A lot of studies on the summary measures of predictive strength of categorical response models consider the likelihood ratio index (LRI), also known as the McFadden-$R^2$, a better option than many other measures. We propose a simple modification of the LRI that adjusts for the effect of the number of response categories on the measure and that also rescales its values, mimicking an underlying latent measure. The modified measure is applicable to both binary and ordinal response models fitted by maximum likelihood. Results from simulation studies and a real data example on the olfactory perception of boar taint show that the proposed measure outperforms most of the widely used goodness-of-fit measures for binary and ordinal models. The proposed $R^2$ interestingly proves quite invariant to an increasing number of response categories of an ordinal model.

翻译：大量关于分类响应模型预测强度汇总度量的研究认为，似然比指数（LRI），即麦克法登$R^2$，是优于许多其他度量的选择。我们提出了对LRI的一个简单修正，该修正调整了响应类别数量对度量的影响，并重新缩放其数值，以模拟潜在的潜变量度量。该修正度量适用于通过极大似然法拟合的二元和有序响应模型。模拟研究以及关于公猪膻味嗅觉感知的真实数据实例结果表明，所提出的度量在多数情况下优于广泛使用的二元和有序模型拟合优度度量。有趣的是，该提出的$R^2$对有序模型响应类别数量的增加表现出相当的不变性。