Exact local Moran's Ii tests

localmoran.exact provides exact local Moran's Ii tests under the null hypothesis, while localmoran.exact.alt provides exact local Moran's Ii tests under the alternative hypothesis. In this case, the model may be a fitted model derived from a model fitted by spatialreg::errorsarlm, with the covariance matrix can be passed through the Omega= argument.

Usage

localmoran.exact(model, select, nb, glist = NULL, style = "W", 
 zero.policy = NULL, alternative = "two.sided", spChk = NULL, 
 resfun = weighted.residuals, save.Vi = FALSE, useTP=FALSE, truncErr=1e-6, 
 zeroTreat=0.1)
localmoran.exact.alt(model, select, nb, glist = NULL, style = "W",
 zero.policy = NULL, alternative = "two.sided", spChk = NULL,
 resfun = weighted.residuals, Omega = NULL, save.Vi = FALSE,
 save.M = FALSE, useTP=FALSE, truncErr=1e-6, zeroTreat=0.1)
# S3 method for class 'localmoranex'
print(x, ...)
# S3 method for class 'localmoranex'
as.data.frame(x, row.names=NULL, optional=FALSE, ...)

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
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
save.Vi: if TRUE, return the star-shaped weights lists for each zone tested
save.M: if TRUE, save a list of left and right M products
useTP: default FALSE, if TRUE, use truncation point in integration rather than upper=Inf, see Tiefelsdorf (2000), eq. 6.7, p.69
truncErr: when useTP=TRUE, pass truncation error to truncation point function
zeroTreat: when useTP=TRUE, pass zero adjustment to truncation point function
x: object to be printed
row.names: ignored argument to as.data.frame.localmoranex; row names assigned from localmoranex object
optional: ignored argument to as.data.frame.localmoranex; row names assigned from localmoranex object
...: arguments to be passed through

Value

A list with class localmoranex containing "select" lists, each with class moranex with the following components:

statistic: the value of the exact standard deviate of global Moran's I.
p.value: the p-value of the test.
estimate: the value of the observed local Moran's Ii.
method: a character string giving the method used.
alternative: a character string describing the alternative hypothesis.
gamma: eigenvalues (two extreme values for null, vector for alternative)
oType: usually set to "E", but set to "N" if the integration leads to an out of domain value for qnorm, when the Normal assumption is substituted. This only occurs when the output p-value would be very close to zero
data.name: a character string giving the name(s) of the data.
df: degrees of freedom
i: zone tested
Vi: zone tested

When the alternative is being tested, a list of left and right M products in attribute M.

References

Bivand RS, Müller W, Reder M (2009) Power calculations for global and local Moran’s I. Comput Stat Data Anal 53:2859–2872; Bivand RS, Wong DWS 2018 Comparing implementations of global and local indicators of spatial association. TEST, 27(3), 716–748 doi:10.1007/s11749-018-0599-x

Author

Markus Reder and Roger Bivand

Examples

eire <- st_read(system.file("shapes/eire.gpkg", package="spData")[1])
#> Reading layer `eire' from data source 
#>   `/home/rsb/lib/r_libs/spData/shapes/eire.gpkg' using driver `GPKG'
#> Simple feature collection with 26 features and 10 fields
#> Geometry type: MULTIPOLYGON
#> Dimension:     XY
#> Bounding box:  xmin: -4.12 ymin: 5768 xmax: 300.82 ymax: 6119.25
#> Projected CRS: Undefined Cartesian SRS with unknown unit
row.names(eire) <- as.character(eire$names)
eire.nb <- poly2nb(eire)
e.lm <- lm(OWNCONS ~ ROADACC, data=eire)
localmoran.sad(e.lm, nb=eire.nb)
#>              Local Morans I Saddlepoint    Pr. (Sad)
#> 1 Carlow         0.21699668  0.95074844 3.417321e-01
#> 2 Cavan         -0.37257361 -1.00603119 3.144006e-01
#> 3 Clare          0.23197510  0.67166518 5.017969e-01
#> 4 Cork           0.78193548  1.74761575 8.053059e-02
#> 5 Donegal       -1.69064059 -1.72031078 8.537596e-02
#> 6 Dublin        -0.16069692 -0.35212627 7.247436e-01
#> 7 Galway         1.31371473  2.66849536 7.619183e-03
#> 8 Kerry          0.36534866  0.78073279 4.349597e-01
#> 9 Kildare       -0.02557544  0.04167665 9.667565e-01
#> 10 Kilkenny      0.57684331  1.70897697 8.745521e-02
#> 11 Laoghis      -0.05951798 -0.12155465 9.032517e-01
#> 12 Leitrim       0.38484587  1.47227033 1.409479e-01
#> 13 Limerick      0.11817987  0.45727712 6.474719e-01
#> 14 Longford      1.41643200  2.51113769 1.203427e-02
#> 15 Louth         0.56242920  1.07441571 2.826364e-01
#> 16 Mayo          0.87572704  2.05251226 4.011990e-02
#> 17 Meath         0.00367856  0.12813539 8.980418e-01
#> 18 Monaghan      0.55098311  1.23999193 2.149784e-01
#> 19 Offaly        0.15155556  0.80786519 4.191682e-01
#> 20 Roscommon     2.04368839  4.53187292 5.846302e-06
#> 21 Sligo        -0.47579871 -0.94578114 3.442602e-01
#> 22 Tipperary    -0.03454106 -0.06919691 9.448329e-01
#> 23 Waterford     0.85723423  1.91385108 5.563919e-02
#> 24 Westmeath     0.45138572  1.36017204 1.737755e-01
#> 25 Wexford       0.64371834  1.63188492 1.027037e-01
#> 26 Wicklow       0.02441950  0.21197000 8.321304e-01
localmoran.exact(e.lm, nb=eire.nb)
#>              Local Morans I    Exact SD  Pr. (exact)
#> 1 Carlow         0.21699668  1.02706083 3.043918e-01
#> 2 Cavan         -0.37257361 -1.04864802 2.943401e-01
#> 3 Clare          0.23197510  0.76362894 4.450884e-01
#> 4 Cork           0.78193548  1.77727656 7.552275e-02
#> 5 Donegal       -1.69064059 -1.74428756 8.110896e-02
#> 6 Dublin        -0.16069692 -0.44236119 6.582279e-01
#> 7 Galway         1.31371473  2.69199974 7.102500e-03
#> 8 Kerry          0.36534866  0.85742696 3.912090e-01
#> 9 Kildare       -0.02557544 -0.03475476 9.722753e-01
#> 10 Kilkenny      0.57684331  1.74146177 8.160267e-02
#> 11 Laoghis      -0.05951798 -0.21824035 8.272419e-01
#> 12 Leitrim       0.38484587  1.51434641 1.299380e-01
#> 13 Limerick      0.11817987  0.56922630 5.692026e-01
#> 14 Longford      1.41643200  2.53491837 1.124735e-02
#> 15 Louth         0.56242920  1.12775107 2.594251e-01
#> 16 Mayo          0.87572704  2.08125803 3.741029e-02
#> 17 Meath         0.00367856  0.16372685 8.699462e-01
#> 18 Monaghan      0.55098311  1.28435459 1.990179e-01
#> 19 Offaly        0.15155556  0.89537870 3.705847e-01
#> 20 Roscommon     2.04368839  4.55244870 5.302509e-06
#> 21 Sligo        -0.47579871 -0.98101752 3.265841e-01
#> 22 Tipperary    -0.03454106 -0.16132608 8.718366e-01
#> 23 Waterford     0.85723423  1.94188723 5.215075e-02
#> 24 Westmeath     0.45138572  1.40091422 1.612397e-01
#> 25 Wexford       0.64371834  1.66488051 9.593660e-02
#> 26 Wicklow       0.02441950  0.29717701 7.663314e-01
localmoran.exact(e.lm, nb=eire.nb, useTP=TRUE)
#>              Local Morans I    Exact SD  Pr. (exact)
#> 1 Carlow         0.21699668  1.02706127 3.043916e-01
#> 2 Cavan         -0.37257361 -1.04864834 2.943400e-01
#> 3 Clare          0.23197510  0.76362936 4.450881e-01
#> 4 Cork           0.78193548  1.77727666 7.552273e-02
#> 5 Donegal       -1.69064059 -1.74428741 8.110899e-02
#> 6 Dublin        -0.16069692 -0.44236158 6.582276e-01
#> 7 Galway         1.31371473  2.69199162 7.102673e-03
#> 8 Kerry          0.36534866  0.85742736 3.912087e-01
#> 9 Kildare       -0.02557544 -0.03540278 9.717586e-01
#> 10 Kilkenny      0.57684331  1.74146192 8.160264e-02
#> 11 Laoghis      -0.05951798 -0.21823772 8.272439e-01
#> 12 Leitrim       0.38484587  1.51434680 1.299379e-01
#> 13 Limerick      0.11817987  0.56922674 5.692023e-01
#> 14 Longford      1.41643200  2.53491383 1.124750e-02
#> 15 Louth         0.56242920  1.12775145 2.594249e-01
#> 16 Mayo          0.87572704  2.08125778 3.741032e-02
#> 17 Meath         0.00367856  0.15714487 8.751307e-01
#> 18 Monaghan      0.55098311  1.28435495 1.990178e-01
#> 19 Offaly        0.15155556  0.89537914 3.705844e-01
#> 20 Roscommon     2.04368839  4.53362873 5.797890e-06
#> 21 Sligo        -0.47579871 -0.98101780 3.265840e-01
#> 22 Tipperary    -0.03454106 -0.16116590 8.719627e-01
#> 23 Waterford     0.85723423  1.94188711 5.215077e-02
#> 24 Westmeath     0.45138572  1.40091457 1.612396e-01
#> 25 Wexford       0.64371834  1.66488069 9.593656e-02
#> 26 Wicklow       0.02441950  0.29790569 7.657751e-01
run <- FALSE
if (requireNamespace("spatialreg", quietly=TRUE)) run <- TRUE
if (run) {
e.errorsar <- spatialreg::errorsarlm(OWNCONS ~ ROADACC, data=eire,
 listw=nb2listw(eire.nb))
lm.target <- lm(e.errorsar$tary ~ e.errorsar$tarX - 1)
localmoran.exact.alt(lm.target, nb=eire.nb)
}
#>              Local Morans I    Exact SD Pr. (exact)
#> 1 Carlow         0.17958462  0.83313548 0.404768327
#> 2 Cavan         -0.24752628 -0.82861508 0.407322251
#> 3 Clare         -0.27901334 -0.72313220 0.469598670
#> 4 Cork           0.37808655  1.15083938 0.249798320
#> 5 Donegal       -1.01894688 -1.50937607 0.131202704
#> 6 Dublin        -0.18171297 -0.50995181 0.610085224
#> 7 Galway         1.02193390  2.27605549 0.022842689
#> 8 Kerry         -0.94967914 -1.46954733 0.141684395
#> 9 Kildare        0.07005053  0.56174770 0.574287931
#> 10 Kilkenny      0.43022231  1.39078833 0.164289622
#> 11 Laoghis      -0.12239133 -0.45750046 0.647311369
#> 12 Leitrim      -0.24203970 -0.84925558 0.395739092
#> 13 Limerick     -0.03214546 -0.13098232 0.895789292
#> 14 Longford      0.38307454  1.17909760 0.238359315
#> 15 Louth         0.21301552  0.62068302 0.534808204
#> 16 Mayo          0.93971200  1.94079901 0.052282661
#> 17 Meath         0.12484415  0.78747315 0.431004937
#> 18 Monaghan     -0.16109919 -0.48277561 0.629255068
#> 19 Offaly       -0.00632492  0.03008421 0.975999892
#> 20 Roscommon     1.02089429  2.64333471 0.008209384
#> 21 Sligo        -2.01629233 -2.54211789 0.011018300
#> 22 Tipperary    -0.10810709 -0.51063241 0.609608478
#> 23 Waterford     0.44099279  1.28152280 0.200010096
#> 24 Westmeath    -0.06329661 -0.26575204 0.790430182
#> 25 Wexford       0.30764883  1.02115599 0.307180540
#> 26 Wicklow      -0.01696406 -0.04109885 0.967217095
if (run) {
Omega <- spatialreg::invIrW(nb2listw(eire.nb), rho=e.errorsar$lambda)
Omega1 <- tcrossprod(Omega)
localmoran.exact.alt(lm.target, nb=eire.nb, Omega=Omega1)
}
#>              Local Morans I     Exact SD  Pr. (exact)
#> 1 Carlow         0.17958462  0.009413917 0.9924888919
#> 2 Cavan         -0.24752628 -1.416786314 0.1565454123
#> 3 Clare         -0.27901334 -1.512813055 0.1303271517
#> 4 Cork           0.37808655  0.078686494 0.9372819884
#> 5 Donegal       -1.01894688 -1.935314161 0.0529517556
#> 6 Dublin        -0.18171297 -1.226543250 0.2199943101
#> 7 Galway         1.02193390  1.477322295 0.1395892645
#> 8 Kerry         -0.94967914 -2.427948363 0.0151845050
#> 9 Kildare        0.07005053 -0.042522018 0.9660825597
#> 10 Kilkenny      0.43022231  0.488471215 0.6252161096
#> 11 Laoghis      -0.12239133 -1.091910863 0.2748722827
#> 12 Leitrim      -0.24203970 -1.631825829 0.1027161824
#> 13 Limerick     -0.03214546 -0.921751557 0.3566581813
#> 14 Longford      0.38307454  0.183273193 0.8545836671
#> 15 Louth         0.21301552 -0.184065307 0.8539622227
#> 16 Mayo          0.93971200  1.182398031 0.2370478040
#> 17 Meath         0.12484415 -0.060499926 0.9517574750
#> 18 Monaghan     -0.16109919 -1.342787509 0.1793407892
#> 19 Offaly       -0.00632492 -0.409598788 0.6821002863
#> 20 Roscommon     1.02089429  1.238387417 0.2155724441
#> 21 Sligo        -2.01629233 -3.623597237 0.0002905339
#> 22 Tipperary    -0.10810709 -1.227333133 0.2196974073
#> 23 Waterford     0.44099279  0.433358039 0.6647546796
#> 24 Westmeath    -0.06329661 -0.956519228 0.3388099748
#> 25 Wexford       0.30764883  0.141381248 0.8875687671
#> 26 Wicklow      -0.01696406 -0.783977878 0.4330530911
if (run) {
localmoran.exact.alt(lm.target, nb=eire.nb, Omega=Omega1, useTP=TRUE)
}
#>              Local Morans I     Exact SD  Pr. (exact)
#> 1 Carlow         0.17958462  0.009414171 0.9924886892
#> 2 Cavan         -0.24752628 -1.416786809 0.1565452674
#> 3 Clare         -0.27901334 -1.512813519 0.1303270338
#> 4 Cork           0.37808655  0.078686628 0.9372818822
#> 5 Donegal       -1.01894688 -1.935314436 0.0529517219
#> 6 Dublin        -0.18171297 -1.226543734 0.2199941280
#> 7 Galway         1.02193390  1.477322399 0.1395892367
#> 8 Kerry         -0.94967914 -2.427949220 0.0151844691
#> 9 Kildare        0.07005053 -0.042526751 0.9660787872
#> 10 Kilkenny      0.43022231  0.488471349 0.6252160152
#> 11 Laoghis      -0.12239133 -1.091912026 0.2748717711
#> 12 Leitrim      -0.24203970 -1.631826543 0.1027160320
#> 13 Limerick     -0.03214546 -0.921785589 0.3566404259
#> 14 Longford      0.38307454  0.183273325 0.8545835636
#> 15 Louth         0.21301552 -0.184065128 0.8539623631
#> 16 Mayo          0.93971200  1.182398127 0.2370477661
#> 17 Meath         0.12484415 -0.060499243 0.9517580190
#> 18 Monaghan     -0.16109919 -1.342788191 0.1793405681
#> 19 Offaly       -0.00632492 -0.407816672 0.6834082724
#> 20 Roscommon     1.02089429  1.238387488 0.2155724178
#> 21 Sligo        -2.01629233 -3.623590126 0.0002905419
#> 22 Tipperary    -0.10810709 -1.227335007 0.2196967033
#> 23 Waterford     0.44099279  0.433358168 0.6647545858
#> 24 Westmeath    -0.06329661 -0.956505669 0.3388168214
#> 25 Wexford       0.30764883  0.141381402 0.8875686460
#> 26 Wicklow      -0.01696406 -0.784451761 0.4327750768