Positive and negative predictive values

Key takeaways

Positive and negative predictive values belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Positive and negative predictive values to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Positive and negative predictive values from memory before moving on to harder problems.

Reference excerpt

The positive and negative predictive values (PPV and NPV respectively) are the proportions of positive and negative results in statistics and diagnostic tests that are true positive and true negative results, respectively. The PPV and NPV describe the performance of a diagnostic test or other statistical measure. A high result can be interpreted as indicating the accuracy of such a statistic. The PPV and NPV are not intrinsic to the test (as true positive rate and true negative rate are); they depend also on the prevalence. Both PPV and NPV can be derived using Bayes' theorem. Although sometimes used synonymously, a positive predictive value generally refers to what is established by control groups, while a post-test probability refers to a probability for an individual. Still, if the individual's pre-test probability of the target condition is the same as the prevalence in the control group used to establish the positive predictive value, the two are numerically equal. In information retrieval, the PPV statistic is often called the precision.

Definition

Positive predictive value (PPV) The positive predictive value (PPV), or precision, is defined as

PPV = Number of true positives Number of true positives + Number of false positives = Number of true positives Number of positive calls {\displaystyle {\text{PPV}}={\frac {\text{Number of true positives}}{{\text{Number of true positives}}+{\text{Number of false positives}}}}={\frac {\text{Number of true positives}}{\text{Number of positive calls}}}}

where a "true positive" is the event that the test makes a positive prediction, and the subject has a positive result under the gold standard, and a "false positive" is the event that the test makes a positive prediction, and the subject has a negative result under the gold standard. The ideal value of the PPV, with a perfect test, is 1 (100%), and the worst possible value would be zero. The PPV can also be computed from sensitivity, specificity, and the prevalence of the condition:

PPV = sensitivity × prevalence sensitivity × prevalence + ( 1 − specificity ) × ( 1 − prevalence ) {\displaystyle {\text{PPV}}={\frac {{\text{sensitivity}}\times {\text{prevalence}}}{{\text{sensitivity}}\times {\text{prevalence}}+(1-{\text{specificity}})\times (1-{\text{prevalence}})}}}

cf. Bayes' theorem The complement of the PPV is the false discovery rate (FDR):

FDR = 1 − PPV = Number of false positives Number of true positives + Number of false positives = Number of false positives Number of positive calls {\displaystyle {\text{FDR}}=1-{\text{PPV}}={\frac {\text{Number of false positives}}{{\text{Number of true positives}}+{\text{Number of false positives}}}}={\frac {\text{Number of false positives}}{\text{Number of positive calls}}}}

Negative predictive value (NPV) The negative predictive value is defined as:

NPV = Number of true negatives Number of true negatives + Number of false negatives = Number of true negatives Number of negative calls {\displaystyle {\text{NPV}}={\frac {\text{Number of true negatives}}{{\text{Number of true negatives}}+{\text{Number of false negatives}}}}={\frac {\text{Number of true negatives}}{\text{Number of negative calls}}}}

where a "true negative" is the event that the test makes a negative prediction, and the subject has a negative result under the gold standard, and a "false negative" is the event that the test makes a negative prediction, and the subject has a positive result under the gold standard. With a perfect test, one which returns no false negatives, the value of the NPV is 1 (100%), and with a test which returns no true negatives the NPV value is zero. The NPV can also be computed from sensitivity, specificity, and prevalence:

NPV = specificity × ( 1 − prevalence ) specificity × ( 1 − prevalence ) + ( 1 − sensitivity ) × prevalence {\displaystyle {\text{NPV}}={\frac {{\text{specificity}}\times (1-{\text{prevalence}})}{{\text{specificity}}\times (1-{\text{prevalence}})+(1-{\text{sensitivity}})\times {\text{prevalence}}}}}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Positive and negative predictive values

Start with the simplest possible case. Write down what Positive and negative predictive values claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Positive and negative predictive values before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Positive and negative predictive values ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Positive and negative predictive values

In research: Positive and negative predictive values appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Positive and negative predictive values in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Positive and negative predictive values is common in secondary-school and first-year university syllabi. It links to neighbouring topics Biostatistics, Categorical data, Statistical ratios, so understanding it makes those chapters shorter.
In everyday life: Look for Positive and negative predictive values outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Positive and negative predictive values in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Positive and negative predictive values means in your own words.
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Frequently asked questions

What is Positive and negative predictive values in simple terms?

The positive and negative predictive values (PPV and NPV respectively) are the proportions of positive and negative results in statistics and diagnostic tests that are true positive and true negative results, respectively. The PPV and NPV describe the performance of a diagnostic test or other stati…

Why does Positive and negative predictive values matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Positive and negative predictive values?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Positive and negative predictive values.