R2 for binomial data is sensitive on the grouping scheme

 Try to use these data to calculate Nagelkerke R2:

library(fmsb)
data_grouped <- data.frame(x=c(3, 4, 5), y=c(2, 1, 0))
cof_grouped <- c(10, 7, 2)
res_grouped <- glm(cbind(x,y) ~ cof_grouped, data=data_grouped, family=binomial())
summary(res_grouped)
or

data_splited <- data.frame(x=c(1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1), 
                           y=1-c(1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1))
cof_splited <- rep(c(10, 7, 2), each=5)
res_splited <- glm(cbind(x,y) ~ cof_splited, data=data_splited, family=binomial())
summary(res_splited)

These data are exactly the same and indeed the fitted model is the same.

But...

> NagelkerkeR2(res_grouped)$R2

[1] 0.9538362

> NagelkerkeR2(res_splited)$R2

[1] 0.2879462

You will conclude for a very strong link and a weaker link in the second case.

Note that you have the same problem with a common R2:

> cor(x = data_grouped$x/(data_grouped$x+data_grouped$y), y=res_grouped$fitted.values)^2

[1] 0.964555

> cor(x = data_splited$x/(data_splited$x+data_splited$y), y=res_splited$fitted.values)^2

[1] 0.1607592


This problem is also indicated here:

https://thestatsgeek.com/2014/02/08/r-squared-in-logistic-regression/

The conclusion of the author is:

The low R squared for the individual binary data model reflects the fact that the covariate x does not enable accurate prediction of the individual binary outcomes. In contrast, x can give a good prediction for the number of successes in a large group of individuals.

See also:

Mittlböck M, Heinzl H (2001) A note on R2 measures for Poisson and logistic regression models when both models are applicable. Journal of Clinical Epidemiology 54: 99-103 doi 10.1016/S0895-4356(00)00292-4


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