I'm doing Geographically Weighted Regression (GWR) analysis in ArcMap and need the degrees of freedom for the model in order to calculate p values.

The summary does not list degrees of freedom, but lists an "effective number".

Is this the same thing?

  • 1
    GWR is a local regression so, I do not believe that "degrees of freedom" are relevant here. Technically, the degrees of freedom would be dependent on each local fit and thus, variable. May 27, 2015 at 23:24
  • Might be a dumb question, but are you using a fixed or adaptive bandwidth? May 28, 2015 at 0:49
  • I'm using an adaptive bandwidth with 18 neighbors.
    – Zoe
    May 28, 2015 at 1:00
  • Just as a comment, I've found the implementation of GWR in Arc to be really quite painful at times, and prone to fall over at points. I've had more success running it using the standalone version (geodacenter.asu.edu/software) ... which has the bonus of having an associated manual and test data with it.... at worst, have a look at the manual! May 28, 2015 at 5:54

1 Answer 1


No, it is not the same thing. This question has gone unanswered so I dug into it a bit,

Based on the below excerpt from the ESRI website coupled with an excerpt from Fotheringham. It seems to me that the effective number is the effective number of parameters, per explicit mention in Fotheringham et al. 2002 (p. 92) as well as the fact that ESRI worked or contracted with Fotheringham and friends to develop the built-in tools that come with ArcGIS.

The effective degrees of freedom would then be n - effective number of parameters. Where n is the population or number of datapoints.

See the below excerpt from the diagnostic output of the GWModel GWR package in R from my own GWR:

Number of data points: 62 
Effective number of parameters (2trace(S) - trace(S'S)): 39.07396 
Effective degrees of freedom (n-2trace(S) + trace(S'S)): 22.92604 

That being said, I do not know to what extent one may or may not calculate p-values or determine cut-offs for other statistical tests using the effective degrees of freedom, but I believe this answers the question of whether they are the same.

"The effective number of parameters in a GWR is often not an integer but varies between k (when the bandwidth tends to infinity) and n (when the bandwidth tends to zero)." (Fotheringham et al. 2002, p 92)

ESRI excerpt:

"EffectiveNumber: this value reflects a tradoff between the variance of the fitted values and the bias in the coefficient estimates, and is related to the choice of bandwidth. As the bandwidth approaches infinity, the geographical weights for every observation approach 1, and the coefficient estimates will be very close to those for a global OLS model. For very large bandwidths, the effective number of coefficients approaches the actual number; local coefficient estimates will have a small variance but will be quite biased. Conversely, as the bandwidth approaches zero, the geographical weights for every observation approach zero with the exception of the regression point itself. For extremely small bandwidths, the effective number of coefficients is the number observations, and the local coefficient estimates will have a large variance but low bias. The effective number is used to compute a number of diagnostic measures."

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    A global evaluation of the degrees of freedom just do not make sense to me in a local regression. It seems like an approach that evaluated the degrees of freedom in the residual error would be more prudent and honest. Feb 19, 2018 at 21:11
  • @JeffreyEvans I agree, however, I'm currently grappling with the concepts. How and why do you think the residual degrees of freedom would be more prudent and honest?
    – bwp8nt
    Feb 20, 2018 at 0:32

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