Table 1 Diagnostic accuracy of a single symptom to diagnose the disease cause in equations.

From: Diagnostic accuracy of symptoms for an underlying disease: a simulation study

Disease status

Disease present

Disease absent

Proportions diseased

\(d\)

\(1-d\)

Symptom present

\(d \times ir \times rr\)

\((1-d)\times ir\)

Symptom absent

\(d\times (1-ir\times rr)\)

\((1-d)\times (1-ir)\)

Derived statistics if the symptom present

Sensitivity11

\(\frac{d\times ir\times rr}{d}=ir\times rr\)

Specificity11

\(\frac{\left(1-d\right)\times \left(1-ir\right)}{1-d} =1-ir\)

Positive predictive value11

\(\frac{d\times ir\times rr}{d\times ir\times rr+\left(1-d\right)\times ir}\) = \(\frac{d\times ir\times rr}{d\times ir\times rr+ir-d\times ir}\)

Negative predictive value11

\(\frac{(1-d)\times (1-ir)}{d\times \left(1-ir\times rr\right)+(1-d)\times (1-ir)}\) = \(\frac{1-d-ir+d\times ir}{1-d\times ir\times rr-ir+d\times ir}\)

Observed ratios of developing symptoms

\(\frac{d\times ir\times rr+(1-d)\times ir}{d\times \left(1-ir\times rr\right)+(1-d)\times (1-ir)}\) = \(\frac{d\times ir\times rr+ir-d\times ir}{1-d\times ir\times rr-ir+d\times ir}\)

Derived statistics if the incidence reaching 1 among those diseased (\({\varvec{i}}{\varvec{r}}\times {\varvec{r}}{\varvec{r}}\,{\mathbf{=}}\,{\mathbf{1}})\)

Sensitivity11

\(\frac{d\times ir\times rr}{d}=ir\times rr=1\)

Specificity11

\(\frac{\left(1-d\right)\times \left(1-ir\right)}{1-d} =1-ir\)

Positive predictive value11

\(\frac{d\times ir\times rr}{d\times ir\times rr+\left(1-d\right)\times ir}\) = \(\frac{d\times ir\times rr}{d\times ir\times rr+ir-d\times ir}\) = \(\frac{d}{d+ir-d\times ir}\)

Negative predictive value11

\(\frac{(1-d)\times (1-ir)}{d\times \left(1-ir\times rr\right)+(1-d)\times (1-ir)}\) = \(\frac{1-d-ir+d\times ir}{1-d\times ir\times rr-ir+d\times ir}\) = \(\frac{1-d-ir+d\times ir}{1-d-ir+d\times ir}\) = 1

Observed ratios of developing symptoms

\(\frac{d\times ir\times rr+(1-d)\times ir}{d\times \left(1-ir\times rr\right)+(1-d)\times (1-ir)}\) = \(\frac{d\times ir\times rr+ir-d\times ir}{1-d\times ir\times rr-ir+d\times ir}\) = \(\frac{d+ir-d\times ir}{1-d-ir+d\times ir}\)

  1. Equations are derived based on Baratloo et al.’s definitions11.
  2. d proportions in a population with the disease, ir symptom incidence rate, rr risk ratios.