Bivariate Moran's I

Whether high values of one variable sit next to high values of another (Wartenberg 1985): with zx and zy the centred values, I = n / S0 sum_ij w_ij zx_i zy_j / sqrt(sum zx^2 sum zy^2), which is Moran's I (gmoran()) when y is x (proved in Lean: see proof("bvmor")) and, with row-standardised weights, spdep::moran_bv()'s statistic. Inference is by permutation: y's values are shuffled over the units nsim times from seed while x stays, so under the null its expectation is 0. The statistic mixes the two variables' correlation at a unit with their spatial association; it is not a test of spillover from x to y.

Usage

bvmor(x, y, w, nsim = 999, seed = NULL)

Arguments

x, y

The two variables: finite numbers, one per unit, neither constant.

w

The weights: spwt()'s list, an spdep listw object, or an n x n matrix of weights from 0 (zero diagonal).

nsim

The number of permutations: a whole number from 0.

seed

NULL to draw the simulations' seed from R's generator (so set.seed() fixes them), or one whole number from 0.

Value

A named numeric vector: i, mean and sd (of the permuted statistics), z ((i - mean) / sd), p ((1 + #{I* >= I}) / (nsim + 1)) and ptwo ((1 + min(#{I* >= I}, #{I* <= I})) / (nsim + 1)); NaN without permutations.

References

Wartenberg, D. (1985). Multivariate spatial correlation: a method for exploratory geographical analysis. Geographical Analysis, 17(4), 263-283. doi:10.1111/j.1538-4632.1985.tb00849.x

Examples

set.seed(1)
p <- cbind(x = runif(60), y = runif(60))
a <- p[, 1] + rnorm(60, sd = 0.3)
b <- p[, 1] + rnorm(60, sd = 0.3)
bvmor(a, b, spwt(p, k = 5), nsim = 199, seed = 1)
#>            i         mean           sd            z            p         ptwo 
#> 0.4279403762 0.0002540543 0.0982550720 4.3528167375 0.0050000000 0.0050000000