Including in the approximated formulae the contribution of the uncertainty on $p_s$ due to sampling

Next section will be dedicated to the effect of sampling $n_s$ individuals from a population. However, having taken some confidence with the approximated formulae, we can already extend them in order to see how the uncertain $p_s$, characterized by its expected value E$(p_s)$ and standard uncertainty $\sigma(p_s)$, whose evaluation will be the subject of Sec. , affects our prediction about the number of individuals resulting positive in the test. In the approximated expression for the expected value of $n_P$ (Eq. ) we have to replace $p_s$ by its expected value E$(p_s)$, while in the variance we have to add a term again obtained by linearization,³³ thus getting

$$(n_P) \approx (\pi_1)\cdot (p_s)\cdot n_s + (\pi_2)\cdot (1- (p_s))\cdot n_s$$ (57)

$$\sigma^2(n_P) \approx (\pi_1)\cdot (1- (\pi_1)) \cdot (p_s)\cdot n_s + (\pi_2)\cdot (1- (\pi_2))\cdot (1- (p_s))\cdot n_s$$

$$+ \,\sigma^2(\pi_1)\cdot ^2(p_s)\cdot n_s^2 + \sigma^2(\pi_2)\cdot (1- (p_s))^2\cdot n_s^2$$

$$+ \,\sigma^2(p_s)\cdot ( (\pi_1) - (\pi_2))^2\cdot n_s^2\,.$$ (58)

As far as the fraction of positives is concerned, we have the following four contributions to the global uncertainty,

$$\sigma(f_P) \approx \sigma_R(f_P) \oplus \sigma_{\pi_1}(f_P) \oplus \sigma_{\pi_2}(f_P) \oplus \sigma_{p_s}(f_P) \,,$$ (59)

the first three given by Eqs. (-), in which $p_s$ has to be replaced by its expected value E$(p_s)$, and the fourth term being

$$\sigma_{p_s}(f_P) = \sigma(p_s)\cdot \vert (\pi_1) - (\pi_2)\vert\,.$$ (60)

(Note that also the fourth term is of `random nature', although, from the `perspective' we are now seeing the problem it could be considered as a third contribution to systematics.³⁴)