Which statistic is a version of the coefficient of determination for logistic regression, based on log-likelihoods, but does not reach 1?

Which statistic is a version of the coefficient of determination for logistic regression, based on log-likelihoods, but does not reach 1?

• A. Nagelkerke's
• B. Cronbach's α
• C. Cox and Snell's
• D. Cross-validation

Answer: (C)

In logistic regression, we assess fit with pseudo-R^2 statistics that come from comparing how likely the observed data are under the fitted model versus a baseline model. One such measure is Cox and Snell’s R^2, which is defined from the likelihoods of the null model (no predictors) and the full model. Specifically, it is 1 minus the ratio of these likelihoods raised to the power 2 divided by the sample size. A key point is that, because of the way likelihoods work in binary outcomes, this statistic cannot reach 1 even with a perfect predictor set; its maximum value is inherently less than 1. That’s why it’s described as a version of the coefficient of determination that does not reach 1. Nagelkerke’s R^2, by contrast, adjusts Cox and Snell’s so the maximum is 1, but that adjustment changes the interpretation. The other options aren’t based on log-likelihood comparisons: Cronbach’s alpha measures internal consistency reliability, and cross-validation assesses predictive performance through data partitioning rather than a single fit statistic.

In logistic regression, we assess fit with pseudo-R^2 statistics that come from comparing how likely the observed data are under the fitted model versus a baseline model. One such measure is Cox and Snell’s R^2, which is defined from the likelihoods of the null model (no predictors) and the full model. Specifically, it is 1 minus the ratio of these likelihoods raised to the power 2 divided by the sample size. A key point is that, because of the way likelihoods work in binary outcomes, this statistic cannot reach 1 even with a perfect predictor set; its maximum value is inherently less than 1. That’s why it’s described as a version of the coefficient of determination that does not reach 1. Nagelkerke’s R^2, by contrast, adjusts Cox and Snell’s so the maximum is 1, but that adjustment changes the interpretation. The other options aren’t based on log-likelihood comparisons: Cronbach’s alpha measures internal consistency reliability, and cross-validation assesses predictive performance through data partitioning rather than a single fit statistic.