Mean-Correlation Parameterization of Beta-Binomial Models

Why Reparameterize?

A beta-binomial model describes grouped successes when the trials in a group share an unobserved success probability. Its usual beta shape parameters determine the model, but they are not immediately easy to interpret: changing either shape can change both the average success probability and the amount of within-group dependence. A mean-correlation parameterization separates these roles.

Write \(Y\) for the number of successes among \(m\) trials in one group. The model assumes a group-specific probability \(P\), with \(P\) following a beta distribution, and then \(Y\mid P\) following a binomial distribution. The marginal mean success probability is denoted by \(\mu\), and \(\rho\) denotes the correlation between two distinct trial outcomes in the same group.

From Beta Shapes to Mean and Correlation

Let \(\kappa=a+b\), the total beta concentration. The beta distribution has mean \(a/\kappa\) and variance \(ab/\{\kappa^2(\kappa+1)\}\). Conditional on \(P\), the trial indicators are independent Bernoulli variables with success probability \(P\). Marginally, however, they are correlated because they share the same random \(P\).

For distinct trials \(j\) and \(k\), the conditional expectation of \(Z_jZ_k\) is \(P^2\). Thus the covariance of their outcomes is the variance of the shared probability. The individual trial variance is \(\mu(1-\mu)\), so their correlation is the variance of \(P\) divided by \(\mu(1-\mu)\). This gives the mapping below.

Parameter Constraints and Boundary Interpretations

A proper beta distribution requires both shape parameters to be strictly positive. In the mean-correlation formulation, this is exactly the restriction \(0<\mu<1\) and \(0<\rho<1\). For example, when \(\mu\) is strictly between zero and one, both \(\mu(1-\rho)/\rho\) and \((1-\mu)(1-\rho)/\rho\) are positive if and only if \(\rho\) lies strictly between zero and one.

The lower boundary \(\rho=0\) is interpreted as a limit, not as a proper beta distribution with finite shape parameters. Holding \(\mu\) fixed and letting \(\rho\) decrease to zero makes \(\kappa\) grow without bound. The random probability \(P\) then concentrates around \(\mu\), and the beta-binomial distribution approaches the ordinary binomial distribution. At the other boundary, as \(\rho\) increases to one, \(\kappa\) tends to zero. The limiting group probability is at an endpoint: a group has all failures with probability \(1-\mu\) or all successes with probability \(\mu\). Neither boundary is represented by a proper beta distribution with \(a,b>0\).

Deriving the Intraclass Correlation and Count Variance

The connection between \(\rho\) and overdispersion follows from the shared probability \(P\). The derivation also explains why the mean-correlation parameterization changes dependence without changing the marginal mean.

Proof. Set \(\kappa=a+b\), and write \(B(a,b)\) for the beta function. The beta integral gives \(E[P^r]=B(a+r,b)/B(a,b)\); using \(B(a+1,b)=aB(a,b)/\kappa\) and \(B(a+2,b)=a(a+1)B(a,b)/\{\kappa(\kappa+1)\}\), we obtain \(E[P]=a/\kappa=\mu\) and \(E[P^2]=a(a+1)/\{\kappa(\kappa+1)\}\). Subtracting the square of the mean gives \[ \operatorname{Var}(P) =\frac{a(a+1)}{\kappa(\kappa+1)}-\frac{a^2}{\kappa^2} =\frac{ab}{\kappa^2(\kappa+1)} =\frac{\mu(1-\mu)}{\kappa+1}. \]

For any trial \(j\), iterated expectation gives \(E[Z_j]=E[E[Z_j\mid P]]=E[P]=\mu\), so \(\operatorname{Var}(Z_j)=\mu(1-\mu)\). Conditional independence gives \(E[Z_jZ_k\mid P]=P^2\) for \(j\ne k\), hence \(\operatorname{Cov}(Z_j,Z_k)=E[P^2]-\{E[P]\}^2=\operatorname{Var}(P)\). Consequently their correlation is \(\operatorname{Var}(P)/\{\mu(1-\mu)\}=1/(\kappa+1)=\rho\). Finally, since \(Y\) is the sum of \(m\) indicators, \[ E[Y]=\sum_{j=1}^m E[Z_j]=m\mu, \] and \[ \operatorname{Var}(Y) =\sum_{j=1}^m\operatorname{Var}(Z_j) +2\sum_{j<k}\operatorname{Cov}(Z_j,Z_k) =m\mu(1-\mu)+m(m-1)\rho\mu(1-\mu). \] Factoring gives the stated variance. \(\square\)

The variance multiplier relative to a binomial count with the same mean is \(1+(m-1)\rho\). It equals one when the binomial limit \(\rho=0\) is taken, and increases with \(\rho\) whenever \(m>1\). Thus \(\mu\) sets the marginal success probability and expected count, while \(\rho\) controls the extra variability caused by within-group dependence.

Beta-Binomial Probabilities in This Parameterization

For a count \(y\in\{0,\ldots,m\}\), integrating the conditional binomial probability over the beta distribution gives \(P(Y=y)=\binom{m}{y}B(a+y,b+m-y)/B(a,b)\). In the mean-correlation parameterization, substitute \(a=\mu(1-\rho)/\rho\) and \(b=(1-\mu)(1-\rho)/\rho\). This expresses the entire count distribution in terms of its marginal risk and its intraclass correlation. The formula is useful for checking that parameter changes affect the full distribution in the intended way, not just its variance.

How the Parameters Work in a Regression Model

In a beta-binomial regression, covariates commonly determine the marginal risk through a link function, for example \(\operatorname{logit}(\mu_i)=x_i^\mathsf{T}\beta\). The dispersion parameter \(\rho\) may be held common across groups. With this specification, changing a regression coefficient changes the fitted risks \(\mu_i\) and hence the expected counts \(m_i\mu_i\). Changing \(\rho\) leaves those marginal fitted risks and expected counts unchanged, while changing the within-group dependence and the count variance.

For a group of size \(m_i\), the variance is \(m_i\mu_i(1-\mu_i)\{1+(m_i-1)\rho\}\). Both parameters appear in this expression, but in different ways: \(\mu_i\) determines the binomial mean-variance component, and \(\rho\) multiplies that component to account for extra variation. The same value of \(\rho\) need not produce the same absolute variance increase in groups with different fitted risks.

Common Mistake and Practical Interpretation

• Mistake: treating \(\rho\) as a multiplier of the fitted mean. A larger \(\rho\) does not increase \(\mu\) or \(m\mu\). It increases the correlation among trials and the variance multiplier \(1+(m-1)\rho\).
• Mistake: assuming any positive number is a valid correlation parameter. The beta-binomial requires \(0<\rho<1\) for proper beta shapes. A proposed value outside that interval cannot be converted into valid positive shapes.
• Mistake: calling the marginal trials independent because each has success probability \(\mu\). Each trial does have marginal success probability \(\mu\), but two trials sharing \(P\) have covariance \(\operatorname{Var}(P)>0\) for a proper beta distribution.
• Mistake: expecting overdispersion to affect every group equally in absolute units. The multiplier is common when \(\rho\) is common, but the binomial component \(m_i\mu_i(1-\mu_i)\) varies with group size and fitted risk.