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I have the following result from running glm function.

How can I interpret the following values:

  • Null deviance
  • Residual deviance
  • AIC

Do they have something to do with the goodness of fit? Can I calculate some goodness of fit measure from these result such as R-square or any other measure?

Call:
glm(formula = tmpData$Y ~ tmpData$X1 + tmpData$X2+ 
        tmpData$X3 + as.numeric(tmpData$X4) + tmpData$X5 + 
    tmpData$X6 + tmpData$X7)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-0.52628  -0.24781  -0.02916   0.25581   0.48509  

Coefficients:
                              Estimate Std. Error  t value Pr(>|t|)    
(Intercept         -1.305e-01  1.391e-01   -0.938   0.3482    
tmpData$X1         -9.999e-01  1.059e-03 -944.580   <2e-16 ***
    tmpData$X2         -1.001e+00  1.104e-03 -906.787   <2e-16 ***
tmpData$X3         -5.500e-03  3.220e-03   -1.708   0.0877 .  
    tmpData$X4         -1.825e-05  2.716e-05   -0.672   0.5017    
tmpData$X5          1.000e+00  5.904e-03  169.423   <2e-16 ***
    tmpData$X6          1.002e+00  1.452e-03  690.211   <2e-16 ***
tmpData$X7          6.128e-04  3.035e-04    2.019   0.0436 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

(Dispersion parameter for gaussian family taken to be 0.08496843)

    Null deviance: 109217.71  on 3006  degrees of freedom
Residual deviance:    254.82  on 2999  degrees of freedom
  (4970 observations deleted due to missingness)
AIC: 1129.8

Number of Fisher Scoring iterations: 2
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migrated from stackoverflow.com Dec 20 '12 at 23:10

This question came from our site for professional and enthusiast programmers.

    
I realize this was migrated from SO, where one would not normally look for information on these statistical terms. You have a great resource here! For example, see what you can learn from a search on some of your terms, like AIC. A little time spent doing this should either fully answer your question or at least guide you to asking a more specific one. –  whuber Dec 20 '12 at 23:56

1 Answer 1

The default error family for a glm model in (the language) R is Gaussian, so with the code submitted you are getting ordinary linear regression where $R^2$ is a widely accepted measure of "goodness of fit". The glm function reports the Nagelkerke-pseudo-"$R^2$". In the case of an OLS model, the Nagelkerke GOF measure will be roughly the same as the $R^2$.

$$R^2_{\mathrm{GLM}}=1-\frac{(\sum_id_{i,\mathrm{model}}^2)^{2/N} }{(\sum_id_{i,\mathrm{null}}^2)^{2/N}} ~~~~~~~~.=.~~~~~~~~ 1-\frac{\mathit{SSE}/n[\mathrm{model}]}{\mathit{SST}/n[\mathrm{total}]} = R^2_{\mathrm{OLS}}$$

There is some debate about how such a measure on the LHS gets interpreted, but only when the models depart from the simpler Gaussian/OLS situation. But in GLMs where the link function may not be "identity", as was here, and the "squared error" may not have the same clear interpretation, so the Akaike Information Criterion is also reported because it appears to be more general. There are several other contenders in the GLM GOF sweepstakes with no clear winner.

You might want to consider not reporting a GOF measure if you are going to be using GLMs with other error structures: Which pseudo-$R^2$ measure is the one to report for logistic regression (Cox & Snell or Nagelkerke)?

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Where exactly is the " Nagelkerke-pseudo-"R2"" in the above output? –  Tom Sep 25 '13 at 2:57
    
It's a transformation of the difference in log-likelihoods from nested generalized linear models to a [0,1] range in an effort to mimic the R^2 that people understand in the context of comparing ordinary linear models. See: palgrave.com/psychology/baguley/students/supplements/… –  DWin Sep 27 '13 at 16:48

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