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Baker 2012 (Journal of Applied Ecology) used similar model. I was asking him and he replied he uses normal glm()! He inspired me to use the following transformation (that he actually used in the linked article):

$$\begin{eqnarray} \mbox{log} ( \mu_{i,j+1} ) &=& \alpha_i + \sum\limits_{t=1}^{j} R_{t} + \sum\limits_{k} \alpha_k \sum\limits_{t=1}^{j}x_{k,t} \\ \mbox{log} ( \mu_{i,1} ) &=& \alpha_i \end{eqnarray}$$

so actually $\mbox{log}(\mu_{i,1})$ acts as a site-specific intercept. It is trivial to see that the transformed equations are equivalent to the original model. I do not trivially see that the whole fit proccess including poisson errors will also be equivalent, but this is probably more limitation of my brain than an actual problem :).

The transformed model is of course very easily fitted using glm()! Including overdispersion using the quasipoisson family.

Baker 2012 (Journal of Applied Ecology) used similar model. I was asking him and he replied he uses normal glm()! He inspired me to use the following transformation (that he actually used in the linked article) - just recursively substitute the $\mbox{log} (\mu_{i,j})$, until you get this:

$$\mbox{log} ( \mu_{i,j+1} ) = \mbox{log} ( \mu_{i,1} ) + \sum\limits_{t=1}^{j} R_{t} + \sum\limits_{k} \alpha_k \sum\limits_{t=1}^{j}x_{k,t}$$

and then, $\mbox{log}(\mu_{i,1})$ can be simply taken as a site-specific intercept:

$$\begin{eqnarray} \mbox{log} ( \mu_{i,j+1} ) &=& \alpha_i + \sum\limits_{t=1}^{j} R_{t} + \sum\limits_{k} \alpha_k \sum\limits_{t=1}^{j}x_{k,t} \\ \mbox{log} ( \mu_{i,1} ) &=& \alpha_i \end{eqnarray}$$

so this can be easily solved by classic GLM. It is trivial to see that the transformed equations are equivalent to the original model. I do not trivially see that the whole fit proccess including poisson errors will also be equivalent, but this is probably more limitation of my brain than an actual problem :).

The transformed model is of course very easily fitted using glm()! Including overdispersion using the quasipoisson family.

Baker 2012 (Journal of Applied Ecology) used similar model. I was asking him and he replied he uses normal glm()! He inspired me to use the following transformation (that he actually used in the linked article):

$$\mbox{log} ( \mu_{i,j+1} ) = \alpha_0 + \sum\limits_{t=1}^{j} R_{t} + \sum\limits_{k} \alpha_k \sum\limits_{t=1}^{j}x_{k,t}$$

$$\mbox{log} ( \mu_{i,1} ) = \alpha_0$$

so actually $\mbox{log}(\mu_{i,1})$ acts as a site-specific intercept. It is trivial to see that the transformed equations are equivalent to the original model. I do not trivially see that the whole fit proccess including poisson errors will also be equivalent, but this is probably more limitation of my brain than an actual problem :).

The transformed model is of course very easily fitted using glm()! Including overdispersion using the quasipoisson family.

Baker 2012 (Journal of Applied Ecology) used similar model. I was asking him and he replied he uses normal glm()! He inspired me to use the following transformation (that he actually used in the linked article):

$$\begin{eqnarray} \mbox{log} ( \mu_{i,j+1} ) &=& \alpha_i + \sum\limits_{t=1}^{j} R_{t} + \sum\limits_{k} \alpha_k \sum\limits_{t=1}^{j}x_{k,t} \\ \mbox{log} ( \mu_{i,1} ) &=& \alpha_i \end{eqnarray}$$

so actually $\mbox{log}(\mu_{i,1})$ acts as a site-specific intercept. It is trivial to see that the transformed equations are equivalent to the original model. I do not trivially see that the whole fit proccess including poisson errors will also be equivalent, but this is probably more limitation of my brain than an actual problem :).

The transformed model is of course very easily fitted using glm()! Including overdispersion using the quasipoisson family.

Baker 2012 (Journal of Applied Ecology) used similar model. I was asking him and he replied he uses normal glm()! He inspired me to use the following transformation (that he actually used in the linked article):

$$\mbox{log} ( \mu_{i,j+1} ) = \alpha_0 + \sum\limits_{t=1}^{j} R_{t} + \sum\limits_{k} \alpha_k \sum\limits_{t=1}^{j}x_{k,t}$$

$$\mbox{log} ( \mu_{i,1} ) = \alpha_0$$

so actually $\mbox{log}(\mu_{i,1})$ act as a site-specific intercept. It is trivial to see that the transformed equations are equivalent to the original model. I do not trivially see that the whole fit proccess including poisson errors will also be equivalent, but this is probably more limitation of my brain than an actual problem :).

The transformed model is of course very easily fitted using glm()! Including overdispersion using the quasipoisson family.

Baker 2012 (Journal of Applied Ecology) used similar model. I was asking him and he replied he uses normal glm()! He inspired me to use the following transformation (that he actually used in the linked article):

$$\mbox{log} ( \mu_{i,j+1} ) = \alpha_0 + \sum\limits_{t=1}^{j} R_{t} + \sum\limits_{k} \alpha_k \sum\limits_{t=1}^{j}x_{k,t}$$

$$\mbox{log} ( \mu_{i,1} ) = \alpha_0$$

so actually $\mbox{log}(\mu_{i,1})$ acts as a site-specific intercept. It is trivial to see that the transformed equations are equivalent to the original model. I do not trivially see that the whole fit proccess including poisson errors will also be equivalent, but this is probably more limitation of my brain than an actual problem :).

The transformed model is of course very easily fitted using glm()! Including overdispersion using the quasipoisson family.