# Where does Quasi-Likelihood formula come from?

In regular likelihood/log likelihood, if there is random variable "$$Y$$" with pdf (probability distribution functions) $$f_Y(y)$$... the likelihood of this can be written as: $$\mathcal{L}(y_i) = \prod_i {f_Y(y_i)}$$.

In Quasi Likelihood, it is no longer based on valid pdf. We write QL as : $$Q(\mu \mid y) = \int_{u,y} \frac{y-t}{\sigma^2 V(t)} ~\mathrm dt$$

We then take its derivative and write QL score formula as: $$U = \frac{Y-\mu}{ \phi\, V(y_i)}$$

I do not understand where this original formula come from. I read that QL require relationship to be specified between mean ($$\mu$$) and variance function $$V(y_i)$$. But I still don't understand why it takes this specific form.

EX: Why is $$Q(\mu \mid y) = \int (u,y) \left\{ [y-t] / [\sigma^2 V(t)]~\mathrm dt \right\}$$ and not $$Q(\mu\mid y) = \int(u,y) \left\{ [y-t] + [\sigma^2 V(t)]~\mathrm dt \right\}$$ or $$Q(\mu \mid y) = \int(u,y) \left\{ [y-t] / \exp[\sigma^2 V(t)] ~\mathrm dt\right\}?$$

It seems that many options could have been chosen... so why exactly is QL written as $$Q(\mu \mid y) = \int(u,y) \left\{ [y-t] / [\sigma^2 V(t)] ~\mathrm dt \right\}$$ and not some other form?

PS:

• Does it have something do with CLT (Central Limit Theorem)? EX: in CLT, $$y-\mu/\sigma \sim\mathcal N(0,1)$$ .... this resembles QL. How come in regular likelihood its just $$f(y_i)$$ and not $$f[y_i-\mu/v(y_i)]?$$

• Regular likelihood is not an integral... but QL is an integral? Why?

– Sycorax
Oct 22, 2023 at 20:52
• can i use ai tool to help write equations... and then copy/paste here? Oct 22, 2023 at 20:59
• Given you are likely to continue to ask questions, it's worth learning to use the tool we have (it works in many other contexts than stackexchange). Looking at code in people's questions and answers will help (as will looking at code used to generate the equations in wikipedia; they use the same LaTeX-based notation). If you use an AI tool to help you construct the code that's inconsequential as long as you make absolutely sure it says what you intended it to say before pasting the MathJax code in. I'll make some edits that should reflect what you're trying to say, but please check carefully. Oct 22, 2023 at 21:18
• ..to clarify, I made some tiny edits to clarify notation along the way, such as distinguishing between $Y$ and $y$ (I took the conflation between the random variable and the values it could take to be an unintentional error) but didn't change the functional forms you had in your original. Oct 22, 2023 at 21:33

In this q&a a motivation for the use of quasi-likelihood functions is given.

Quasi-likelihood functions arise when we generalize GLM by adding a dispersion parameter.

When we incorporate a dispersion parameter, then we are not working with real likelihood functions anymore.

### Motivation, where does it come from

For several one parameter distributions (more precisely from the exponential family) it is possible to describe the derivative of the likelihood function as

$$\frac{\partial \log\mathcal{L}(x,\mu)} {\partial \mu}=\frac{x-\mu}{V(\mu)}$$

Where $$V(\mu)$$ is the variance of the distribution as function of the mean $$\mu$$.

• For instance for the Poisson distribution with mean $$\mu$$ and variance $$V(\mu) = \mu$$ the logarithm of the density is

$$\log[f(x,\mu)] = x \log(\mu) - \mu - \log(x!)$$

and the derivative with respect to location parameter $$\mu$$ is

$$\frac{\partial \log[f(x,\mu)]}{\partial \mu} = x/\mu - 1 = \frac{x-\mu}{\mu}$$

### Including a dispersion parameter

This way to describe the likelihood function can be extended and we add a parameter

$$\frac{\partial \log\mathcal{L}(x,\mu,\phi)} {\partial \mu}= \frac{x-\mu}{\phi V(\mu)}$$

The effect of this parameter $$\phi$$ is that it changes the curvature of the likelihood function and this relates to more/less precise knowledge about the parameter $$\mu$$ depending on whether the likelihood is made sharper/blunt.

A useful resource is

Quasi-Likelihood Functions, Generalized Linear Models, and the Gauss-Newton Method Wedderburn Biometrika Vol. 61, No. 3 (Dec., 1974), pp. 439-447

• thx! In the first link u posted, is "m" the number of parameters in the model? Oct 23, 2023 at 4:06
• In the same link, in the section "Finding the dispersion parameter (alternative)" .... is it possible to see why that E{ Sum_i (x-mu)^2/Var(mu_i) } = phi (n-m) ? Oct 23, 2023 at 4:08