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

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.

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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