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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?

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1 Answer 1

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Ah, this is easier than I thought.

Let $\pi = P(D+)$ the the true prevalence of the disease. The probability a patient is positive on both tests is

$$ P (T_1^+ \cap T_2^+) = P (T_1^+ \cap T_2^+\vert D+)P(D+) + P (T_1^+ \cap T_2^+\vert D-)P(D-) $$

Assuming the tests are independent, then

$$ P (T_1^+ \cap T_2^+\vert D+) = P(T_1^+ \vert D+)P(T_2^+ \vert D+) $$ which is the product of the sensitivity values of the test. The remaining four tests can be computed similarly.

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