I'm facing this new concept: the quasi likelihood. I'm looking for some clear explanation of what it is. I have a very basic knowledge about this, so I need to go step by step very slowly. I discovered this concept in dealing with overdispersed count data.

Quasipoisson models will estimate relative rates with no distribution assumptions on $Y$ and yet this kind of estimator has similar properties to MLE. We're interested in the first two moments:

  1. $E(Y_i)=\mu_i \quad i=1,...,n$
  2. $\mu_i=g^{-1}(X'_i\beta)$ this derives from the link in GLM assumptions
  3. $cov(Y)=\phi V(\mu)=\phi Diag(V(\mu_i),...,V(\mu_n))$

Here I'm stuck.

What I understand is: we're not saying anything about $Y$ (which has some distribution) expect for its mean and variance. For every $y_i$ it comes from something that has mean $\mu_i$and variance= $\phi V(\mu_i)$ and they're independent from others. When the models are estimated, it generates the exact same coefficients from a Poisson regression but changes the standard errors of the coefficient.

Could someone provide me a clear definition of quasi-likelihood/quasi-Poisson and how it works?


2 Answers 2


What happens is that the likelihood equations depend on the distribution of Y only through the mean ($\mu$) and the variance ($V(\mu)$). Other moments of the distribution do not affect the coefficients $\hat \beta$, neither the asymptotic covariance.

The quasi-likelihood approach is based on this fact, requiring that only the mean and variance of the distribution be specified. And then the quasi-likelihood estimates are obtained through the solution of the likelihood equations for GLMs. As focusing in the quasi-Poisson model, a dispersion parameter is included, giving us:

$$ V(\mu) = \phi \mu$$

This new parameter can be estimated with:

$$ \hat \phi = \frac{X^2}{n - p} $$ where $ X^2 = \frac{\sum_{i = 1}^{n}(y_i - \hat \mu_i)^2}{V(\hat \mu_i)}$. This means that que quasi-Poisson model is equivalent to a Poisson model, with the $\hat \beta$'s standard errors multiplied by $\sqrt(\hat\phi)$. But it's not exactly Poisson because we do not have the property of mean = variance. This kind of model is usually considered when one wants to account for overdispersion in count data.


In general, quasi-likelihood has score function $\frac{y-\mu}{\phi V(\mu)}$. Under mild conditions, the estimator has the same asymptotic distribution as the MLE. In your GLM notation, the score function is $D^{T}V^{-1}\frac{y-\mu}{\phi }$, where $D= \partial \mu/ \partial \beta$ is the $n \times p$ derivative matrix.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.