Dependence on our knowledge concerning $\pi_1$ and $\pi_2$

As we have already well understood, the uncertainty on the result is highly dependent of the uncertainty concerning $\pi_1$ and $\pi_2$. Therefore, as we have done in the previous sections, let us also change here our assumptions and see how the main result changes accordingly ($n_I$ is of little interest, at this point, also because of its very high correlation with $p$, and hence we shall not monitor it any longer in further examples).

1. First we start assuming negligible uncertainty on sensitivity and specificity.⁴⁹ As a result, the standard uncertainty $\sigma(p)$ becomes $0.0046$, that is about a factor $\approx 5$ smaller.
2. Then, as we have done in the above sections, we keep $\pi_1$ to its default value ( $r_1= 409.1, s_1 = 9.1$), only reducing the uncertainty of $\pi_2$ to $0.007$.⁵⁰ Also in this case the uncertainty decreases, getting $\sigma(p)=0.0085$.
3. Finally, we make the pdf of $\pi_2$ mirror symmetric with respect to that of $\pi_1$, that is $r_2= 9.1, s_2 = 409.1$. But, obviously we need to change the number of the observed positives, choosing this time 1170, as suggested by our expectations (see Fig. ). As a result we get $p=0.0995\pm 0.0076$, with an uncertainty not differing so much with respect to the previous case. Indeed, as we have have already noted in the previous sections, improving the specificity ($\pi_2$ reduced by a factor five) has only a little effect on the quality of the measurement, being more important the uncertainty with which that test parameter is known. (And we expect that something like that is also true for the sensitivity.)