Odds2PairProbs {mipfp}

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

Note

If we denote P(X_i = 1, X_j = 1) by h_{ij}, and P(X_i = 1) by p_i, then it can be shown that

O_{ij} = \frac{h_{ij} * (1 - p_i - p_j + h_{ij})}{ ((p_i - h_{ij}) * (p_j - h_{ij}))}

Author(s)

Thomas Suesse.

Maintainer: Johan Barthelemy [e-mail].

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

Corr2PairProbs for converting the correlation to pairwise probability.

Examples

# 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)