Instead of the Nagelkerke way of scaling $R^2$ to allow a 1.0 to be attained, I prefer to substitute the effective sample size for $N$ in the $R^2$ formula. This will not reach 1.0 for perfect binary predictions but this approach translates to other settings such as survival analysis where often the effect $N$ is the number of events, and to ordinal regression. See https://hbiostat.org/bib/r2.html.

Of those my favorite is the modified Maddala-Cox-Snell $R^{2}_{m,p}$ which uses effective sample size $m$ and penalizes for $p$ covariates. In the normal linear model this is almost exactly the traditional $R^{2}_{\mathrm{adj}}$.

Instead of the Nagelkerke way of scaling $R^2$ to allow a 1.0 to be attained, I prefer to substitute the effective sample size for $N$ in the $R^2$ formula. This will not reach 1.0 for perfect binary predictions but this approach translates to other settings such as survival analysis where often the effective $N$ is the number of events, and to ordinal regression. See https://hbiostat.org/bib/r2.html.

Of those my favorite is the modified Maddala-Cox-Snell $R^{2}_{m,p}$ which uses effective sample size $m$ and penalizes for $p$ covariates. In the normal linear model this is almost exactly the traditional $R^{2}_{\mathrm{adj}}$.