# Conditional Permutations with sfdep

Source:`vignettes/conditional-permutation.Rmd`

`conditional-permutation.Rmd`

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] 0.8857495
#>
#> $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] 0.955
```

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