A long-term agricultural experiment had $n=90$ grassland plots, each $25 \mathrm{~m} \times 25 \mathrm{~m}$, differing in biomass, soil pH, and species richness (the count of species in the whole plot). While it was well-known that species richness declines with increasing biomass, it was not known how this relationship depends on soil pH. In the experiment, there were 30 plots of "low pH", 30 of "medium pH" and 30 of "high pH". Three lines of the data are reproduced here as an aid.

Briefly explain the commands below. That is, explain the models being fitted.

Let $H_{1}, H_{2}$ and $H_{3}$ denote the hypotheses represented by the three models and fits. Based on the output of the code below, what hypotheses are being tested, and which of the models seems to give the best fit to the data? Why?

Finally, what is the value obtained by the following command?

$$\begin{aligned} & >\operatorname{grass}[\mathrm{c}(1,31,61),] \\ & \text { pH Biomass Species } \\ & 1 \text { high } 0.4692972 \quad 30 \\ & 31 \text { mid } 0.1757627 \quad 29 \\ & 61 \text { low } 0.1008479 \quad 18 \\ & >\text { fit1 <- glm(Species Biomass, family = poisson) } \\ & >\text { fit2 <- glm(Species } ~ \mathrm{pH}+\text { Biomass, family }=\text { poisson) } \\ & >\text { fit3 <- glm(Species } \mathrm{pH} * \text { Biomass, family }=\text { poisson) } \end{aligned}$$