Biology · Biostatistics

Positive and negative predictive values

Predictive values state how often positive or negative test results agree with reference status.

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Formula

PP V = \frac{TP}{TP + FP}, NP V = \frac{TN}{TN + FN}

PPV and NPV as fractions or percentages · TP, FP, TN and FN as counts

Variables & units

Variables & units
Symbol	Meaning	Unit
$PPV$	Reference-positive share among all test-positive people	fraction
$NPV$	Reference-negative share among all test-negative people	fraction
TP, FP, TN, FN	Cells of the diagnostic 2×2 table	people or samples

LaTeX

PPV=\frac{TP}{TP+FP},\quad NPV=\frac{TN}{TN+FN}

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Derivation & background

PPV and NPV are conditional probabilities after a test result and depend strongly on target-population prevalence as well as sensitivity and specificity. A study with an artificially fixed case-control ratio cannot directly transfer them to a population. Interpretation requires reference standard, threshold, population and uncertainty.

Exam blueprint

Validity range

Requires a representative population sample or valid prevalence and reference classification.

Derivation steps

Predictive values condition on test result rather than true disease status.

1Divide TP by all positive tests.
2Divide TN by all negative tests.

Rearrangements

Solve for a quantity

PPV=\frac{Se\cdot Prev}{Se\cdot Prev+(1-Sp)(1-Prev)}

Bayes form using sensitivity, specificity and prevalence.

Task variant

TP=45, FP=5: PPV?

45/(45+5)=0.90=90%.

Common mistakes

Confusing PPV with sensitivity.

PPV starts with test positives; sensitivity starts with reference positives.

Exam context

Interpreting a test result in light of pretest probability.

Formula cluster

Biology and quantitative models

Different prevalence produces different predictive values for the same test.

Sensitivität = TP/(TP+FN); Spezifität = TN/(TN+FP)

Diagnostic Test Sensitivity and Specificity

AUC = Fläche unter der ROC-Kurve

ROC area under the curve

p̂ = x/n; Wilson-Intervall [p_L, p_U]

Wilson Score Confidence Interval for a Proportion

Worked example

With TP=90, FP=20, TN=180 and FN=10, PPV=90/110≈81.8% and NPV=180/190≈94.7%. Check: each denominator includes all positive or all negative tests.

Applications

Interpreting the meaning of positive and negative test results in a target population

How do you calculate with Positive and negative predictive values?

Here is how to work through a typical Positive and negative predictive values (PPV = TP/(TP+FP); NPV = TN/(TN+FN)) task step by step:

1
Task
TP=45, FP=5: PPV?
Solution path
45/(45+5)=0.90=90%.

Frequently asked questions about Positive and negative predictive values

How do you calculate PPV and NPV?+

PPV is true positives divided by all positive tests: TP/(TP+FP). NPV is TN/(TN+FN), true negatives divided by all negative results. With TP=90, FP=20, TN=180 and FN=10, values are about 81.8% and 94.7%. Their denominators are test-result groups, unlike sensitivity and specificity, whose denominators are defined by reference status. Predictive values depend on prevalence, threshold and reference standard. With an artificial case-control fraction, they do not transfer directly to the target population. Recalculate for other prevalence using sensitivity and specificity. The test result itself does not describe prevalence.

Why do predictive values change with prevalence?+

When a condition is rare, many tested people are unaffected. Even good specificity can then produce enough false positives to lower PPV. As prevalence rises, PPV typically rises and NPV falls. Sensitivity and specificity are conditional on true status, though they can vary in practice across disease spectra. Predictive values are therefore population-specific. Predictive values depend on prevalence, threshold and reference standard. With an artificial case-control fraction, they do not transfer directly to the target population. Recalculate for other prevalence using sensitivity and specificity. The test result itself does not describe prevalence.

Can PPV be taken from a case-control study?+

Usually not directly, because case-control designs deliberately select numbers of cases and controls. The diseased fraction in the sample then does not represent target-population prevalence, distorting PPV and NPV. They can be transferred only with suitable prevalence and sensitivity and specificity for the same threshold and population. Otherwise predictive values from the sample are not population predictions. Predictive values depend on prevalence, threshold and reference standard. With an artificial case-control fraction, they do not transfer directly to the target population. Recalculate for other prevalence using sensitivity and specificity. The test result itself does not describe prevalence.

How do I apply Positive and negative predictive values to measured data?+

Reproduce the example calculation: With TP=90, FP=20, TN=180 and FN=10, PPV=90/110≈81.8% and NPV=180/190≈94.7%. Positive and negative predictive values uses PPV = TP/(TP+FP); NPV = TN/(TN+FN). In plain terms, predictive values state how often positive or negative test results agree with reference status. Unit check: PPV and NPV as fractions or percentages · TP, FP, TN and FN as counts. Requires a representative population sample or valid prevalence and reference classification. Values are comparable only when their definitions, denominators and periods match. Record these details next to the result so the reference group is clear. If the direction is implausible, check data preparation and the calculation basis before discussing biological causes.

Which model limits should I state for Positive and negative predictive values?+

A key limit of Positive and negative predictive values is this: Requires a representative population sample or valid prevalence and reference classification. Do not use the metric without its reference; it is useful for Interpreting the meaning of positive and negative test results in a target population. Predictive values state how often positive or negative test results agree with reference status. When comparing populations or experiments, check whether definitions, denominators and observation periods match. If the assumption is implausible, the result shows only what the simplified model calculates from its inputs; it is not a reliable estimate for the target population. State this limitation in the report, and do not infer a causal effect from the formula alone.

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Subject references and further reading

This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.

[1]Centers for Disease Control and Prevention. Definitions of diagnostic test measures.

More formulas: Biostatistics and epidemiology

RR = \frac{a/(a+c)}{b/(b+d)}

Relative Risk (Risk Ratio)

d = \frac{\bar{x}_1-\bar{x}_2}{s_p}, \quad s_p=\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}

Cohen's d Effect Size

OR = \frac{a/c}{b/d} = \frac{ad}{bc}

Odds Ratio

\hat S(t)=\prod_{t_j\le t}\left(1-\frac{d_j}{n_j}\right)

Kaplan–Meier estimator

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Symbol

Meaning

Unit

PPV

Reference-positive share among all test-positive people

fraction

NPV

Reference-negative share among all test-negative people

fraction

TP, FP, TN, FN

Cells of the diagnostic 2×2 table

people or samples

Derivation & background

Frequently asked questions about Positive and negative predictive values

How do you calculate PPV and NPV?+

Why do predictive values change with prevalence?+

Can PPV be taken from a case-control study?+

How do I apply Positive and negative predictive values to measured data?+

Which model limits should I state for Positive and negative predictive values?+

Retain Positive and negative predictive values for exams

Create a curated FSRS exam set for PPV = TP/(TP+FP); NPV = TN/(TN+FN): formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential