Fit spatial generalized linear models for point-referenced data (i.e., generalized geostatistical models) using a variety of estimation methods, allowing for random effects, anisotropy, partition factors, and big data methods.
Usage
spglm(
formula,
family,
data,
spcov_type,
xcoord,
ycoord,
spcov_initial,
dispersion_initial,
estmethod = "reml",
anisotropy = FALSE,
random,
randcov_initial,
partition_factor,
local,
...
)Arguments
- formula
A two-sided linear formula describing the fixed effect structure of the model, with the response to the left of the
~operator and the terms on the right, separated by+operators.- family
The generalized linear model family describing the distribution of the response variable to be used. Available options
"poisson","nbinomial","binomial","beta","Gamma", and"inverse.gaussian". Can be quoted or unquoted. Note that thefamilyargument only takes a single value, rather than the list structure used by stats::glm. See Details for more.- data
A data frame or
sfobject object that contains the variables infixed,random, andpartition_factoras well as geographical information. If ansfobject is provided withPOINTgeometries, the x-coordinates and y-coordinates are used directly. If ansfobject is provided withPOLYGONgeometries, the x-coordinates and y-coordinates are taken as the centroids of each polygon.- spcov_type
The spatial covariance type. Available options include
"exponential","spherical","gaussian","triangular","circular","cubic","pentaspherical","cosine","wave","jbessel","gravity","rquad","magnetic","matern","cauchy","pexponential", and"none". Parameterizations of each spatial covariance type are available in Details. Multiple spatial covariance types can be provided as a character vector, and thenspglm()is called iteratively for each element and a list is returned for each model fit. The default forspcov_typeis"exponential". Whenspcov_typeis specified, all unknown spatial covariance parameters are estimated.spcov_typeis ignored ifspcov_initialis provided.- xcoord
The name of the column in
datarepresenting the x-coordinate. Can be quoted or unquoted. Not required ifdatais ansfobject.- ycoord
The name of the column in
datarepresenting the y-coordinate. Can be quoted or unquoted. Not required ifdatais ansfobject.- spcov_initial
An object from
spcov_initial()specifying initial and/or known values for the spatial covariance parameters. Multiplespcov_initial()objects can be provided in a list. Thenspglm()is called iteratively for each element and a list is returned for each model fit.- dispersion_initial
An object from
dispersion_initial()specifying initial and/or known values for the dispersion parameter for the"nbinomial","beta","Gamma", and"inverse.gaussian"families.familyis ignored ifdispersion_initialis provided.- estmethod
The estimation method. Available options include
"reml"for restricted maximum likelihood and"ml"for maximum likelihood. The default is"reml".- anisotropy
A logical indicating whether (geometric) anisotropy should be modeled. Not required if
spcov_initialis provided with 1)rotateassumed unknown or assumed known and non-zero or 2)scaleassumed unknown or assumed known and less than one. WhenanisotropyisTRUE, computational times can significantly increase. The default isFALSE.- random
A one-sided linear formula describing the random effect structure of the model. Terms are specified to the right of the
~ operator. Each term has the structurex1 + ... + xn | g1/.../gm, wherex1 + ... + xnspecifies the model for the random effects andg1/.../gmis the grouping structure. Separate terms are separated by+and must generally be wrapped in parentheses. Random intercepts are added to each model implicitly when at least one other variable is defined. If a random intercept is not desired, this must be explicitly defined (e.g.,x1 + ... + xn - 1 | g1/.../gm). If only a random intercept is desired for a grouping structure, the random intercept must be specified as1 | g1/.../gm. Note thatg1/.../gmis shorthand for(1 | g1/.../gm). If only random intercepts are desired and the shorthand notation is used, parentheses can be omitted.- randcov_initial
An optional object specifying initial and/or known values for the random effect variances.
- partition_factor
A one-sided linear formula with a single term specifying the partition factor. The partition factor assumes observations from different levels of the partition factor are uncorrelated.
- local
An optional logical or list controlling the big data approximation. If omitted,
localis set toTRUEorFALSEbased on the sample size (the number of non-missing observations indata) -- if the sample size exceeds 3,000,localis set toTRUE. Otherwise it is set toFALSE. IfFALSE, no big data approximation is implemented. If a list is provided, the following arguments detail the big data approximation:index:The group indexes. Observations in different levels ofindexare assumed to be uncorrelated for the purposes of estimation. Ifindexis not provided, it is determined by specifyingmethodand eithersizeorgroups.method: The big data approximation method used to determineindex. Ignored ifindexis provided. Ifmethod = "random", observations are randomly assigned toindexbased onsize. Ifmethod = "kmeans", observations assigned toindexbased on k-means clustering on the coordinates withgroupsclusters. The default is"kmeans". Note that both methods have a random component, which means that you may get different results from separate model fitting calls. To ensure consistent results, specifyindexor set a seed viabase::set.seed().size: The number of observations in eachindexgroup whenmethodis"random". If the number of observations is not divisible bysize, some levels getsize - 1observations. The default is 100.groups:The number ofindexgroups. Ifmethodis"random",sizeis \(ceiling(n / groups)\), where \(n\) is the sample size. Automatically determined ifsizeis specified. Ifmethodis"kmeans",groupsis the number of clusters.var_adjust:The approach for adjusting the variance-covariance matrix of the fixed effects."none"for no adjustment,"theoretical"for the theoretically-correct adjustment,"pooled"for the pooled adjustment, and"empirical"for the empirical adjustment. The default is"theoretical".parallel: IfTRUE, parallel processing via the parallel package is automatically used. The default isFALSE.ncores: Ifparallel = TRUE, the number of cores to parallelize over. The default is the number of available cores on your machine.
When
localis a list, at least one list element must be provided to initialize default arguments for the other list elements. IflocalisTRUE, defaults forlocalare chosen such thatlocalis transformed intolist(size = 100, method = "kmeans", var_adjust = "theoretical", parallel = FALSE).- ...
Other arguments to
esv()orstats::optim().
Value
A list with many elements that store information about
the fitted model object. If spcov_type or spcov_initial are
length one, the list has class spglm. Many generic functions that
summarize model fit are available for spglm objects, including
AIC, AICc, anova, augment, coef,
cooks.distance, covmatrix, deviance, fitted, formula,
glance, glances, hatvalues, influence,
labels, logLik, loocv, model.frame, model.matrix,
plot, predict, print, pseudoR2, summary,
terms, tidy, update, varcomp, and vcov. If
spcov_type or spcov_initial are length greater than one, the
list has class spglm_list and each element in the list has class
spglm. glances can be used to summarize spglm_list
objects, and the aforementioned spglm generics can be used on each
individual list element (model fit).
Details
The spatial generalized linear model for point-referenced data (i.e., generalized geostatistical model) can be written as \(g(\mu) = \eta = X \beta + \tau + \epsilon\), where \(\mu\) is the expectation of the response (\(y\)) given the random errors, \(g(.)\) is called a link function which links together the \(\mu\) and \(\eta\), \(X\) is the fixed effects design matrix, \(\beta\) are the fixed effects, \(\tau\) is random error that is spatially dependent, and \(\epsilon\) is random error that is spatially independent.
There are six generalized linear model
families available: poisson assumes \(y\) is a Poisson random variable
nbinomial assumes \(y\) is a negative binomial random
variable, binomial assumes \(y\) is a binomial random variable,
beta assumes \(y\) is a beta random variable,
Gamma assumes \(y\) is a gamma random
variable, and inverse.gaussian assumes \(y\) is an inverse Gaussian
random variable.
The supports for \(y\) for each family are given below:
family: support of \(y\)
poisson: \(0 \le y\); \(y\) an integer
nbinomial: \(0 \le y\); \(y\) an integer
binomial: \(0 \le y\); \(y\) an integer
beta: \(0 < y < 1\)
Gamma: \(0 < y\)
inverse.gaussian: \(0 < y\)
The generalized linear model families and the parameterizations of their link functions are given below:
family: link function
poisson: \(g(\mu) = log(\eta)\) (log link)
nbinomial: \(g(\mu) = log(\eta)\) (log link)
binomial: \(g(\mu) = log(\eta / (1 - \eta))\) (logit link)
beta: \(g(\mu) = log(\eta / (1 - \eta))\) (logit link)
Gamma: \(g(\mu) = log(\eta)\) (log link)
inverse.gaussian: \(g(\mu) = log(\eta)\) (log link)
The variance function of an individual \(y\) (given \(\mu\)) for each generalized linear model family is given below:
family: \(Var(y)\)
poisson: \(\mu \phi\)
nbinomial: \(\mu + \mu^2 / \phi\)
binomial: \(n \mu (1 - \mu) \phi\)
beta: \(\mu (1 - \mu) / (1 + \phi)\)
Gamma: \(\mu^2 / \phi\)
inverse.gaussian: \(\mu^2 / \phi\)
The parameter \(\phi\) is a dispersion parameter that influences \(Var(y)\).
For the poisson and binomial families, \(\phi\) is always
one. Note that this inverse Gaussian parameterization is different than a
standard inverse Gaussian parameterization, which has variance \(\mu^3 / \lambda\).
Setting \(\phi = \lambda / \mu\) yields our parameterization, which is
preferred for computational stability. Also note that the dispersion parameter
is often defined in the literature as \(V(\mu) \phi\), where \(V(\mu)\) is the variance
function of the mean. We do not use this parameterization, which is important
to recognize while interpreting dispersion estimates.
For more on generalized linear model constructions, see McCullagh and
Nelder (1989).
Together, \(\tau\) and \(\epsilon\) are modeled using a spatial covariance function, expressed as \(de * R + ie * I\), where \(de\) is the dependent error variance, \(R\) is a correlation matrix that controls the spatial dependence structure among observations, \(ie\) is the independent error variance, and \(I\) is an identity matrix. Recall that \(\tau\) and \(\epsilon\) are modeled on the link scale, not the inverse link (response) scale. Random effects are also modeled on the link scale.
spcov_type Details: Parametric forms for \(R\) are given below, where \(\eta = h / range\)
for \(h\) distance between observations:
exponential: \(exp(- \eta )\)
spherical: \((1 - 1.5\eta + 0.5\eta^3) * I(h <= range)\)
gaussian: \(exp(- \eta^2 )\)
triangular: \((1 - \eta) * I(h <= range)\)
circular: \((1 - (2 / \pi) * (m * sqrt(1 - m^2) + sin^{-1}(m))) * I(h <= range), m = min(\eta, 1)\)
cubic: \((1 - 7\eta^2 + 8.75\eta^3 - 3.5\eta^5 + 0.75\eta^7) * I(h <= range)\)
pentaspherical: \((1 - 1.875\eta + 1.25\eta^3 - 0.375\eta^5) * I(h <= range)\)
cosine: \(cos(\eta)\)
wave: \(sin(\eta) / \eta * I(h > 0) + I(h = 0)\)
jbessel: \(Bj(h * range)\), Bj is Bessel-J function
gravity: \((1 + \eta^2)^{-0.5}\)
rquad: \((1 + \eta^2)^{-1}\)
magnetic: \((1 + \eta^2)^{-1.5}\)
matern: \(2^{1 - extra}/ \Gamma(extra) * \alpha^{extra} * Bk(\alpha, extra)\), \(\alpha = (2extra * \eta)^{0.5}\), Bk is Bessel-K function with order \(1/5 \le extra \le 5\)
cauchy: \((1 + \eta^2)^{-extra}\), \(extra > 0\)
pexponential: \(exp(h^{extra}/range)\), \(0 < extra \le 2\)
none: \(0\)
All spatial covariance functions are valid in one spatial dimension. All
spatial covariance functions except triangular and cosine are
valid in two dimensions.
estmethod Details: The various estimation methods are
reml: Maximize the restricted log-likelihood.ml: Maximize the log-likelihood.
Note that the likelihood being optimized is obtained using the Laplace approximation.
anisotropy Details: By default, all spatial covariance parameters except rotate
and scale as well as all random effect variance parameters
are assumed unknown, requiring estimation. If either rotate or scale
are given initial values other than 0 and 1 (respectively) or are assumed unknown
in spcov_initial(), anisotropy is implicitly set to TRUE.
(Geometric) Anisotropy is modeled by transforming a covariance function that
decays differently in different directions to one that decays equally in all
directions via rotation and scaling of the original coordinates. The rotation is
controlled by the rotate parameter in \([0, \pi]\) radians. The scaling
is controlled by the scale parameter in \([0, 1]\). The anisotropy
correction involves first a rotation of the coordinates clockwise by rotate and then a
scaling of the coordinates' minor axis by the reciprocal of scale. The spatial
covariance is then computed using these transformed coordinates.
random Details: If random effects are used, the model
can be written as \(y = X \beta + Z1u1 + ... Zjuj + \tau + \epsilon\),
where each Z is a random effects design matrix and each u is a random effect.
partition_factor Details: The partition factor can be represented in matrix form as \(P\), where
elements of \(P\) equal one for observations in the same level of the partition
factor and zero otherwise. The covariance matrix involving only the
spatial and random effects components is then multiplied element-wise
(Hadmard product) by \(P\), yielding the final covariance matrix.
local Details: The big data approximation works by sorting observations into different levels
of an index variable. Observations in different levels of the index variable
are assumed to be uncorrelated for the purposes of model fitting. Sparse matrix methods are then implemented
for significant computational gains. Parallelization generally further speeds up
computations when data sizes are larger than a few thousand. Both the "random" and "kmeans" values of method
in local have random components. That means you may get slightly different
results when using the big data approximation and rerunning spglm() with the same code. For consistent results,
either set a seed via base::set.seed() or specify index to local.
Observations with NA response values are removed for model
fitting, but their values can be predicted afterwards by running
predict(object).
Note
This function does not perform any internal scaling. If optimization is not stable due to large extremely large variances, scale relevant variables so they have variance 1 before optimization.
References
McCullagh P. and Nelder, J. A. (1989) Generalized Linear Models. London: Chapman and Hall.
Examples
spgmod <- spglm(presence ~ elev,
family = "binomial", data = moose,
spcov_type = "exponential"
)
summary(spgmod)
#>
#> Call:
#> spglm(formula = presence ~ elev, family = "binomial", data = moose,
#> spcov_type = "exponential")
#>
#> Deviance Residuals:
#> Min 1Q Median 3Q Max
#> -1.5249 -0.8114 0.5600 0.8306 1.5757
#>
#> Coefficients (fixed):
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) -0.874038 1.140953 -0.766 0.444
#> elev 0.002365 0.003184 0.743 0.458
#>
#> Pseudo R-squared: 0.00311
#>
#> Coefficients (exponential spatial covariance):
#> de ie range
#> 3.746e+00 4.392e-03 3.203e+04
#>
#> Coefficients (Dispersion for binomial family):
#> dispersion
#> 1