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Suppose that the probabilities of the positive outcomes of B and C condition on the positive outcome of event A are:

$p_1 = Pr(B+|A+) = 0.8 $

$p_2 = Pr(C+|A+) = 0.9 $

And we also know that the correlation coefficient between $p_1$ and $p_2$ is 0.75 (i.e. B and C are paired binary outcomes, e.g. a same cohort of patients tested for COVID-19 at day0 and day3). My question is what the joint probability of $Pr(C+,B+|A+)$ is in a closed form and the value in this case.

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  • $\begingroup$ That's insufficient information unless $\Pr(A+)=1.$ $\endgroup$
    – whuber
    Commented Aug 22, 2022 at 16:29
  • $\begingroup$ Thanks for your comment. Let assume $Pr(A+) = 0.3$. $\endgroup$
    – David Z
    Commented Aug 22, 2022 at 19:09
  • $\begingroup$ That doesn't help. You need information equivalent to the conditional correlation to obtain a unique answer. $\endgroup$
    – whuber
    Commented Aug 22, 2022 at 21:10

1 Answer 1

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Let $B|A\sim Bernoulli(p_B)$ and $C|A\sim Bernoulli(p_C)$, and let $$ Cor(B,C|A) = \rho. $$ We know that $$ \begin{aligned} Cor(B,C|A) &= \frac{Cov(B,C|A)}{\sqrt{Var(B|A)Var(C|A)}}\\ &=\frac{E[BC|A]-E[B|A]E[C|A]}{\sqrt{Var(B|A)Var(C|A)}}\\ &=\frac{E[BC|A]-p_Bp_C}{\sqrt{p_B(1-p_B)p_C(1-p_C)}}\\ \end{aligned} $$ Note that $E[BC|A] = P(B = 1, C = 1|A)$, for in any other case, the product $BC$ is equal to zero. This is, if $C = 0$ or $B = 0$ then $BC = 0$. Only when both are one we have that $BC = 1$. Thus, using the above and solving for $E[BC|A]$ we have that $$ \begin{aligned} P(B = 1, C = 1|A) &= E[BC|A]\\ &= \rho \sqrt{p_B(1-p_B)p_C(1-p_C)} + p_Bp_C. \end{aligned} $$ Given the data, $\rho = 0.75$, $p_B = 0.8$ and $p_C = 0.9$. Thus, $$ P(B,C|A) = 0.75\sqrt{0.8\times 0.2 \times 0.9\times 0.1} + 0.8\times 0.9 = 0.81. $$

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  • $\begingroup$ The question states the correlation coefficient is known, not the conditional correlation. $\endgroup$
    – whuber
    Commented Aug 22, 2022 at 21:10
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    $\begingroup$ It states that the correlation between $p_1$ and $p_2$, the conditional probabilities, is 0.75, so I assumed the author refers to the conditional correlation. But indeed, if it is not, then we do not have enough information. $\endgroup$ Commented Aug 22, 2022 at 21:56
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    $\begingroup$ Thanks @JesúsA.! Your assumption was correct! $\endgroup$
    – David Z
    Commented Aug 23, 2022 at 3:18

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