I am confused with the definition of the weights in glm and lm.
Using the McCullagh and Nelder (1989)'s notation, If random variable $y_i$ is from the Generalized Linear Model (GLM), then its density is modelled in the form:
\begin{equation} f(y_i) = exp\Big(\frac{m_i}{\phi} [\theta_i y_i - b(\theta_i) ] + c(y_i;\phi)\Big) \end{equation}
where $\theta_i$ is the canonical parameter, $\kappa$ is the dispersion parameter and $m$ is the known prior "weight". I would like to know that this prior "weight" is NOT the weight specified in glm. help(glm) says that:
Non-NULL weights can be used to indicate that different observations have different dispersions (with the values in weights being inversely proportional to the dispersions); or equivalently, when the elements of weights are positive integers w_i, that each response y_i is the mean of w_i unit-weight observations. For a binomial GLM prior weights are used to give the number of trials when the response is the proportion of successes: they would rarely be used for a Poisson GLM.
Therefore, in my understanding, what "weight" $w_i$ does is to re-parameterize the dispersion parameter as
$$\phi=\frac{\phi^*}{w_i},$$
where $\phi^*$ is the redefined dispersion parameter. This means that for example, when $y_i$ is modelled only with an intercept term $\beta_0$, lm function with non NULL "weight" specification maximizes the sum of the weighted likelihood of $y_i$ with respect to $\phi^*$ and $\beta_0$ where:
$$ f(y_i)=\sqrt{ \frac{w_i}{2\pi \phi^*} } \exp\Big(-\frac{1}{2}\frac{w_i (y-\beta_0)^2}{\phi^*}\Big), $$ where the identity link is used $\beta_0=\theta_i$.
Similarly, glm function with family = "poisson" with non NULL "weight" maximizes the sum of the weighted likelihood of $y_i$ with respect to $\beta_0$ where:
$$ f(y_i)=\frac{\beta_0^{w_i y_i}}{y_{i}!} exp(-w_i \beta_0), $$
where the log link is used $\beta_0=exp(\theta_i)$.
Similarly, glm function with family = "binomial" with non NULL "weight" maximizes the sum of the weighted likelihood of $y_i$ with respect to $\phi^*$ and $\beta_0$ where:
$$ f(y_i)= \begin{pmatrix} m\\ y_i \end{pmatrix} \beta_0^{w_iy_i}(1-\beta_0)^{w_i(m-y_i)} $$
where logit link is used $\beta_0 = logit^{-1}(\theta_i)$.
Is my understanding correct?
Reference:
C.E. McCulloch and J.A. Nelder. Generalized Linear Models. Chapman and Hall, London, 1989.
