sdmTMB {sdmTMB}

Fit a spatial or spatiotemporal GLMM with TMB

Description

Fit a spatial or spatiotemporal generalized linear mixed effects model (GLMM) with the TMB (Template Model Builder) R package and the SPDE (stochastic partial differential equation) approximation to Gaussian random fields.

Usage

sdmTMB(
  formula,
  data,
  mesh,
  time = NULL,
  family = gaussian(link = "identity"),
  spatial = c("on", "off"),
  spatiotemporal = c("iid", "ar1", "rw", "off"),
  share_range = TRUE,
  time_varying = NULL,
  time_varying_type = c("rw", "rw0", "ar1"),
  spatial_varying = NULL,
  weights = NULL,
  offset = NULL,
  extra_time = NULL,
  reml = FALSE,
  silent = TRUE,
  anisotropy = FALSE,
  control = sdmTMBcontrol(),
  priors = sdmTMBpriors(),
  knots = NULL,
  bayesian = FALSE,
  previous_fit = NULL,
  do_fit = TRUE,
  do_index = FALSE,
  predict_args = NULL,
  index_args = NULL,
  experimental = NULL
)

Arguments

formula

Model formula. IID random intercepts are possible using lme4 syntax, e.g., + (1 | g) where g is a column of class character or factor representing groups. Penalized splines are possible via mgcv with s(). Optionally a list for delta (hurdle) models. See examples and details below.

data

A data frame.

mesh

An object from make_mesh().

time

An optional time column name (as character). Can be left as NULL for a model with only spatial random fields; however, if the data are actually spatiotemporal and you wish to use get_index() or get_cog() downstream, supply the time argument.

family

The family and link. Supports gaussian(), Gamma(), binomial(), poisson(), Beta(), nbinom2(), truncated_nbinom2(), nbinom1(), truncated_nbinom1(), censored_poisson(), gamma_mix(), lognormal_mix(), student(), tweedie(), and gengamma(). Supports the delta/hurdle models: delta_beta(), delta_gamma(), delta_gamma_mix(), delta_lognormal_mix(), delta_lognormal(), and delta_truncated_nbinom2(), For binomial family options, see 'Binomial families' in the Details section below.

spatial

Estimate spatial random fields? Options are 'on' / 'off' or TRUE / FALSE. Optionally, a list for delta models, e.g. list('on', 'off').

spatiotemporal

Estimate the spatiotemporal random fields as 'iid' (independent and identically distributed; default), stationary 'ar1' (first-order autoregressive), a random walk ('rw'), or fixed at 0 'off'. Will be set to 'off' if time = NULL. If a delta model, can be a list. E.g., list('off', 'ar1'). Note that the spatiotemporal standard deviation represents the marginal steady-state standard deviation of the process in the case of the AR1. I.e., it is scaled according to the correlation. See the TMB documentation. If the AR1 correlation coefficient (rho) is estimated close to 1, say > 0.99, then you may wish to switch to the random walk 'rw'. Capitalization is ignored. TRUE gets converted to 'iid' and FALSE gets converted to 'off'.

share_range

Logical: estimate a shared spatial and spatiotemporal range parameter (TRUE, default) or independent range parameters (FALSE). If a delta model, can be a list. E.g., list(TRUE, FALSE).

time_varying

An optional one-sided formula describing covariates that should be modelled as a time-varying process. Set the type of process with time_varying_type. See the help for time_varying_type for warnings about modelling the first time step. Structure shared in delta models.

time_varying_type

Type of time-varying process to apply to time_varying formula. 'rw' indicates a random walk with the first time step estimated independently (included for legacy reasons), 'rw0' indicates a random walk with the first time step estimated with a mean-zero normal prior, 'ar1' indicates a stationary first-order autoregressive process with the first time step estimated with a mean-zero prior. In the case of 'rw', be careful not to include covariates (including the intercept) in both the main and time-varying formula since the first time step is estimated independently. I.e., in this case, at least one should have ~ 0 or ~ -1. Structure shared in delta models.

spatial_varying

An optional one-sided formula of coefficients that should vary in space as random fields. Note that you likely want to include a fixed effect for the same variable to improve interpretability since the random field is assumed to have a mean of 0. If a (scaled) time column is used, it will represent a local-time-trend model. See doi:10.1111/ecog.05176 and the spatial trends vignette. Note this predictor should usually be centered to have mean zero and have a standard deviation of approximately 1. The spatial intercept is controlled by the spatial argument; therefore, include or exclude the spatial intercept by setting spatial = 'on' or 'off'. The only time when it matters whether spatial_varying excludes an intercept is in the case of factor predictors. In this case, if spatial_varying excludes the intercept (~ 0 or ~ -1), you should set spatial = 'off' to match. Structure must be shared in delta models.

weights

A numeric vector representing optional likelihood weights for the conditional model. Implemented as in glmmTMB: weights do not have to sum to one and are not internally modified. Can also be used for trials with the binomial family; the weights argument needs to be a vector and not a name of the variable in the data frame. See the Details section below.

offset

A numeric vector representing the model offset or a character value representing the column name of the offset. In delta/hurdle models, this applies only to the positive component. Usually a log transformed variable.

extra_time

Optional extra time slices (e.g., years) to include for interpolation or forecasting with the predict function. See the Details section below.

reml

Logical: use REML (restricted maximum likelihood) estimation rather than maximum likelihood? Internally, this adds the fixed effects to the list of random effects to integrate over.

silent

Silent or include optimization details? Helpful to set to FALSE for models that take a while to fit.

anisotropy

Logical: allow for anisotropy (spatial correlation that is directionally dependent)? See plot_anisotropy(). Must be shared across delta models.

control

Optimization control options via sdmTMBcontrol().

priors

Optional penalties/priors via sdmTMBpriors(). Must currently be shared across delta models.

knots

Optional named list containing knot values to be used for basis construction of smoothing terms. See mgcv::gam() and mgcv::gamm(). E.g., ⁠s(x, bs = 'cc', k = 4), knots = list(x = c(1, 2, 3, 4))⁠

bayesian

Logical indicating if the model will be passed to tmbstan. If TRUE, Jacobian adjustments are applied to account for parameter transformations when priors are applied.

previous_fit

A previously fitted sdmTMB model to initialize the optimization with. Can greatly speed up fitting. Note that the model must be set up exactly the same way. However, the data and weights arguments can change, which can be useful for cross-validation.

do_fit

Fit the model (TRUE) or return the processed data without fitting (FALSE)?

do_index

Do index standardization calculations while fitting? Saves memory and time when working with large datasets or projection grids since the TMB object doesn't have to be rebuilt with predict.sdmTMB() and get_index(). If TRUE, then predict_args must have a newdata element supplied and area can be supplied to index_args. Most users can ignore this option. The fitted object can be passed directly to get_index().

predict_args

A list of arguments to pass to predict.sdmTMB() if do_index = TRUE. Most users can ignore this option.

index_args

A list of arguments to pass to get_index() if do_index = TRUE. Currently, only area is supported. Bias correction can be done when calling get_index() on the resulting fitted object. Most users can ignore this option.

experimental

A named list for esoteric or in-development options. Here be dragons.

Details

Model description

See the model description vignette or the relevant appendix of the preprint on sdmTMB: doi:10.1101/2022.03.24.485545

Binomial families

Following the structure of stats::glm() and glmmTMB, a binomial family can be specified in one of 4 ways: (1) the response may be a factor (and the model classifies the first level versus all others), (2) the response may be binomial (0/1), (3) the response can be a matrix of form cbind(success, failure), and (4) the response may be the observed proportions, and the 'weights' argument is used to specify the Binomial size (N) parameter (⁠prob ~ ..., weights = N⁠).

Smooth terms

Smooth terms can be included following GAMs (generalized additive models) using + s(x), which implements a smooth from mgcv::s(). sdmTMB uses penalized smooths, constructed via mgcv::smooth2random(). This is a similar approach implemented in gamm4 and brms, among other packages. Within these smooths, the same syntax commonly used in mgcv::s() or mgcv::t2() can be applied, e.g. 2-dimensional smooths may be constructed with + s(x, y) or + t2(x, y); smooths can be specific to various factor levels, + s(x, by = group); the basis function dimensions may be specified, e.g. + s(x, k = 4); and various types of splines may be constructed such as cyclic splines to model seasonality (perhaps with the knots argument also be supplied).

Threshold models

A linear break-point relationship for a covariate can be included via + breakpt(variable) in the formula, where variable is a single covariate corresponding to a column in data. In this case, the relationship is linear up to a point and then constant (hockey-stick shaped).

Similarly, a logistic-function threshold model can be included via + logistic(variable). This option models the relationship as a logistic function of the 50% and 95% values. This is similar to length- or size-based selectivity in fisheries, and is parameterized by the points at which f(x) = 0.5 or 0.95. See the threshold vignette.

Note that only a single threshold covariate can be included and the same covariate is included in both components for the delta families.

Extra time: forecasting or interpolating

Extra time slices (e.g., years) can be included for interpolation or forecasting with the predict function via the extra_time argument. The predict function requires all time slices to be defined when fitting the model to ensure the various time indices are set up correctly. Be careful if including extra time slices that the model remains identifiable. For example, including + as.factor(year) in formula will render a model with no data to inform the expected value in a missing year. sdmTMB() makes no attempt to determine if the model makes sense for forecasting or interpolation. The options time_varying, spatiotemporal = "rw", spatiotemporal = "ar1", or a smoother on the time column provide mechanisms to predict over missing time slices with process error.

extra_time can also be used to fill in missing time steps for the purposes of a random walk or AR(1) process if the gaps between time steps are uneven.

extra_time can include only extra time steps or all time steps including those found in the fitted data. This latter option may be simpler.

Regularization and priors

You can achieve regularization via penalties (priors) on the fixed effect parameters. See sdmTMBpriors(). You can fit the model once without penalties and look at the output of print(your_model) or tidy(your_model) or fit the model with do_fit = FALSE and inspect head(your_model$tmb_data$X_ij[[1]]) if you want to see how the formula is translated to the fixed effect model matrix. Also see the Bayesian vignette.

Delta/hurdle models

Delta models (also known as hurdle models) can be fit as two separate models or at the same time by using an appropriate delta family. E.g.: delta_gamma(), delta_beta(), delta_lognormal(), and delta_truncated_nbinom2(). If fit with a delta family, by default the formula, spatial, and spatiotemporal components are shared. Some elements can be specified independently for the two models using a list format. These include formula, spatial, spatiotemporal, and share_range. The first element of the list is for the binomial component and the second element is for the positive component (e.g., Gamma). Other elements must be shared for now (e.g., spatially varying coefficients, time-varying coefficients). Furthermore, there are currently limitations if specifying two formulas as a list: the two formulas cannot have smoothers, threshold effects, or random intercepts. For now, these must be specified through a single formula that is shared across the two models.

The main advantage of specifying such models using a delta family (compared to fitting two separate models) is (1) coding simplicity and (2) calculation of uncertainty on derived quantities such as an index of abundance with get_index() using the generalized delta method within TMB. Also, selected parameters can be shared across the models.

See the delta-model vignette.

Index standardization

For index standardization, you may wish to include 0 + as.factor(year) (or whatever the time column is called) in the formula. See a basic example of index standardization in the relevant package vignette. You will need to specify the time argument. See get_index().

Value

An object (list) of class sdmTMB. Useful elements include:

• sd_report: output from TMB::sdreport()

• gradients: marginal log likelihood gradients with respect to each fixed effect

• model: output from stats::nlminb()

• data: the fitted data

• mesh: the object that was supplied to the mesh argument

• family: the family object, which includes the inverse link function as family$linkinv()

• tmb_params: The parameters list passed to TMB::MakeADFun()

• tmb_map: The 'map' list passed to TMB::MakeADFun()

• tmb_data: The data list passed to TMB::MakeADFun()

• tmb_obj: The TMB object created by TMB::MakeADFun()

References

Main reference introducing the package to cite when using sdmTMB:

Anderson, S.C., E.J. Ward, P.A. English, L.A.K. Barnett. 2022. sdmTMB: an R package for fast, flexible, and user-friendly generalized linear mixed effects models with spatial and spatiotemporal random fields. bioRxiv 2022.03.24.485545; doi:10.1101/2022.03.24.485545.

Reference for local trends:

Barnett, L.A.K., E.J. Ward, S.C. Anderson. 2021. Improving estimates of species distribution change by incorporating local trends. Ecography. 44(3):427-439. doi:10.1111/ecog.05176.

Further explanation of the model and application to calculating climate velocities:

English, P., E.J. Ward, C.N. Rooper, R.E. Forrest, L.A. Rogers, K.L. Hunter, A.M. Edwards, B.M. Connors, S.C. Anderson. 2021. Contrasting climate velocity impacts in warm and cool locations show that effects of marine warming are worse in already warmer temperate waters. Fish and Fisheries. 23(1) 239-255. doi:10.1111/faf.12613.

Discussion of and illustration of some decision points when fitting these models:

Commander, C.J.C., L.A.K. Barnett, E.J. Ward, S.C. Anderson, T.E. Essington. 2022. The shadow model: how and why small choices in spatially explicit species distribution models affect predictions. PeerJ 10: e12783. doi:10.7717/peerj.12783.

Application and description of threshold/break-point models:

Essington, T.E., S.C. Anderson, L.A.K. Barnett, H.M. Berger, S.A. Siedlecki, E.J. Ward. 2022. Advancing statistical models to reveal the effect of dissolved oxygen on the spatial distribution of marine taxa using thresholds and a physiologically based index. Ecography. 2022: e06249 doi:10.1111/ecog.06249.

Application to fish body condition:

Lindmark, M., S.C. Anderson, M. Gogina, M. Casini. Evaluating drivers of spatiotemporal individual condition of a bottom-associated marine fish. bioRxiv 2022.04.19.488709. doi:10.1101/2022.04.19.488709.

Several sections of the original TMB model code were adapted from the VAST R package:

Thorson, J.T. 2019. Guidance for decisions using the Vector Autoregressive Spatio-Temporal (VAST) package in stock, ecosystem, habitat and climate assessments. Fish. Res. 210:143–161. doi:10.1016/j.fishres.2018.10.013.

Code for the family R-to-TMB implementation, selected parameterizations of the observation likelihoods, general package structure inspiration, and the idea behind the TMB prediction approach were adapted from the glmmTMB R package:

Brooks, M.E., K. Kristensen, K.J. van Benthem, A. Magnusson, C.W. Berg, A. Nielsen, H.J. Skaug, M. Maechler, B.M. Bolker. 2017. glmmTMB Balances Speed and Flexibility Among Packages for Zero-inflated Generalized Linear Mixed Modeling. The R Journal, 9(2):378-400. doi:10.32614/rj-2017-066.

Implementation of geometric anisotropy with the SPDE and use of random field GLMMs for index standardization:

Thorson, J.T., A.O. Shelton, E.J. Ward, H.J. Skaug. 2015. Geostatistical delta-generalized linear mixed models improve precision for estimated abundance indices for West Coast groundfishes. ICES J. Mar. Sci. 72(5): 1297–1310. doi:10.1093/icesjms/fsu243.

Examples

library(sdmTMB)

# Build a mesh to implement the SPDE approach:
mesh <- make_mesh(pcod_2011, c("X", "Y"), cutoff = 20)

# - this example uses a fairly coarse mesh so these examples run quickly
# - 'cutoff' is the minimum distance between mesh vertices in units of the
#   x and y coordinates
# - 'cutoff = 10' might make more sense in applied situations for this dataset
# - or build any mesh in 'fmesher' and pass it to the 'mesh' argument in make_mesh()`
# - the mesh is not needed if you will be turning off all
#   spatial/spatiotemporal random fields

# Quick mesh plot:
plot(mesh)

# Fit a Tweedie spatial random field GLMM with a smoother for depth:
fit <- sdmTMB(
  density ~ s(depth),
  data = pcod_2011, mesh = mesh,
  family = tweedie(link = "log")
)
fit

# Extract coefficients:
tidy(fit, conf.int = TRUE)
tidy(fit, effects = "ran_par", conf.int = TRUE)

# Perform several 'sanity' checks:
sanity(fit)

# Predict on the fitted data; see ?predict.sdmTMB
p <- predict(fit)

# Predict on new data:
p <- predict(fit, newdata = qcs_grid)
head(p)


# Visualize the depth effect with ggeffects:
ggeffects::ggpredict(fit,  "depth [all]") |> plot()

# Visualize depth effect with visreg: (see ?visreg_delta)
visreg::visreg(fit, xvar = "depth") # link space; randomized quantile residuals
visreg::visreg(fit, xvar = "depth", scale = "response")
visreg::visreg(fit, xvar = "depth", scale = "response", gg = TRUE, rug = FALSE)

# Add spatiotemporal random fields:
fit <- sdmTMB(
  density ~ 0 + as.factor(year),
  time = "year", #<
  data = pcod_2011, mesh = mesh,
  family = tweedie(link = "log")
)
fit

# Make the fields AR1:
fit <- sdmTMB(
  density ~ s(depth),
  time = "year",
  spatial = "off",
  spatiotemporal = "ar1", #<
  data = pcod_2011, mesh = mesh,
  family = tweedie(link = "log")
)
fit

# Make the fields a random walk:
fit <- sdmTMB(
  density ~ s(depth),
  time = "year",
  spatial = "off",
  spatiotemporal = "rw", #<
  data = pcod_2011, mesh = mesh,
  family = tweedie(link = "log")
)
fit

# Depth smoothers by year:
fit <- sdmTMB(
  density ~ s(depth, by = as.factor(year)), #<
  time = "year",
  spatial = "off",
  spatiotemporal = "rw",
  data = pcod_2011, mesh = mesh,
  family = tweedie(link = "log")
)
fit

# 2D depth-year smoother:
fit <- sdmTMB(
  density ~ s(depth, year), #<
  spatial = "off",
  data = pcod_2011, mesh = mesh,
  family = tweedie(link = "log")
)
fit

# Turn off spatial random fields:
fit <- sdmTMB(
  present ~ poly(log(depth)),
  spatial = "off", #<
  data = pcod_2011, mesh = mesh,
  family = binomial()
)
fit

# Which, matches glm():
fit_glm <- glm(
  present ~ poly(log(depth)),
  data = pcod_2011,
  family = binomial()
)
summary(fit_glm)
AIC(fit, fit_glm)

# Delta/hurdle binomial-Gamma model:
fit_dg <- sdmTMB(
  density ~ poly(log(depth), 2),
  data = pcod_2011, mesh = mesh,
  spatial = "off",
  family = delta_gamma() #<
)
fit_dg

# Delta model with different formulas and spatial structure:
fit_dg <- sdmTMB(
  list(density ~ depth_scaled, density ~ poly(depth_scaled, 2)), #<
  data = pcod_2011, mesh = mesh,
  spatial = list("off", "on"), #<
  family = delta_gamma()
)
fit_dg

# Delta/hurdle truncated NB2:
pcod_2011$count <- round(pcod_2011$density)
fit_nb2 <- sdmTMB(
  count ~ s(depth),
  data = pcod_2011, mesh = mesh,
  spatial = "off",
  family = delta_truncated_nbinom2() #<
)
fit_nb2

# Regular NB2:
fit_nb2 <- sdmTMB(
  count ~ s(depth),
  data = pcod_2011, mesh = mesh,
  spatial = "off",
  family = nbinom2() #<
)
fit_nb2

# IID random intercepts by year:
pcod_2011$fyear <- as.factor(pcod_2011$year)
fit <- sdmTMB(
  density ~ s(depth) + (1 | fyear), #<
  data = pcod_2011, mesh = mesh,
  family = tweedie(link = "log")
)
fit

# Spatially varying coefficient of year:
pcod_2011$year_scaled <- as.numeric(scale(pcod_2011$year))
fit <- sdmTMB(
  density ~ year_scaled,
  spatial_varying = ~ 0 + year_scaled, #<
  data = pcod_2011, mesh = mesh, family = tweedie(), time = "year"
)
fit

# Time-varying effects of depth and depth squared:
fit <- sdmTMB(
  density ~ 0 + as.factor(year),
  time_varying = ~ 0 + depth_scaled + depth_scaled2, #<
  data = pcod_2011, time = "year", mesh = mesh,
  family = tweedie()
)
print(fit)
# Extract values:
est <- as.list(fit$sd_report, "Estimate")
se <- as.list(fit$sd_report, "Std. Error")
est$b_rw_t[, , 1]
se$b_rw_t[, , 1]

# Linear break-point effect of depth:
fit <- sdmTMB(
  density ~ breakpt(depth_scaled), #<
  data = pcod_2011,
  mesh = mesh,
  family = tweedie()
)
fit