`ubms`

is an R package for fitting models of wildlife occurrence and abundance with datasets where animals are not individually marked. It provides a nearly identical interface to the popular `unmarked`

package [1]. Instead of using maximum likelihood to fit models (as with `unmarked`

), models are fit in a Bayesian framework using Stan [2]. It is generally expected that you are already familiar with `unmarked`

when using `ubms`

. You can download `ubms`

, report issues, or help with development on Github.

There are several advantages to using `ubms`

over `unmarked`

. First, it is possible to include random effects in `ubms`

models, which is not currently possible in `unmarked`

. These are specified using the familiar syntax of lme4 [3]. Second, `ubms`

generates posterior distributions for model parameters, including latent occupancy and abundance parameters. These can be useful for post-hoc analyses and diagnostics. Finally, fitting models with Stan gives you access to the large ecosystem of Stan-related tools, such as LOO (leave-one-out-cross validation; [4]).

Another alternative to `ubms`

would be to fit models in an existing modeling language such as BUGS, JAGS, or directly in Stan. `ubms`

abstracts away the complex process of defining and writing custom models in these languages which need to be updated each time you make changes to your model. It also provides many useful helper functions (e.g. `predict`

) which would otherwise require custom code. Finally, because of Stan’s efficient sampler [5] and because the underlying likelihoods in `ubms`

are marginalized, `ubms`

will often fit models much faster than equivalent models in BUGS or JAGS [6].

Relative to `unmarked`

, `ubms`

has fewer types of models available. For example, models that incorporate temporary emigration (like `gdistsamp`

) [7] are currently not available in `ubms`

. Models should run much faster in `unmarked`

as well. If you do not need one of the specific benefits of `ubms`

described above, it makes sense to stick with `unmarked`

. Even if you do plan to use `ubms`

, it makes sense to test the models in `unmarked`

first. The similar interface between the two packages makes this very easy, as you will see in the next section.

Relative to BUGS/JAGS/Stan, `ubms`

is less flexible because you cannot customize your model structure. You are limited to the provided model types. Furthermore, you cannot currently customize prior distributions (although I plan to add this in the future in some form). Finally, writing your own BUGS/JAGS model can be valuable for gaining a deeper understanding of how a model works; `ubms`

, like `unmarked`

, is essentially a black box.

To summarize the advantages and disadvantages: I see `ubms`

as an intermediate step along the continuum from `unmarked`

to custom models in BUGS/JAGS. It is not meant to replace either approach but rather to supplement them, for situations when a Bayesian framework is needed and “off-the-shelf” model structures are adequate.

Occupancy models estimate the probability \(\psi\) that a species occupies a site, while accounting for detection probability \(p < 1\) [8]. In order to estimate both \(p\) and \(\psi\), repeated observations (detection/non-detection data) at each site are required.

First, load the dataset we’ll be using, which comes with `unmarked`

:

The `crossbill`

dataset is a `data.frame`

with many columns. It contains detection/non-detection data for the European crossbill (*Loxia curvirostra*) in Switzerland [9].

`## [1] 267 58`

```
## [1] "id" "ele" "forest" "surveys" "det991" "det992" "det993"
## [8] "det001" "det002" "det003" "det011" "det012" "det013" "det021"
## [15] "det022" "det023" "det031" "det032" "det033" "det041" "det042"
## [22] "det043" "det051" "det052" "det053" "det061" "det062" "det063"
## [29] "det071" "det072" "det073" "date991" "date992" "date993" "date001"
## [36] "date002" "date003" "date011" "date012" "date013" "date021" "date022"
## [43] "date023" "date031" "date032" "date033" "date041" "date042" "date043"
## [50] "date051" "date052" "date053" "date061" "date062" "date063" "date071"
## [57] "date072" "date073"
```

Check `?crossbill`

for details about each column. The first three columns `id`

, `ele`

, and `forest`

are site covariates.

The following 27 columns beginning with `det`

are the binary detection/non-detection data; 9 years with 3 observations per year. For this example we want to fit a single-season occupancy model; thus we will use only the first three columns (year 1) of `det`

as our response variable `y`

.

```
## det991 det992 det993
## 1 0 0 0
## 2 0 0 0
## 3 NA NA NA
## 4 0 0 0
## 5 0 0 0
## 6 NA NA NA
```

Note that missing values are possible. The final 27 columns beginning with `date`

are the Julian dates for each observation. As with `y`

we want only the first three columns corresponding to year 1.

Finally, we build our `unmarkedFrame`

object holding our detection/non-detection data, site covariates, and observation covariates. Since we will conduct a single-season occupancy analysis, we need to use `unmarkedFrameOccu`

specifically. The resulting `unmarkedFrame`

can be used by both `unmarked`

and `ubms`

.

```
## Data frame representation of unmarkedFrame object.
## y.1 y.2 y.3 id ele forest date.1 date.2 date.3
## 1 0 0 0 1 450 3 34 59 65
## 2 0 0 0 2 450 21 17 33 65
## 3 NA NA NA 3 1050 32 NA NA NA
## 4 0 0 0 4 950 9 29 59 65
## 5 0 0 0 5 1150 35 24 45 65
## 6 NA NA NA 6 550 2 NA NA NA
## 7 0 0 0 7 750 6 26 54 74
## 8 0 0 0 8 650 60 23 43 71
## 9 0 0 0 9 550 5 21 36 56
## 10 0 0 0 10 550 13 37 62 75
```

First, we fit a null model (no covariates) in `unmarked`

using the `occu`

function. The `occu`

function requires as input a double formula (for detection and occupancy, respectively) along with our `unmarkedFrame`

.

```
##
## Call:
## occu(formula = ~1 ~ 1, data = umf)
##
## Occupancy:
## Estimate SE z P(>|z|)
## -0.546 0.218 -2.51 0.0121
##
## Detection:
## Estimate SE z P(>|z|)
## -0.594 0.208 -2.86 0.00426
##
## AIC: 511.2538
```

Next, we fit the same model in `ubms`

. The equivalent to `occu`

in `ubms`

is `stan_occu`

. Functions in `ubms`

generally use this `stan_`

prefix, based on the approach used in package rstanarm for GLMs. We need to provide the same arguments to `stan_occu`

. In addition, we will specify that we want 3 MCMC chains (`chains=3`

), with 500 iterations per chain (`iter=500`

) of which the first half will be warmup iterations. It is beyond the scope of this vignette to discuss the appropriate number or length of chains; see the Stan user’s guide for more details. Generally 4 chains of 2000 iterations each is recommended (of which 1000 per chain are warmups). Thus, 500 iterations per chain is probably not enough, but to keep things running quickly it is sufficient for this vignette. Note that if you are more familiar with BUGS or JAGS, Stan generally requires a smaller number of iterations to reach convergence thanks to its default NUTS sampler [5]. If you have a good multi-core CPU, you can run chains in parallel. Tell Stan how many cores you want to use by setting the `mc.cores`

option.

```
##
## Call:
## stan_occu(formula = ~1 ~ 1, data = umf, chains = 3, iter = 1000,
## refresh = 0)
##
## Occupancy (logit-scale):
## Estimate SD 2.5% 97.5% n_eff Rhat
## -0.506 0.231 -0.922 -0.0334 1237 0.999
##
## Detection (logit-scale):
## Estimate SD 2.5% 97.5% n_eff Rhat
## -0.625 0.208 -1.04 -0.215 1422 1
##
## LOOIC: 511.771
```

The structure of the output from `unmarked`

and `ubms`

is intentionally similar. Estimates of the occupancy and detection parameters are also similar, but not identical. For a more direct comparison, call the `coef`

function on both model objects:

```
## unmarked stan
## psi(Int) -0.5461203 -0.5059667
## p(Int) -0.5939612 -0.6245178
```

Let’s look at the output from our `fit_stan`

model again:

```
##
## Call:
## stan_occu(formula = ~1 ~ 1, data = umf, chains = 3, iter = 1000,
## refresh = 0)
##
## Occupancy (logit-scale):
## Estimate SD 2.5% 97.5% n_eff Rhat
## -0.506 0.231 -0.922 -0.0334 1237 0.999
##
## Detection (logit-scale):
## Estimate SD 2.5% 97.5% n_eff Rhat
## -0.625 0.208 -1.04 -0.215 1422 1
##
## LOOIC: 511.771
```

The first part (under `Call:`

) is the command we used to get this model output. Underneath are two tables, one per submodel, corresponding to the occupancy and detection parts of the model. Within each table there is one row per parameter in the submodel. Since `fit_stan`

had no covariates, there is only an intercept term for each submodel. Model parameters in this summary table are always shown on the appropriate transformed scale, in this case logit. To get the corresponding probabilities, you can use the `predict`

function, which we will demonstrate later.

For each parameter, the mean and standard deviation of the posterior distribution are given. Unlike `unmarked`

output, there is no \(Z\) or \(p\)-value. Instead, there is a 95% uncertainty interval provided.

The final two columns in each summary table `n_eff`

and `Rhat`

are MCMC diagnostics. We will discuss their meaning later.

To extract summary values into an R table for easy manipulation, use the `summary`

method. Note that you have to specify which submodel you want (`"state"`

for occupancy or `"det"`

for detection).

`## [1] -0.5059667`

To extract the entire posterior for a parameter, use the `extract`

method. To avoid name collisions you need to use the full name of the parameter (which contains both the submodel and the parameter name) when extracting. To see a list of the available full parameter names, use the `names`

method.

`## [1] "beta_state[(Intercept)]" "beta_det[(Intercept)]"`

Now we’ll fit a series of candidate models to the `crossbill`

data in `ubms`

and compare them.

Along with our previous null model, we’ll fit a model with site covariates (elevation and forest), a model with an observation covariate (date), and a “global” model with both site and observation covariates. This is just an example; perhaps other models should also be considered if we were preparing this analysis for publication. In our model formulas, we have normalized all covariates with `scale`

so they have a mean of 0 and a standard deviation of 1. This can help improve model convergence and is generally a good idea.

```
fit_null <- fit_stan
fit_sc <- stan_occu(~1~scale(forest)+scale(ele), data=umf,
chains=3, iter=500)
fit_oc <- stan_occu(~scale(date)~1, data=umf, chains=3, iter=500)
fit_global <- stan_occu(~scale(date)~scale(forest)+scale(ele), data=umf,
chains=3, iter=500)
```

```
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
```

```
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
```

The `fit_global`

model gave us some warnings about the effective sample size (`n_eff`

) along with a suggested solution. We will ignore this warning for now but normally it is a good idea to pay close attention to these warnings.

First we combine the models into a `fitList`

:

Then we generate a model selection table:

```
## elpd nparam elpd_diff se_diff weight
## fit_global -237.431 7.012 0.000 0.000 0.812
## fit_sc -241.782 5.599 -4.351 3.566 0.142
## fit_oc -249.894 3.418 -12.463 5.158 0.046
## fit_null -255.886 2.418 -18.454 6.622 0.000
```

Instead of AIC, models are compared using leave-one-out cross-validation (LOO) [4] via the `loo`

package. Based on this cross-validation, the expected predictive accuracy (`elpd`

) for each model is calculated. The model with the largest `elpd`

(`fit_global`

) performed best. The `elpd_diff`

column shows the difference in `elpd`

between a model and the top model; if this difference is several times larger than the standard error of the difference (`se_diff`

), we are confident the model with the larger `elpd`

performed better. LOO model weights, analogous to AIC weights, are also calculated. We can see that the `fit_global`

model is clearly the best performing model.

You can obtain LOO information for a single model using the `loo`

method:

```
##
## Computed from 1500 by 245 log-likelihood matrix
##
## Estimate SE
## elpd_loo -237.4 19.3
## p_loo 7.0 0.7
## looic 474.9 38.6
## ------
## Monte Carlo SE of elpd_loo is 0.2.
##
## All Pareto k estimates are good (k < 0.5).
## See help('pareto-k-diagnostic') for details.
```

The `looic`

value is analogous to AIC.

You can also obtain the WAIC (Widely Applicable Information Criterion) if you prefer [4]:

```
##
## Computed from 1500 by 245 log-likelihood matrix
##
## Estimate SE
## elpd_waic -237.4 19.3
## p_waic 7.0 0.7
## waic 474.8 38.6
```

We’ll define the global model as our top model:

```
##
## Call:
## stan_occu(formula = ~scale(date) ~ scale(forest) + scale(ele),
## data = umf, chains = 3, iter = 1000, refresh = 0)
##
## Occupancy (logit-scale):
## Estimate SD 2.5% 97.5% n_eff Rhat
## (Intercept) -0.494 0.665 -1.150 1.72 59.9 1.03
## scale(forest) 1.231 0.658 0.631 3.45 60.7 1.03
## scale(ele) 0.566 0.252 0.049 1.03 453.9 1.00
##
## Detection (logit-scale):
## Estimate SD 2.5% 97.5% n_eff Rhat
## (Intercept) -0.791 0.282 -1.470 -0.339 143 1.01
## scale(date) 0.563 0.172 0.237 0.906 978 1.00
##
## LOOIC: 474.862
```

Again looking at the summary of `fit_top`

, it appears all chains have adequately converged based on \(\hat{R}\) values. Ideally, you want all \(\hat{R} > 1.05\). To visualize convergence, look at the traceplots: