费雪变换(英语:Fisher transformation)是统计学中用于相关系数假设检验的一种方法。对样本相关系数进行费雪变换后,可以用来检验关于总体相关系数ρ的假设。[1][2] 费雪变换 定义 已知N组双变量样本(Xi, Yi), i = 1, ..., N,样本相关系数r为 r = ∑ i = 1 N ( X i − X ¯ ) ( Y i − Y ¯ ) ∑ i = 1 N ( X i − X ¯ ) 2 ∑ i = 1 N ( Y i − Y ¯ ) 2 {\displaystyle r={\frac {\sum _{i=1}^{N}(X_{i}-{\bar {X}})(Y_{i}-{\bar {Y}})}{{\sqrt {\sum _{i=1}^{N}(X_{i}-{\bar {X}})^{2}}}{\sqrt {\sum _{i=1}^{N}(Y_{i}-{\bar {Y}})^{2}}}}}} 于是,r的费雪变换可定义为 z := 1 2 ln ( 1 + r 1 − r ) = arctanh ( r ) . {\displaystyle z:={1 \over 2}\ln \left({1+r \over 1-r}\right)=\operatorname {arctanh} (r).} 当(X, Y)为二元正态分布且(Xi, Yi)对相互独立时,z近似为正态分布。其均值为 1 2 ln ( 1 + ρ 1 − ρ ) , {\displaystyle {1 \over 2}\ln \left({{1+\rho } \over {1-\rho }}\right),} 标准差为 1 N − 3 , {\displaystyle {1 \over {\sqrt {N-3}}},} 其中N是样本大小,ρ是变量X与Y的总体相关系数。 费雪变换及其逆变换 r = exp ( 2 z ) − 1 exp ( 2 z ) + 1 = tanh ( z ) , {\displaystyle r={{\exp(2z)-1} \over {\exp(2z)+1}}=\operatorname {tanh} (z),} 可以用于构造ρ的置信区间。 参考文献 [1]Fisher, R.A. Frequency distribution of the values of the correlation coefficient in samples of an indefinitely large population. Biometrika (Biometrika Trust). 1915, 10 (4): 507–521. JSTOR 2331838. doi:10.2307/2331838. [2]Fisher, R.A. On the `probable error' of a coefficient of correlation deduced from a small sample (PDF). Metron. 1921, 1: 3–32 [2015-09-03]. (原始内容存档 (PDF)于2021-02-12). Wikiwand - on Seamless Wikipedia browsing. On steroids.