---
title: "Fitting occupancy models with flocker"
author: Jacob Socolar & Simon Mills
date: "2023-10-20"
output: rmarkdown::html_vignette
vignette: >
%\VignetteIndexEntry{Fitting occupancy models with flocker}
%\VignetteEngine{knitr::rmarkdown}
%\VignetteEncoding{UTF-8}
---
`flocker` is an R package for fitting [occupancy models](https://jsocolar.github.io/closureOccupancy/) that incorporate
sophisticated effects structures using simple formula-based syntax. `flocker` is
built on R package `brms`, which in turn is a front-end for `Stan`.
This vignette is intended as a companion to [Socolar & Mills 2023](https://www.biorxiv.org/content/10.1101/2023.10.26.564080v1),
where we provide details of the models and post-processing functionality
available in `flocker` in greater detail. Here, we provide illustrative R code
for several types of model, demonstrating data simulation, model fitting,
and model post-processing. We also showcase the `brms` syntax
that `flocker` can use to fit a variety of sophisticated effect
structures.
## Terms and definitions
Socolar & Mills (2023) introduce several terms that figure importantly in this
vignette, including:
* **closure-unit**: The groupings of observations over which
[closure](https://jsocolar.github.io/closureOccupancy/) is assumed. In
single-species models, a closure-unit corresponds to a "site" or "point". In
multi-species models, a closure-unit is a species-site combination. In dynamic
(multi-season) models, a closure-unit is a site-season combination (or
species-site-season in a multi-species dynamic model).
* **rep-constant**, **rep-varying**: We refer to models that assume constant
detection probabilities across repeat visits within closure-units as
*rep-constant models*, as contrasted with *rep-varying models* that incorporate
event-specific detection covariates. It turns out that rep-constant models
enable a more efficient parametrization of the likelihood than rep-varying models.
* **unit covariates**, **event covariates**: We refer to any covariate that does
not vary across sampling events within closure-units as a "unit covariate".
This includes covariates that are intrinsically properties of single
closure-units (e.g. the elevations of sites in a single-species model),
covariates that are intrinsically properties of groups of closure units (e.g.
elevations of sites in a multi-species model), and covariates that are
intrinsically properties of sampling events but happen to be constant within
all closure-units (e.g. observer in a sampling design where every site is
visited by exactly one observer). We refer to any covariate that varies across
sampling events within covariates as an "event covariate". Note that while unit
covariates may appear in either the occupancy or the detection formula, event
covariates are restricted to the detection formula. Models that incorporate
event covariates are *rep-varying* (see above); those that do not are
*rep-constant*.
## Installation and feedback
[Installation instructions are available here](https://jsocolar.github.io/flocker/).
To request features or report bugs (much appreciated!), please [open an issue on GitHub](https://github.com/jsocolar/flocker/issues).
To make `flocker` and `brms` functions globally available within an R session
run:
```r
library(flocker)
library(brms)
set.seed(1)
```
## Data simulation
General purpose data simulation is provided via `simulate_flocker_data()`, which
by default will simulate a dataset with 30 species sampled at 50 sites using
four replicate surveys (i.e. a single-season multi-species dataset). Non-default
arguments will simulate example data for other likelihoods, including
multi-season and data-augmented occupancy models.
```r
d <- simulate_flocker_data()
```
The simulated data `d` are in list form, with elements for the
detection/non-detection observations `d$obs`, unit covariates
`d$unit_covs`, and event covariates `d$event_covs`.
`d$obs` is a matrix where rows are species-site combinations,
columns are replicate visits, and entries are `1` (detection), `0`
(nondetection), or `NA` (no visit). `d$unit_covs` is a dataframe
containing covariates that vary across the rows of obs (i.e. by closure-unit)
but not across the columns within any given row (i.e. do not vary across
replicate visits). `event_covs` is a named list of matrices, with each matrix
having the same dimensions as the observation matrix. Each list element
corresponds to a covariate that varies across the columns of `d$obs` (i.e.
varies between replicate visits).
## Data formatting
`flock()`, the main function in `flocker` for fitting occupancy models,
expects a highly specific data format that we [describe more fully here](https://jsocolar.github.io/flocker/articles/flocker_format.html). The function `make_flocker_data()`
formats data for use with `flock()` automatically. For single-season models,
`make_flocker_data()` takes as input a matrix or dataframe of
detection/non-detection data. Rows represent closure-units, columns represent
repeated sampling events within closure-units, and entries must be `0`
(nondetection), `1` (detection), or `NA` (no corresponding sampling event). The
data must be formatted so that all `NA`s are trailing within their rows. For
example, if some units were sampled four times and other three times, the three
sampling events must be treated as events 1, 2, and 3 (with the fourth event
`NA`) rather than as events 1, 3, and 4 (with the second event `NA`) or any
other combination.
Many occupancy models also include covariates that influence occupancy or
detection probabilities. Unit covariates (see [Terms and definitions]) can
be passed to `make_flocker_data()` as a dataframe with the same number of rows
as the observation matrix and data in the same order as the rows of the
observation matrix. Columns are covariates, and we recommend using informative
column names. *Event covariates* (see [Terms and definitions]) can be
passed as a named list of matrices whose elements `[i, j]` are the covariate
values for the sampling event represented by the corresponding position of the
observation matrix. Again, we recommend using informative names for the list
elements. If the corresponding observation is `NA`, then the value of the event
covariate does not matter.
To pass data to `flocker`, we first pass the output from
`simulate_flocker_data()` to `make_flocker_data()`, which will repackage data
and apply the necessary formatting:
```r
fd_rep_varying <- make_flocker_data(
obs = d$obs,
unit_covs = d$unit_covs,
event_covs = d$event_covs
)
#> Formatting data for a single-season occupancy model. For details, see make_flocker_data_static. All warnings and error messages should be interpreted in the context of make_flocker_data_static
```
The function `make_flocker_data()` outputs an object of class `flocker_data`
that we can pass to flocker's model fitting function `flock()`. Note that this
is the general workflow users will need to follow with real data. Alternative
inputs to `make_flocker_data()` and `flock()` enable the user to readily fit
multi-season models as well as multi-species models with data augmentation
(see below).
## Model fitting
### The single-season rep-varying model
To fit a model, in this case a single-season multi-species occupancy model, we
use the function `flock()`. By supplying different arguments to this function,
all flavors of occupancy model available in `flocker` can be fitted. Formulas
for the different distributional parameters in the model (occupancy, detection,
colonization, extinction, and autologistic terms as applicable) are provided
as one-sided formulas to the relevant arguments of `flock()` (`f_occ`, `f_det`,
`f_col`, `f_ex`, and `f_auto` as applicable).
```r
rep_varying <- flock(
f_occ = ~ uc1 + (1 + uc1 | species),
f_det = ~ uc1 + ec1 + (1 + uc1 + ec1 | species),
flocker_data = fd_rep_varying,
cores = 4,
silent = 2,
refresh = 0
)
```
Arguments supplied to `flock()` define formulas using `brms` syntax for the
occupancy (`f_occ`) and detection (`f_det`) components, and also provide the
formatted data. At this stage, the full flexibility and power of `brms` formula
syntax are available to the user (see following sections for some examples).
`rep_varying` is a `brmsfit` object from package `brms` and also a `flockerfit`
object from package `flocker`. Post-processing functions from `brms` will
typically not work with this object and are instead replaced by `flocker`
equivalents.
### The single-season rep-constant model
`make_flocker_data()` will automatically format the data for a rep-constant
model when `event_covs = NULL` and the desired model is a single-season model
without data augmentation. To take advantage of the efficiency gains and
post-processing functionality of the rep-constant model, it is necessary to
supply `event_covs = NULL` to `make_flocker_data()` at the moment of data
formatting; it is insufficient to omit event covariates from the detection
formula supplied to `flock()` after formatting the data for a rep-varying model.
```r
fd_rep_constant <- make_flocker_data(
obs = d$obs,
unit_covs = d$unit_covs
)
#> Formatting data for a single-season occupancy model. For details, see make_flocker_data_static. All warnings and error messages should be interpreted in the context of make_flocker_data_static
rep_constant <- flock(
f_occ = ~ uc1 + (1 + uc1 | species),
f_det = ~ uc1 + (1 + uc1 | species),
flocker_data = fd_rep_constant,
save_pars = save_pars(all = TRUE), # for loo with moment matching
silent = 2,
refresh = 0,
cores = 4
)
```
Note that within-chain parallelization is available (uniquely so) for the
rep-constant mode:
```r
rep_constant <- flock(
f_occ = ~ uc1 + (1 + uc1 | species),
f_det = ~ uc1 + (1 + uc1 | species),
flocker_data = fd_rep_constant,
silent = 2,
refresh = 0,
chains = 2,
cores = 2,
threads = 2
)
```
### Multi-season models
Here we provide code examples to complement the [companion publication](https://www.biorxiv.org/content/10.1101/2023.10.26.564080v1).
For a more complete vignette on multi-season models in `flocker`, see the [multiseason models vignette](https://jsocolar.github.io/flocker/articles/multiseason_models.html).
First, we simulate some data that are valid for use with multi-season models.
Here, we will simulate data for three seasons with one unit covariate and one
event covariate. The data will be simulated under a colonization-extinction
model with explicit inits, but we will be able to fit other models
(autologistic, equilibrium inits) to the same data (note that
`simulate_flocker_data()` can also simulate directly from these other model
types).
```r
multi_data <- simulate_flocker_data(
n_season = 3,
n_pt = 300,
n_sp = 1,
multiseason = "colex",
multi_init = "explicit",
seed = 1
)
fd_multi <- make_flocker_data(
multi_data$obs,
multi_data$unit_covs,
multi_data$event_covs,
type = "multi",
quiet = TRUE
)
```
Below, we fit the colonization-extinction model with an explicit model for
occupancy in the first timestep. Depending on hardware, fitting this model might
take several minutes.
```r
multi_colex <- flock(
f_occ = ~ uc1,
f_det = ~ uc1 + ec1,
f_col = ~ uc1,
f_ex = ~ uc1,
flocker_data = fd_multi,
multiseason = "colex",
multi_init = "explicit",
cores = 4,
silent = 2,
refresh = 0
)
```
Here is the colonization-extinction model using equilibrium occupancy
probabilities in the first timestep:
```r
multi_colex_eq <- flock(
f_det = ~ uc1 + ec1,
f_col = ~ uc1,
f_ex = ~ uc1,
flocker_data = fd_multi,
multiseason = "colex",
multi_init = "equilibrium",
cores = 4,
silent = 2,
refresh = 0
)
```
Here is the autologistic model with explicit occupancy probabilities in the
first timestep. To reflect the stereotypical autologistic model with a constant
logit-scale offset separating colonization and persistence probabilities, we use
the formula `f_auto = ~ 1`, but it is fine to relax this constraint and use,
e.g. `f_auto = ~ uc1`.
```r
multi_auto <- flock(
f_occ = ~ uc1,
f_det = ~ uc1 + ec1,
f_col = ~ uc1,
f_auto = ~ 1,
flocker_data = fd_multi,
multiseason = "autologistic",
multi_init = "explicit",
cores = 4,
silent = 2,
refresh = 0
)
```
And the autologistic model with equilibrium occupancy probabilities in the
first timestep:
```r
multi_auto_eq <- flock(
f_det = ~ uc1 + ec1,
f_col = ~ uc1,
f_auto = ~ 1,
flocker_data = fd_multi,
multiseason = "autologistic",
multi_init = "equilibrium",
cores = 4,
silent = 2,
refresh = 0
)
```
### Data-augmented multi-species models
Here we provide a simple example of code for a data augmented model.
For a more complete unified vignette on data-augmented models in `flocker`, see
the [data-augmented models vignette](https://jsocolar.github.io/flocker/articles/augmented_models.html).
Fitting the data-augmented model in `flocker` requires passing the observed
data as a three-dimensional array with sites along the first dimension, visits
along the second, and species along the third. Additionally, we must supply the
`n_aug` argument to `make_flocker_data()`, specifying how many all-zero
pseudospecies to augment the data with.
```r
augmented_data <- simulate_flocker_data(
augmented = TRUE
)
fd_augmented <- make_flocker_data(
augmented_data$obs, augmented_data$unit_covs, augmented_data$event_covs,
type = "augmented", n_aug = 100,
quiet = TRUE
)
augmented <- flock(
f_occ = ~ (1 | ff_species),
f_det = ~ uc1 + ec1 + (1 + uc1 + ec1 | ff_species),
augmented = TRUE,
flocker_data = fd_augmented,
cores = 4,
silent = 2,
refresh = 0
)
#> 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
#> https://mc-stan.org/misc/warnings.html#bulk-ess
```
Here, the random effect of species is specified using the special grouping
keyword `ff_species` (names beginning with `ff_` are reserved in `flocker` and
are not allowed as names for user-supplied covariates).
## Post-processing
`flocker` provides functions for four main types of bespoke post-processing
for occupancy models. `fitted_flocker()` computes (and optionally summarizes)
posterior distributions of fitted values at the locations of the data
used in model fitting or of new data. `get_Z()` provides the posterior distribution for the
latent occupancy state. `predict_flocker()` provides posterior predictions at
the observed points (e.g. for use in posterior predictive checking) or for new
data. `loo_flocker()` and `loo_compare_flocker()` both provide functionality for
model comparison. See below for details on all four types of post-processing.
Both posterior predictions and model comparison rely on subtle aspects of the
occupancy model likelihood that we explain in more detail [here](https://jsocolar.github.io/likelihoodOccupancy/).
### brms-native post-processing
All post-processing functions from `brms` work on single-season rep-constant
models, but do not work on any other model types. For example:
```r
predictions_rep_constant <- brms::posterior_predict(rep_constant)
loo_rep_constant <- brms::loo(rep_constant, moment_match = TRUE)
brms::conditional_effects(rep_constant)
```
The following functions work on all model types available in `flocker`.
### Fitted values
Fitted values for any of the distributional parameter (one or more of occupancy,
detection, colonization, extinction, autologistic, and/or Omega, the fitted
probability that a given (pseudo)species occurs in the metacommunity) are
available via `fitted_flocker`. For example:
```r
fitted_flocker(rep_constant)
fitted_flocker(rep_varying)
fitted_flocker(multi_colex)
fitted_flocker(augmented)
```
`fitted_flocker` provides a replacement for
`brms::posterior_linpred()`. While the `brms`-native function executes on
any `flocker` model, it returns in an opaque shape related to
[the flocker data format](https://jsocolar.github.io/flocker/articles/flocker_format.html). `fitted_flocker()` returns in the shape of the observations passed
to `make_flocker_data()`, with posterior iterations stacked along its final
dimension.
### The posterior occupancy state
The function `get_Z()` returns the posterior distribution of occupancy probabilities across the closure-units. The shape of the output depends on the class of model, and is an array in the shape of the first visit in `obs` as passed to `make_flocker_data`, with posterior iterations stacked along the final dimension. Thus, for a single-season rep-varying model, the output is a matrix where rows are posterior iterations, columns are closure-units, and values are draws from the posterior distribution of occupancy probabilities:
```r
get_Z(rep_varying)
```
For all model types, `get_Z()` accepts an optional `new_data` argument. Leaving
the default `new_data = NULL` supplies the posterior for the true occupancy state
at the locations of the data used to fit the model. Otherwise, the posterior is
computed over the new data. For single-season models, `new_data` can be supplied
as a dataframe of unit covariate values or as a `flocker_data` object. For
multi-season models, only a `flocker_data` object is allowed. Note that if
predictions are desired at sites without observations, it is acceptable to pass
an array of dummy observations (e.g. all zeros) to `make_flocker_data()` and
then to set `history_condition = FALSE` in the call to `get_Z()`.
`get_Z()` accepts several additional arguments that control the way that posterior is obtained and the values returned. See the [companion paper](https://www.biorxiv.org/content/10.1101/2023.10.26.564080v1) and
`?get_Z` for details.
### Posterior prediction
The function `predict_flocker()` provides posterior predictions. By default,
predictions are provided for the covariate data to which the model were fit, but predictions to new data are also possible via the `new_data` argument. The
output differs by model type. For single-season rep-constant models, the return
is a matrix where rows are iterations, columns are units, and values are the
number of detections. For single-season rep-varying models, the return is an
array whose first dimension is units, second dimension is sampling events,
third dimension is iterations, and values are `1`, `0`, or `NA`, representing
detection, nondetection, and no corresponding sampling event. For example:
```r
predict_flocker(rep_varying)
```
`predict_flocker()` accepts several additional arguments that control the way
that posterior is obtained and the values of returned. See the [companion paper](https://www.biorxiv.org/content/10.1101/2023.10.26.564080v1) and `?predict_flocker` for details.
### Model comparison
The most straightforward way to compare models fit with `flocker` is the
function `loo_compare_flocker()`. This function takes a list of flocker_fit
objects as its argument and returns a model comparison table based on the
difference in the expected log predictive density (elpd) between models. This
table is a `compare.loo` object from `loo::loo_compare()`. The "leave-one-out"
holdouts consist of entire closure-units (single-season models), series
(multi-season models), or species (augmented models), not single sampling events
(see the [companion paper](https://www.biorxiv.org/content/10.1101/2023.10.26.564080v1) and [here](https://jsocolar.github.io/likelihoodOccupancy/) for details of why).
`loo_compare_flocker()` accepts as input a list of `flockerfit` objects
and outputs a model comparison table. For example, we can compare the
rep-constant and rep-varying models that we fit to the same initial data. Recall
that the data were simulated with event-covariate effects on detection, and as
expected the rep-varying model performs best. Note that we ensure that these
comparisons between rep-constant and rep-varying models are valid by omitting
the binomial coefficient when computing the log-likelihood for the rep-constant
model.
```r
loo_compare_flocker(
list(rep_constant, rep_varying)
)
#> Warning: Some Pareto k diagnostic values are too high. See help('pareto-k-diagnostic') for details.
#> Warning: Some Pareto k diagnostic values are too high. See help('pareto-k-diagnostic') for details.
#> elpd_diff se_diff
#> model2 0.0 0.0
#> model1 -171.1 17.2
```
Likewise, we can compare the four flavors of multi-season model that we fit
above. Recall that the data were simulated under colonization-extinction
dynamics (rather than autologistic) and under explicit initial occupancy
probabilities (rather than equilibrium). As expected, the `multi_colex` model
performs best:
```r
loo_compare_flocker(
list(multi_colex, multi_colex_eq, multi_auto, multi_auto_eq)
)
#> elpd_diff se_diff
#> model1 0.0 0.0
#> model3 -7.8 4.1
#> model2 -27.1 7.2
#> model4 -32.9 7.9
```
Flocker also provides the function `loo_flocker()` to return a table of
`elpd_loo`, `p_loo`, and `looic` estimates from `loo::loo()` or `brms::loo()`
(the latter for single-season rep-constant models only).
## `brms` tips and tricks
Mastering advanced occupancy modeling via `flocker` is mostly a matter of
mastering the syntax available in `brms`. Here are some useful pieces of syntax:
### Priors
Priors can be implemented as they would with any `brms` model. Priors can be
specified using `set_prior()`, with priors specified for groups of parameters
(via `class`) or individual parameters (via `coef`). The priors used for a
particular model can be retrieved using `brms::prior_summary()`, and the
names of the parameters and their default priors can be displayed prior to
model fitting using `get_flocker_prior()` which is a drop-in replacement for
`brms::get_prior()`.
```r
get_flocker_prior(
f_occ = ~ uc1 + (1 + uc1 | species),
f_det = ~ uc1 + ec1 + (1 + uc1 + ec1 | species),
flocker_data = fd_rep_varying
)
#> prior class coef group resp dpar nlpar lb ub source
#> (flat) b default
#> (flat) b ec1 (vectorized)
#> (flat) b uc1 (vectorized)
#> lkj(1) cor default
#> lkj(1) cor species (vectorized)
#> student_t(3, 0, 2.5) Intercept default
#> student_t(3, 0, 2.5) sd 0 default
#> student_t(3, 0, 2.5) sd species 0 (vectorized)
#> student_t(3, 0, 2.5) sd ec1 species 0 (vectorized)
#> student_t(3, 0, 2.5) sd Intercept species 0 (vectorized)
#> student_t(3, 0, 2.5) sd uc1 species 0 (vectorized)
#> (flat) b occ default
#> (flat) b uc1 occ (vectorized)
#> (flat) Intercept occ default
#> student_t(3, 0, 2.5) sd occ 0 default
#> student_t(3, 0, 2.5) sd species occ 0 (vectorized)
#> student_t(3, 0, 2.5) sd Intercept species occ 0 (vectorized)
#> student_t(3, 0, 2.5) sd uc1 species occ 0 (vectorized)
brms::prior_summary(rep_varying)
#> prior class coef group resp dpar nlpar lb ub source
#> (flat) b default
#> (flat) b ec1 (vectorized)
#> (flat) b uc1 (vectorized)
#> (flat) b occ default
#> (flat) b uc1 occ (vectorized)
#> student_t(3, 0, 2.5) Intercept default
#> (flat) Intercept occ default
#> lkj_corr_cholesky(1) L default
#> lkj_corr_cholesky(1) L species (vectorized)
#> student_t(3, 0, 2.5) sd 0 default
#> student_t(3, 0, 2.5) sd occ 0 default
#> student_t(3, 0, 2.5) sd species 0 (vectorized)
#> student_t(3, 0, 2.5) sd ec1 species 0 (vectorized)
#> student_t(3, 0, 2.5) sd Intercept species 0 (vectorized)
#> student_t(3, 0, 2.5) sd uc1 species 0 (vectorized)
#> student_t(3, 0, 2.5) sd species occ 0 (vectorized)
#> student_t(3, 0, 2.5) sd Intercept species occ 0 (vectorized)
#> student_t(3, 0, 2.5) sd uc1 species occ 0 (vectorized)
```
Note that in examples like the above, with covariates shared between both the occupancy and detection model formulas (`uc1` in this example), then the prior table will contain two entries
associated with the covariate, one for the parameter governing occupancy and
one for the parameter governing detection. Specifying priors for parameters in formulas
other than detection can be done with reference to the `dpar` column, e.g.:
```r
user_prior <- c(brms::set_prior("normal(0, 1)", coef = "uc1"),
brms::set_prior("normal(0, 3)", coef = "uc1", dpar = "occ"))
```
where the `uc1` parameter in the occupancy component is specified by the
addition of the `dpar` argument, and the `uc1` parameter in the detection
component is specified without reference to `dpar`.
For more on priors in `brms`, see `?brms::set_prior`.
Users should understand the implications of the default `brms` behavior to
internally center the design matrix, which affects how the prior on the intercept
gets set (see `?brms::set_prior`). Here is an example, based on a
single-season rep-varying model, wherein we set a logistic prior on the value of
the intercepts (flat on the probability scale) when all predictors are held at
their means and a moderately regularizing prior on the coefficients:
```r
rep_varying_prior1 <- flock(
f_occ = ~ uc1,
f_det = ~ ec1,
flocker_data = fd_rep_varying,
prior =
brms::set_prior("logistic(0,1)", class = "Intercept") +
brms::set_prior("logistic(0,1)", class = "Intercept", dpar = "occ") +
brms::set_prior("normal(0,2)", class = "b") +
brms::set_prior("normal(0,2)", class = "b", dpar = "occ"),
cores = 4,
silent = 2,
refresh = 0
)
brms::prior_summary(rep_varying_prior1)
#> prior class coef group resp dpar nlpar lb ub source
#> normal(0,2) b user
#> normal(0,2) b ec1 (vectorized)
#> normal(0,2) b occ user
#> normal(0,2) b uc1 occ (vectorized)
#> logistic(0,1) Intercept user
#> logistic(0,1) Intercept occ user
```
Here is an example where we set informative priors on the intercepts when all covariates are fixed to zero and the same moderately regularizing prior on the coefficients:
```r
rep_varying_prior2 <- flock(
f_occ = ~ 0 + Intercept + uc1,
f_det = ~ 0 + Intercept + ec1,
flocker_data = fd_rep_varying,
prior =
brms::set_prior("normal(0,2)", class = "b") +
brms::set_prior("normal(0,2)", class = "b", dpar = "occ") +
brms::set_prior("normal(1, 1)", class = "b", coef = "Intercept") +
brms::set_prior("normal(-1, 1)", class = "b", coef = "Intercept", dpar = "occ"),
cores = 4,
silent = 2,
refresh = 0
)
brms::prior_summary(rep_varying_prior2)
#> prior class coef group resp dpar nlpar lb ub source
#> normal(0,2) b user
#> normal(0,2) b ec1 (vectorized)
#> normal(1, 1) b Intercept user
#> normal(0,2) b occ user
#> normal(-1, 1) b Intercept occ user
#> normal(0,2) b uc1 occ (vectorized)
```
### Model formulas
Simple formulas follow the same syntax as R's `lm()` function. For example:
```r
mod1 <- flock(
f_occ = ~ uc1 + (1|species),
f_det = ~ 1,
flocker_data = fd_rep_constant
)
```
### Random effects
Simple random effects follow `lme4` syntax, including advanced `lme4` syntax
like `||` for uncorrelated effects and `/` and `:` for expansion of multiple
grouping terms. Here's a simple example:
```r
mod2 <- flock(
f_occ = ~ uc1 + (1|species),
f_det = ~ 1,
flocker_data = fd_rep_constant
)
```
When a model includes multiple random effects with the same grouping term, by
default they are modeled as correlated *within* the occupancy or detection
formulas, but as uncorrelated *between* formulas. For example, the code below
estimates a single correlation for the intercept and slope in the occupancy
sub-model.
```r
mod3 <- flock(
f_occ = ~ uc1 + (1 + uc1 | species),
f_det = ~ ec1 + (1 | species),
flocker_data = fd_rep_varying
)
```
However, this assumption can easily be relaxed using the `||` syntax from
`brms`. The `` is an arbitrary character string representing a group of
terms to model as correlated. The below code, for example, models correlated
intercepts in the occupancy and detection sub-models, and correlated effects of
`sc1` on occupancy and `vc1` on detection, but no correlations between the
intercepts and the slopes in either sub-model:
```r
mod4 <- flock(
f_occ = ~ uc1 + (1 |g1| species) + (0 + uc1 |g2| species),
f_det = ~ ec1 + (1 |g1| species) + (0 + ec1 |g2| species),
flocker_data = fd_rep_varying
)
```
For more on `brms` syntax for random effects syntax, see the [documentation here](https://journal.r-project.org/archive/2018/RJ-2018-017/index.html).
### Nonlinear models
Via `brms`, `flocker` supports Gaussian processes of arbitrary dimensionality
(`brms::gp()`) as well as `mgcv` syntax for thin-plate regression splines
(`brms::s()`) and tensor product smooths (`brms::t2()`), and `brms` syntax for
monotonic effects of ordinal factors via `brms::mo()` ([see here](https://paul-buerkner.github.io/brms/articles/brms_monotonic.html)). For
example:
```r
mod5 <- flock(
f_occ = ~ s(uc1),
f_det = ~ t2(uc1, ec1),
flocker_data = fd_rep_varying
)
mod6 <- flock(
f_occ = ~ 1,
f_det = ~ gp(uc1, ec1),
flocker_data = fd_rep_varying
)
```
In addition, `brms` provides the ability to estimate models wherein the
predictors (e.g. for occupancy and detection) are parametric nonlinear functions
whose parameters have their own covariate-based linear predictors. For more
details and an example, see the [nonlinear models vignette](https://jsocolar.github.io/flocker/articles/nonlinear_models.html).
### Phylogenetic models
Phylogenetic effects can be included by providing a covariance matrix as a
`data2` argument and using the `brms::gr()` function to link species identities
in `flocker_data` with the supplied covariance matrix. Note that phylogenetic
effects can be included in either the occupancy component, the detection
component, or both! In our experience, it can be computationally tractable to
include multiple phylogenetic effects within a single occupancy model (see
[Mills et al. 2022](https://doi.org/10.1002/ecy.3867)).
```r
# simulate an example phylogeny
phylogeny <- ape::rtree(30, tip.label = paste0("sp_", 1:30))
# calculate covariance matrix
A <- ape::vcv.phylo(phylogeny)
mod8 <- flock(
f_occ = ~ 1 + (1|gr(species, cov = A)),
f_det = ~ 1 + ec1 + (1|species),
flocker_data = fd_rep_varying,
data2 = list(A = A)
)
mod9 <- flock(
f_occ = ~ 1 + (1|gr(species, cov = A)),
f_det = ~ 1 + ec1 + (1|gr(species, cov = A)),
flocker_data = fd_rep_varying,
data2 = list(A = A)
)
```
[See here](https://paul-buerkner.github.io/brms/articles/brms_phylogenetics.html) for further details about specifying phylogenetic effects in `brms`.
### Spatial and autoregressive structures
In addition to spatial Gaussian processes, `brms` provides a variety of
autoregressive structures, both one-dimensional (see `brms::ar()`,
`brms::arma()`) and two-dimensional (see `brms::car()`, `brms::sar()`. [See here](https://paul-buerkner.github.io/brms/reference/car.html) for details about conditional autoregressive (CAR) models in `brms`, and note that `flock()`
accepts a `data2` argument that it can pass to `brms` as necessary.
Our principle caution for users is that these autoregressive structures might
lead to degenerate models when applied at the visit level (in detection
formulas) or at the closure-unit level (in occupancy, colonization, extinction,
or autologistic formulas) because observation-level random effects are often
degenerate in regressions with Bernoulli responses. Thus we recommend applying
autoregressive terms to groupings of multiple visits (detection formula) or
multiple closure-units (other formulas). However, we note that `flocker` *does*
provide a well-identified one-dimensional first-order autoregressive structure
for occupancy across closure-units in a single-season model. This is achieved by
co-opting the autologistic parameterization of the multi-season model and
applying it instead to closure-units arranged along a one-dimensional spatial
transect, yielding a one-dimensional analog of a spatial autologistic occupancy
model.
A second caution is to remind users that in multi-species models, users will
likely want to fit separate spatial terms by species ([Doser et al 2022](https://besjournals.onlinelibrary.wiley.com/doi/10.1111/2041-210X.13897)).
For Gaussian processes, this can be achieved via the `by` argument to
`brms::gp()`. For some conditional autoregressive structures (those that allow
disconnected islands), this can be achieved by passing a block-diagonal
adjacency matrix wherein species are disconnected components.
We note that gaussian process priors for spatially varying coefficients are
readily achieved via the nonlinear formula syntax of `brms`, though they may require large volumes of data to successfully fit. For more
details and an example, see the [nonlinear models vignette](https://jsocolar.github.io/flocker/articles/nonlinear_models.html).
### Measurement error in covariates
[See here](http://paul-buerkner.github.io/brms/reference/me.html) for relevant
`brms` documentation.
## Additional fitting arguments
`flock` will pass any relevant parameters forward to `brms::brm()`, giving the
user important control over the algorithmic details of how the model is fit. See
`?brms::brm` for details. To speed up the execution, we recommend supplying the
argument `backend = "cmdstanr"`. This requires the `cmdstanr` package and a
working installation of `cmdstan`; [see here](https://mc-stan.org/cmdstanr/) for
instructions to get started and further details.
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