Two Questions That Sound Similar
Suppose a medical screening test is positive for many people who have a particular disease. That fact alone does not tell us what proportion of people with a positive result actually have the disease. The first statement asks about test results among people with the disease; the second asks about disease among people with positive results. They use different denominators, so their proportions can be very different.
This distinction applies well beyond screening. “What proportion of people who passed received tutoring?” is not the same question as “What proportion of tutored people passed?” Both are valid conditional comparisons, but they condition on different groups. In a two-way table, the quickest safeguard is to state the denominator in words before calculating.
For a screening example, let \(D\) mean “has the disease” and \(+\) mean “has a positive test result.” Then \(P(+ \mid D)\) is the proportion with positive results among people who have the disease. In contrast, \(P(D \mid +)\) is the proportion who have the disease among people with positive results. The vertical bar means “given” or “among.”
A Screening Table: The Denominator Changes the Answer
Consider a hypothetical population of 10,000 people. Suppose 1% have a disease, the test is positive for 90% of those who have it, and 5% of those who do not have it also test positive. The 90% is the test’s sensitivity, or positive-test rate among people with the disease. The 5% is its false-positive rate, or positive-test rate among people without the disease. These rates are specified for this example; the table below makes their implications visible.
Worked Example: What Does a Positive Test Mean?
Find (1) the proportion of people with the disease who test positive and (2) the proportion of positive-test results that occur in people with the disease.
Step 1: Convert the base rate to counts. One percent of 10,000 is \(0.01 \times 10{,}000 = 100\), so 100 people have the disease and 9,900 do not.
Step 2: Fill in the test-result counts. Of the 100 people with the disease, 90% test positive: \(0.90 \times 100 = 90\). The other 10 test negative. Of the 9,900 people without the disease, 5% test positive: \(0.05 \times 9{,}900 = 495\). The other 9,405 test negative.
| Positive test | Negative test | Total | |
|---|---|---|---|
| Disease | 90 | 10 | 100 |
| No disease | 495 | 9,405 | 9,900 |
| Total | 585 | 9,415 | 10,000 |
Step 3: Answer the first question. Among the 100 people with the disease, 90 test positive:
Step 4: Answer the reversed question. There are 585 positive results altogether, but only 90 of those are from people with the disease:
Thus, the test is positive for 90% of people who have the disease, while only about 15.38% of positive results in this population are from people who have the disease. Those figures do not conflict: the first denominator is the 100 people with disease, and the second is the 585 people with positive tests. In this hypothetical population, most positive results—495 of 585—are false positives because the group without the disease is much larger.
Why the Base Rate Matters
A test result is interpreted against the population from which the tested person comes. If a condition is uncommon, even a modest false-positive rate can produce many positive results among people without the condition: the false-positive rate is applied to a large group. The positive-test rate among people with the disease and the false-positive rate among people without it do not, by themselves, determine the proportion of positive results that indicate disease. The base rate is also needed.
The calculation can be expressed with proportions rather than counts. The joint proportion \(P(D \text{ and } +)\) is the proportion of the population that both has the disease and tests positive. It can be calculated either as a proportion of the whole population or by multiplying the proportion with disease by the positive-test rate among that group.
The equation follows directly from the definition of a conditional proportion. Among all members of the population, the fraction who both have the disease and test positive is \(P(D \text{ and } +)\). Within the group with disease, the fraction who test positive is \(P(+ \mid D)\). Since the disease group makes up a fraction \(P(D)\) of the population, the joint fraction is the product of these two fractions.
Now, among everyone with a positive result, the fraction who have the disease is the joint fraction with both disease and a positive result divided by the overall fraction with positive results. Substituting the product above gives the stated relationship. This reasoning requires no assumption that disease status and test result are independent; it uses the actual conditional proportion \(P(+ \mid D)\).
Worked Example: The Same Test in a Population with a Higher Base Rate
Imagine a second hypothetical population of 10,000 people. The disease prevalence is 20%, while the test still has a 90% positive-test rate among people with the disease and a 5% positive-test rate among people without it. Calculate the proportion of positive results that occur in people with the disease.
Step 1: Calculate the group sizes. There are \(0.20 \times 10{,}000 = 2{,}000\) people with disease and \(10{,}000 - 2{,}000 = 8{,}000\) without it.
Step 2: Calculate positive results in each group. Among people with disease, \(0.90 \times 2{,}000 = 1{,}800\) test positive. Among people without disease, \(0.05 \times 8{,}000 = 400\) test positive. The total number of positive results is \(1{,}800 + 400 = 2{,}200\).
| Positive test | Negative test | Total | |
|---|---|---|---|
| Disease | 1,800 | 200 | 2,000 |
| No disease | 400 | 7,600 | 8,000 |
| Total | 2,200 | 7,800 | 10,000 |
Step 3: Divide by the positive-result total.
The proportion of positive results that occur in people with disease is about 81.82% in this population, compared with 15.38% in the first example. The stipulated test rates are unchanged in both populations, but the base rate is higher here. Consequently, the count of true positives is much larger relative to the count of false positives. This comparison shows why a positive result cannot be interpreted without knowing the relevant population’s base rate.
Reversed Comparisons in an Everyday Table
The same denominator issue appears when neither variable is a medical test. Reading the two conditional proportions separately prevents a frequent error: reporting a group’s share of outcome cases as though it were that group’s outcome rate.
Worked Example: Tutoring and Passing
In a hypothetical group of 1,000 students, 200 received tutoring and 800 did not. Of the tutored students, 160 passed an assessment; of the students without tutoring, 480 passed. Find the pass rate among tutored students and the proportion of passing students who received tutoring.
| Passed | Did not pass | Total | |
|---|---|---|---|
| Received tutoring | 160 | 40 | 200 |
| No tutoring | 480 | 320 | 800 |
| Total | 640 | 360 | 1,000 |
Pass rate among tutored students. The denominator is all 200 tutored students, so \(160/200 = 0.80\), or 80%. This answers: “What proportion of tutored students passed?”
Proportion of passing students who received tutoring. The denominator is all 640 students who passed, so \(160/640 = 0.25\), or 25%. This answers: “What proportion of students who passed received tutoring?”
The 80% and 25% are both correct, but they describe different conditional proportions. The second does not say that tutoring had a 25% pass rate, and the first does not say that 80% of passing students were tutored. This table describes a group comparison; it does not, by itself, establish that tutoring caused a difference in passing.
A Reliable Way to Read the Questions
When a question uses words such as “among,” “of those who,” or “given,” identify the group after those words. That group supplies the denominator. A useful sequence is to name the outcome, name the conditioning group, locate the corresponding row or column total, and then divide the appropriate cell count by that total.
For “What proportion of positive tests are in people with disease?”, the group is positive tests; for “What proportion of people with disease test positive?”, the group is people with disease.
Use the total for the group named in the question, not the total for the outcome category you happen to be discussing.
Divide the count in the relevant cell by that group total, then report the result with its meaning and context.
The base rate is important because it determines how large the groups are before the result is considered. With a rare disease, the no-disease group may be so large that a small false-positive percentage still produces many false positives. With a more common disease, the same test characteristics can yield a much larger proportion of positive results that come from people with the disease. A conditional percentage should therefore be reported with its denominator or an explicit description of the group it is “among.”
Common Mistake
- Wrong reasoning: “The test detects 90% of people with the disease, so a positive result means there is a 90% chance the person has the disease.”
- Correction: The 90% describes \(P(+ \mid D)\), the positive-test proportion among people with disease. The proportion of positive results that occur in people with disease is \(P(D \mid +)\). It must be calculated using the positive-result total and depends on the base rate as well as the test’s rates.
- Wrong reasoning: “Twenty-five percent of students who passed were tutored, so the pass rate for tutored students was 25%.”
- Correction: The 25% uses all students who passed as its denominator. The pass rate among tutored students uses all tutored students as its denominator; in the example it is \(160/200 = 80\%\).
Check Your Understanding
For each question, identify the denominator before calculating or explaining.
- In a screening population, 120 people have a disease and 108 of them test positive. What conditional proportion does \(108/120\) describe?
- If 540 people test positive in that same population and 108 of them have the disease, what does \(108/540\) describe? Why is it not the same quantity as your answer to question 1?
- Explain in words how a low disease base rate can lead to many false-positive results even when the false-positive rate is small.
- A table has 300 people in Group A, of whom 210 experience an outcome. Among all 500 people with the outcome, 210 are in Group A. State the two different conditional proportions represented by \(210/300\) and \(210/500\).
- Why can the proportion of positive tests that occur in people with disease change between populations even when the sensitivity and false-positive rate remain the same?