Simulate data and fit 2-species occupancy model à la Rota et al. (2016) w/ the R package unmarked

We consider a two-species static occupancy model à la Rota et al. (2016). We simulate data from this model, and fit the model to these data using Unmarked. With the Rmd file, you can run the code in RStudio and reproduce the results.

Setting the scene

Ignoring the site index, we use the following notation for the occupancy probabilities:

$\psi_{11}$ is the prob. that species 1 and species 2 are both present;
$\psi_{10}$ is the prob. that species 1 is present and species 2 is absent;
$\psi_{01}$ is the prob. that species 1 is absent and species 2 is present;
$\psi_{00}$ is the prob. that species 1 and species 2 are both absent, with avec $\psi_{11} + \psi_{10} + \psi_{01} + \psi_{00} = 1.$

The marginal probabilities of occupancy are:

And the conditional probabilities given that $\Pr(\text{A|B}) = \Pr(\text{A and B})/\Pr(\text{B})$ :

Data simulation

We will use the package mipfb to simulate occupancy state as a multivariate Bernoulli random variable; more about the multivariate Bernoulli can be found in Dai et al. (2013):

library(mipfp)

For reproducibility, we set the seed:

set.seed(2020)

Choose the number of species, the number of sites, and the number of visits:

S <- 2 # nb species 
N <- 500 # nb sites
J <- 5 # nb visits

Let's consider a scenario in which species 2 avoids species 1 while species 1 does not care about species 2 and its presence or absence. To specify this scenario, we will work out the conditional probabilities with, for example:

$\Pr(z_2 = 1 | z_1 = 0) = 0.6$ , species 2 is present with high probability whenever species 1 is absent
$\Pr(z_2 = 1 | z_1 = 1) = 0.1$ , species 2 avoids species 1 when it is present
$\Pr(z_{1} = 1 | z_2 = 0) = \Pr(z_1 = 1 | z_2 = 1) = 0.4$ , species 1 does not care about presence/absence of species 2

Now we need to go back to the occupancy probabilities. Let $x = \psi_{01}$ , $y = \psi_{10}$ , $z = \psi_{11}$ so that $\psi_{00} = 1 - x - y - z$ , then we have a system of 3 equations with 3 unknowns:

$0.6 = / (x + 1 - x - y - z) \Leftrightarrow x + 0.6y + 0.6z = 0.6$

$0.1 = z / (z + y) \Leftrightarrow -0.1y + 0.9z = 0$

$0.4 = y / (y + 1 - x - y - z) \Leftrightarrow 0.4x + y + 0.4z = 0.4$

which can be solved with the Mathematica online solver:

psi01 <- 81/175
psi10 <- 36/175
psi11 <- 4/175
psi00 <- 1 - (psi01 + psi10 + psi11) # 54/175

In R, we could use

a <- matrix(c(1,0.6,0.6,0,-0.1,0.9,0.4,1,0.4),ncol=3,byrow=T)
b <- matrix(c(0.6,0,0.4),ncol=1)
solve(a,b)

## 0.46285714
## 0.20571429
## 0.02285714

We then obtain the marginal occupancy probabilities:

psiS1 <- psi10 + psi11
psiS2 <- psi01 + psi11

Now we're ready to simulate data from a multivariate Bernoulli (check out ?RMultBinary and ?ObtainMultBinaryDist).

First, we calculate the odds ratios:

or <- matrix(c(1, (psiS1*(1-psiS2))/(psiS2*(1-psiS1)), 
               (psiS2*(1-psiS1))/(psiS1*(1-psiS2)), 1), nrow = 2, ncol = 2, byrow = TRUE)
rownames(or) <- colnames(or) <- c("sp1", "sp2")

Then the marginal probabilities:

marg.probs <- c(psiS1, psiS2)

And we estimate the joint probability:

p.joint <- ObtainMultBinaryDist(odds = or, marg.probs = marg.probs)

At last, we generate $N$ random samples from a bivariate Bernoulli (2 species) with relevant parameters

z <- RMultBinary(n = N, mult.bin.dist = p.joint)$binary.sequences

Now we add on top the observation. First, we fix the detection probability for each species:

ps <- c(0.5,0.9)

Then we generate the detection and non-detections for each species, which we store in a list:

y <- list()
for (i in 1:S){
  y[[i]] <- matrix(NA,N,J)
  for (j in 1:N){
    for (k in 1:J){
      y[[i]][j,k] <- rbinom(1,1,z[j,i]*ps[i])
    }
  }
}
names(y) <- c('sp1','sp2')

Model fitting

Now let us fit a 2-species static occupancy model to the data we have simulated. We need to load the package unmarked:

library(unmarked)

We format the data as required:

data <- unmarkedFrameOccuMulti(y=y)

Let's have a look to the data:

summary(data)

## unmarkedFrame Object
## 
## 500 sites
## 2 species: sp1 sp2 
## Maximum number of observations per site: 5 
## Mean number of observations per site:
## sp1: 5  sp2: 5  
## Sites with at least one detection:
## sp1: 113  sp2: 238  
## Tabulation of y observations:
## sp1:
##    0    1 
## 2210  290 
## sp2:
##    0    1 
## 1425 1075

And in particular the detections and non-detections:

plot(data)

Now we specify the effects we would like to consider on the occupancy and detection probabilities. The thing is that the function occuMulti doesn't work directly on the occupancy probabilities but on the so-called natural parameters (in that specific order):

that is:

occFormulas <- c('~1','~1','~1')

To specify the effects on detection, there is no difficulty:

detFormulas <- c('~1','~1')

We fit a model with constant natural parameters and constant detection probabilities

fit <- occuMulti(detFormulas,occFormulas,data)

Display the result:

fit

## 
## Call:
## occuMulti(detformulas = detFormulas, stateformulas = occFormulas, 
##     data = data)
## 
## Occupancy:
##                       Estimate    SE      z  P(>|z|)
## [sp1] (Intercept)      -0.8862 0.139 -6.355 2.09e-10
## [sp2] (Intercept)       0.0636 0.103  0.619 5.36e-01
## [sp1:sp2] (Intercept)  -0.7042 0.225 -3.136 1.71e-03
## 
## Detection:
##                   Estimate     SE      z   P(>|z|)
## [sp1] (Intercept)  -0.0132 0.0906 -0.146  8.84e-01
## [sp2] (Intercept)   2.2351 0.0981 22.777 7.72e-115
## 
## AIC: 2758.236

Get the natural parameter and detection estimates:

mle <- fit@opt$par
names(mle) <- c('f1','f2','f12','lp1','lp2')

Get the occupancy estimates:

den <- 1 + exp(mle['f1'])+exp(mle['f2'])+exp(mle['f1']+mle['f2']+mle['f12'])
psi11hat <- exp(mle['f1']+mle['f2']+mle['f12'])/den
psi10hat <- exp(mle['f1'])/den
psi01hat <- exp(mle['f2'])/den

I do it by hand to understand how unmarked works. The easy way is to use predict(fit,'state').

Get the detection estimates:

p1hat <- plogis(mle['lp1'])
p2hat <- plogis(mle['lp2'])

Again I do it by hand, but unmarked can do it for you with predict(fit,'det').

Now compare the parameters we used to simulate the data (left column) to the parameter estimates (right column)

res <- data.frame(real = c(psiS1,
                           psiS2,
                           psi01,
                           psi10,
                           psi11,
                           ps[1],
                           ps[2]),
                  estim = c(psi10hat+psi11hat,
                            psi01hat+psi11hat,
                            psi01hat,
                            psi10hat,
                            psi11hat,
                            p1hat,
                            p2hat))
rownames(res) <- c('marginal_occ1','marginal_occ2','psi01','psi10','psi11','det1','det2')
res

##                     real      estim
## marginal_occ1 0.22857143 0.23354481
## marginal_occ2 0.48571429 0.47600454
## psi01         0.46285714 0.39540621
## psi10         0.20571429 0.15294648
## psi11         0.02285714 0.08059832
## det1          0.50000000 0.49669782
## det2          0.90000000 0.90335332

If you just want to get the parameter estimates directly:

# detection
predict(fit,'det',species=1)[1,]

##   Predicted         SE     lower     upper
## 1 0.4966978 0.02264026 0.4523237 0.5410719

predict(fit,'det',species=2)[1,]

##   Predicted          SE     lower     upper
## 1 0.9033533 0.008567074 0.8865622 0.9201445

# marginal occupancy
predict(fit,'state',species=1)[1,]

## Bootstrapping confidence intervals with 100 samples

##   Predicted         SE     lower     upper
## 1 0.2335448 0.02867384 0.1867353 0.2919506

predict(fit,'state',species=2)[1,]

## Bootstrapping confidence intervals with 100 samples

##   Predicted        SE     lower     upper
## 1 0.4760045 0.0268335 0.4320817 0.5302099

# conditional occupancy
predict(fit,'state',species=1,cond='sp2')[1,] # species 1 | species 2 present

## Bootstrapping confidence intervals with 100 samples

##   Predicted         SE     lower     upper
## 1 0.1693226 0.03534907 0.1130592 0.2308998

predict(fit,'state',species=1,cond='-sp2')[1,] # species 1 | species 2 absent

## Bootstrapping confidence intervals with 100 samples

##   Predicted         SE     lower     upper
## 1 0.2918851 0.02951294 0.2430333 0.3499646

predict(fit,'state',species=2,cond='sp1')[1,] # species 2 | species 1 present

## Bootstrapping confidence intervals with 100 samples

##   Predicted         SE     lower     upper
## 1 0.3451086 0.05456903 0.2511465 0.4491831

predict(fit,'state',species=2,cond='-sp1')[1,] # species 2 | species 1 absent

## Bootstrapping confidence intervals with 100 samples

##   Predicted         SE     lower     upper
## 1 0.5158895 0.02033788 0.4754084 0.5544719

R version used

sessionInfo()

## R version 3.6.2 (2019-12-12)
## Platform: x86_64-apple-darwin15.6.0 (64-bit)
## Running under: macOS Catalina 10.15.4
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
## 
## locale:
## [1] fr_FR.UTF-8/fr_FR.UTF-8/fr_FR.UTF-8/C/fr_FR.UTF-8/fr_FR.UTF-8
## 
## attached base packages:
## [1] parallel  stats     graphics  grDevices utils     datasets  methods  
## [8] base     
## 
## other attached packages:
## [1] unmarked_0.13-2     Rcpp_1.0.4          lattice_0.20-38    
## [4] mipfp_3.2.1         numDeriv_2016.8-1.1 Rsolnp_1.16        
## [7] cmm_0.12           
## 
## loaded via a namespace (and not attached):
##  [1] codetools_0.2-16 digest_0.6.25    MASS_7.3-51.4    truncnorm_1.0-8 
##  [5] plyr_1.8.6       grid_3.6.2       magrittr_1.5     evaluate_0.14   
##  [9] rlang_0.4.5      stringi_1.4.6    sp_1.4-1         raster_3.0-12   
## [13] Matrix_1.2-18    rmarkdown_2.1    tools_3.6.2      stringr_1.4.0   
## [17] xfun_0.12        yaml_2.2.1       compiler_3.6.2   htmltools_0.4.0 
## [21] knitr_1.28

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Simulate data and fit 2-species occupancy model à la Rota et al. (2016) w/ the R package unmarked

Setting the scene

Data simulation

Model fitting

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Simulate data and fit 2-species occupancy model à la Rota et al. (2016) w/ the R package unmarked

Setting the scene

Data simulation

Model fitting

R version used

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