Diagnostic Tests and Bayes — Part II: PPV, NPV, and Why Prevalence Matters

Before you begin, take a few minutes to check what you already know and how confident you feel. You will see the same questions again at the end of the module — this helps both you and us measure what you have learned. Click an option for every question and confidence rating, then click Next to continue.

In Part I you built the 2x2 diagnostic table and read sensitivity and specificity down its disease columns — fixed properties of the test itself. But a patient does not know their true status; they hold a result and want the truth behind it. This part answers their question: my test came back positive, so how likely is it that I really have the disease?

This part of the module covers four steps, each one building on the last:

• PPV and NPV — what a result means for a patient, read across the test rows as P(disease | positive) and P(no disease | negative), never to be confused with sensitivity.
• Prevalence — why predictive values rise and fall with how common the disease is, even though the test never changes.
• Natural frequencies — imagining 100,000 whole people to walk from a prevalence to a post-test probability with counting, not algebra.
• Metrics in R — computing every measure at once, each with a confidence interval, using epi.tests() from the epiR package.

By the end of this part you will be able to compute PPV and NPV across the test rows, explain why the chance of disease given a positive test is not the same as sensitivity, reason in natural frequencies to find a PPV from a prevalence, and run epi.tests() to report every metric with a confidence interval — and say plainly why a positive screen in a low-prevalence setting means far less than students expect.

Try every snippet in the R Scratchpad on the right. This part needs no data file — you will build the small screening tables yourself with matrix() and c(), and load the epiR package with library(epiR).

Sensitivity and specificity start from the truth and look at the test. A patient sits at the opposite end: they have a result in hand and want to know the truth. For that you read ACROSS the test rows instead of down the disease columns.

The positive predictive value (PPV) is the chance a patient truly has the disease given a positive test: P(disease | positive). You condition on the test, so you read across the Test = Positive row: TP / (TP + FP). The negative predictive value (NPV) is the chance a patient is truly disease-free given a negative test: P(no disease | negative), read across the Test = Negative row as TN / (TN + FN).

Now the trap that catches every beginner, and the whole reason this part exists. P(positive | disease) and P(disease | positive) are different questions with different answers. Sensitivity reads down the disease column; PPV reads across the test row. Swapping them — assuming a positive result means the patient almost certainly has the disease — is the classic diagnostic error.

Compute both predictive values by hand from the same four counts — reading ACROSS the test rows this time, not down the disease columns.

tp <- 90
fp <- 7
fn <- 10
tn <- 93
ppv <- tp / (tp + fp)
npv <- tn / (tn + fn)
c(PPV = ppv, NPV = npv)

Unlike sensitivity, PPV and NPV depend on prevalence — how common the disease is in the population you are testing. The reason is the false positives. In a low-prevalence setting there are very few diseased people but a vast pool of healthy people, so even a small false-positive rate produces a flood of false alarms that swamp the true ones.

This is why the same excellent test gives a high PPV in a specialist clinic full of sick patients but a low PPV when used to screen the general population. A good sensitivity and specificity do NOT guarantee a high PPV — prevalence has the final say. We make this concrete with natural frequencies next.

The cleanest way to find a PPV is not algebra — it is to imagine a large, round group of people and count. This is the natural-frequency method, and it turns a confusing probability problem into simple arithmetic on whole patients.

The recipe has three steps. First, the pre-test probability is just the prevalence — the chance a patient has the disease before any test. Imagine a population of that size, say 100,000. Split it into diseased and healthy using the prevalence. Then apply sensitivity to the diseased group and specificity to the healthy group to fill the four cells. The post-test probability after a positive result is then just the PPV you read off the counts.

Work an example. A test has sensitivity 0.90 and specificity 0.92, used to screen a population where prevalence is 1% (so 1 in 100). Out of 100,000 people, 1,000 have the disease and 99,000 do not.

• Of the 1,000 diseased, sensitivity 0.90 makes 900 test positive (true positives) and 100 test negative (false negatives).
• Of the 99,000 healthy, specificity 0.92 clears 91,080 as negative (true negatives), leaving 8% — that is 7,920 — testing positive (false positives).

Now find the PPV by reading across the positive row. The positives are 900 true plus 7,920 false, a total of 8,820. The PPV is 900 / 8,820, which is about 0.10. Even with a 90% sensitivity, a positive screen here carries only a 1-in-10 chance of real disease — because the 7,920 false alarms vastly outnumber the 900 real cases.

n <- 100000
prevalence <- 0.01
sensitivity <- 0.90
specificity <- 0.92
diseased <- n * prevalence
healthy <- n - diseased
tp <- diseased * sensitivity
fp <- healthy * (1 - specificity)
ppv <- tp / (tp + fp)
ppv

Sit with that number. Even an excellent test gives a low PPV when prevalence is low, because false positives drawn from the huge healthy majority drown out the few true positives. This is the single most important idea in screening, and the reason a positive screen is usually followed by a confirmatory test, not a diagnosis.

Raise the prevalence and the PPV climbs. Take the very same test to a clinic where 10% of patients have the disease. Out of 100,000 there are now 10,000 diseased (9,000 true positives) and 90,000 healthy (7,200 false positives), so the PPV jumps to 9000 / (9000 + 7200), about 0.56. Same test, prevalence up tenfold, PPV from one-in-ten to better than even.

n <- 100000
prevalence <- 0.10
sensitivity <- 0.90
specificity <- 0.92
diseased <- n * prevalence
healthy <- n - diseased
tp <- diseased * sensitivity
fp <- healthy * (1 - specificity)
ppv <- tp / (tp + fp)
ppv

Hand arithmetic is good for understanding, but in practice you let R compute every metric at once, complete with confidence intervals. The tool is epi.tests() from the epiR package.

The epiR package is a collection of veterinary and human epidemiology functions; switch it on with library(epiR). Its epi.tests function takes a 2x2 table and returns sensitivity, specificity, PPV, NPV and more, each with a 95% confidence interval.

There is one rule you must obey, and getting it wrong is the most common mistake with this function. epi.tests() expects the test outcome in the ROWS and the true disease status in the COLUMNS, with the positive/diseased cell at the top left. Feed it a table in the wrong orientation and it will silently compute the metrics for the wrong cells. Always print the table and check its dimnames before you trust the output.

Build the screening table from the natural-frequency counts — 900 true positives, 7,920 false positives, 100 false negatives, 91,080 true negatives — with the test in rows and disease in columns, then pass it in.

library(epiR)
dat <- c(900, 7920, 100, 91080)
screen_tab <- matrix(dat, nrow = 2, byrow = TRUE,
              dimnames = list(Test = c("Positive", "Negative"),
                              Disease = c("Yes", "No")))
screen_tab

Now run the analysis. Read the output table for the rows labelled Sensitivity, Specificity, Pos. predict. value and Neg. predict. value — each shows the estimate with its confidence interval. The PPV will read about 0.10, exactly the number you counted by hand.

library(epiR)
dat <- c(900, 7920, 100, 91080)
screen_tab <- matrix(dat, nrow = 2, byrow = TRUE,
              dimnames = list(Test = c("Positive", "Negative"),
                              Disease = c("Yes", "No")))
epi.tests(screen_tab)

Notice what just happened. epi.tests() agreed with your natural-frequency count: a PPV near 0.10 for a test with 0.90 sensitivity. That is the punchline of the whole part — even an excellent test gives a positive result that means little in a low-prevalence screening setting, and R confirms it with a confidence interval to report.

Now run the full chain on a fresh screening scenario: turn a prevalence and a test's sensitivity and specificity into natural-frequency counts, feed them to epi.tests(), and read the PPV. The worked example fades its support — study the full solution, fill the gap, then solve one alone.

library(epiR)
dat <- c(900, 7920, 100, 91080)
tab <- matrix(dat, nrow = 2, byrow = TRUE,
              dimnames = list(Test = c("Positive", "Negative"),
                              Disease = c("Yes", "No")))
epi.tests(tab)

You have reached the end of the module. Try the same questions again — your answers here, paired with your pre-test answers, are how we measure what the module taught you. Answer every question and confidence rating, then click Submit and see results to view your score.

Here is how your post-test answers compare with your pre-test answers. The pre/post pairing is the most reliable way to see what this module actually taught you.