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My data consists of data measured at different positions (250 measurements in a 5x5 grid). It has been proposed that results may be somewhat dependent on the position. Looking at the data I don't think this is the case. What statistical test can I do to show this?

My current idea is to do an F-test between the set of all the data and the results averaged at each position with a null result indicating that they can be described by the same distribution and so any position dependent component is small. Is this approach correct and is there a better way of showing this?

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  • $\begingroup$ Usually what matters more than the possible presence of spatial dependence is its potential effect on your analysis. In many cases the effects are inconsequential (suggesting, if that's your situation, that the best answer is "who cares?"). What kind of analyses of these data do you propose? $\endgroup$
    – whuber ♦
    Commented Oct 22, 2014 at 14:26
  • $\begingroup$ @whuber This data is to test the repeatability of my system. The measurements are the same artifact at different points in the field of view. If the effect of position is small compared to the other sources of noise then I could just ignore it, which would be nice. If it was bigger I would probably try and do some sort of correction to apply to my real data. The overall point of doing these tests is to be able to determine when doing measurements on different samples whether the values are significantly different and what is the distribution due to manufacturing. $\endgroup$
    – nivag
    Commented Oct 22, 2014 at 15:45
  • $\begingroup$ I am tempted to read "significantly different" as meaning "important for the manufacturing process." Those are two completely different things, though! It sounds like you only want to see whether there exist two positions exhibiting materially important differences. If (a) two such positions do not exist and (b) you have enough measurements to be confident in the results at any individual position, then wouldn't that be enough to settle the issue without having to assess spatial correlations? $\endgroup$
    – whuber ♦
    Commented Oct 22, 2014 at 15:49

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Yes, there is a better way to evaluate spatial dependence.

A common and easy way is to calculate a measure of spatial autocorrelation, such as Moran's I (http://en.wikipedia.org/wiki/Moran%27s_I). There is multiple estimators of spatial autocorrelation available. I think Fortin and Dale (2005) is a good reference book. Here's an example of Moran's I calculated in R: http://www.ats.ucla.edu/stat/r/faq/morans_i.htm

I understand that you are interested in spatial dependence and not necessarely spatial autocorrelation. You have to know that both are estimated by measures such as Moran's I and can't be distinguished.

Fortin, M.-J. et M. R. T. Dale (2005). Spatial analysis: a guide for ecologist. New York, NY, USA, Cambridge University Press.

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