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When working with the binomial distribution, it's sometimes useful to compute the probability of "at least one", which is 1 - P(none) or, after setting x to 0 in the formula for the PDF of the binomial distribution:

$$ P(x|n) = \frac{n!}{x!(n-x)!}(p)^x(1-p)^{n-x} $$

$$ P(0|n) = \frac{n!}{0!(n-0)!}(p)^0(1-p)^{n-0} $$

$$ P(none) = (1 - p)^n $$

$$ P(at least one) = 1 - (1 - p)^n $$

The similar formula for the beta-binomial is:

$$ P(x|n) = \frac{n!}{x!(n-x)!}\frac{B(x+\frac{\pi}{\theta},n+\frac{1-\pi}{\theta}-x)}{B(\frac{\pi}{\theta},\frac{1-\pi}{\theta})} $$

So a similar process for the formula for a beta-binomial "at least one" should be:

$$ P(0|n) = \frac{n!}{0!(n-0)!}\frac{B(0+\frac{\pi}{\theta},n+\frac{1-\pi}{\theta}-0)}{B(\frac{\pi}{\theta},\frac{1-\pi}{\theta})} $$

$$ P(none) = \frac{B(\frac{\pi}{\theta},n+\frac{1-\pi}{\theta})}{B(\frac{\pi}{\theta},\frac{1-\pi}{\theta})} $$

$$ P(at least one) = 1-\frac{B(\frac{\pi}{\theta},n+\frac{1-\pi}{\theta})}{B(\frac{\pi}{\theta},\frac{1-\pi}{\theta})} $$

But when I try to apply this to my data, I'm not getting the expected results. Given some amount of overdispersion, I would expect, as n increases from 1 to 5, for example, for the beta-binomial with $\pi=0.519215045$ and $\theta=0.015210626$ to approach 1 more slowly than the standard binomial, but instead it approaches 1 more quickly (Binomial: n=1: .519; n=2: 0.769; n=3: 0.889; n=4: 0.947; n=5: 0.974 -- Beta-binomial: n=1: 0.519; n=2: 0.882; n=3: 0.971; n=4: 0.993; n=5: 0.998).

Is my reasoning sound in deriving this formula, or have I done something wrong? Can someone point me to a proper worked-out example that compares the binomial and beta-binomial in this way?

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I cannot reproduce your calculations for the beta-binomial distribution.

library(VGAM)
pi <- 0.519215045
theta <- 0.015210626
1-pbinom(0,1:5,prob = pi)

1-beta(pi/theta,1:5+(1-pi)/theta)/(beta(pi/theta,(1-pi)/theta))     # your formula
1-pbetabinom.ab(0, size=1:5, shape1=pi/theta, shape2=(1-pi)/theta)  # according to the VGAM package

The latter both produce

[1] 0.5192150 0.7651057 0.8834657 0.9413315 0.9700460

This seems more in line with your intuition.

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  • $\begingroup$ For my original calculations of the beta-binomial I used Excel gamma functions. I'm not sure what I did wrong, but it isn't worth the time to spend on it because you have clearly demonstrated the right way (or at least a right way) to do what I was trying to do. Thank you! $\endgroup$
    – Jim Lewis
    Mar 6, 2017 at 18:55

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