Minimum X 2 and Maximum Likelihood Solution in Terms of a Linear Transform, with Particular Reference to Bio-Assay
以线性变换求最小卡方与最大似然解：侧重生物测定AI 解读 ↓
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以线性变换求最小卡方与最大似然解：侧重生物测定
- 研究问题
- 通过变量变换拟合直线时，如何调整变换后的拟合，使结果满足原始观测尺度上的最小卡方或最大似然准则？
- 方法与思路
- 作者沿用生物测定中 probit、logit 等变换的调整思路，给出若干常用函数的最小卡方处理，并概述相应最大似然调整。
- 材料陈述的贡献
- 摘要将贡献定位为补充若干函数在线性变换框架下的最小卡方解法，同时汇总同类函数已有的最大似然调整方法。
- 阅读关注点
- 应核对所覆盖函数、调整步骤、权重和分布条件，以及变换尺度上的线性拟合如何对应原始尺度的目标准则。
- 阅读边界
- 仅据版本未注明的索引摘要；未给出调整公式或收敛条件，不能把生物测定中的 logit 直接解释为经济离散选择模型。
原文摘要
An old and commonly used device for fitting a curve y = F(x, α, β) is to find a function of y which is linearly related to x, or a function of x which is linearly related to y, or functions of y and x which are linearly related to each other. The linearly related functions are plotted against one another, and a line is fitted to the points, usually by eye, sometimes by “least squares” in terms of the transforms. Texts dealing with empirical curve-fitting characteristically use this scheme [6]. Before Bliss and Fisher [2], no attempt was made to adjust the transforms systematically in order to achieve a fit that fulfilled defined criteria in terms of the original measures y and x, nor was it known that such adjustments were possible. These authors presented for the bio-assay experiment a method in terms of probits, which as shown by Garwood, [5] accomplished a maximum likelihood estimate of the integrated normal curve, when the observations were distributed binomially. Following the procedures used by Bliss and Fisher for the integrated normal curve, I [1] formulated the similar adjustments applicable to “logits” for a maximum likelihood solution of the logistic function, but did not advocate using this solution. Finney [4] has recently presented the adjustments for a maximum likelihood solution of several functions of interest in bio-assay. So far as I know the similar method for a solution fulfilling the criteria of minimum X2 rather than maximum likelihood has not been presented. In the following paper this is given for several functions in common statistical use, and at the same time a résumé is given of the maximum likelihood adjustments for the same functions.