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