mipfp (version 3.2.1)

Odds2Corr: Converting odds ratio to correlation

Description

For \(K\) binary (Bernoulli) random variables \(X_1\), ..., \(X_K\), this function transforms the odds ratios measure of association \(O_{ij}\) between every pair \((X_i, X_j)\) to the correlation \(C_{ij}\) 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

Odds2Corr(odds, marg.probs)

Arguments

odds

A \(K \times K\) matrix where the \(i\)-th row and the \(j\)-th column represents the odds ratio \(O_{ij}\) 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.

corr

A matrix of the same dimension as odds containing the correlations

pair.proba

A matrix of the same dimension as odds 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.

See Also

Corr2Odds for converting correlation to odds ratio.

Examples

# NOT RUN {
# from Qaqish et al. (2012)
or <- matrix(c(Inf, 0.281, 2.214, 2.214,
               0.281, Inf, 2.214, 2.214,
               2.214, 2.214, Inf, 2.185,
               2.214, 2.214, 2.185, Inf), nrow = 4, ncol = 4, byrow = TRUE)
rownames(or) <- colnames(or) <- c("Parent1", "Parent2", "Sibling1", "Sibling2")

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

# converting odds ratio to correlation
corr <- Odds2Corr(odds = or, marg.probs = p)
print(corr)
# }