# How can I determine the multinomial probabilities when applying two different tests to the same population?

I have a population of $$N$$ individuals and am interested in determining the prevalence $$\pi$$ of some mental health disorder. To do this, I give the population two tests, each with a sensitivity of $$Se_1$$, $$Se_2$$, and a specificity of $$Sp_1$$, $$Sp_2$$ respectively.

I can arrange the outcome of these tests into a 2x2 table

$$\begin{bmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{bmatrix}$$

Such that $$y_{11}$$ were identified as disease positive in both tests, $$y_{12}$$ were identified as positive on the first test, but negative on the second and so on.

I'm interested in the multinomial probabilites for this table. Assuming I knew what the prevalence was exactly, how could I determine the expected frequencies in this table?

I'm thinking I need to compute $$\operatorname{Pr}(\mbox{+ on Test 1} \cap \mbox{+ on test 2})$$ for isntance. Assuming the tests are independent, this would just factor into the product of the two probabilities.

In section 3.2 of this paper, the authors identify the probabilities as $$\begin{array}{l} p_{11 k}=\pi_{k} \operatorname{Se}_{1} \operatorname{Se}_{2}+\left(1-\pi_{k}\right)\left(1-\operatorname{Sp}_{1}\right)\left(1-\operatorname{Sp}_{2}\right) \\ p_{12 k}=\pi_{k} \operatorname{Se}_{1}\left(1-\operatorname{Se}_{2}\right)+\left(1-\pi_{k}\right)\left(1-\operatorname{Sp}_{1}\right) \operatorname{Sp}_{2} \\ p_{21 k}=\pi_{k}\left(1-\operatorname{Se}_{1}\right) \operatorname{Se}_{2}+\left(1-\pi_{k}\right) \operatorname{Sp}_{1}\left(1-\operatorname{Sp}_{2}\right) \\ p_{22 k}=\pi_{k}\left(1-\operatorname{Se}_{1}\right)\left(1-\operatorname{Se}_{2}\right)+\left(1-\pi_{k}\right) \operatorname{Sp}_{1} \operatorname{Sp}_{2} \end{array}$$

Are these the probabilites I'm seeking? If so, how were they derived?

Let $$\pi = P(D+)$$ the the true prevalence of the disease. The probability a patient is positive on both tests is
$$P (T_1^+ \cap T_2^+) = P (T_1^+ \cap T_2^+\vert D+)P(D+) + P (T_1^+ \cap T_2^+\vert D-)P(D-)$$
$$P (T_1^+ \cap T_2^+\vert D+) = P(T_1^+ \vert D+)P(T_2^+ \vert D+)$$ which is the product of the sensitivity values of the test. The remaining four tests can be computed similarly.