How to calculate sensitivity, specificity, positive predictive value, negative predictive value, positive likelihood ratio and negative likelihood ratio and their 95 % confidence interval with cross tabulation?

You can calculate 95 % confidence interval of probability as formula below.

$\displaystyle p \pm 1.96 \sqrt{\frac{p(1 - p)}{n}}$

In this article, I’d like to describe how to calculate sensitivity, specificity, positive predictive value (PPV), negative predictive value (NPV), positive likelihood ratio (LR+) and negative likelihood ratio (LR-) and their 95 % confidence interval with cross tabulation.

	TRUE	FALSE	Marginal total
POSITIVE	a	b	a + b
NEGATIVE	c	d	c + d
Marginal total	a + c	b + d	N

Sensitivity, specificity, PPV, NPV, LR+ and LR- and their 95 % confidence interval are shown as following table. ‘Exp()’ means exponential with Napier’s constant and ‘LN()’ means natural logarithmic function, respectively.

	Formula	95% Confidence Interval
Sensitivity	$\displaystyle \frac{a}{a + c}$	$\displaystyle Sensitivity\pm1.96\sqrt{\frac{Sensitivity(1 - Sensitivity)}{a + c}}$
Specificity	$\displaystyle \frac{d}{b + d}$	$\displaystyle Specificity\pm1.96\sqrt{\frac{Specificity(1 - Specificity)}{b + d}}$
PPV	$\displaystyle \frac{a}{a + b}$	$\displaystyle PPV \pm 1.96\sqrt{\frac{PPV(1 - PPV)}{a + b}}$
NPV	$\displaystyle \frac{d}{c + d}$	$\displaystyle NPV \pm 1.96\sqrt{\frac{NPV(1 - NPV)}{c + d}}$
LR+	$\displaystyle \frac{a}{a + c}\bigg/\frac{b}{b + d}$	$\displaystyle Exp \left[LN\left( \frac{Sensitivity}{1 - Specificity}\right)\pm1.96\left(\frac{1 - Sensitivity}{a} + \frac{Specificity}{b}\right) \right]$
LR-	$\displaystyle \frac{c}{a + c}\bigg/\frac{d}{b + d}$	$\displaystyle Exp\left[ LN\left(\frac{1 - Sensitivity}{Specificity}\right) \pm1.96\left( \frac{Sensitivity}{c} + \frac{1 - Specificity}{d} \right)\right]$