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I would like to perform a Geographically Weighted Regression, however my Jarque–Bera test is significant? What is the ideal solution to overcome this? Transform the data? Use a semi-parametric GWR? Or, may I still proceed?

This is what I have read about this test:

Ensuring that the residuals of over/under predictions are the result of random noise using the Jarque–Bera (JBT) test, this test result should not be significant.

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    $\begingroup$ It is difficult to imagine any real dataset whose deviations would be so perfectly Normal in all locations that the GWR residuals would pass any test of normality. Consider reading over our many discussions of normality tests of residuals (and why they are almost always pointless). $\endgroup$ – whuber Oct 14 '14 at 15:20
  • $\begingroup$ I have read now that the GWR is a non-parametric test. Does this not imply that my GWR results are valid, despite having some variables without normal distributions? $\endgroup$ – I Heart Beats Oct 14 '14 at 17:05
  • $\begingroup$ It depends on what results you refer to and what you mean by "valid"! I hate to sound so slippery, but GWR produces entire maps of outputs, including estimated coefficients, measures of linearity, of correlation, and so on; and "valid" depends on what you are actually using GWR for and how you interpret its results. I suspect GWR is very robust to non-normality for the purpose of identifying non-stationarity in a spatial random field, for instance, but I am sure it is sensitive to non-normality if you are using any "p-values" that it might output. $\endgroup$ – whuber Oct 14 '14 at 17:14
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    $\begingroup$ What an extraordinary number of decimal places! I'd regard such output as a really bad sign in terms of taking the software, whatever it is, fully seriously. $\endgroup$ – Nick Cox Oct 14 '14 at 18:10
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    $\begingroup$ It's very bad taste, computationally and statistically. Where there is one kind of bad taste others often follow. You are right; it's not logic, but a prejudice I have often found matched by experience. $\endgroup$ – Nick Cox Oct 14 '14 at 18:57
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The normal distribution is sometimes (maybe even often) a reasonable simplification of reality, but it is almost never going to be true in a strict sense. So, your Jarque-Bera test being significant did not tell you anything you did not already know before you perfomed that test.

The real question you should ask yourself is whether or not the deviations are large enough such that the normal distribution is no longer a reasonable simplification of reality, and statistical tests don't answer that question. Instead you should just look at the residuals (I like QQ-plots for that) and make your own subjective judgement. You should then document how you came to that decision, so that others can decide for themselves if they would have made the same judgement call as you did. This being able to trace back how you came to a decision is what makes your work "scientific".

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  • $\begingroup$ Thank you @maarten. I appreciate your answer, I was hoping for one, more specifically related to the realm of the GWR analysis and what it means to work with normal data in the GWR environment. I know there are semi-parametric versions this analysis and was hoping to generate some discussion. $\endgroup$ – I Heart Beats Oct 14 '14 at 14:54
  • $\begingroup$ (+1) This is excellent advice, especially considering that GWR is considered an exploratory technique. Did you know, though, that GWR typically generates hundreds (to millions) of regressions, each with its own set of residuals? Recommending QQ plots does not seem practicable in this setting. $\endgroup$ – whuber Oct 14 '14 at 15:24
  • $\begingroup$ I have read mixed opinions on the exploratory nature of GWR. If it is in fact simply exploratory, once completed, and best explanatory variables are determined, what then is the modelling to technique to solidify the relationships established using GWR? $\endgroup$ – I Heart Beats Oct 14 '14 at 17:08
  • $\begingroup$ It varies. GWR could be a useful prelude to a geostatistical analysis that might end with kriging, a spatially lagged regression, or applying a hierarchical Bayesian model, for instance. $\endgroup$ – whuber Oct 14 '14 at 17:16
  • $\begingroup$ A simple point that needs emphasis is that these residuals are most unlikely to be spatially independent, or even uncorrelated. They are for geographical data.... I suspect that messes up all P-value calculations mightily. Jarque-Bera as I understand it involves using asymptotic results for skewness and kurtosis in a situation where convergence to those results is very slow for small or moderate sample sizes. These problems compound all the usual reasons for not taking the test very seriously. $\endgroup$ – Nick Cox Oct 14 '14 at 18:07

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