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In the lattice approach to spatial analysis, calculating p-values is often done based on a conditional permutation approach as outlined by Anselin 1995.

Conditional permutation can be summed up with the question “if I were to hold this observation constant, and change it’s neighbors, will my statistic be the same?”

To understand this a bit better, lets first obtain our neighbors and weights lists, as well as our numeric variable.

library(sfdep)

nb <- st_contiguity(guerry)
wt <- st_weights(nb)
x <- guerry$crime_pers

sfdep exports a function cond_permute_nb() to allow users to create conditional permutations of their nb lists. This works by first identifying the cardinality (number of neighbors) of each observation. Then, for each location from i through n—where i is the position index and n is the number of observations—we sample k values, where k is the number of neighbors for observation i, from the set of values containing 1 through n minus i \(\{1, ..., n\}\setminus i\).

Inference using conditional permutation

With spatial data, we always start from the assumption of spatial randomness. Traditional hypothesis testing like t-test often use an assumption of normality. This assumption, however, is often broken in spatial data. So analytical p-values (p-values that come from a reference distribution such as the normal distribution) are often unreliable, or inaccurate measures of significance. For this reason, in ESDA, we often use simulated, or pseudo, p-values.

Simulated p-values are calculated by creating a reference distribution and comparing our observed statistic to the reference distribution. The reference is created by making M conditional permutations of our data set and calculating a test statistic. Then the ratio of times the simulated statistic is greater than the observed statistic (in either direction) to the number of simulations becomes our simulated p-value.

Worked Example

We can use cond_permute_nb() to create a conditional permutation of the neighbors list. From that permuted neighbor list we can then create a new weights list and calculate the global Geary C for each of the permutation.

 p_nb <- cond_permute_nb(nb)
 p_wt <- st_weights(p_nb)
 observed <- global_c(x, p_nb, p_wt)
 
 observed
#> $C
#> [1] 0.9821733
#> 
#> $K
#> [1] 2.400641

If we did this say, 199 times, we can make a fairly robust reference distribution. We will do this by putting the above code in a replicate() call. replicate() execute some code a number of times as determined by the first argument. This is how I calculate simulated p-values for many measures in sfdep.

reps <- replicate(199, {
  p_nb <- cond_permute_nb(nb)
  p_wt <- st_weights(p_nb)
  global_c(x, p_nb, p_wt)[["C"]]
})

Here we can plot the reference distribution and the observed value over the reference distribution.

library(ggplot2)

ggplot(data.frame(sim_val = reps), 
       aes(sim_val)) +
  geom_histogram(bins = 20) +
  geom_vline(xintercept = observed[["C"]], 
             color = "#6fb381", 
             lty = 2,
             ) +
  theme_light()

We can see that the observed Geary C value is not very extreme and falls some what close to the center of our distribution.

We can now calculate the pseudo p-value using the formula \((M + 1) / (R + 1)\)

# simulated p-value
(sum(observed[["C"]] <= reps) + 1) / (199 + 1)
#> [1] 0.57

This is the approach taken by Pysal and by sfdep where other methods do not apply or are not provided by spdep.