# What log-likelihood function do you use when doing a Poisson regression with continuous response?

I am given $N$ observations of pairs of covariates and response $(\mathbf{x}_i, y_i)$. When the response are non-negative integers, by doing Poisson regression I am modelling $y_i \sim \mathrm{Pois}(\mu_i)$ as Poisson random variables with mean $\mu_i$, such that $\ln(\mu_i)$ is a linear function of the covariate $\mathbf{x}_i$. A maximum likelihood estimator for the coefficients of $\mathbf{x}_i$ maximises the Poisson log-likelihood:

$$\sum_{i=1}^N (y_i \ln(\mu_i) - \mu_i)$$

I have seen references to doing Poisson regression with non-negative, non-integers, e.g. How does a Poisson distribution work when modeling continuous data and does it result in information loss?

In this case, what log-likelihood function is used? Do you still use the above function but allow $y_i$ to take the non-integer values?

You are right. In fact, since proper Poisson model would be incorrect in here because of dealing with continuous outcome, you'll be using quasi-Poisson model.

It is called quasi-likelihood and was described for the first time by Wedderburn (1974):

To define a likelihood we have to specify the form of distribution of the observations, but to define a quasi-likelihood function we need only specify a relation between the mean and variance of the observations and the quasi-likelihood can then be used for estimation.

You can find some more description and examples in paper by McCullagh (1983) and handbooks on GLM's.

In case of quasi-Poisson model, the quasi-likelihood is

$$y \log \mu - \mu$$

where $y \ge 0$ and $\mu > 0$ as described in McCullagh (1983).

Wedderburn, R. W. (1974). Quasi-likelihood functions, generalized linear models, and the Gauss—Newton method. Biometrika, 61(3), 439-447.

McCullagh, P. (1983). Quasi-likelihood functions. The Annals of Statistics, 59-67.