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Correlation Ratio

Let there be $N_i$ observations of the $i$th phenomenon, where $i=1$, ..., $p$ and

$\displaystyle N$ $\textstyle \equiv$ $\displaystyle \sum N_i$ (1)
$\displaystyle \bar y_i$ $\textstyle \equiv$ $\displaystyle {1\over N_i} \sum_\alpha y_{i\alpha }$ (2)
$\displaystyle \bar y$ $\textstyle \equiv$ $\displaystyle {1\over N} \sum_i \sum_\alpha y_{i\alpha }.$ (3)

Then
\begin{displaymath}
{E_{yx}}^2 \equiv {\sum_i N_i(\bar y_i-\bar y)^2\over \sum_i \sum_\alpha
(y_{i\alpha}-\bar y)^2}.
\end{displaymath} (4)

Let $\eta_{yx}$ be the population correlation ratio. If $N_i=N_j$ for $i\not=j$, then
\begin{displaymath}
f(E^2) = {e^{-\lambda}(E^2)^{a-1}(1-E^2)^{b-1}{}_1F_1(a,b;\lambda E^2)\over B(a,b)},
\end{displaymath} (5)

where
$\displaystyle \lambda$ $\textstyle \equiv$ $\displaystyle {N\eta^2\over 2(1-\eta^2)}$ (6)
$\displaystyle a$ $\textstyle \equiv$ $\displaystyle {n_1\over 2}$ (7)
$\displaystyle b$ $\textstyle \equiv$ $\displaystyle {n_2\over 2},$ (8)

and ${}_1F_1(a,b;z)$ is the Confluent Hypergeometric Limit Function. If $\lambda=0$, then
\begin{displaymath}
f(E^2)=\beta(a,b)
\end{displaymath} (9)

(Kenney and Keeping 1951, pp. 323-324).

See also Correlation Coefficient, Regression Coefficient


References

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.




© 1996-9 Eric W. Weisstein
1999-05-25