Fit a spatial or spatiotemporal GLMM with TMB

Fit a spatial or spatiotemporal Gaussian random field generalized linear mixed effects model (GLMM) with the TMB (Template Model Builder) R package and the SPDE (stochastic partial differential equation) approach. This can be useful for (dynamic) species distribution models and relative abundance index standardization among many other uses.

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(), and tweedie(). 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 be centered to have mean zero and have a standard deviation of approximately 1 and should likely also be included as a main effect. Structure must currently 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.
predict_args: A list of arguments to pass to predict.sdmTMB() if do_index = TRUE.
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.
experimental: A named list for esoteric or in-development options. Here be dragons.

Value

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

sd_report: output from TMB::sdreport()
gradients: 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()

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, + s(month, bs = "cc", k = 12) (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 their inclusion makes the gaps between time steps even.

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().

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, parameters can be shared across the models.

See the delta-model vignette.

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. 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. In press at Fish and Fisheries. doi:10.1111/faf.12613 .

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

Commander, C.J.C., Barnett, L.A.K., Ward, E.J., Anderson, S.C., and Essington, T.E. 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. Advancing statistical models to reveal the effect of dissolved oxygen on the spatial distribution of marine taxa using thresholds and a physiologically based index. In press at Ecography. 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 .

A number of 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:

Mollie E. Brooks, Kasper Kristensen, Koen J. van Benthem, Arni Magnusson, Casper W. Berg, Anders Nielsen, Hans J. Skaug, Martin Maechler and Benjamin 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., Shelton, A.O., Ward, E.J., and Skaug, H.J. 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 an SPDE mesh with INLA:
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` or `cutoff = 15` might make more sense in applied situations
#   for this dataset
# * or build any mesh in INLA and pass it to the `mesh` argument in `make_mesh()`
# * 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
#> Spatial model fit by ML ['sdmTMB']
#> Formula: density ~ s(depth)
#> Mesh: mesh (isotropic covariance)
#> Data: pcod_2011
#> Family: tweedie(link = 'log')
#>  
#>             coef.est coef.se
#> (Intercept)     2.16    0.34
#> sdepth        -20.26   32.62
#> 
#> Smooth terms:
#>            Std. Dev.
#> sds(depth)     13.07
#> 
#> Dispersion parameter: 13.68
#> Tweedie p: 1.58
#> Matérn range: 16.84
#> Spatial SD: 2.20
#> ML criterion at convergence: 2937.789
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.

# Extract coefficients:
tidy(fit, conf.int = TRUE)
#> # A tibble: 1 × 5
#>   term        estimate std.error conf.low conf.high
#>   <chr>          <dbl>     <dbl>    <dbl>     <dbl>
#> 1 (Intercept)     2.16     0.340     1.50      2.83
tidy(fit, effects = "ran_par", conf.int = TRUE)
#> Standard errors intentionally omitted because they have been calculated in log
#> space.
#> # A tibble: 4 × 5
#>   term      estimate std.error conf.low conf.high
#>   <chr>        <dbl> <lgl>        <dbl>     <dbl>
#> 1 range        16.8  NA           3.40      83.3 
#> 2 phi          13.7  NA          12.4       15.0 
#> 3 sigma_O       2.20 NA           0.735      6.59
#> 4 tweedie_p     1.58 NA           1.55       1.61

# Perform several 'sanity' checks:
sanity(fit)
#> ✔ Non-linear minimizer suggests successful convergence
#> ✔ Hessian matrix is positive definite
#> ✔ No extreme or very small eigenvalues detected
#> ✔ No gradients with respect to fixed effects are >= 0.001
#> ✔ No fixed-effect standard errors are NA
#> ✔ No standard errors look unreasonably large
#> ✔ No sigma parameters are < 0.01
#> ✔ No sigma parameters are > 100
#> ✔ Range parameter doesn't look unreasonably large

# Visualize depth effect: (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)


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

# Predict on new data:
p <- predict(fit, newdata = qcs_grid)
head(p)
#>     X    Y    depth depth_scaled depth_scaled2       est est_non_rf      est_rf
#> 1 456 5636 347.0834    1.5608122    2.43613479 -4.726638  -4.567385 -0.15925308
#> 2 458 5636 223.3348    0.5697699    0.32463771  2.342470   2.368314 -0.02584421
#> 3 460 5636 203.7408    0.3633693    0.13203724  3.087513   2.979948  0.10756466
#> 4 462 5636 183.2987    0.1257046    0.01580166  3.878560   3.637586  0.24097353
#> 5 464 5636 182.9998    0.1220368    0.01489297  4.020914   3.646532  0.37438240
#> 6 466 5636 186.3892    0.1632882    0.02666303  4.050895   3.543104  0.50779127
#>       omega_s
#> 1 -0.15925308
#> 2 -0.02584421
#> 3  0.10756466
#> 4  0.24097353
#> 5  0.37438240
#> 6  0.50779127

# Add spatiotemporal random fields:
fit <- sdmTMB(
  density ~ 0 + as.factor(year),
  time = "year", #<
  data = pcod_2011, mesh = mesh,
  family = tweedie(link = "log")
)
fit
#> Spatiotemporal model fit by ML ['sdmTMB']
#> Formula: density ~ 0 + as.factor(year)
#> Mesh: mesh (isotropic covariance)
#> Time column: year
#> Data: pcod_2011
#> Family: tweedie(link = 'log')
#>  
#>                     coef.est coef.se
#> as.factor(year)2011     2.76    0.36
#> as.factor(year)2013     3.10    0.35
#> as.factor(year)2015     3.21    0.35
#> as.factor(year)2017     2.47    0.36
#> 
#> Dispersion parameter: 14.83
#> Tweedie p: 1.57
#> Matérn range: 13.31
#> Spatial SD: 3.16
#> Spatiotemporal SD: 1.79
#> ML criterion at convergence: 3007.552
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.

# 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
#> Spatiotemporal model fit by ML ['sdmTMB']
#> Formula: density ~ s(depth)
#> Mesh: mesh (isotropic covariance)
#> Time column: year
#> Data: pcod_2011
#> Family: tweedie(link = 'log')
#>  
#>             coef.est coef.se
#> (Intercept)     1.84    0.33
#> sdepth        -20.49   34.07
#> 
#> Smooth terms:
#>            Std. Dev.
#> sds(depth)      13.6
#> 
#> Dispersion parameter: 12.84
#> Tweedie p: 1.55
#> Spatiotemporal AR1 correlation (rho): 0.67
#> Matérn range: 12.22
#> Spatiotemporal SD: 3.28
#> ML criterion at convergence: 2914.393
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.

# 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
#> Spatiotemporal model fit by ML ['sdmTMB']
#> Formula: density ~ s(depth)
#> Mesh: mesh (isotropic covariance)
#> Time column: year
#> Data: pcod_2011
#> Family: tweedie(link = 'log')
#>  
#>             coef.est coef.se
#> (Intercept)     1.95    0.34
#> sdepth        -20.46   33.21
#> 
#> Smooth terms:
#>            Std. Dev.
#> sds(depth)     13.22
#> 
#> Dispersion parameter: 12.84
#> Tweedie p: 1.56
#> Matérn range: 14.66
#> Spatiotemporal SD: 2.17
#> ML criterion at convergence: 2919.181
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.

# 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
#> Spatiotemporal model fit by ML ['sdmTMB']
#> Formula: density ~ s(depth, by = as.factor(year))
#> Mesh: mesh (isotropic covariance)
#> Time column: year
#> Data: pcod_2011
#> Family: tweedie(link = 'log')
#>  
#>                             coef.est coef.se
#> (Intercept)                     1.76    0.34
#> sdepth):as.factor(year)2011    -0.71   42.82
#> sdepth):as.factor(year)2013   -48.95   35.00
#> sdepth):as.factor(year)2015   -63.69   64.09
#> sdepth):as.factor(year)2017    21.05   34.33
#> 
#> Smooth terms:
#>                                 Std. Dev.
#> sds(depth):as.factor(year)2011)     16.62
#> sds(depth):as.factor(year)2013)     13.57
#> sds(depth):as.factor(year)2015)     28.24
#> sds(depth):as.factor(year)2017)     18.65
#> 
#> Dispersion parameter: 12.70
#> Tweedie p: 1.55
#> Matérn range: 8.62
#> Spatiotemporal SD: 3.14
#> ML criterion at convergence: 2924.193
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.

# 2D depth-year smoother:
fit <- sdmTMB(
  density ~ s(depth, year), #<
  spatial = "off",
  data = pcod_2011, mesh = mesh,
  family = tweedie(link = "log")
)
fit
#> Model fit by ML ['sdmTMB']
#> Formula: density ~ s(depth, year)
#> Mesh: mesh (isotropic covariance)
#> Data: pcod_2011
#> Family: tweedie(link = 'log')
#>  
#>              coef.est coef.se
#> (Intercept)      2.55    0.24
#> sdepthyear_1    34.27   24.22
#> sdepthyear_2     1.79    1.07
#> 
#> Smooth terms:
#>                 Std. Dev.
#> sds(depth,year)      6.08
#> 
#> Dispersion parameter: 14.95
#> Tweedie p: 1.60
#> ML criterion at convergence: 2974.143
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.

# Turn off spatial random fields:
fit <- sdmTMB(
  present ~ poly(log(depth)),
  spatial = "off", #<
  data = pcod_2011, mesh = mesh,
  family = binomial()
)
fit
#> Model fit by ML ['sdmTMB']
#> Formula: present ~ poly(log(depth))
#> Mesh: mesh (isotropic covariance)
#> Data: pcod_2011
#> Family: binomial(link = 'logit')
#>  
#>                  coef.est coef.se
#> (Intercept)         -0.16    0.07
#> poly(log(depth))   -13.19    2.14
#> 
#> ML criterion at convergence: 648.334
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.

# Which, matches glm():
fit_glm <- glm(
  present ~ poly(log(depth)),
  data = pcod_2011,
  family = binomial()
)
summary(fit_glm)
#> 
#> Call:
#> glm(formula = present ~ poly(log(depth)), family = binomial(), 
#>     data = pcod_2011)
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -1.6551  -1.0607  -0.8424   1.1975   1.4785  
#> 
#> Coefficients:
#>                   Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)       -0.16433    0.06583  -2.496   0.0126 *  
#> poly(log(depth)) -13.18981    2.14179  -6.158 7.35e-10 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> (Dispersion parameter for binomial family taken to be 1)
#> 
#>     Null deviance: 1337.2  on 968  degrees of freedom
#> Residual deviance: 1296.7  on 967  degrees of freedom
#> AIC: 1300.7
#> 
#> Number of Fisher Scoring iterations: 4
#> 
AIC(fit, fit_glm)
#>         df      AIC
#> fit      2 1300.668
#> fit_glm  2 1300.668

# 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
#> Model fit by ML ['sdmTMB']
#> Formula: density ~ poly(log(depth), 2)
#> Mesh: mesh (isotropic covariance)
#> Data: pcod_2011
#> Family: delta_gamma(link1 = 'logit', link2 = 'log')
#> 
#> Delta/hurdle model 1: -----------------------------------
#> Family: binomial(link = 'logit') 
#>                      coef.est coef.se
#> (Intercept)             -0.48    0.09
#> poly(log(depth), 2)1   -23.06    3.15
#> poly(log(depth), 2)2   -48.79    4.45
#> 
#> 
#> Delta/hurdle model 2: -----------------------------------
#> Family: Gamma(link = 'log') 
#>                      coef.est coef.se
#> (Intercept)              4.24    0.08
#> poly(log(depth), 2)1    -5.49    3.50
#> poly(log(depth), 2)2   -13.26    3.23
#> 
#> Dispersion parameter: 0.64
#> 
#> ML criterion at convergence: 2936.579
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.

# 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
#> Spatial model fit by ML ['sdmTMB']
#> Formula: list(density ~ depth_scaled, density ~ poly(depth_scaled, 2))
#> Mesh: mesh (isotropic covariance)
#> Data: pcod_2011
#> Family: delta_gamma(link1 = 'logit', link2 = 'log')
#> 
#> Delta/hurdle model 1: -----------------------------------
#> Family: binomial(link = 'logit') 
#>              coef.est coef.se
#> (Intercept)     -0.17    0.07
#> depth_scaled    -0.43    0.07
#> 
#> 
#> Delta/hurdle model 2: -----------------------------------
#> Family: Gamma(link = 'log') 
#>                        coef.est coef.se
#> (Intercept)                4.08    0.14
#> poly(depth_scaled, 2)1    -6.15    4.52
#> poly(depth_scaled, 2)2   -12.58    4.14
#> 
#> Dispersion parameter: 0.72
#> Matérn range: 0.01
#> Spatial SD: 2149.18
#> 
#> ML criterion at convergence: 2861.783
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.
#> 
#> **Possible issues detected! Check output of sanity().**

# 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
#> Model fit by ML ['sdmTMB']
#> Formula: count ~ s(depth)
#> Mesh: mesh (isotropic covariance)
#> Data: pcod_2011
#> Family: delta_truncated_nbinom2(link1 = 'logit', link2 = 'log')
#> 
#> Delta/hurdle model 1: -----------------------------------
#> Family: binomial(link = 'logit') 
#>             coef.est coef.se
#> (Intercept)    -0.69    0.21
#> sdepth         -2.98   24.71
#> 
#> Smooth terms:
#>            Std. Dev.
#> sds(depth)       9.6
#> 
#> 
#> Delta/hurdle model 2: -----------------------------------
#> Family: truncated_nbinom2(link = 'log') 
#>             coef.est coef.se
#> (Intercept)     4.18    0.22
#> sdepth          3.83   18.43
#> 
#> Smooth terms:
#>            Std. Dev.
#> sds(depth)      6.73
#> 
#> Dispersion parameter: 0.49
#> 
#> ML criterion at convergence: 2915.733
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.

# Regular NB2:
fit_nb2 <- sdmTMB(
  count ~ s(depth),
  data = pcod_2011, mesh = mesh,
  spatial = "off",
  family = nbinom2() #<
)
fit_nb2
#> Model fit by ML ['sdmTMB']
#> Formula: count ~ s(depth)
#> Mesh: mesh (isotropic covariance)
#> Data: pcod_2011
#> Family: nbinom2(link = 'log')
#>  
#>             coef.est coef.se
#> (Intercept)     2.49    0.27
#> sdepth        -37.71   41.31
#> 
#> Smooth terms:
#>            Std. Dev.
#> sds(depth)      16.4
#> 
#> Dispersion parameter: 0.14
#> ML criterion at convergence: 3006.939
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.

# 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
#> Spatial model fit by ML ['sdmTMB']
#> Formula: density ~ s(depth) + (1 | fyear)
#> Mesh: mesh (isotropic covariance)
#> Data: pcod_2011
#> Family: tweedie(link = 'log')
#>  
#>             coef.est coef.se
#> (Intercept)     2.13    0.37
#> sdepth        -18.95   31.25
#> 
#> Smooth terms:
#>            Std. Dev.
#> sds(depth)     12.51
#> 
#> Random intercepts:
#>       Std. Dev.
#> fyear       0.3
#> 
#> Dispersion parameter: 13.55
#> Tweedie p: 1.58
#> Matérn range: 16.66
#> Spatial SD: 2.21
#> ML criterion at convergence: 2933.138
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.

# 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
#> Spatiotemporal model fit by ML ['sdmTMB']
#> Formula: density ~ year_scaled
#> Mesh: mesh (isotropic covariance)
#> Time column: year
#> Data: pcod_2011
#> Family: tweedie(link = 'log')
#>  
#>             coef.est coef.se
#> (Intercept)     2.86    0.33
#> year_scaled    -0.06    0.15
#> 
#> Dispersion parameter: 14.79
#> Tweedie p: 1.56
#> Matérn range: 20.56
#> Spatial SD: 2.38
#> Spatially varying coefficient SD (year_scaled): 0.81
#> Spatiotemporal SD: 1.11
#> ML criterion at convergence: 3008.886
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.

# 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)
#> Spatiotemporal model fit by ML ['sdmTMB']
#> Formula: density ~ 0 + as.factor(year)
#> Mesh: mesh (isotropic covariance)
#> Time column: year
#> Data: pcod_2011
#> Family: tweedie(link = 'log')
#>  
#>                     coef.est coef.se
#> as.factor(year)2011     3.73    0.30
#> as.factor(year)2013     3.64    0.28
#> as.factor(year)2015     4.00    0.29
#> as.factor(year)2017     3.31    0.32
#> 
#> Time-varying parameters:
#>                    coef.est coef.se
#> depth_scaled-2011     -0.87    0.16
#> depth_scaled-2013     -0.81    0.13
#> depth_scaled-2015     -0.75    0.13
#> depth_scaled-2017     -1.11    0.23
#> depth_scaled2-2011    -1.92    0.29
#> depth_scaled2-2013    -0.92    0.14
#> depth_scaled2-2015    -1.59    0.22
#> depth_scaled2-2017    -2.20    0.35
#> 
#> Dispersion parameter: 12.80
#> Tweedie p: 1.56
#> Matérn range: 0.01
#> Spatial SD: 2573.54
#> Spatiotemporal SD: 2088.43
#> ML criterion at convergence: 2911.371
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.
#> 
#> **Possible issues detected! Check output of sanity().**
# Extract values:
est <- as.list(fit$sd_report, "Estimate")
se <- as.list(fit$sd_report, "Std. Error")
est$b_rw_t[, , 1]
#>            [,1]       [,2]
#> [1,] -0.8738038 -1.9199722
#> [2,] -0.8115162 -0.9195934
#> [3,] -0.7514766 -1.5858307
#> [4,] -1.1064060 -2.1992575
se$b_rw_t[, , 1]
#>           [,1]      [,2]
#> [1,] 0.1619744 0.2944087
#> [2,] 0.1287059 0.1386946
#> [3,] 0.1346182 0.2197914
#> [4,] 0.2305927 0.3514650

# Linear break-point effect of depth:
fit <- sdmTMB(
  density ~ breakpt(depth_scaled), #<
  data = pcod_2011,
  mesh = mesh,
  family = tweedie()
)
fit
#> Spatial model fit by ML ['sdmTMB']
#> Formula: density ~ breakpt(depth_scaled)
#> Mesh: mesh (isotropic covariance)
#> Data: pcod_2011
#> Family: tweedie(link = 'log')
#>  
#>                      coef.est coef.se
#> (Intercept)              4.11    0.33
#> depth_scaled-slope       1.07    0.22
#> depth_scaled-breakpt    -1.30    0.25
#> 
#> Dispersion parameter: 15.19
#> Tweedie p: 1.58
#> Matérn range: 23.57
#> Spatial SD: 1.92
#> ML criterion at convergence: 2997.241
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.