Model for overdispersion in success/failure data
The most commonly used model that generalises the binomial distribution to allow for overdispersion is the beta-binomial distribution.
Definition
A random variable, \(X\), has a beta-binomial distribution if its probability function is
\[ p(x) \;\;=\;\; \begin{cases} \displaystyle {n \choose x} \frac {B(x + \alpha, n - x + \beta)}{B(\alpha, \beta)} &\text{for }x = 0, 1, 2, \dots, n \\[0.4em] 0 & \text{otherwise} \end{cases} \]where \(\alpha \gt 0\), \(\beta \gt 0\) and
\[ B(a, b) \;\;=\;\; \frac {\Gamma(a)\Gamma(b)}{\Gamma(a+b)} \]This distribution can be derived as the distribution of the number of successes in a binomial experiment in which the probability of success varies (with a distribution called a beta distribution). However we will just use it here as an empirical distribution whose two parameters, \(\alpha\) and \(\beta\) give it flexibility to cope with overdispersion.
The probability function for the beta-binomial distribution looks complex, but its probabilities can be found in Excel using.
Maths function | In Excel |
---|---|
\(\displaystyle {n \choose x}\) | =COMBIN(n, x) |
\(\Gamma(k)\) | =EXP(GAMMALN(k)) |
The mean and variance of the beta-binomial distribution are stated without proof below.
Mean and variance
The mean and variance of the beta-binomial distribution are
\[ E[X] = \frac {n\alpha}{\alpha + \beta} \spaced{and} \Var(X) = \frac {n\alpha\beta}{(\alpha + \beta)^2}\times \frac {\alpha + \beta + n} {\alpha + \beta + 1} \]
If we write
\[ \pi = \frac {\alpha}{\alpha + \beta} \spaced{so} (1-\pi) = \frac {\beta}{\alpha + \beta} \]
then
\[ E[X] = n\pi \spaced{and} \Var(X) = n\pi (1 - \pi) \times \frac {\alpha + \beta + n} {\alpha + \beta + 1} \]
The variance of the beta-binomial distribution is therefore \(\frac {\alpha + \beta + n} {\alpha + \beta + 1}\) times the variance of the binomial distribution with the same mean. Since this factor is greater than 1, the beta-binomial distribution can be used as a model when there is overdispersion.
Relationship to binomial distribution
The beta-binomial distribution can be made arbitrarily close to a binomial distribution with suitable choice of \(\alpha\) and \(\beta\). The following result is stated without proof.
Asymptotic distribution
If \(\alpha \to \infty\) and \(\beta \to \infty\) simultaneously with \(\dfrac {\alpha}{\alpha + \beta} = \pi\), the beta-binomial distribution approaches a \(\BinomDistn(n, \pi)\) distribution.
Shape of beta-binomial distribution
The following diagram shows the possible shapes of the beta-binomial distribution and illustrates the properties stated above.
Shape of beta-binomial distribution
The diagram below initially shows the probability function for a binomial distribution with \(n = 10\); the top slider can be used to adjust its probability of success, \(\pi\).
For any binomial distribution with probability of success \(\pi\) and mean \(E[X] = n \pi\) the variance is \(\Var(X) = n \pi (1 - \pi)\). The beta-binomial has two parameters, allowing it more flexibility. Drag the bottom slider to make the variance a greater multiple of the corresponding binomial distribution's variance. Observe that whatever the value of \(\pi\),