localmoran.sad: Saddlepoint approximation of local Moran's Ii tests

In spdep: Spatial Dependence: Weighting Schemes, Statistics

Saddlepoint approximation of local Moran's Ii tests

Description

The function implements Tiefelsdorf's application of the Saddlepoint approximation to local Moran's Ii's reference distribution. If the model object is of class "lm", global independence is assumed; if of class "sarlm", global dependence is assumed to be represented by the spatial parameter of that model. Tests are reported separately for each zone selected, and may be summarised using summary.localmoransad. Values of local Moran's Ii agree with those from localmoran(), but in that function, the standard deviate - here the Saddlepoint approximation - is based on the randomisation assumption.

Usage

localmoran.sad(model, select, nb, glist=NULL, style="W",
 zero.policy=NULL, alternative="two.sided", spChk=NULL,
 resfun=weighted.residuals, save.Vi=FALSE,
 tol = .Machine$double.eps^0.5, maxiter = 1000, tol.bounds=0.0001,
 save.M=FALSE, Omega = NULL)

## S3 method for class 'localmoransad'
print(x, ...)
## S3 method for class 'localmoransad'
summary(object, ...)
## S3 method for class 'summary.localmoransad'
print(x, ...)
listw2star(listw, ireg, style, n, D, a, zero.policy=attr(listw, "zero.policy"))

Arguments

model	an object of class lm returned by lm (assuming no global spatial autocorrelation), or an object of class sarlm returned by a spatial simultaneous autoregressive model fit (assuming global spatial autocorrelation represented by the model spatial coefficient); weights may be specified in the lm fit, but offsets should not be used
select	an integer vector of the id. numbers of zones to be tested; if missing, all zones
nb	a list of neighbours of class nb
glist	a list of general weights corresponding to neighbours
style	can take values W, B, C, and S
zero.policy	default attr(listw, "zero.policy") as set when listw was created, if attribute not set, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA
alternative	a character string specifying the alternative hypothesis, must be one of greater (default), less or two.sided.
spChk	should the data vector names be checked against the spatial objects for identity integrity, TRUE, or FALSE, default NULL to use get.spChkOption()
resfun	default: weighted.residuals; the function to be used to extract residuals from the lm object, may be residuals, weighted.residuals, rstandard, or rstudent
save.Vi	if TRUE, return the star-shaped weights lists for each zone tested
tol	the desired accuracy (convergence tolerance) for uniroot
maxiter	the maximum number of iterations for uniroot
tol.bounds	offset from bounds for uniroot
save.M	if TRUE, save a list of left and right M products in a list for the conditional tests, or a list of the regression model matrix components
Omega	A SAR process matrix may be passed in to test an alternative hypothesis, for example Omega <- invIrW(listw, rho=0.1); Omega <- tcrossprod(Omega), chol() is taken internally
x	object to be printed
object	object to be summarised
...	arguments to be passed through
listw	a listw object created for example by nb2listw
ireg	a zone number
n	internal value depending on listw and style
D	internal value depending on listw and style
a	internal value depending on listw and style

Details

The function implements the analytical eigenvalue calculation together with trace shortcuts given or suggested in Tiefelsdorf (2002), partly following remarks by J. Keith Ord, and uses the Saddlepoint analytical solution from Tiefelsdorf's SPSS code.

If a histogram of the probability values of the saddlepoint estimate for the assumption of global independence is not approximately flat, the assumption is probably unjustified, and re-estimation with global dependence is recommended.

No n by n matrices are needed at any point for the test assuming no global dependence, the star-shaped weights matrices being handled as listw lists. When the test is made on residuals from a spatial regression, taking a global process into account. n by n matrices are necessary, and memory constraints may be reached for large lattices.

Value

A list with class localmoransad containing "select" lists, each with class moransad with the following components:

statistic	the value of the saddlepoint approximation of the standard deviate of local Moran's Ii.
p.value	the p-value of the test.
estimate	the value of the observed local Moran's Ii.
alternative	a character string describing the alternative hypothesis.
method	a character string giving the method used.
data.name	a character string giving the name(s) of the data.
internal1	Saddlepoint omega, r and u
df	degrees of freedom
tau	maximum and minimum analytical eigenvalues
i	zone tested

Author(s)

Roger Bivand [e-mail]

References

Tiefelsdorf, M. 2002 The Saddlepoint approximation of Moran's I and local Moran's Ii reference distributions and their numerical evaluation. Geographical Analysis, 34, pp. 187–206.

See Also

localmoran, lm.morantest, lm.morantest.sad, errorsarlm

Examples

eire <- st_read(system.file("shapes/eire.gpkg", package="spData")[1])
row.names(eire) <- as.character(eire$names)
eire.nb <- poly2nb(eire)
lw <- nb2listw(eire.nb)
e.lm <- lm(OWNCONS ~ ROADACC, data=eire)
e.locmor <- summary(localmoran.sad(e.lm, nb=eire.nb))
e.locmor
mean(e.locmor[,1])
sum(e.locmor[,1])/Szero(lw)
lm.morantest(e.lm, lw)
# note equality for mean() only when the sum of weights equals
# the number of observations (thanks to Juergen Symanzik)
hist(e.locmor[,"Pr. (Sad)"])
e.wlm <- lm(OWNCONS ~ ROADACC, data=eire, weights=RETSALE)
e.locmorw1 <- summary(localmoran.sad(e.wlm, nb=eire.nb, resfun=weighted.residuals))
e.locmorw1
e.locmorw2 <- summary(localmoran.sad(e.wlm, nb=eire.nb, resfun=rstudent))
e.locmorw2
run <- FALSE
if (requireNamespace("spatialreg", quietly=TRUE)) run <- TRUE
if (run) {
e.errorsar <- spatialreg::errorsarlm(OWNCONS ~ ROADACC, data=eire,
  listw=lw)
if (packageVersion("spatialreg") < "1.1.7")
  spatialreg::print.sarlm(e.errorsar)
else
  print(e.errorsar)
}
if (run) {
lm.target <- lm(e.errorsar$tary ~ e.errorsar$tarX - 1)
Omega <- tcrossprod(spatialreg::invIrW(lw, rho=e.errorsar$lambda))
e.clocmor <- summary(localmoran.sad(lm.target, nb=eire.nb, Omega=Omega))
e.clocmor
}
if (run) {
hist(e.clocmor[,"Pr. (Sad)"])
}

spdep documentation built on Sept. 13, 2024, 5:07 p.m.