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