# Bootstrapping-based test for local spatial heteroscedasticity

`LOSH.mc.Rd`

The function draws inferences about local spatial heteroscedasticity (LOSH) by means of the randomisation-based Monte-Carlo bootstrap proposed by Xu et al. (2014).

## Usage

```
LOSH.mc(x, listw, a = 2, nsim = 99, zero.policy = attr(listw, "zero.policy"),
na.action = na.fail, spChk = NULL, adjust.n = TRUE, p.adjust.method = "none")
```

## Arguments

- x
a numeric vector of the same length as the neighbours list in listw

- listw
a

`listw`

object created for example by`nb2listw`

- a
the exponent applied to the local residuals; the default value of 2 leads to a measure of heterogeneity in the spatial variance

- nsim
the number of randomisations used in the bootstrap

- zero.policy
default

`attr(listw, "zero.policy")`

as set when`listw`

was created, if attribute not set, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA- na.action
a function (default

`na.fail`

), can also be`na.omit`

or`na.exclude`

- in these cases the weights list will be subsetted to remove NAs in the data. It may be necessary to set zero.policy to TRUE because this subsetting may create no-neighbour observations. Note that only weights lists created without using the glist argument to`nb2listw`

may be subsetted. If`na.pass`

is used, zero is substituted for NA values in calculating the spatial lag. (Note that na.exclude will only work properly starting from R 1.9.0, na.omit and na.exclude assign the wrong classes in 1.8.*)- spChk
should the data vector names be checked against the spatial objects for identity integrity, TRUE, or FALSE, default NULL to use

`get.spChkOption()`

- adjust.n
default TRUE, if FALSE the number of observations is not adjusted for no-neighbour observations, if TRUE, the number of observations is adjusted

- p.adjust.method
a character string specifying the probability value adjustment for multiple tests, default "none"; see

`p.adjustSP`

. Note that the number of multiple tests for each region is only taken as the number of neighbours + 1 for each region, rather than the total number of regions.

## Details

The test calculates LOSH (see `LOSH`

) and estimates pseudo p-values from a conditional bootstrap. Thereby, the i-th value in each location is held fixed, whereas all other values are permuted `nsim`

times over all other spatial units.

## Value

- Hi
LOSH statistic

- E.Hi
expectation of LOSH

- Var.Hi
variance of LOSH

- Z.Hi
the approximately chi-square distributed test statistics

- x_bar_i
local spatially weighted mean values

- ei
residuals about local spatially weighted mean values

- Pr()
p-values for

`Hi`

obtained from a conditional bootstrap distribution

## References

Ord, J. K., & Getis, A. 2012. Local spatial heteroscedasticity (LOSH), The Annals of Regional Science, 48 (2), 529–539; Xu, M., Mei, C. L., & Yan, N. 2014. A note on the null distribution of the local spatial heteroscedasticity (LOSH) statistic. The Annals of Regional Science, 52 (3), 697–710.

## Author

René Westerholt rene.westerholt@tu-dortmund.de

## See also

`LOSH`

, `LOSH.mc`

## Examples

```
data(columbus, package="spData")
resLOSH_mc <- LOSH.mc(columbus$CRIME, nb2listw(col.gal.nb), 2, 100)
summary(resLOSH_mc)
#> Hi x_bar_i ei Pr()
#> Min. :0.03438 Min. :13.85 Min. : 0.0298 Min. :0.009901
#> 1st Qu.:0.23838 1st Qu.:24.71 1st Qu.: 7.0114 1st Qu.:0.079208
#> Median :0.66689 Median :35.90 Median : 52.1094 Median :0.653465
#> Mean :1.06592 Mean :34.88 Mean : 151.9232 Mean :0.552233
#> 3rd Qu.:1.59680 3rd Qu.:45.39 3rd Qu.: 105.0551 3rd Qu.:0.900990
#> Max. :4.68765 Max. :54.91 Max. :2455.2201 Max. :0.990099
resLOSH_cs <- LOSH.cs(columbus$CRIME, nb2listw(col.gal.nb))
summary(resLOSH_cs)
#> Hi E.Hi Var.Hi Z.Hi
#> Min. :0.03438 Min. :1 Min. :0.4972 Min. :0.04356
#> 1st Qu.:0.23838 1st Qu.:1 1st Qu.:0.9136 1st Qu.:0.35069
#> Median :0.66689 Median :1 Median :1.4342 Median :1.19175
#> Mean :1.06592 Mean :1 Mean :1.4758 Mean :1.89447
#> 3rd Qu.:1.59680 3rd Qu.:1 3rd Qu.:1.9547 3rd Qu.:2.65120
#> Max. :4.68765 Max. :1 Max. :2.9958 Max. :7.25681
#> x_bar_i ei Pr()
#> Min. :13.85 Min. : 0.0298 Min. :0.0190
#> 1st Qu.:24.71 1st Qu.: 7.0114 1st Qu.:0.2050
#> Median :35.90 Median : 52.1094 Median :0.5264
#> Mean :34.88 Mean : 151.9232 Mean :0.4507
#> 3rd Qu.:45.39 3rd Qu.: 105.0551 3rd Qu.:0.6529
#> Max. :54.91 Max. :2455.2201 Max. :0.9540
plot(resLOSH_mc[,"Pr()"], resLOSH_cs[,"Pr()"])
```