Odds2PairProbs: Converting odds ratio to pairwise probability

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 pairwise probability $P(X_i = 1, X_j = 1)$, where $O_{ij}$ is defined as $$ O_{ij} = \frac{P(X_i = 1, X_j = 1) * P(X_i = 0, X_j = 0)} {P(X_i = 1, X_j = 0) * P(X_i = 0, X_j = 1)}. $$

Usage

Odds2PairProbs(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

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.

Examples

Run this code

# 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)

# getting the pairwise probabilities
pp <- Odds2PairProbs(odds = or, marg.probs = p)
print(pp)
# }