# Function for Fitting Latent Factor Multivariate Abundance GLMMs

`lfMsAbund.Rd`

Function for fitting multivariate generalized linear (mixed) models with species correlations (i.e., an abundance-based joint species distribution model). We use a factor modeling approach for dimension reduction.

## Usage

```
lfMsAbund(formula, data, inits, priors, tuning, n.factors,
n.batch, batch.length, accept.rate = 0.43, family = 'Poisson',
n.omp.threads = 1, verbose = TRUE, n.report = 100,
n.burn = round(.10 * n.batch * batch.length), n.thin = 1, n.chains = 1,
save.fitted = TRUE, ...)
```

## Arguments

- formula
a symbolic description of the model to be fit for the model using R's model syntax. Only right-hand side of formula is specified. See example below. Random intercepts and slopes are allowed using lme4 syntax (Bates et al. 2015).

- data
a list containing data necessary for model fitting. Valid tags are

`y`

,`covs`

,`coords`

,`z`

, and`offset`

.`y`

is a two or three-dimensional array of observed count data. The first dimension of the array is equal to the number of species and the second dimension is equal to the number of sites. If specified as a three-dimensional array, the third dimension corresponds to replicate observations at each site (e.g., sub-samples, repeated sampling over multiple seasons).`covs`

is a list or data frame containing the variables used in the model. If a data frame, each row of`covs`

is a site and each column is a variable. If specified as a list, each list element is a different covariate, which can be site-level or observation-level. Site-level covariates are specified as a vector of length \(J\), while observation-level covariates are specified as a matrix or data frame with the number of rows equal to \(J\) and number of columns equal to the maximum number of replicate observations at a given site.`coords`

is a matrix or data frame with two columns that contain the spatial coordinates of each site. Note that`spAbundance`

assumes coordinates are specified in a projected coordinate system. For zi-Gaussian models, the tag`z`

is used to specify the binary component of the zi-Gaussian model and should have the same dimensions as`y`

.`offset`

is an offset to use in the abundance model (e.g., an area offset). This can be either a single value, a vector with an offset for each site (e.g., if survey area differed in size), or a site x replicate matrix if more than one count is available at a given site.- inits
a list with each tag corresponding to a parameter name. Valid tags are

`beta.comm`

,`beta`

,`tau.sq.beta`

,`sigma.sq.mu`

,`kappa`

,`lambda`

,`w`

,`tau.sq`

.`kappa`

is only specified if`family = 'NB'`

,`tau.sq`

is only specified for Gaussian and zi-Gaussian models, and`sigma.sq.mu`

is only specified if random effects are included in`formula`

. The value portion of each tag is the parameter's initial value. See`priors`

description for definition of each parameter name. Additionally, the tag`fix`

can be set to`TRUE`

to fix the starting values across all chains. If`fix`

is not specified (the default), starting values are varied randomly across chains.- priors
a list with each tag corresponding to a parameter name. Valid tags are

`beta.comm.normal`

,`tau.sq.beta.ig`

,`sigma.sq.mu`

,`kappa.unif`

,`tau.sq.ig`

. Community-level (`beta.comm`

) regression coefficients are assumed to follow a normal distribution. The hyperparameters of the normal distribution are passed as a list of length two with the first and second elements corresponding to the mean and variance of the normal distribution, which are each specified as vectors of length equal to the number of coefficients to be estimated or of length one if priors are the same for all coefficients. If not specified, prior means are set to 0 and prior variances to 100. Community-level variance parameters (`tau.sq.beta`

) are assumed to follow an inverse Gamma distribution. The hyperparameters of the inverse gamma distribution are passed as a list of length two with the first and second elements corresponding to the shape and scale parameters, which are each specified as vectors of length equal to the number of coefficients to be estimated or a single value if priors are the same for all parameters. If not specified, prior shape and scale parameters are set to 0.1.`sigma.sq.mu`

are the random effect variances random effects, respectively, and are assumed to follow an inverse Gamma distribution. The hyperparameters of the inverse-Gamma distribution are passed as a list of length two with first and second elements corresponding to the shape and scale parameters, respectively, which are each specified as vectors of length equal to the number of random intercepts or of length one if priors are the same for all random effect variances.`kappa`

is the negative binomial dispersion parameter for each species and is assumed to follow a uniform distribution. The hyperparameters of the uniform distribution are passed as a list of length two with first and second elements corresponding to the lower and upper bounds of the uniform distribution, respectively, which are each specified as vectors of length equal to the number of species or of length one if priors are the same for all species-specific dispersion parameters.`tau.sq`

is the species-specific residual variance for Gaussian (or zi-Gaussian) models, and it is assigned an inverse-Gamma prior. The hyperparameters of the inverse-Gamma are passed as a list of length two, with the first and second element corresponding to the shape and scale parameters, respectively, which are each specified as vectors of length equal to the number of species or a single value if priors are the same for all species.- tuning
a list with each tag corresponding to a parameter name, whose whose value defines the initial variance of the adaptive sampler. Valid tags are

`beta`

,`beta.star`

(the abundance random effect values),`kappa`

,`lambda`

(the latent factor loadings), and`w`

(the latent factors). See Roberts and Rosenthal (2009) for details. Note that no tuning is necessary for Gaussian or zi-Gaussian models.- n.factors
the number of factors to use in the latent factor model approach. Typically, the number of factors is set to be small (e.g., 4-5) relative to the total number of species in the community, which will lead to substantial decreases in computation time. However, the value can be anywhere between 1 and N (the number of species in the community).

- n.batch
the number of MCMC batches in each chain to run for the adaptive MCMC sampler. See Roberts and Rosenthal (2009) for details.

- batch.length
the length of each MCMC batch to run for the adaptive MCMC sampler. See Roberts and Rosenthal (2009) for details.

- accept.rate
target acceptance rate for adaptive MCMC. Defaul is 0.43. See Roberts and Rosenthal (2009) for details.

- family
the distribution to use for the latent abundance process. Currently supports

`'NB'`

(negative binomial),`'Poisson'`

,`'Gaussian'`

, and`'zi-Gaussian'`

.- n.omp.threads
a positive integer indicating the number of threads to use for SMP parallel processing. The package must be compiled for OpenMP support. For most Intel-based machines, we recommend setting

`n.omp.threads`

up to the number of hyperthreaded cores. Note,`n.omp.threads`

> 1 might not work on some systems.- verbose
if

`TRUE`

, messages about data preparation, model specification, and progress of the sampler are printed to the screen. Otherwise, no messages are printed.- n.report
the interval to report Metropolis sampler acceptance and MCMC progress. Note this is specified in terms of batches and not overall samples for spatial models.

- n.burn
the number of samples out of the total

`n.samples`

to discard as burn-in for each chain. By default, the first 10% of samples is discarded.- n.thin
the thinning interval for collection of MCMC samples. The thinning occurs after the

`n.burn`

samples are discarded. Default value is set to 1.- n.chains
the number of chains to run in sequence.

- save.fitted
logical value indicating whether or not fitted values and likelihood values should be saved in the resulting model object. If

`save.fitted = FALSE`

, the components`y.rep.samples`

,`mu.samples`

, and`like.samples`

will not be included in the model object, and subsequent functions for calculating WAIC, fitted values, and posterior predictive checks will not work, although they all can be calculated manually if desired. Setting`save.fitted = FALSE`

can be useful when working with very large data sets to minimize the amount of RAM needed when fitting and storing the model object in memory.- ...
currently no additional arguments

## References

Roberts, G.O. and Rosenthal J.S. (2009) Examples of adaptive MCMC.
*Journal of Computational and Graphical Statistics*, 18(2):349-367.

Bates, Douglas, Martin Maechler, Ben Bolker, Steve Walker (2015). Fitting Linear Mixed-Effects Models Using lme4. Journal of Statistical Software, 67(1), 1-48. doi:10.18637/jss.v067.i01 .

## Author

Jeffrey W. Doser doserjef@msu.edu,

Andrew O. Finley finleya@msu.edu,

## Value

An object of class `lfMsAbund`

that is a list comprised of:

- beta.comm.samples
a

`coda`

object of posterior samples for the community level regression coefficients.- tau.sq.beta.samples
a

`coda`

object of posterior samples for the abundance community variance parameters.- beta.samples
a

`coda`

object of posterior samples for the species level abundance regression coefficients.- kappa.samples
a

`coda`

object of posterior samples for the species level abundance dispersion parameters. Only included when`family = 'NB'`

.- tau.sq.samples
a

`coda`

object of posterior samples for the Gaussian residual variance parameter. Only included when`family = 'Gaussian'`

or`family = 'zi-Gaussian'`

.- lambda.samples
a

`coda`

object of posterior samples for the latent factor loadings.- w.samples
a three-dimensional array of posterior samples for the latent effects for each latent factor. Array dimensions correspond to MCMC sample, latent factor, then site.

- y.rep.samples
a three or four-dimensional array of posterior samples for the fitted (replicate) values for each species with dimensions corresponding to MCMC sample, species, site, and replicate.

- mu.samples
a three or four-dimensional array of posterior samples for the expected abundance values for each species with dimensions corresponding to MCMC samples, species, site, and replicate.

- sigma.sq.mu.samples
a

`coda`

object of posterior samples for variances of random effects included in the abundance portion of the model. Only included if random effects are specified in`abund.formula`

.- beta.star.samples
a

`coda`

object of posterior samples for the abundance random effects. Only included if random effects are specified in`abund.formula`

.- like.samples
a three-dimensional array of posterior samples for the likelihood value associated with each site and species. Used for calculating WAIC.

- rhat
a list of Gelman-Rubin diagnostic values for some of the model parameters.

- ESS
a list of effective sample sizes for some of the model parameters.

- run.time
MCMC sampler execution time reported using

`proc.time()`

.

The return object will include additional objects used for subsequent prediction and/or model fit evaluation.

## Examples

```
set.seed(408)
J.x <- 8
J.y <- 8
J <- J.x * J.y
n.rep <- sample(3, size = J, replace = TRUE)
n.sp <- 6
# Community-level covariate effects
beta.mean <- c(-2, 0.5)
p.abund <- length(beta.mean)
tau.sq.beta <- c(0.2, 1.2)
# Random effects (two random intercepts)
mu.RE <- list(levels = c(10, 15),
sigma.sq.mu = c(0.43, 0.5))
# Draw species-level effects from community means.
beta <- matrix(NA, nrow = n.sp, ncol = p.abund)
for (i in 1:p.abund) {
beta[, i] <- rnorm(n.sp, beta.mean[i], sqrt(tau.sq.beta[i]))
}
sp <- FALSE
kappa <- runif(n.sp, 0.1, 1)
factor.model <- TRUE
n.factors <- 3
dat <- simMsAbund(J.x = J.x, J.y = J.y, n.rep = n.rep,
n.sp = n.sp, beta = beta, mu.RE = mu.RE,
sp = sp, kappa = kappa, family = 'NB')
y <- dat$y
X <- dat$X
X.re <- dat$X.re
coords <- dat$coords
# Package all data into a list
covs <- list(int = X[, , 1],
abund.cov.1 = X[, , 2],
abund.factor.1 = X.re[, , 1],
abund.factor.2 = X.re[, , 2])
data.list <- list(y = y, covs = covs, coords = coords)
prior.list <- list(beta.comm.normal = list(mean = 0, var = 100),
kappa.unif = list(a = 0, b = 10),
tau.sq.beta.ig = list(a = .1, b = .1))
inits.list <- list(beta.comm = 0, beta = 0, kappa = 0.5,
tau.sq.beta = 1)
tuning.list <- list(kappa = 0.3, beta = 0.1, beta.star = 0.1,
lambda = 0.5, w = 0.5)
# Small
n.batch <- 2
batch.length <- 25
n.burn <- 20
n.thin <- 1
n.chains <- 1
out <- lfMsAbund(formula = ~ abund.cov.1 + (1 | abund.factor.1) +
(1 | abund.factor.2),
data = data.list,
n.batch = n.batch,
inits = inits.list,
priors = prior.list,
tuning = tuning.list,
batch.length = batch.length,
n.factors = n.factors,
n.omp.threads = 3,
verbose = TRUE,
n.report = 1,
n.burn = n.burn,
n.thin = n.thin,
n.chains = n.chains)
#> ----------------------------------------
#> Preparing to run the model
#> ----------------------------------------
#> No prior specified for sigma.sq.mu.ig.
#> Setting prior shape to 0.1 and prior scale to 0.1
#> sigma.sq.mu is not specified in initial values.
#> Setting initial values to random values between 0.05 and 1
#> lambda is not specified in initial values.
#> Setting initial values of the lower triangle to 0
#> w is not specified in initial values.
#> Setting initial value to 0
#> ----------------------------------------
#> Model description
#> ----------------------------------------
#> Latent Factor Multi-species Poisson Abundance
#> model fit with 64 sites and 6 species.
#>
#> Samples per Chain: 50
#> Burn-in: 20
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 30
#>
#> Using 3 latent factors.
#>
#> Source compiled with OpenMP support and model fit using 3 thread(s).
#>
#> Adaptive Metropolis with target acceptance rate: 43.0
#> ----------------------------------------
#> Chain 1
#> ----------------------------------------
#> Sampling ...
#> Batch: 1 of 2, 50.00%
#> Number Parameter Acceptance Tuning
#> 1 beta[1] 68.0 0.10000
#> 2 beta[1] 80.0 0.10202
#> 3 beta[1] 80.0 0.10202
#> 4 beta[1] 76.0 0.10202
#> 5 beta[1] 72.0 0.10202
#> 6 beta[1] 88.0 0.10202
#> -------------------------------------------------
#> Batch: 2 of 2, 100.00%
summary(out)
#>
#> Call:
#> lfMsAbund(formula = ~abund.cov.1 + (1 | abund.factor.1) + (1 |
#> abund.factor.2), data = data.list, inits = inits.list, priors = prior.list,
#> tuning = tuning.list, n.factors = n.factors, n.batch = n.batch,
#> batch.length = batch.length, n.omp.threads = 3, verbose = TRUE,
#> n.report = 1, n.burn = n.burn, n.thin = n.thin, n.chains = n.chains)
#>
#> Samples per Chain: 50
#> Burn-in: 20
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 30
#> Run Time (min): 0.0014
#>
#> ----------------------------------------
#> Community Level
#> ----------------------------------------
#> Abundance Means (log scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept) -0.9062 0.3729 -1.7337 -0.9159 -0.2994 NA 30
#> abund.cov.1 0.1621 0.5262 -0.5723 0.0852 1.1068 NA 30
#>
#> Abundance Variances (log scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept) 0.5043 0.6322 0.0578 0.2145 2.3187 NA 16
#> abund.cov.1 1.6466 1.9955 0.2719 0.8109 7.3439 NA 14
#>
#> Abundance Random Effect Variances (log scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept)-abund.factor.1 0.7308 0.1284 0.5067 0.7199 0.9379 NA 30
#> (Intercept)-abund.factor.2 1.1655 0.2577 0.7931 1.1439 1.7077 NA 30
#>
#> ----------------------------------------
#> Species Level
#> ----------------------------------------
#> Abundance (log scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept)-sp1 -0.9565 0.3015 -1.5302 -0.9764 -0.5807 NA 2
#> (Intercept)-sp2 -0.8741 0.1729 -1.1303 -0.9094 -0.5522 NA 3
#> (Intercept)-sp3 -0.4595 0.0733 -0.5863 -0.4571 -0.2990 NA 26
#> (Intercept)-sp4 -0.7235 0.1304 -0.9658 -0.7043 -0.5386 NA 4
#> (Intercept)-sp5 -1.1951 0.2557 -1.6054 -1.1776 -0.7155 NA 2
#> (Intercept)-sp6 -1.3668 0.2598 -1.7863 -1.4053 -0.9374 NA 2
#> abund.cov.1-sp1 -0.6581 0.2326 -1.0060 -0.6710 -0.3082 NA 2
#> abund.cov.1-sp2 0.8438 0.1065 0.6626 0.8540 1.0114 NA 2
#> abund.cov.1-sp3 -0.6539 0.0596 -0.7415 -0.6607 -0.5528 NA 12
#> abund.cov.1-sp4 -0.5865 0.2868 -1.0500 -0.4911 -0.2699 NA 2
#> abund.cov.1-sp5 0.7035 0.1049 0.5286 0.6748 0.9413 NA 6
#> abund.cov.1-sp6 1.3891 0.1822 1.0229 1.4248 1.6339 NA 2
```