Linear Regression When the Standard Deviations of Arrays are Not All Equal
各条件组标准差不相等时的线性回归AI 解读 ↓
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各条件组标准差不相等时的线性回归
- 研究问题
- 当一个变量在另一个变量不同取值下的标准差发生变化时，如何摆脱等标准差假设来讨论线性回归？
- 方法与思路
- 所给文字通过生活支出、农业产量及生长测量等例子说明条件组离散程度可能变化，强调回归需要考虑这种异质性。
- 材料陈述的贡献
- 所给文字明确提出忽视标准差变化可能影响实际回归分析，但没有展示具体估计方法、推断程序或性能结果。
- 阅读关注点
- 应核对全文如何定义各条件组、刻画变化的标准差，以及是否据此调整回归估计、标准误或其他推断步骤。
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- 仅据 OpenAlex 索引所给文字，稿件版本未注明；内容主要是动机说明，未展示方法或结果，不能补造全文结论。
原文摘要
M ANY TEXTBOOKS on statistics consider regression only for the case when the standard deviations of all arrays are the same. The implication is that the formulas given are adequate for the cases met in practice and that, even though the standard deviations are not strictly equal, the usual formulas are still valid. Consequently many applications have been made to date without much attention to the equality of the standard deviations of the arrays.' It has been recognized since the days of Gauss that the standard deviations may vary from point to point. The methods for handling varying standard deviations are found in many writings on least squares and are used by some.2 Nevertheless, the ignorance of varying standard deviations is very widespread as well as the indifference to their importance. In any set of two-variable data in which the mean of one variable increases from zero or a small value to a large value for values of the other variable there is opportunity for the spread of the arrays to increase as the means of the arrays increase. An excellent example of this situation has been given by Dr. Brady3 who has shown that the standard deviations of the amounts expended on different phases of living have increased with income. Dr. S. C. Salmon has recently forcibly called attention4 to the increase of the standard deviations of agricultural yields with the increase in yield. Anything that grows, as a peach embryo, say, must start with size zero and zero standard deviation. As the size increases, the opportunity for variation comes into existence. Thus, peach embryos measured at different times show different standard deviations. In view of the many cases of practical importance where the standard deviations change from array to array, it becomes very important to consider regression freed of the assumption that the standard deviations df arrays are equal.