topic Re: Is it possible to get confidence intervals with Moran's I? in Spatial Statistics Questions

Is it possible to get confidence intervals with Moran's I?

Charle — Thu, 24 Sep 2020 20:56:19 GMT

I have been asked by a colleague if ArcGIS estimates confidence intervals (CI) to Moran's index coefficient. In the results window and report for Spatial Autocorrelation (Global Moran's I), I see it only provides p-values and z-scores, but no CI. The reviewer of the manuscript is asking for CI. Does anyone know if/how this can be done?

Re: Is it possible to get confidence intervals with Moran's I?

DanPatterson — Thu, 24 Sep 2020 21:01:37 GMT

You reviewed this?

How Spatial Autocorrelation (Global Moran's I) works—ArcGIS Pro | Documentation

Re: Is it possible to get confidence intervals with Moran's I?

Charle — Thu, 24 Sep 2020 21:08:23 GMT

Yes, thank you, but it is not relevant to my colleague who only has ArcGIS Desktop 10.8. The Pro documentation also does not contain any info about confidence intervals (CI). I suspect the reviewer they are dealing with is asking for something that may not be standard practice for Moran's I. But hopefully a GeoNet member knows for sure?

Re: Is it possible to get confidence intervals with Moran's I?

EricKrause — Thu, 24 Sep 2020 21:30:32 GMT

Hi Charlene,

Yes, it's possible to construct confidence intervals for the Moran's I index. The statistical test is based on a classical Z-distribution test, so you can build a confidence interval using some of the numbers that appear in the messages window. When the tool runs, you should see something like this:

The numbers you need to use are the Moran's Index and Variance. First, take the square root of the variance to calculate the standard deviation. Next, you need to decide a confidence level and look up the associated Z-value for that confidence level. For 95% confidence, this Z-value is 1.96, but you can look up other values in Z-tables online. The confidence interval is then the Moran's Index plus/minus the Z-value times the standard deviation:

95% Confidence Interval: (Moran's Index) +/- 1.96 * (Standard Deviation)

So you can check your work, the numbers in the image above, a 95% confidence interval is:

0.304063 +/- 1.96 * SquareRoot(0.000411) = (0.264328, 0.343798)

Please let me know if you have any other questions.

-Eric

Re: Is it possible to get confidence intervals with Moran's I?

Charle — Thu, 24 Sep 2020 21:58:53 GMT

Of course! I've done this with standard errors for other output. So beautiful, thank you, Eric!

Re: Is it possible to get confidence intervals with Moran's I?

EricKrause — Thu, 24 Sep 2020 22:42:11 GMT

I also want to clarify that, yes, it's not usually standard practice to create confidence intervals for the Moran's I Index. There's nothing incorrect about doing it, but it just isn't very meaningful. Confidence intervals are most effective when they are in meaningful units. For example, some political position has 56% support, plus or minus 3%. Similarly with dollars, you could project a cost of $5000 dollars, plus or minus $400. But since the Moran's I Index isn't really in a meaningful unit, it's hard to interpret what the confidence interval actually means. That's why usually just z-scores and p-values are calculated for it, but again, there's nothing incorrect about creating confidence intervals for it.

Re: Is it possible to get confidence intervals with Moran's I?

Charle — Thu, 24 Sep 2020 22:53:41 GMT

Thank you for the additional helpful and meaningful clarification.

Re: Is it possible to get confidence intervals with Moran's I?

WilliamMeyer — Thu, 24 Sep 2020 23:31:50 GMT

I strongly agree with Eric about not applying confidence intervals to Moran's I it would be misleading to do so. That said confidence intervals do not translate into an absolute certainty no matter where they are used. This is statistics and absolute certainties do not exist.

Re: Is it possible to get confidence intervals with Moran's I?

Charle — Fri, 25 Sep 2020 00:07:05 GMT

My colleague is following your advice. He lives in the world of epidemiology where everyone thinks the CI reigns supreme. We thank you all for your support.