Epidemiology & Diagnostic Accuracy formula solvers

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A C H I N O P R S Y

Attack Rate

\mathrm{AR} = \frac{I}{N_{e}}

Epidemiology & Diagnostic AccuracyThe share of an exposed group who fell ill during an outbreak. The CDC's own Principles of Epidemiology notes that this is a misnomer: it is a proportion, not a rate, because there is no time in the denominator. The name has stuck anyway.

Case Fatality Ratio (CFR)

\mathrm{CFR} = \frac{D}{C}

Epidemiology & Diagnostic AccuracyThe share of diagnosed cases that ended in death. Long called the case fatality RATE, and the CDC flags that name as a misnomer for the same reason as the attack rate: no time appears in the denominator, so it is a proportion.

Cohen's Kappa (Inter-Rater Agreement)

\kappa = \frac{p_o - p_e}{1 - p_e}

Epidemiology & Diagnostic AccuracyCohen's 1960 measure of how far two raters agree beyond what chance alone would have produced, given how often each of them uses each label.

Herd Immunity Threshold

H_{c} = 1 - \frac{1}{R_{0}}

Epidemiology & Diagnostic AccuracyThe share of a population that must be immune before each infection replaces itself with fewer than one new infection, so that transmission fades rather than grows.

Incidence Rate (Person-Time)

\mathrm{IR} = 1000 \cdot \frac{C}{\mathrm{PT}}

Epidemiology & Diagnostic AccuracyNew cases divided by the person-time at risk, reported here per 1,000 person-years. The denominator counts time rather than people, so someone followed for two years contributes twice what someone followed for one does.

Negative Likelihood Ratio (LR−)

\mathrm{LR^-} = \frac{1 - \mathrm{Se}}{\mathrm{Sp}}

Epidemiology & Diagnostic AccuracyHow many times more often a negative result turns up in someone with the condition than in someone without it. Below 1 it lowers the odds; the further below, the more a negative result rules out.

Negative Predictive Value (NPV)

\mathrm{NPV} = \frac{\mathrm{TN}}{\mathrm{TN} + \mathrm{FN}}

Epidemiology & Diagnostic AccuracyOf everyone the test called negative, the share who genuinely do not have the condition — read along the test-negative ROW of a 2×2 table.

Number Needed to Treat (NNT)

\mathrm{NNT} = \frac{1}{\mathrm{CER} - \mathrm{EER}}

Epidemiology & Diagnostic AccuracyHow many people have to receive the treatment for one of them to avoid the outcome — the reciprocal of the absolute risk reduction. It is a trial summary, not a prescription: it counts people, and this shard computes no dose.

Odds Ratio (Cross-Product)

\mathrm{OR} = \frac{a\,d}{b\,c}

Epidemiology & Diagnostic AccuracyThe odds of exposure among cases divided by the odds among controls, which collapses to the cross-product of a 2×2 table. It is the measure a case-control study can produce, because such a study fixes its own denominators and cannot report a risk.

Positive Likelihood Ratio (LR+)

\mathrm{LR^+} = \frac{\mathrm{Se}}{1 - \mathrm{Sp}}

Epidemiology & Diagnostic AccuracyHow many times more often a positive result turns up in someone with the condition than in someone without it. A multiplier on the odds, not a probability.

Positive Predictive Value (PPV)

\mathrm{PPV} = \frac{\mathrm{TP}}{\mathrm{TP} + \mathrm{FP}}

Epidemiology & Diagnostic AccuracyOf everyone the test called positive, the share who genuinely have the condition — read along the test-positive ROW of a 2×2 table rather than down a column, which is what makes it move with prevalence.

Post-Test Odds (Odds × Likelihood Ratio)

O_{post} = O_{pre} \cdot \mathrm{LR}

Epidemiology & Diagnostic AccuracyBayes' theorem with the arithmetic taken out of it: in odds, an update is one multiplication. Pre-test odds times the likelihood ratio gives post-test odds, and O/(1 + O) turns that back into a probability.

Prevalence

P = \frac{C}{N}

Epidemiology & Diagnostic AccuracyThe share of a population that has the condition at a given moment. Existing cases, not new ones — a stock, not a flow.

Relative Risk (Risk Ratio)

\mathrm{RR} = \frac{R_{e}}{R_{u}}

Epidemiology & Diagnostic AccuracyThe risk in the exposed group divided by the risk in the unexposed group. A multiplier: 3 means three times the risk, and it says nothing at all about how large either risk was.

Sensitivity (True Positive Rate)

\mathrm{Se} = \frac{\mathrm{TP}}{\mathrm{TP} + \mathrm{FN}}

Epidemiology & Diagnostic AccuracyThe share of people who genuinely have the condition that the test correctly calls positive — read straight down the disease-present column of a 2×2 table.

Specificity (True Negative Rate)

\mathrm{Sp} = \frac{\mathrm{TN}}{\mathrm{TN} + \mathrm{FP}}

Epidemiology & Diagnostic AccuracyThe share of people who do not have the condition that the test correctly calls negative — read straight down the disease-absent column of a 2×2 table.

Youden's J Index

J = \mathrm{Se} + \mathrm{Sp} - 1

Epidemiology & Diagnostic AccuracyYouden's 1950 single-number summary of a test: sensitivity plus specificity minus one. It runs from 0 for a test no better than a coin to 1 for a perfect one, and its maximum is a common rule for picking a cut-off.