Corr2Odds: Converting correlation to odds ratio

Description

For $K$ binary (Bernoulli) random variables $X_1$, ..., $X_K$, this function transforms the correlation measure of association $C_{ij}$ between every pair $(X_i, X_j)$ to the odds ratio $O_{ij}$ where $$C_{ij} = \frac{cov(X_i, X_j)}{\sqrt{var(X_i) * var(X_j)}}$$ and $$ O_{ij} = \frac{P(X_i = 1, X_j = 1) * P(X_i = 0, X_j = 0)} {P(X_i = 1, X_j = 0) * P(X_i = 0, X_j = 1)}. $$

Usage

Corr2Odds(corr, marg.probs)

Arguments

corr

A $K \times K$ matrix where the $i$-th row and the $j$-th column represents the correlation $C_{ij}$ between variables $i$ and $j$.

marg.probs

A vector with $K$ elements of marginal probabilities where the $i$-th entry refers to $P(X_i = 1)$.

Value

The function return a list with the correlations and the pairwise probabilities.

odds

A matrix of the same dimension as corr containing the correlations

pair.proba

A matrix of the same dimension as corr containing the pairwise probabilities.

References

Lee, A.J. (1993). Generating Random Binary Deviates Having Fixed Marginal Distributions and Specified Degrees of Association The American Statistician 47 (3): 209-215.

Qaqish, B. F., Zink, R. C., and Preisser, J. S. (2012). Orthogonalized residuals for estimation of marginally specified association parameters in multivariate binary data. Scandinavian Journal of Statistics 39, 515-527.

Examples

Run this code

# NOT RUN {
# correlation matrix from Qaqish et al. (2012)
cr <- matrix(c( 1.000, -0.215, 0.144, 0.107,
               -0.215,  1.000, 0.184, 0.144,
                0.144,  0.184, 1.000, 0.156,
                0.107,  0.144, 0.156, 1.000), nrow = 4, ncol = 4, byrow = TRUE)
rownames(cr) <- colnames(cr) <- c("Parent1", "Parent2", "Sibling1", "Sibling2")

# hypothetical marginal probabilities
p <- c(0.2, 0.4, 0.6, 0.8)

# converting correlation to odds ratio and getting pairwise probabilities
or <- Corr2Odds(corr = cr, marg.probs = p)
print(or)
# }