[R-sig-Geo] Moran's I vs. spatial rho

[Larry Layne]

I am using spdep_0.3-12 to compute Moran's I and spatial rho. To compute 
Moran's I, I am using the function 'moran.test' and the row stochastic 
definition of the connectivity matrix (the W matrix). Here is the full line 
of code for Moran's I:

a <- moran.test(NMmap$att.data$y,y,NMlistw,randomisation=FALSE, 
zero.policy=TRUE,alternative="two.sided",rank=FALSE)

I am using the function lagsarlm to compute spatial rho using an 
intercept-only model: Y = pWY + XB + e, and using the row stochastic 
definition of the connectivity matrix (the W matrix). Here is the full line 
of code for this:

x <- lagsarlm(y ~ 1,data=NMpop,NMlistw,type="lag",method="eigen", 
quiet=FALSE,zero.policy=TRUE)

I am having difficulty figuring out why the Moran's I estimates are very 
different from the spatial rho estimates. Specifically (n is number of 
areal units):

Var  n        Moran's I             spatial rho
A  3109    0.365187132203878    0.573153392466612
B  3109    0.360858943229591    0.562977003813789
C  3109    0.140015456674040    0.291046543475327
D  3109    0.613850771465824    0.797930036214143

A   49     0.261942390553635    0.411076873069647
B   49     0.328416006893752    0.526902446982636
C   49     0.341110258614797    0.535113540423957
D   49     0.239118528840316    0.371896460431941

A    4    -0.412023041115369   -1.08397184420832
B    4    -0.577311366437566   -1.21286625541679
C    4    -0.623070319848968   -1.27502649289533
D    4    -0.63499460958309     0.371896460431941

Any ideas why there would be such a discrepancy between Moran's I values 
and spatial rho values (especially when n=4) when both are using the same 
row stochastic connectivity matrix?

Larry Layne
ljlayne at unm.edu