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I have data on plant diversity in response to a fully crossed treatments of fertilizer and light in grassland systems:

>dput(plants)
structure(list(Fertilizer = structure(c(1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L), .Label = c("F-", "F+"
), class = "factor"), Light = structure(c(1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L), .Label = c("L-", 
"L+"), class = "factor"), FL = structure(c(1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L), .Label = c("F-L-", 
"F-L+", "F+L-", "F+L+"), class = "factor"), LF = structure(c(1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L), .Label = c("L-F-", 
"L-F+", "L+F-", "L+F+"), class = "factor"), Diversity = c(6L, 
7L, 10L, 9L, 5L, 9L, 7L, 6L, 10L, 9L, 9L, 11L, 9L, 9L, 7L, 7L, 
4L, 4L, 6L, 4L, 5L, 5L, 4L, 4L, 7L, 8L, 9L, 9L, 10L, 10L, 10L, 
7L)), class = "data.frame", row.names = c(NA, -32L))

In these grassland systems, there is typically a loss of species diversity in response to fertilization, which may be due to light competition. My goal is to estimate whether the loss of species can be prevented by restoring light to these areas. There were a total of 32 plots, 8 received fertilizer and light(F+L+), 8 received fertilizer and no light (F+L-), 8 received no fertilizer and light (F-L+), and 8 received no fertilizer and no light(F-L-). The number of species was counted in each plot.

This is a toy data set given to me by a colleague (which I thought was actually to practice the Poisson distribution...) , so I know virtually no more than what has been presented above. I want to know if I can use the Poisson distribution to model this data. Thinking about the assumptions of the Poisson distribution as they relate to this example, I have a few questions:

  1. The Poisson distribution assumes the mean is equal to the variance: In this case, if I group the data by the FL column, which would be the saturated model (grouping the plots by both main effects Fertilizer and Light and their interaction, totaling 4 groups), if this were to follow a poisson distribution, would I expect: a. The mean of each level of FL to equal the variance of that same level, or b. The mean of all values of Diversity to be equal the variance of all values

  2. This question is related to the following questions but explains what I am confused about. When we talk about "events" under the poison distribution, in my case, would I think about an "event" as: a. each species within each plot, or b. individual plots, ignoring their grouping, or c. individual plots, according to their grouping *when I say grouping, I mean whether they are grouped by the main effects: Light alone or Fertilizer alone (both of which would create 2 groups) or whether they are grouped by FL (which would create 4 groups because it takes both main effects into account)

  3. The Poisson distribution assumes the probability of two events occurring in the same narrow interval is negligible. In my case does that mean the probability of 2 plots occurring in the same physical location (which is not possible) is negligible?

  4. The Poisson distribution assumes that the probability of an event within a certain interval does not change over different intervals. If I am interpreting this correctly, my design significantly violates this because the probability of having "n" number of species in a plot changes depending on the treatment, as you are going to have a different number of species in plots depending on the treatment they received?

  5. The Poisson distribution assumes the probability of an event in one interval is independent of the probability of an event in any other non-overlapping interval. Again, If I am interpreting an this correctly, this assumption is violated on several levels; assuming we don't know how internal or external factors (e.g. seed dispersal, influence of plants around these plots, if there is interaction between plots because of animals, ect...) other than Fertilizer andLight` influence the "probability" of having "n" number of species in a plot, we cant assume any independence, right?

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  • $\begingroup$ Is your variable Diversitya count of number of species? I can see no reason to think that should have a Poisson distribution (number of individuals of some species would be ...), and cardinal diversity is not a very good measure of diversity ... $\endgroup$ – kjetil b halvorsen Mar 19 at 20:36
  • $\begingroup$ yes Diversity is a count of the number of species $\endgroup$ – Ryan Mar 20 at 17:40
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First, you should visualize your data:

Boxplots by group of Diversity with overlaid raw data

made in R by:

library(tidyverse)
ggplot(plants, aes(x=Fertilizer:Light, y=Diversity)) + 
geom_boxplot(notch=TRUE, notchwidth=0.9) + 
geom_jitter(position=position_jitter(0.03), color="blue", cex=2) + 
ggtitle("Cardinal Diversity")

which shows clearly at least some differences between the groups.

Then your question, if we can assume a Poisson distribution for these data. I would answer with a clear no. The numbers given for Diversity is number of species observed. If your data where, for each of the species in some list, the number of individuals observed from that species, a Poisson model could make sense. Let $X_1, X_2, \dotsc, X_k$ be such counts for $k$ species (say, in one of the groups.) Then the number of species observed is given by $$ \text{Number of species} = \sum_{i=1}^k \mathbb{I}(X_i>0) $$ which would have a Poisson-Binomial distribution, not a Poisson distribution. See the tag .

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