# Constructing diversity profiles with Hill numbers

## 2019-12-05

Hill numbers represent a shared form of diversity indices. The order (q) of a Hill number determines its sensitivity to rare vs. abundant species, by modifying how the weighted mean of the species proportional abundances is calculated. Common diversity indices are special cases of Hill numbers:

q Diversity index
0 Species richness
1 Exponential Shannon index
2 Inverse Simpsons index

Hill numbers show the “effective number of species”. That is, the number of equally abundant species needed to produce the observed value of diversity. Compared to traditional diversity indices, the relationship between Hill numbers and diversity is geometric. If you double the number of species present with the same abundance, the value of the Hill number will also double.

To investigate the contributions of rare and abundant species in a community it is sometimes desirable to plot a diversity profile, showing the value of diversity calculated along a continuum of the order `q`. I wrote some R functions to do this easily.

``````# Load data
library(vegan)
data(BCI)
##' A species (columns) by site (rows) matrix of abundance values

# Calculate diversity for any order q
qd <- function(data, q = 1){
# Convert to matrix
data <- drop(as.matrix(data))

# get relative abundance
data <- sweep(data, 1, apply(data, 1, sum), "/")

# Calculate hill numbers
if (q == 0) { # Richness
hill <- apply(data > 0, 1, sum, na.rm = TRUE)
} elseif (q==1) { # Shannon
data <- -data * log(data)
hill <- exp(apply(data, 1, sum, na.rm = TRUE))
} else { # Other Hill number
data <- data^q # p_i^q
hill <- (apply(data, 1, sum, na.rm = TRUE))^(1/(1 - q))
}
return(hill)
}

# Calculate hill numbers over range of q
qd_curve <- function(data, qmin = 0, qmax = 5) {
# Define range of q
qrange <- seq(from = qmin , to = qmax , by = 0.01)

# For each value of q, calculate hill numbers for each site
qdf <- sapply(qrange , function(x){
qd(data, x)
})

# Transpose to clean dataframe
qclean <- data.frame(cbind(qrange , t(qdf)))
names(qclean) <- "qrange"

return(qclean)
}
``````

`qd_curve()` returns a dataframe of sites (columns) by Hill number order `q` (rows). This dataframe can then be used to plot a diversity profile with a line for each site.

``````library(vegan)
library(ggplot2)
library(dplyr)
library(tidyr)

data(BCI)

qd_curve(BCI) %>%
gather("id", "value", -qrange) %>%
ggplot(., aes(x = qrange, y = value)) +
geom_line(aes(group = id))
``````