Real-World Mathematics: Healthcare Screening, Diagnostic Tests, Risk and Rates

Application of Mathematics in Real-World Usage · Guide 25 · BTT Mathematics Hub

A test can be highly sensitive and still produce many false positives. A treatment can halve relative risk while changing absolute risk by only a few percentage points. A hospital can report more cases while a population-adjusted rate falls. Healthcare Mathematics is full of ratios whose meaning changes when the denominator changes.

This is a Mathematics teaching guide, not medical advice. Every condition, test result, treatment effect and population below is fictional. Real screening and treatment decisions require clinical evidence, professional interpretation and the individual patient’s circumstances. The mathematical task here is to understand how counts become rates, how rates return to counts, and how conditional probabilities change when the population base changes.

The 2×2 table keeps diagnostic states visible

Suppose 1,000 fictional people are tested. One hundred truly have condition C and 900 do not. The test identifies 90 of the 100 cases and correctly returns negative for 810 of the 900 non-cases.

• True positives: 90
• False negatives: 10
• False positives: 90
• True negatives: 810

These four counts are the raw objects. Sensitivity, specificity and predictive values are different ways of conditioning on different parts of this table.

Sensitivity conditions on true cases

Sensitivity = TP/(TP+FN). Here it is 90/100=90%.

The denominator is all people who truly have the condition. Sensitivity therefore answers how often the test is positive among true cases. It does not answer the probability of disease given a positive test.

Specificity conditions on true non-cases

Specificity = TN/(TN+FP)=810/900=90%.

The denominator is all people who truly do not have the condition. The corresponding false-positive rate is 1−specificity=10%.

Positive predictive value conditions on positive tests

Positive predictive value is TP/(TP+FP). In the 10% prevalence example it is 90/(90+90)=50%.

Now keep sensitivity and specificity at 90% but reduce prevalence to 1% in 10,000 people. There are 100 true cases, of whom 90 test positive. There are 9,900 non-cases, of whom 990 test positive falsely. PPV becomes 90/(90+990)=8.33%.

The test characteristics did not change. The predictive value changed because the base rate changed.

Negative predictive value uses the negative-test population

In the first 1,000-person example, NPV=TN/(TN+FN)=810/(810+10)≈98.78%.

Predictive values therefore belong to the test-plus-population context. They are not fixed properties of a test detached from prevalence.

Bayes’ rule gives the same result algebraically

Let prevalence P(C)=0.10, sensitivity P(+|C)=0.90 and false-positive rate P(+|not C)=0.10. Then:

P(C|+)=[0.90×0.10]/[0.90×0.10+0.10×0.90]=0.50.

The 2×2 table and Bayes’ rule are two representations of the same conditional-probability structure. The table often helps readers see the denominator; Bayes’ formula makes the general relationship compact.

Likelihood ratios separate test evidence from prevalence

Positive likelihood ratio LR+ = sensitivity/(1−specificity). With both at 90%, LR+=0.9/0.1=9.

Negative likelihood ratio LR−=(1−sensitivity)/specificity=0.1/0.9≈0.111.

Likelihood ratios multiply prior odds to obtain posterior odds. A prior probability is still required; the test does not erase the population base rate.

Risk is a proportion over a defined period

Suppose 20 of 1,000 people in Group A experience an outcome during one year. Risk is 2%. In Group B, 40 of 1,000 do so, giving 4%.

The risk ratio A/B is 0.02/0.04=0.5. Under this fictional comparison, Group A has half the risk over the declared period.

Absolute and relative risk changes answer different questions

Using 4% control risk and 2% treatment risk, absolute risk reduction is 4%−2%=2 percentage points.

Relative risk reduction is (4%−2%)/4%=50%.

The same data therefore support “a 2-percentage-point absolute reduction” and “a 50% relative reduction.” Both are mathematically correct, but the scale of the communicated effect is different.

Number needed to treat is based on the absolute difference

With ARR=0.02, NNT=1/0.02=50 under the simplified model. It should normally be rounded upward to a whole person when used as a count-like summary.

NNT depends on the outcome, comparison group and time horizon. It is not a property of a treatment independent of context.

Rates include time in the denominator

Suppose 12 events occur during 600 person-years of observation. Incidence rate is 12/600=0.02 per person-year, or 20 per 1,000 person-years.

This is not automatically the same as a one-year risk. A rate measures event occurrence per unit of observed person-time; risk measures the proportion experiencing an event over a stated period.

Person-time allows unequal follow-up

Ten people followed for one year contribute 10 person-years. Twenty people followed for six months each also contribute 10 person-years.

Person-time aggregates exposure duration, but it discards some individual timing structure. Two studies with the same total person-time can have different follow-up patterns.

Crude rates can differ because population structures differ

Region A has 900 younger people with event rate 1% and 100 older people with event rate 10%, producing 19 events among 1,000 people, or 1.9%.

Region B has 500 younger and 500 older people with the same age-specific rates, producing 55 events, or 5.5%.

The age-specific rates are identical. The crude rates differ because the weights of the age groups differ.

Direct standardisation applies common weights

Using a standard population that is 80% younger and 20% older, the standardised rate for either region is 0.8×1%+0.2×10%=2.8%.

Standardisation creates a hypothetical comparison under a common population structure. It does not replace the observed crude rate.

Odds and probability are related but not identical

If probability p=0.20, odds are p/(1−p)=0.25, or 1:4. If odds are 3:2, probability is 3/(3+2)=0.60.

An odds ratio should not automatically be read as a risk ratio, particularly when outcomes are common.

Threshold choice trades false positives against false negatives

Suppose lowering a fictional threshold changes sensitivity from 80% to 95% while specificity falls from 95% to 75%.

The lower threshold catches more true cases but also labels more non-cases positive. Mathematics can display the trade-off. Deciding which trade-off is acceptable is a clinical, ethical and policy question outside this page.

Expected counts turn probabilities back into operational quantities

In 50,000 people with 2% prevalence, expected true cases are 1,000. At 90% sensitivity, expected true positives are 900. Among 49,000 non-cases, a 5% false-positive rate produces 2,450 false positives.

Expected positive tests total 3,350, of which only 900 are true positives under the model. A large tested population can therefore produce a large false-positive count even when specificity is high.

Confidence intervals describe sampling uncertainty in an estimate

Suppose a sample proportion is 0.20 from n=400. A simple normal-approximation standard error is √[0.2×0.8/400]=0.02. An illustrative 95% interval is 0.20±1.96×0.02, or about 16.08% to 23.92%.

This interval concerns uncertainty in a population proportion estimate under the stated model. It is not a probability interval for an individual person’s outcome.

Missing data can silently change the denominator

A study enrolls 1,000 people but outcome status is known for only 900. If 45 events are observed, the complete-case proportion is 45/900=5%.

Dividing by all 1,000 gives 4.5%, a different quantity unless the missing outcomes are explicitly treated in a justified way.

Verification means returning every rate to its count base

If sensitivity is 90% among 100 true cases, expect 90 true positives. If specificity is 95% among 49,000 non-cases, expect 46,550 true negatives and 2,450 false positives. Returning percentages to expected counts is one of the strongest checks against denominator confusion.

A rate that produces impossible counts—more positive cases than people in its conditioning group, for example—signals an arithmetic or definition error.

A complete healthcare calculation states population, period and conditioning

For a test statistic, state whether the denominator is true cases, true non-cases, positive tests or negative tests. For a risk, state the population and period. For a rate, state the person-time. For comparisons, report absolute counts beside relative measures whenever possible.

The denominator is part of the quantity’s identity. Keeping it visible prevents a familiar percentage from being attached to the wrong question.

Practice: twenty healthcare Mathematics questions

1. TP=90 and FN=10. Find sensitivity.
2. TN=810 and FP=90. Find specificity.
3. TP=90 and FP=90. Find PPV.
4. TN=810 and FN=10. Find NPV.
5. With sensitivity 90% and specificity 90%, find LR+.
6. Find LR−.
7. At 1% prevalence in 10,000 people with 90% sensitivity and specificity, find expected true positives.
8. Find expected false positives for question 7.
9. Find PPV from questions 7–8.
10. Risks are 4% and 2%. Find risk ratio.
11. Find absolute risk reduction.
12. Find relative risk reduction.
13. Find NNT.
14. 12 events occur over 600 person-years. Find rate per 1,000 person-years.
15. Convert probability 0.20 to odds.
16. Convert odds 3:2 to probability.
17. Find a standardised rate using weights 0.8 and0.2 with age-specific rates 1% and10%.
18. In 50,000 people at 2% prevalence, find number with the condition.
19. With 95% specificity, find false positives among the non-cases in question 18.
20. Why can predictive value change when sensitivity and specificity stay fixed?

Worked answers

1. 90%.
2. 90%.
3. 50%.
4. About 98.78%.
5. 9.
6. About 0.111.
7. 90.
8. 990.
9. About 8.33%.
10. 0.5.
11. 2 percentage points.
12. 50%.
13. 50.
14. 20 per 1,000 person-years.
15. 0.25, or 1:4.
16. 60%.
17. 2.8%.
18. 1,000.
19. 2,450.
20. Because predictive value depends on the population base rate as well as the test’s conditional sensitivity and specificity.

Sources and connected applications

For standard diagnostic-test definitions and evidence interpretation, see NCBI Bookshelf: Diagnostic Testing. All populations, treatments and outcomes on this page are fictional teaching constructions.

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