Moran's I test for residual spatial autocorrelation

Moran's I test for spatial autocorrelation in residuals from an estimated linear model (lm()). The helper function listw2U() constructs a weights list object corresponding to the sparse matrix \(\frac{1}{2} ( \mathbf{W} + \mathbf{W}'\)

lm.morantest(model, listw, zero.policy=NULL, alternative = "greater",
 spChk=NULL, resfun=weighted.residuals, naSubset=TRUE)
listw2U(listw)

Arguments

model

an object of class lm returned by lm; weights may be specified in the lm fit, but offsets should not be used

listw

a listw object created for example by nb2listw

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

naSubset

default TRUE to subset listw object for omitted observations in model object (this is a change from earlier behaviour, when the model$na.action component was ignored, and the listw object had to be subsetted by hand)

Value

A list with class htest containing the following components:

statistic

the value of the standard deviate of Moran's I.

p.value

the p-value of the test.

estimate

the value of the observed Moran's I, its expectation and variance under the method assumption.

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.

References

Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion, p. 203,

Author

Roger Bivand Roger.Bivand@nhh.no

See also

lm.LMtests, lm

Examples

data(oldcol)
oldcrime1.lm <- lm(CRIME ~ 1, data = COL.OLD)
oldcrime.lm <- lm(CRIME ~ HOVAL + INC, data = COL.OLD)
lm.morantest(oldcrime.lm, nb2listw(COL.nb, style="W"))
#> 
#> 	Global Moran I for regression residuals
#> 
#> data:  
#> model: lm(formula = CRIME ~ HOVAL + INC, data = COL.OLD)
#> weights: nb2listw(COL.nb, style = "W")
#> 
#> Moran I statistic standard deviate = 2.9539, p-value = 0.001569
#> alternative hypothesis: greater
#> sample estimates:
#> Observed Moran I      Expectation         Variance 
#>      0.235638354     -0.033302866      0.008289408 
#> 
lm.LMtests(oldcrime.lm, nb2listw(COL.nb, style="W"))
#> 
#> 	Lagrange multiplier diagnostics for spatial dependence
#> 
#> data:  
#> model: lm(formula = CRIME ~ HOVAL + INC, data = COL.OLD)
#> weights: nb2listw(COL.nb, style = "W")
#> 
#> LMErr = 5.7231, df = 1, p-value = 0.01674
#> 
lm.morantest(oldcrime.lm, nb2listw(COL.nb, style="S"))
#> 
#> 	Global Moran I for regression residuals
#> 
#> data:  
#> model: lm(formula = CRIME ~ HOVAL + INC, data = COL.OLD)
#> weights: nb2listw(COL.nb, style = "S")
#> 
#> Moran I statistic standard deviate = 3.1745, p-value = 0.0007504
#> alternative hypothesis: greater
#> sample estimates:
#> Observed Moran I      Expectation         Variance 
#>      0.239317561     -0.033431740      0.007381982 
#> 
lm.morantest(oldcrime1.lm, nb2listw(COL.nb, style="W"))
#> 
#> 	Global Moran I for regression residuals
#> 
#> data:  
#> model: lm(formula = CRIME ~ 1, data = COL.OLD)
#> weights: nb2listw(COL.nb, style = "W")
#> 
#> Moran I statistic standard deviate = 5.6754, p-value = 6.92e-09
#> alternative hypothesis: greater
#> sample estimates:
#> Observed Moran I      Expectation         Variance 
#>      0.510951264     -0.020833333      0.008779831 
#> 
moran.test(COL.OLD$CRIME, nb2listw(COL.nb, style="W"),
 randomisation=FALSE)
#> 
#> 	Moran I test under normality
#> 
#> data:  COL.OLD$CRIME  
#> weights: nb2listw(COL.nb, style = "W")    
#> 
#> Moran I statistic standard deviate = 5.6754, p-value = 6.92e-09
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>       0.510951264      -0.020833333       0.008779831 
#> 
oldcrime.wlm <- lm(CRIME ~ HOVAL + INC, data = COL.OLD,
 weights = I(1/AREA_PL))
lm.morantest(oldcrime.wlm, nb2listw(COL.nb, style="W"),
 resfun=weighted.residuals)
#> 
#> 	Global Moran I for regression residuals
#> 
#> data:  
#> model: lm(formula = CRIME ~ HOVAL + INC, data = COL.OLD, weights =
#> I(1/AREA_PL))
#> weights: nb2listw(COL.nb, style = "W")
#> 
#> Moran I statistic standard deviate = 3.0141, p-value = 0.001289
#> alternative hypothesis: greater
#> sample estimates:
#> Observed Moran I      Expectation         Variance 
#>      0.241298974     -0.032224366      0.008235091 
#> 
lm.morantest(oldcrime.wlm, nb2listw(COL.nb, style="W"),
 resfun=rstudent)
#> 
#> 	Global Moran I for regression residuals
#> 
#> data:  
#> model: lm(formula = CRIME ~ HOVAL + INC, data = COL.OLD, weights =
#> I(1/AREA_PL))
#> weights: nb2listw(COL.nb, style = "W")
#> 
#> Moran I statistic standard deviate = 2.822, p-value = 0.002387
#> alternative hypothesis: greater
#> sample estimates:
#> Observed Moran I      Expectation         Variance 
#>      0.223860298     -0.032224366      0.008235091 
#>