Model notation in methods draft

[christophe-rd]

@lizzieinvancouver as you requested, starting an issue with the equation text:

$$\ln(\text{ringwidth})_i \sim \text{Normal}(\mu, \sigma_y)$$

$$ \mu = \alpha + \alpha_{\text{species}} + \alpha_{\text{site}} + \alpha_{\text{treeid}} + \alpha_{\text{year}} + \beta_{\text{species}} \times \text{season predictor} $$ $$ \alpha \sim \text{Normal}(1, 3), \quad $$ $$\alpha_{\text{species}} \sim \text{Normal}(0, 6), \quad $$ $$\alpha_{\text{site}} \sim \text{Normal}(0, \sigma_{\alpha,\text{site}})$$ $$\alpha_{\text{treeid}} \sim \text{Normal}(0, \sigma_{\alpha,\text{treeid}}), \quad $$ $$\beta_{\text{species}} \sim \text{Normal}(0, 0.5) $$ $$ \sigma_{\alpha,\text{site}} \sim \text{Normal}(0, 1), \quad$$ $$\sigma_{\alpha,\text{treeid}} \sim \text{Normal}(0, 1), \quad $$ $$\sigma_y \sim \text{Normal}+(0, 3)$$

I wasn't sure if you wanted me to also add the text that describes the equation. In case you did, here it is (I didn't include the model diagnostics and package details):

where $i$ is each observation of the dataset. Intercepts on species ($\alpha_{species}$) and year ($\alpha_{year}$) and slope on species ($\beta_{species}$) were not pooled (fixed effects), and we partially pooled (random effects) site ($\alpha_{site}$) and treeid ($\alpha_{treeid}$) to account for the similarity of the measurements within these groups.

Also, you pointed to $\sigma_{y}$ with "missing its normal". I'm not sure what you mean by that...

[lizzieinvancouver]

@christophe-rd Please move the priors that are all normal (number, number) to supp or remove -- what's more important for this model is that you say that the priors are weakly informative and check you get the same answer at 2X priors.

Your sigma y is fine, my mistake!

Start working towards text that describes the model more biologically so ...

We modeled ring width observed for each tree in each year:

... equations ...

where i denotes each observation. This model yields a species-specific effect of each seasonal predictor (e.g. GDD) while also allowing other variation due to species (alpha sp), alongside variation due to site, tree id and year.