Pseudo R squared formula for GLMs

Question

I found a formula for pseudo $R^2$ in the book Extending the Linear Model with R, Julian J. Faraway (p. 59).

$$1-\frac{\text{ResidualDeviance}}{\text{NullDeviance}}$$.

Is this a common formula for pseudo $R^2$ for GLMs?

Answer 1

There are a large number of pseudo-$R^2$s for GLiMs. The excellent UCLA statistics help site has a comprehensive overview of them here. The one you list is called McFadden's pseudo-$R^2$. Relative to UCLA's typology, it is like $R^2$ in the sense that it indexes the improvement of the fitted model over the null model. Some statistical software, notably SPSS, if I recall correctly, print out McFadden's pseudo-$R^2$ by default with the results from some analyses like logistic regression, so I suspect it is quite common, although the Cox & Snell and Nagelkerke pseudo-$R^2$s may be even more so. However, McFadden's pseudo-$R^2$ does not have all of the properties of $R^2$ (no pseudo-$R^2$ does). If someone is interested in using a pseudo-$R^2$ to understand a model, I strongly recommend reading this excellent CV thread: Which pseudo-$R^2$ measure is the one to report for logistic regression (Cox & Snell or Nagelkerke)? (For what it's worth, $R^2$ itself is slipperier than people realize, a great demonstration of which can be seen in @whuber's answer here: Is $R^2$ useful or dangerous?)

Answer 2

R gives null and residual deviance in the output to glm so that you can make exactly this sort of comparison (see the last two lines below).

> x = log(1:10)

> y = 1:10

> glm(y ~ x, family = poisson)

>Call:  glm(formula = y ~ x, family = poisson)

Coefficients:
(Intercept)            x  
  5.564e-13    1.000e+00  

Degrees of Freedom: 9 Total (i.e. Null);  8 Residual
Null Deviance:      16.64 
Residual Deviance: 2.887e-15    AIC: 37.97

You can also pull these values out of the object with model$null.deviance and model$deviance

Answer 3

The formula you proposed have been proposed by Maddala (1983) and Magee (1990) to estimate R squared on logistic model. Therefore I don't think it's applicable to all glm model (see the book Modern Regression Methods by Thomas P. Ryan on page 266).

If you make a fake data set, you will see that it's underestimate the R squared...for gaussian glm per example.

I think for a gaussian glm you can use the basic (lm) R squared formula...

R2gauss<- function(y,model){
    moy<-mean(y)
    N<- length(y)
    p<-length(model$coefficients)-1
    SSres<- sum((y-predict(model))^2)
    SStot<-sum((y-moy)^2)
    R2<-1-(SSres/SStot)
    Rajust<-1-(((1-R2)*(N-1))/(N-p-1))
    return(data.frame(R2,Rajust,SSres,SStot))
}

And for the logistic (or binomial family in r ) I would use the formula you proposed...

    R2logit<- function(y,model){
    R2<- 1-(model$deviance/model$null.deviance)
    return(R2)
    }

So far for poisson glm I have used the equation from this post.

https://stackoverflow.com/questions/23067475/how-do-i-obtain-pseudo-r2-measures-in-stata-when-using-glm-regression

There is also a great article on pseudo R2 available on researchs gates...here is the link:

https://www.researchgate.net/publication/222802021_Pseudo_R-squared_measures_for_Poisson_regression_models_with_over-_or_underdispersion

I hope this help.

Answer 4

The R package modEvA calculates D-Squared as 1 - (mod$deviance/mod$null.deviance) as mentioned by David J. Harris

set.seed(1)
data <- data.frame(y=rpois(n=10, lambda=exp(1 + 0.2 * x)), x=runif(n=10, min=0, max=1.5))

mod <- glm(y~x,data,family = poisson)

1- (mod$deviance/mod$null.deviance)
[1] 0.01133757
library(modEvA);modEvA::Dsquared(mod)
[1] 0.01133757

The D-Squared or explained Deviance of the model is introduced in (Guisan & Zimmermann 2000) https://doi.org/10.1016/S0304-3800(00)00354-9