For the willow tit data, both the habitat covariates were continuous measures: elevation and percentage forest cover. We often want to use a categorical covariate such as forest type, and it’s easy to incorporate one such covariate in our JAGS code.

## The example data set

For this module we’ll use camera trap data for Cape leopards from the Boland area in South Africa. Thanks to Rajan Amin and Zoological Society of London (ZSL) for making the data available. After a first survey, the area was affected by wildfires, so the team returned after the fires to compare leopard occupancy – or rather habitat use – in areas burnt vs unburnt.

You can load and inspect the data with the following code:

**# Read in data
wt <- read.csv("http://bcss.org.my/data/Leopards.csv",
comment="#")
str(leo)
# 'data.frame': 47 obs. of 7 variables:
# $ site: int 1 2 3 4 5 6 7 8 9 10 ...
# $ n1 : int 130 130 130 130 130 130 130 130 130 130 ...
# $ y1 : int 8 5 4 0 2 2 0 0 0 5 ...
# $ n2 : int 104 130 85 130 130 130 130 39 112 130 ...
# $ y2 : int 7 12 1 0 1 1 1 0 0 6 ...
# $ fire: chr "Unburnt" "Unburnt" "Unburnt" "Unburnt" ...
# $ elev: int 286 894 426 1044 405 807 496 1174 653 793 ...
# Convert 'fire' to a factor
leo$fire <- factor(leo$fire)
summary(leo)
# site n1 y1 n2
# Min. : 1.0 Min. : 0.0 Min. : 0.000 Min. : 18.0
# 1st Qu.:12.5 1st Qu.:113.0 1st Qu.: 0.000 1st Qu.: 95.5
# Median :24.0 Median :130.0 Median : 2.000 Median :115.0
# Mean :24.0 Mean :114.1 Mean : 3.128 Mean :108.0
# 3rd Qu.:35.5 3rd Qu.:130.0 3rd Qu.: 5.000 3rd Qu.:130.0
# Max. :47.0 Max. :130.0 Max. :20.000 Max. :130.0
# y2 fire elev
# Min. : 0.000 Burnt :20 Min. : 285.0
# 1st Qu.: 0.000 Unburnt:27 1st Qu.: 414.0
# Median : 2.000 Median : 592.0
# Mean : 2.617 Mean : 641.8
# 3rd Qu.: 3.000 3rd Qu.: 800.0
# Max. :16.000 Max. :1487.0**

The rows correspond to 47 camera trap locations. `n1`

is the number of days that the camera trap was operating at the site during the first survey and `y1`

is the number of days when leopards were detected; n2 and y2 give the same data for the second survey. The fires occurred between the first and second surveys, and `fire`

indicates whether or not the site was affected by the fire. `elev`

is the elevation of the site in metres.

Let’s check the distribution of burnt sites across elevation to see if they may be confounded.

**
range(leo$elev) # 285 1487
breaks <- seq(200, 1500, by=100)
op <- par(mfrow=2:1)
hist(leo$elev[leo$fire=="Burnt"], breaks=breaks, xlim=c(200, 1500),
xlab="Elevation", main="Burnt")
hist(leo$elev[leo$fire=="Unburnt"], breaks=breaks, xlim=c(200, 1500),
xlab="Elevation", main="Unburnt")
par(op)**

Below 1100m there’s a good mix of burnt and unburnt sites, but all 5 sites above 1100m were burnt. If leopards avoid those 5 sites after the fires, we won’t be able to tell whether it’s because of the burning or the elevation. Fortunately we have data from before the fires, which will tell us about the effect of elevation alone.

For the next section we’ll only use the data from the second survey, after the fires.

## Just one categorical covariate

With one covariate, which is categorical, we can simplify the model by estimating one value for occupancy for each category. We do not need to use a linear model.

This is almost like fitting a separate model to each category, and we need enough observations for each category to be able to do this. With a single model fit, we have the advantage of using the data from all the sites to estimate probability of detection.

Here’s the JAGS code for this model:

**# File name "burn1.jags"
model{
# likelihood
for(i in 1:nSite) {
z[i] ~ dbern(psi[fire[i]])
y[i] ~ dbin(p * z[i], nOcc[i])
}
# priors
for(i in 1:nCat) {
psi[i] ~ dbeta(1,1)
}
p ~ dbeta(1,1)
}**

Now we prepare the data. We need to convert the `fire`

covariate to a vector with integers for the categories: 1 for burnt, 2 for unburnt (in alphabetical order, by default). If you had more categories, the numbers would be 1, 2, 3, …

**fireI <- as.integer(leo$fire)
z <- ifelse(leo$y2 > 0, 1, NA) # Known occupancy
bdata <- with(leo, list(nSite=47, nOcc=n2, y=y2, fire=fireI,
nCat=2, z=z))
str(bdata)
# List of 6
# $ nSite: num 47
# $ nOcc : int [1:47] 104 130 85 130 130 130 130 39 112 130 ...
# $ y : int [1:47] 7 12 1 0 1 1 1 0 0 6 ...
# $ fire : int [1:47] 2 2 2 2 2 2 2 1 2 2 ...
# $ nCat : num 2
# $ z : num [1:47] 1 1 1 NA 1 1 1 NA NA 1 ...**

Now we can run the model:

**library(jagsUI)
library(mcmcOutput)
wanted <- c("p", "psi")
outA1 <- jags(bdata, NULL, wanted, "burn1.jags", DIC=FALSE,
n.chains=3, n.iter=1e4)
mcoA1 <- mcmcOutput(outA1)
diagPlot(mcoA1)
summary(mcoA1)
# mean sd median l95 u95 Rhat MCEpc
# p 0.032 0.003 0.032 0.026 0.038 0.999 0.810
# psi[1] 0.595 0.111 0.596 0.377 0.806 1.000 0.689
# psi[2] 0.848 0.071 0.856 0.706 0.971 1.001 0.701
plot(mcoA1)**

That ran very quickly and converged well. And it seems clear that leopards do prefer the unburnt sites (`psi[2]`

) to the burnt sites (`psi[1]`

). But let’s look at the effect of elevation before discussing the results further.

## Categorical and continuous covariates

Once we have multiple covariates, things become a bit less simple. We need to use a linear predictor and a link function, usually the logistic (“logit”) link. But we can still use a different intercept for each category. And provided we standardise the continuous covariates, the intercept is the logit of the probability of occupancy at the mean values of all the continuous covariates. So we can use a prior on the probability. Here’s the JAGS code for this model:

**# File name "burn2.jags"
model{
# likelihood
for(i in 1:nSite) {
logit(psi[i]) <- b0[fire[i]] + bElev * elev[i]
z[i] ~ dbern(psi[i])
y[i] ~ dbin(p * z[i], nOcc[i])
}
# priors
for(i in 1:nCat) {
psi0[i] ~ dbeta(1,1)
b0[i] <- logit(psi0[i])
}
bElev ~ dunif(-5, 5)
p ~ dbeta(1,1)
}**

We standardise the elevation to mean 0 and SD 1, then bundle the data and run the model:

**library(wiqid)
bdata <- with(leo, list(nSite=47, nOcc=n2, y=y2, fire=fireI,
nCat=2, elev=standardize(elev), z=z))
str(bdata)
# List of 7
# $ nSite: num 47
# $ nOcc : int [1:47] 104 130 85 130 130 130 130 39 112 130 ...
# $ y : int [1:47] 7 12 1 0 1 1 1 0 0 6 ...
# $ fire : int [1:47] 2 2 2 2 2 2 2 1 2 2 ...
# $ nCat : num 2
# $ elev : num [1:47] -1.227 0.87 -0.744 1.388 -0.817 ...
# $ z : num [1:47] 1 1 1 NA 1 1 NA NA NA 1 ...
wanted = c("p", "psi0", "bElev")
outA2 <- jags(bdata, NULL, wanted, "burn2.jags", DIC=FALSE,
n.chains=3, n.iter=1e4)
mcoA2 <- mcmcOutput(outA2)
diagPlot(mcoA2)
plot(mcoA2)
summary(mcoA2)
# mean sd median l95 u95 Rhat MCEpc
# p 0.032 0.003 0.032 0.026 0.038 0.999 0.739
# psi0[1] 0.661 0.125 0.666 0.419 0.899 1.000 1.105
# psi0[2] 0.850 0.076 0.860 0.702 0.983 0.999 0.965
# bElev -1.020 0.532 -0.975 -2.074 -0.018 1.000 1.212**

The elevation coefficient is definitely negative – occupancy declines with elevation. We’ll plot the curves for both burnt and unburnt sites. The code is similar to that for the willow tits, but we have to convert the two intercepts, psi0[1] and psi0[2], back to the logit scale.

**# Convert psi0 back to b0
b0 <- qlogis(mcoA2$psi0)
# Create vector of elevations for the x axis
hist(leo$elev) # only 1 site above 1200m
xx <- seq(250, 1200, , 101)
# Standardise in the same way that we standardised the original data
xxS <- standardize2match(xx, leo$elev)
# Get the values of the posterior mean and CrI for each value in xx
burnt <- unburnt <- matrix(NA, nrow=101, ncol=3)
for(i in 1:101) {
logitPsi1 <- b0[,1] + mcoA2$bElev * xxS[i]
psi1 <- plogis(logitPsi1)
burnt[i, ] <- c(mean(psi1), hdi(psi1))
logitPsi2 <- b0[,2] + mcoA2$bElev * xxS[i]
psi2 <- plogis(logitPsi2)
unburnt[i, ] <- c(mean(psi2), hdi(psi2))
}
plot(xx, unburnt[,1], type='n', ylim=range(burnt, unburnt),
xlab="Elevation, m", ylab="Occupancy", las=1)
polygon(c(xx, rev(xx)), c(burnt[,2], rev(burnt[,3])), border=NA,
col=adjustcolor('gray', 0.5))
polygon(c(xx, rev(xx)), c(unburnt[,2], rev(unburnt[,3])), border=NA,
col=adjustcolor('skyblue', 0.5))
lines(xx, burnt[,1], lwd=2)
lines(xx, unburnt[,1], lwd=2, col='blue')
abline(v=mean(leo$elev), lty=3)
text(610, 0.3, 'mean elevation', pos=2)
arrows(600, 0.3, 641, 0.3, length=0.1)
legend('bottomleft', c("burnt", "unburnt"),
col=c('black', 'blue'), lwd=2, bty='n')**

A couple of things are clear from the graph. First, the credible intervals are very wide; that should not be surprising given that we only have 20 sites in the burnt area and 27 in the unburnt area. Second, the effect of burning is modest compared to elevation: burnt sites at 250m have higher occupancy than unburnt sites at 1000m.

## Results from before the fires

Now let’s turn to the data from before the fires, and see how occupancy compared between the soon-to-be-burnt sites and the unburnt sites.

Three of the 47 sites used for the second survey did not have camera traps during the first survey. They are the last 3 sites in the data table, and have `n1 = 0`

and `y1 = 0`

. We can leave them in the data set even though they provide no information, or we can set nsites = 44, so that JAGS ignores the last 3 sites. We chose the latter for the code below.

**z <- ifelse(leo$y1 > 0, 1, NA) # Known occupancy
bdata <- with(leo, list(nSite=44, nOcc=n1, y=y1, fire=fireI,
nCat=2, elev=standardize(elev), z=z))
str(bdata)
# List of 7
# $ nSite: num 44
# $ nOcc : int [1:47] 130 130 130 130 130 130 130 130 130 130 ...
# $ y : int [1:47] 8 5 4 0 2 2 0 0 0 5 ...
# $ fire : int [1:47] 2 2 2 2 2 2 2 1 2 2 ...
# $ nCat : num 2
# $ elev : num [1:47] -1.227 0.87 -0.744 1.388 -0.817 ...
# $ z : num [1:47] 1 1 1 NA 1 1 NA NA NA 1 ...**

The code to run the model is the same, convergence is good, and here are the posterior plots:

The results are very similar, with higher occupancy at the burned sites even before the fires! Let’s get MCMC chains for the differences before and after and plot them together:

**diffA2 <- mcoA2$psi0[,2] - mcoA2$psi0[,1]
diffB2 <- mcoB2$psi0[,2] - mcoB2$psi0[,1]
differences <- cbind(before=diffB2, after=diffA2)
postPlot(differences, compVal=0)**

So, sure, occupancy was higher in the unburnt sites after the fires, but it was also higher in those sites before the fires. Indeed, the difference in occupancy has not changed as a result of the fires. Whatever is causing the difference, we can’t blame the fire.

The major lesson: don’t draw conclusions just by comparing impacted sites with controls. Look at data from before and after the impact. This is Before-After-Control-Impact (BACI) design.

Download a ZIP file with the code and data here.

Hi, Mike:

I appreciate a lot all your tutorial. All of them have been very helpful in my research (I am a master´s degree student from Colombia). I have a quick question, how should incorporate more than one categorical covariate in the model? My hipothesis included some continuous covariates and some categorical covs (presence of humans and presence of dogs). With these covariates would be better to use a Two species occupancy model or can I use them as categorical covariates? Thank you in advance.

Hi Sebastian,

If you know for sure where humans and dogs are absent, you can put that in as 0/1 in the expression for psi for the focal species. But if you are working with detection data for them (ie, there may be places where humans/dogs are present but not detected), you need a joint occupancy model, which is straightforward if the interaction is only one-way. See Bischof et al (2014, 10.1111/jzo.12100) and Waddle et al (2010, 10.1890/09-0850.1) for examples and code, also section 8.3 of Kéry and Royle (2021, ISBN: 978-0-12-809585-0) when that appears.

Do you have dogs around without humans? If the dogs are only there where humans are around, you have a 3-state model with states (1) no humans or dogs, (2) humans but no dogs, (3) humans and dogs. For your target species, that would be a covar with 3 categories, but would have to consider non-detection of humans and/or dogs.

Thank you for your quick answer.

Actually, I can not be sure about the absences of humans and dogs. I collected that information through the search of signs and marks around my survey stations and also checking the images from the camera traps.

I was thinking to use it as a categorical covariate because I am developing a multi-scale occupancy model and I am not sure how to integrate the “two species occupancy model” into the two states for occurrence in the multi-scale model (I have searched and haven´t found any paper using those approaches together). What would you recommend me to do? I do not want to discard the humans and dogs data, I think they are important.

And I have data from dogs and humans alone and together.

Thank you for your help.

Mike,

I don’t know if you still respond to these posts or not, but I figured I’d give this a shot.

From reading over your post, I would assume that the categorical variable in the example (fire) is, in essence, being treated as a random effect. Thus, occupancy would be modeled as a random intercept in the model. Is that inference correct?

I’m sorry if I seem wildly ignorant. I’m very new to the whole Bayesian thing and trying to select the appropriate model for an occupancy analysis, and this seems to be the model I’d want to use.

Thanks!

-Josh

Mike,

Quick question. Based on my understanding from your post, I would assume that fire in the model could be thought of as a random effect. Thus, the analyst could examine how the effects of the continuous covariates affect occupancy for the two different levels of fire. Is that logic correct?

Thanks,

Josh

Hi Josh,

Burnt and Unburnt are not being treated as random effects; we aren’t thinking of some population of categories from which these 2 are drawn, and not attempting to estimate parameters for such a population.

If instead of Burnt/Unburnt our categories were, say, 6 different protected areas, it would make sense to think of the distribution of intercepts across the parks, and to have another level to model that. We could model the 6 b0’s as drawn from a normal distribution and estimate the mean and SD of that.

If you thought that elevation had a different effect in Burnt vs Unburnt areas, you could have different values for bElev for each category. With 6 parks, it might make sense to have a higher level model for the 6 bElev’s.

Mike,

Wow, that was a very rapid reply. Thanks.

Ok, so just to be clear: the fact that fire cannot be thought of as a true random effect (and thus psi[1] and psi[2] cannot be construed as random intercepts from the jags output in the example) in the model is due to a lack of levels (i.e., there’s only two in this case).

This tutorial has helped me address a lot of my “beginner” questions about Bayesian hierarchical models. I didn’t have the opportunity to pursue a course in Bayesian statistics at my PhD program (currently new post-doc at Mizzou). However, I’m hoping to incorporate some of these approaches into my present research.

Thanks again!

-Josh

Dear Mike,

Thank you very much for your tutorials. They are very useful. Apologies if I’m asking too simple a question. Based on the model with the both categorical and continuous variables, how can I find out the eventual mean psi value & credible intervals across all the sites?

Would it be just taking mean(psi0)?

Thank you.

Max

`mean(psi0)`

won’t do it, that gives some sort of “mid-point” of the two categories at mean elevation.Each site has its own value for

`psi`

, depending on the covariates. To get the mean across all the sites in the sample, you need to extract all the`psi`

s. To do that, just add`psi`

to the list of parameters in`wanted`

and rerun JAGS.`outA2$sims.list$psi`

now has a column for each site, and you can get the mean with`rowMeans(outA2$sims.list$psi)`

, which will give an MCMC chain for`mean(psi)`

. Use`postPlot`

to see the histogram with the posterior mean and CrI.Dear Mike,

Thank you very much for your quick reply and for providing the code to this.

I am using the code to analyse camera trap data for 3 species with covariates to obtain modeled occupancy estimates. I have obtained the posterior mean using the code, but this value is lower than the naïve occupancy for 2 of the 3 species.

Will this ever be the case? Or would there likely to have been some error on my part somewhere?

Thank you for your help.

Max

I replied by email.