Overdispersion

Overdispersion occurs when $\text{Var}(Y)>\mathbb{E}[Y|X]$ for the assumed distribution family.

Consequences

• Larger estimated confidence interval

Poisson Overdispersion

Standard Poisson: $\mathbb{E}[Y|X]=\text{Var}(Y|X)=\mu =\exp (X\beta )$

Overdispersed: $\text{Var}(Y|X)=\varphi \mu \text{where }\varphi >1$

Detection: $\hat{\varphi }=\frac{X^2}{n-p}\text{or}\hat{\varphi }=\frac{D}{n-p}$

where $X^2$ is Pearson chi-square, $D$ is deviance, $n$ = sample size, $p$ = parameters.

If we force to model our data even with overdispersion, the standard error will be underestimated. In this situation, we should use a zero-inflated Poisson model instead

Binomial Overdispersion

Standard binomial:

$\mathbb{E}[Y|X]=n\pi ,\text{Var}(Y|X)=n\pi (1-\pi )$

Notice that $(1-\mu )\le 1$. Thus, $\mathrm{Var}(Y|X)\le E[Y|X]$

Overdispersed: $\text{Var}(Y|X)=\varphi n\pi (1-\pi )\text{where }\varphi >1$

Mathematical Models

Quasi-likelihood Approach

Modifies variance function: $\text{Var}(Y|X)=\varphi V(\mu )$ where $V(\mu )$ is the variance function.

Negative Binomial for Counts

$Y\sim \text{NB}(\mu ,k)\text{with}\text{Var}(Y)=\mu +\frac{{\mu }^2}{k}$

As $k\to \infty$, reduces to Poisson.

Beta-binomial for Proportions

$Y\sim \text{BetaBin}(n,\alpha ,\beta )$

with excess variance: $\text{Var}(Y)=n\pi (1-\pi )[1+(n-1)\rho ]$

where $\rho =\frac{1}{1+\alpha +\beta }$ is intracluster correlation.

Likelihood Adjustments

Standard likelihood: $\ell (\beta )={\sum }_{i=1}^n\log f(y_i|x_i,\beta )$

Quasi-likelihood: $Q(\beta ,\varphi )={\sum }_{i=1}^n\frac{y_i{\theta }_i-b({\theta }_i)}{\varphi }$

where ${\theta }_i$ is the canonical parameter.

Statistical Tests

Score Test

$T=\frac{X^2-\text{df}}{\sqrt{2\text{df}}}\sim N(0,1)\text{under }{H}_0:\varphi =1$

Likelihood Ratio Test

For nested models: $-2({\ell }_0-{\ell }_1)\sim {\chi }^2$

Common Solutions

• Quasi-Poisson: $\text{Var}(Y)=\varphi \mu$
• Negative Binomial: $f(y|\mu ,k)=(\genfrac{}{}{0ex}{}{y+k-1}{y}){\left(\frac{k}{k+\mu }\right)}^k{\left(\frac{\mu }{k+\mu }\right)}^y$
• Random Effects: $\log ({\mu }_{ij})=X_{ij}\beta +u_i$ where $u_i\sim N(0,{\sigma }_u^2)$