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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?

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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.

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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.

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