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

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localmoran.sad(model, select, nb, glist=NULL, style="W",
 zero.policy=NULL, alternative="greater", 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=NULL)

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 NULL, 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 Roger.Bivand@nhh.no

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

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require(maptools)
eire <- readShapePoly(system.file("etc/shapes/eire.shp", package="spdep")[1],
  ID="names", proj4string=CRS("+proj=utm +zone=30 +ellps=airy +units=km"))
eire.nb <- poly2nb(eire)
lw <- nb2listw(eire.nb)
#data(eire)
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
e.errorsar <- errorsarlm(OWNCONS ~ ROADACC, data=eire,
  listw=lw)
e.errorsar
lm.target <- lm(e.errorsar$tary ~ e.errorsar$tarX - 1)
Omega <- tcrossprod(invIrW(lw, rho=e.errorsar$lambda))
e.clocmor <- summary(localmoran.sad(lm.target, nb=eire.nb, Omega=Omega))
e.clocmor
hist(e.clocmor[,"Pr. (Sad)"])

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