Are there Two Regressions?
是否存在两条回归线？AI 解读 ↓
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是否存在两条回归线？
- 研究问题
- 从既有总体抽样与控制一个变量的实验，如何影响最小二乘回归的方向，以及自变量测量误差造成的偏差？
- 方法与思路
- 作者比较总体抽样与受控实验两种情形，讨论最小二乘残差最小化的对象，以及变量观测误差对拟合直线的影响。
- 材料陈述的贡献
- 摘要称总体抽样中的自变量测量误差会导致偏差，而受控实验具有不同性质；这些结论的具体适用条件未说明。
- 阅读关注点
- 应先核对原文变量记号与回归方向，再检查受控变量的定义、测量误差机制，以及所谓只有一条回归线的含义。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明，不能推断刊本变化；变量方向表述需查原文，无偏结论不能无条件推广。
原文摘要
If, in sampling from an existent population, we wish to know the regression of u on v, a line is fitted by least squares, minimizing the sum of the squared residuals of v (or y). If there is no error in measuring the independent variate u, regardless of whether the dependent variate has been measured without error as v or with error as y, the regression is an estimate of the true regression. If, however, the independent variate is measured with error in x, the regression obtained is a biassed one. The bias will characterize the fitted line, and will be present whether the sample is a random one of the entire population or has been taken at preassigned selected values of x. If, however, the experiment is one in which one of the variates is a controlled observation, it differs from sampling from an existent population in that, 1), the line estimated by least squares, minimizing the sum of the squared residuals of the dependent uncontrolled variate, is the same, whether x or y is the controlled variate, that is, there is only one regression; and 2), the estimated line is not biassed by the existence of an error of observation in the independent controlled variate, despite our taking no account of it in the least squares fit.