 9 May 2020:
- Added an Excel workbook.
- Added the sections Repeating a test and About COVID-19.
- In the tables, changed the totals row to “Actual status” and moved it first; also repeated the assumptions right in each table.
- Added a table of contents.

- 3 Feb 2013: Added Houston, we have a problem! and the breast-cancer example.
- (intervening changes suppressed)
- 2 June 2002: New article.

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