DV predicted by two IVs, vs. triangular pyramid

-- Diagram from Wiki

It is easier to imagine relation in three spatial vectors by their angles, than by their correlations. For standardized and s , , cosines of three angles of the triangular pyramid determinate the correlation matrix, thus, all statistics of the regressions and . Unexpected but imaginative results on the impact of introducing are --

1. Both s are nearly independent of . Togethor they predict almost perfectly ( and ).

2. Both s are almost perfectly correlated with . Togethor, one of the regressive coefficient is significantly negative (, and respectively).

3. Redundancy (Cohen, Cohen, West, & Aiken, 2003) increases to full and then decreases to zero and even negative (, and closes from to then to ).




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Cohen, J., Cohen, P., West, S. G., & Aiken, L. S.  (2003). Applied multiple regression/correlation analysis for the behavioral sciences(3rd ed.) Mahwah, NJ: Lawrence Erlbaum Associates.

自由度的几何:对截距项投影残差向量的长度平方

这是《相关系数的几何:对截距投影的残差向量之间交角余弦》示意图,恰好可以用于解释为什么满足的分布dfn-1而不是n

其中n维空间中的标准正态随机向量。那么,容易知道有。这个表达式就是向量长度的平方。我们已经知道,就是在截距向量(日晷指针)上的投影。自然,就是对截距项投影残差向量,也就是在日晷盘上的投影。

日晷所处空间的n是3。如果我们对抽样许多次,就会看到三维空间中各个方向对称的标准正态分布散点图。这些散点图在日晷盘上的投影就是二维空间标准正态分布散点图。日晷盘中这些点对应向量的长度平方自然是的抽样。