Correlation Ratio

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

$$N \equiv \sum N_i$$ (1)

$$\bar y_i \equiv {1\over N_i} \sum_\alpha y_{i\alpha }$$ (2)

$$\bar y \equiv {1\over N} \sum_i \sum_\alpha y_{i\alpha }.$$ (3)

Then

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

Let $\eta_{yx}$ be the population correlation ratio. If $N_i=N_j$ for $i\not=j$, then

$$f(E^2) = {e^{-\lambda}(E^2)^{a-1}(1-E^2)^{b-1}{}_1F_1(a,b;\lambda E^2)\over B(a,b)},$$ (5)

where

$$\lambda \equiv {N\eta^2\over 2(1-\eta^2)}$$ (6)

$$a \equiv {n_1\over 2}$$ (7)

$$b \equiv {n_2\over 2},$$ (8)

and ${}_1F_1(a,b;z)$ is the Confluent Hypergeometric Limit Function. If $\lambda=0$, then

$$f(E^2)=\beta(a,b)$$ (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.