10.2 A focus on medical diagnostics
Imagine a disease with a prevalence of 1 in 1,000. The diagnostic test to detect it has a sensitivity and specificity of 90%. What is the chance that if you test positive, you actually have the disease?
Medical Diagnostics
Julian Pakay is a biochemist and science educator. He has a keen interest in promoting quantitative literacy. Watch an interview with Julian to learn why we need to be careful interpreting data around diagnostic tests and to see a solution to the question above. You will also learn some tips to minimise mistakes when performing calculations.
We can calculate the sensitivity of a diagnostic test using the formula:
Sensitivity = (true positives / all diseased) × 100
or
Sensitivity = true positives / (true positives + false negatives) × 100
And we can calculate the specificity of a diagnostic test using the formula:
Specificity= (true negatives / all non-diseased) × 100
or
Specificity = true negatives / (true negatives + false positives) × 100
It is important to remember that if a diagnostic test for a disease has a sensitivity and specificity of 99%, and you test positive, the probability that you actually have the disease is not 99%! In fact, the rarer the disease, the lower the probability that a positive result indicates that you have it.
Diagnostic test ‘accuracy’
You may see the term ‘accuracy’ used in the popular media to describe diagnostic tests. This term can be very misleading and should be avoided. For example, imagine a disease which has a high prevalence, for example, 10 out of 100 people suffer from it. A diagnostic test that fails to detect the disease at all would still be correct 90% of the time! It would correctly be negative for the 90% of people who do not have the disease. Therefore, it would be 90% accurate even though useless.In biomedical literature ‘accuracy’ is usually defined as all the accurate results (i.e. the sum of the true positive and true negatives) divided by the sum of all test results. This means that a diagnostic test for a disease with a very low prevalence could have a high accuracy even if it fails to detect any patients with the disease!So, it is much better to qualify diagnostic tests with sensitivity and specificity but even this can be misleading.
Example 1
A disease has a prevalence of 0.1%. This means that 1 in 1,000 people have it. So, from 100,000 people you would expect 100 people to have it. There is a diagnostic test for the disease with a sensitivity and specificity of 99%. If you test positive for this disease what is the chance that you have it?
The best way to answer this question is by constructing a table of test results versus actually having the disease or not. In this case we test 100,000 people. We know that the prevalence is 0.1% so we expect that 100 of them will be sick. This leaves 99,900 as healthy.
| Sick | Healthy | Totals | |
| Test positive | |||
| Test negative | |||
| Totals | 100 | 99,900 | 100,000 |
If the test has 99% sensitivity, then true positives should make up 99% of the individuals with the disease. That is, 99 should be sick and test positive, which leaves 1 person who is sick but tests negative (false negative).
| Sick | Healthy | Totals | |
| Test positive | 99 | ||
| Test negative | 1 | ||
| Totals | 100 | 99,900 | 100,000 |
If the test has 99% specificity, then true negatives should make up 99% of all the individuals without the disease. That is, 99% of the 99,900 should be sick and test negative. This is equal to 98,901, which leaves 999 people who are healthy but test positive (false positives).
| Sick | Healthy | Totals | |
| Test positive | 99 | 999 | 1,098 |
| Test negative | 1 | 98,901 | 98,902 |
| Totals | 100 | 99,900 | 100,000 |
Therefore, if you test positive for this disease, you have a 99/1,098 or 9% chance of having the disease.
In the case above everyone was screened for the disease regardless of having symptoms or not, but most people only get tested for a disease if they have symptoms, are known to be at risk of having the disease or otherwise exhibit an indicator of the disease. In these cases, positive results will be more likely to be true.
Example 2
A breast cancer diagnostic test is conducted by biopsy. About 30% of women tested have breast cancer. The false positive rate is 2% and the false negative rate is 14%. What is the chance that an individual testing positive has breast cancer?
The way to solve this is again to create a table and put in some theoretical numbers.
If 100,00 people are tested and we expect 30% to have breast cancer then we will have 30,000 with breast cancer, leaving 70,000 without. As the false positive rate is 2%, we expect that 1,400 of the 70,000 without breast cancer will test positive. As the false negative rate is 14%, we expect that 4,200 of the 30,000 with breast cancer will test negative.
| Have breast cancer | Don’t have breast cancer | Totals | |
| Test positive | 25,800 | 1,400 | 27,200 |
| Test negative | 4,200 | 68,600 | 72,800 |
| Totals | 30,000 | 70,000 | 100,000 |
In this case, the probability that an individual testing positive has breast cancer:
= 25,800/27,200 = 95%
