The Cost of Generalizing Logistic Regression
推广 Logistic 回归的代价AI 解读 ↓
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推广 Logistic 回归的代价
- 研究问题
- 在二元响应回归中加入未知链接形状参数，预测概率、目标概率对应的协变量值及系数比的估计方差会增加多少？
- 方法与思路
- 作者研究包含常用链接的对称与非对称单参数族，以渐近方法计算额外参数引起的方差膨胀，并用小样本示例和模拟说明。
- 材料陈述的贡献
- 摘要量化了推广链接函数的方差代价，指出代价随设计点和外推位置变化，并强调评估时还应考虑潜在的偏差下降。
- 阅读关注点
- 应核对方差膨胀的比较基准、跨设计点平均的定义，以及协变量维数、数据范围和链接错设如何影响偏差与方差权衡。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要，不能推断刊本变化；平均膨胀结果针对所研究模型，不代表任意二元响应模型或设计的固定代价。
原文摘要
Binary-response regression models in which the link function is a family defined by one or more unknown shape parameters are considered. Detailed attention is given to the two single-parameter families proposed by Aranda-Ordaz (1981) that incorporate the logistic, linear, and complementary log-log link functions as special cases. One model is a symmetric family of alternatives to the logistic shape and the second model is an asymmetric family. The increase in variance of quantities of interest due to the addition of an extra parameter is calculated using asymptotic methods. This inflation in variance is interpreted as the cost of adding an additional parameter to the model. In considering the seriousness of this cost the reduction in bias due to the estimation of the additional parameter should also be taken into account. Three quantities of interest—predicted probabilities for a fixed value of the explanatory variable, predicted value of the explanatory variable to obtain a specific probability, and the ratio of regression parameters—are considered. It is found that when there is a single explanatory variable the cost of adding the extra parameter in terms of inflation in variance (of either predicted probabilities or predicted explanatory-variable values) is 50% on the average across the design points, although it can be appreciably larger or smaller. The inflation in variance for predicted probabilities or covariate values tends to be very large outside the range of the data and closer to 1 within the range of the observations. Nevertheless, the inflation in variance is not typically at a minimum in the center of the data. When there are q – 1 explanatory variables the average inflation in variance is 1 + 1/q, for both models. The small-sample situation is illustrated with an example and some Monte Carlo simulations; the conclusions based on the asymptotic calculations carry over to small samples.