Clarification on chance-corrected adequacy measure for a subset of predictors

data analysis variable selection
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Prof. Frank Harrell writes in his book Regression Modeling Strategies (2nd ed.) in section 9.8.4 (page 207):

A chance-corrected adequacy measure could be derived by squaring the ratio of the R-index for the subset to the R-index for the whole set.

As far as I’m aware, this is not further developed in the book. I’m guessing that the R-index is Nagelkerke’s R^{2} but I’m not sure.

So the chance-corrected adequacy measure would be \left(\frac{\sqrt{R^{2}_{S}}}{\sqrt{R^{2}_{F}}}\right)^{2} where R^{2}_{S} is Nagelkerke’s R^{2} for the model containing a subset S of the predictors and R^{2}_{F} is Nagelkerke’s R^{2} from the full model.

I’d be grateful if somebody could explain if I’m on the right track and what the rationale and interpretation of this adequacy measure is. Thank you.

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There is some potential with those ideas but a chance-corrected adequacy index is easier to deal with. Compute model likelihood ratio \chi^2 minus the number of degrees of freedom spent to achieve that \chi^2. Get \chi^2 for both the all-inclusive model and a submodel excluding the panel of new biomarkers (for example).

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