[R-sig-Geo] Inference of local Gi*

[Roger Bivand]

On Sat, 25 Apr 2020, Jose Ramon Martinez Batlle wrote:

> Dear Anaïs.
>
> I am sure more experienced members will give you a better answer, but until
> that I will try to help.
>
> 1) If I understood correctly, the spatial objects have 15 000 and 30 000
> points in each case study, respectively. If this is the case, I am afraid
> that nb objects of such large datasets surely would have an impact on the
> system performance when used in subsequent tasks. The best I can suggest is
> to try some sort of spatial binning if possible (e.g. hexbins), but at the
> same time accounting for the modifiable areal unit problem.
>
> 2) The spdep:localG help page states that "For inference, a Bonferroni-type
> test is suggested in the references, where tables of critical values may be
> found". The source mentioned is free access, and can be found here:
>
> Ord, J. K. and Getis, A. 1995 Local spatial autocorrelation statistics:
> distributional issues and an application. Geographical Analysis, 27, 286–306
> https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1538-4632.1995.tb00912.x
>
> Standard measures (critical values) for selected percentiles and number of
> entities, are included in Table 3 of the cited reference. Since the values
> returned from localG are Z-values, you can use them to determine whether
> the critical value chosen is exceeded and thus infer significant local
> spatial association for each entity.

Thanks, José, you are quite correct that false discovery rate problems are 
among the main reasons why so-called "hot-spot" analyses may be very 
misleading, in appearing to give an inferential basis for apparent map 
pattern.

In our survey paper with David Wong referenced on ?localG, 
https://doi.org/10.1007/s11749-018-0599-x, we show that the analytical and 
bootstrap-based inferences are similar - the normality is related not to 
the underlying variable seen globally, but the the local behaviour of the 
statistic. For this reason, bootstrap permutation implementations are not 
included in spdep, though the code is available if need be. Please 
indicate whether users would like this code included for comparative 
purposes here or in a github issue on 
https://github.com/r-spatial/spdep/issues/.

Further, the LOSH statistic, which is a measure of local spatial 
heteroscedasticity building on local G, provides a little insight into the 
problems raised for so-called "hot-spot" analyses by variability across 
the study area in the behaviour of the variable of interest. If, for 
example, the variable of interest is influenced by a background variable 
with a spatial pattern, we will probably find "hot-spots" which look like 
the omitted background variable on a map.

While local G cannot take residuals of a linear model, local Moran's I can 
do so. For local G, we do not have exact case-by-case standard deviates; 
we do have these for local Moran's I as discussed in the article with 
David Wong, and they very typically reduce strongly the counts of 
apparently significant local statistcs even before adjusting p-values for 
FDR. Finally, only some local measures can adjust for global 
autocorrelation - unadjusted local measures also respond to the presence 
of global autocorrelation.

On balance, judicious choice of class intervals in mapping a variable of 
interest may prove more helpful than trying to present wobbly inferences 
from ESDA.

Hope this isn't too discouraging,

Roger


>
> Kind regards.
> José
>
> El vie., 24 abr. 2020 a las 14:00, Anaïs Ladoy (<anais.ladoy using epfl.ch>)
> escribió:
>
>> Dear list members,
>>
>> I'm currently working on a point dataset, from which I want to conduct
>> a Hot Spot Analysis with local Gi* statistics (Getis-Ord).
>>
>> I'm trying to find a way of computing its significance. I see two ways
>> of computing significance in this case:
>>
>> 1) Compare the obtained local Gi from spdep::localG to a normal
>> distribution. But here I have several questions :
>> a) In my first case study (BMI value of 15 000 participants in a cohort
>> study), the distribution of local Gi is far from normal (it is bimodal
>> with a mode around very negative values and a mode around 0). However,
>> I do need a normal distribution of Gi in order to compare it with a
>> normal distribution, right? Or am I missing something here? What should
>> I do in this case?
>> b) In my second case study (Years of life lost for 30 000 individuals),
>> the distribution of Gi returned by spdep::localG is approximately
>> normal but the standard deviation is far from 1. In fact, in
>> spdep::localG, the Gi values are supposedly standardized (from what I
>> understood using an analytical mean and variance). Should I use these
>> to compare to a normal distribution, or should I use raw G values
>> (using return_internals=TRUE) and standardize them with the observed
>> mean and variance of Gi? Does it cause a problem that my observed
>> variance does not match the analytical variance?
>>
>> 2) Compute permutations
>> However this is not implemented in R for localG. I tried using PySAL
>> but the initial file is big and the weight file is huge, and my
>> computer crashes. Any thoughts to solve this issue?
>>
>> Thank you for any feedback.
>> Kind regards,
>> Anaïs
>>
>> --
>> Anaïs Ladoy
>> PhD student, Laboratory of Geographic Information Systems, Swiss
>> Federal Institute of Technology in Lausanne (EPFL), Switzerland.
>>
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>
>
>

-- 
Roger Bivand
Department of Economics, Norwegian School of Economics,
Helleveien 30, N-5045 Bergen, Norway.
voice: +47 55 95 93 55; e-mail: Roger.Bivand using nhh.no
https://orcid.org/0000-0003-2392-6140
https://scholar.google.no/citations?user=AWeghB0AAAAJ&hl=en