Singh Gurbakhshash, Hilton Fick Gordon
Department of Mathematical Sciences, Central Connecticut State University, New Britain, Connecticut.
Department of Community Health Sciences, University of Calgary, Alberta, Canada.
Stat Med. 2020 Apr 30;39(9):1343-1361. doi: 10.1002/sim.8479. Epub 2020 Feb 5.
We present here a study of ordinal outcomes with a cumulative probability model. In particular, we consider the log link with the assumption of proportionality. The logit link is currently the most widely used link with ordinal outcomes in the health research literature. With the logit link, one obtains regression coefficients that are functions of odds. When the log link is used, one obtains regression coefficients that are functions of probabilities. While odds might be preferred with certain studies that are retrospective, many health researchers may prefer to have direct statements about probabilities. There are two classes of models with the log link analogous to those with the logit link. We will call these two classes the Proportional Probability Model (PPM) and the Log Cumulative Probability Model (LCPM). These models introduce a challenge not seen with the logit link models. The log link models have constraints on the parameter space. We must insist that the maximum likelihood estimate (MLE) satisfy these constraints. We present conditions for the uniqueness of the MLE and we present other features of the MLE. Several examples and several closed form expressions for the MLE are presented. While this paper is primarily about the PPM, our R package lcpm contains functions to fit both the PPM and the LCPM.
我们在此展示一项使用累积概率模型对有序结果的研究。特别地，我们考虑在比例假设下的对数链接。对数几率链接是目前健康研究文献中用于有序结果最广泛使用的链接。使用对数几率链接时，会得到作为几率函数的回归系数。当使用对数链接时，会得到作为概率函数的回归系数。虽然在某些回顾性研究中几率可能更受青睐，但许多健康研究人员可能更倾向于直接陈述概率。存在两类具有对数链接的模型，类似于具有对数几率链接的模型。我们将这两类模型称为比例概率模型（PPM）和对数累积概率模型（LCPM）。这些模型带来了对数几率链接模型中未见到的挑战。对数链接模型在参数空间上有约束。我们必须坚持最大似然估计（MLE）满足这些约束。我们给出了MLE唯一性的条件，并展示了MLE的其他特征。给出了几个MLE的例子和几个闭式表达式。虽然本文主要关于PPM，但我们的R包lcpm包含用于拟合PPM和LCPM的函数。
