A test described as “90% sensitive” does not give every person with a positive result a 90% chance of having the target condition. The two statements condition on different groups. Confusing them can make an impressive statistic sound more informative than it is.
The easiest repair is to build a two-by-two table and read the denominator in words. The FDA’s statistical guidance defines diagnostic accuracy measures against a reference standard and discusses bias in their estimation. Here it is a source for statistical concepts, not a statement about Canadian physiotherapy regulation. FDA diagnostic-test guidance.
Everything in the examples below is fictional. No named clinical test, condition or treatment decision is being recommended.
Sort condition status and test result separately
Imagine 1,000 people in a study. A specified reference standard classifies 100 as having a target condition and 900 as not having it. The test under study produces these results:
| Test result | Condition present | Condition absent | Total |
|---|---|---|---|
| Positive | 90 | 90 | 180 |
| Negative | 10 | 810 | 820 |
| Total | 100 | 900 | 1,000 |
The 90 people in the top-left cell are true positives for this exercise. The 90 in the top-right cell are false positives. The bottom-left cell contains false negatives, and the bottom-right contains true negatives.
Do not identify a cell by its position alone. Some papers reverse rows and columns. Read the labels every time. A memorized formula attached to the wrong orientation gives a neat-looking wrong answer.
Ask the denominator question four times
Sensitivity: Of those with the condition, how many tested positive? The denominator is the 100 people classified as having it. Sensitivity is 90/100 = 90%.
Specificity: Of those without the condition, how many tested negative? The denominator is 900. Specificity is 810/900 = 90%.
Positive predictive value: Of those who tested positive, how many had the condition? The denominator is the 180 positive results. PPV is 90/180 = 50%.
Negative predictive value: Of those who tested negative, how many did not have the condition? The denominator is 820. NPV is 810/820, approximately 98.8%.
The result is not contradictory. A positive test result selects a different group from the group used to calculate sensitivity. In this invented population, the positive results divide evenly between people with and without the condition despite 90% sensitivity and specificity.
Change the population while holding the model fixed
Now imagine 1,000 people of whom 500 have the condition. For this arithmetic exercise only, assume sensitivity and specificity stay exactly 90%.
The test identifies 450 true positives and misses 50 people with the condition. Among the 500 without it, it produces 50 false positives and 450 true negatives. PPV is now 450/500 = 90%.
That change from 50% to 90% PPV did not require changing the assumed sensitivity. It followed from changing how common the condition was in the population. The larger group without the condition in the first example generated enough false positives to match the number of true positives.
In real studies, sensitivity and specificity can also differ with the population, threshold, procedures and other study features. Holding them constant here isolates one mathematical relationship; it is not a promise that test performance transfers unchanged between settings.
Check how the table was built
Before applying any reported percentage, ask how condition status was established. A comparison with another test that is itself uncertain does not automatically provide a trustworthy truth label. Ask whether all participants received an appropriate reference assessment and whether the people interpreting one test knew the other result.
Then inspect recruitment. A table assembled from obvious advanced cases and healthy controls may not resemble a setting where people present with uncertain, overlapping findings. If investigators deliberately choose the numbers with and without the condition, the resulting condition proportion is a feature of their sampling design. Its calculated PPV and NPV should not be treated as population predictive values without appropriate adjustment or representative sampling. The earlier tables demonstrate why: changing that proportion changes predictive values even when sensitivity and specificity stay fixed.
These percentages are estimates, so real studies should also report uncertainty. A larger appropriate sample can narrow confidence intervals, but it cannot repair selection bias or an unsuitable reference standard.
Finally, identify the positivity threshold. A score becomes “positive” only after a rule is applied. Do not carry a sensitivity figure from one threshold into a discussion using another.
These questions do not require you to reject every imperfect study. They help you describe what a number can support and where your confidence should stop.
Make a teaching explanation without diagnosing someone
A cautious classroom explanation might be:
In our fictional population, the test found 90 of 100 people with the condition, but it also returned 90 positive results among people without it. Half of the positive results therefore came from people with the condition. We would need the relevant history, assessment context and evidence for the actual test before interpreting a real result.
Notice that the explanation does not turn a group statistic into an individual diagnosis. Its job is to teach how the table works.
Check your understanding
An original exercise gives 200 people: 40 have the condition, 160 do not. There are 30 true positives and 20 false positives. Complete the table.
There must be 10 false negatives and 140 true negatives. Sensitivity is 30/40 = 75%; specificity is 140/160 = 87.5%; PPV is 30/50 = 60%; NPV is 140/150, approximately 93.3%.
If you answered “PPV is 75%,” you probably reused the sensitivity denominator. Write “among positive results” beside PPV and calculate again. For your next research discussion, draw the table before reporting the headline percentage. The labels and denominators are part of the reasoning, not optional working.
Sources and further reading
- FDA — Statistical guidance on diagnostic test studies — Diagnostic two-by-two tables, reference standards and test accuracy definitions; used for statistics, not Canadian regulatory requirements.