Fit spatial linear models for areal data (i.e., spatial autoregressive models) using a variety of estimation methods, allowing for random effects, partition factors, and row standardization.
Usage
spautor(
formula,
data,
spcov_type,
spcov_initial,
estmethod = "reml",
random,
randcov_initial,
partition_factor,
W,
row_st = TRUE,
M,
range_positive = TRUE,
...
)
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, separated by+
operators, on the right.- data
A data frame or
sf
object that contains the variables infixed
,random
, andpartition_factor
, as well as potentially geographical information.- spcov_type
The spatial covariance type. Available options include
"car"
and"sar"
. Parameterizations of each spatial covariance type are available in Details. Whenspcov_type
is specified, relevant spatial covariance parameters are assumed unknown, requiring estimation.spcov_type
is not required (and is ignored) ifspcov_initial
is provided. Multiple values can be provided in a character vector. Thenspautor()
is called iteratively for each element and a list is returned for each model fit. The default forspcov_type
is"car"
.- spcov_initial
An object from
spcov_initial()
specifying initial and/or known values for the spatial covariance parameters. Not required ifspcov_type
is provided. Multiplespcov_initial()
objects can be provided in a list. Thenspautor()
is called iteratively for each element and a list is returned for each model fit.- estmethod
The estimation method. Available options include
"reml"
for restricted maximum likelihood and"ml"
for maximum likelihood The default is"reml"
.- 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 + ... + xn
specifies the model for the random effects andg1/.../gm
is 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/.../gm
is 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.
- W
Weight matrix specifying the neighboring structure used. Not required if
data
is ansf
polygon object, asW
is calculated internally using queen contiguity. If calculated internally,W
is computed usingsf::st_intersects()
.- row_st
A logical indicating whether row standardization be performed on
W
. The default isTRUE
.- M
M
matrix satisfying the car symmetry condition. The car symmetry condition states that \((I - range * W)^{-1}M\) is symmetric, where \(I\) is an identity matrix, \(range\) is a constant that controls the spatial dependence,W
is the weights matrix, and \(^{-1}\) represents the inverse operator.M
is required for car models whenW
is provided androw_st
isFALSE
. WhenM
, is required, the default is the identity matrix.M
must be diagonal or given as a vector or one-column matrix assumed to be the diagonal.- range_positive
Whether the range should be constrained to be positive. The default is
TRUE
.- ...
Other arguments to
stats::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 spautor
. Many generic functions that
summarize model fit are available for spautor
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 spautor_list
and each element in the list has class
spautor
. glances
can be used to summarize spautor_list
objects, and the aforementioned spautor
generics can be used on each
individual list element (model fit).
Details
The spatial linear model for areal data (i.e., spatial autoregressive model)
can be written as
\(y = X \beta + \tau + \epsilon\), where \(X\) 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. 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 matrix that controls the spatial dependence structure among observations,
\(ie\) is the independent error variance, and \(I\) is
an identity matrix. Note that \(de\) and \(ie\) must be non-negative while \(range\)
must be between the reciprocal of the maximum
eigenvalue of W
and the reciprocal of the minimum eigenvalue of
W
.
spcov_type
Details: Parametric forms for \(R\) are given below:
car: \((I - range * W)^{-1}M\), weights matrix \(W\), symmetry condition matrix \(M\)
sar: \([(I - range * W)(I - range * W)^T]^{-1}\), weights matrix \(W\), \(^T\) indicates matrix transpose
If there are observations with no neighbors, they are given a unique variance
parameter called extra
, which must be non-negative.
estmethod
Details: The various estimation methods are
reml
: Maximize the restricted log-likelihood.ml
: Maximize the log-likelihood.
By default, all spatial covariance parameters except ie
as well as all random effect variance parameters
are assumed unknown, requiring estimation. ie
is assumed zero and known by default
(in contrast to models fit using splm()
, where ie
is assumed
unknown by default). To change this default behavior, specify spcov_initial
(an NA
value for ie
in spcov_initial
to assume
ie
is unknown, requiring estimation).
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.
Observations with NA
response values are removed for model
fitting, but their values can be predicted afterwards by running
predict(object)
. This is the only way to perform prediction for
spautor()
models (i.e., the prediction locations must be known prior
to estimation).
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.
Examples
spmod <- spautor(log_trend ~ 1, data = seal, spcov_type = "car")
summary(spmod)
#>
#> Call:
#> spautor(formula = log_trend ~ 1, data = seal, spcov_type = "car")
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.34455 -0.10417 0.04410 0.07338 0.20475
#>
#> Coefficients (fixed):
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) -0.07090 0.02497 -2.839 0.00452 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Coefficients (car spatial covariance):
#> de range extra
#> 0.03252 0.42037 0.02177