mipfp (version 3.2.1)

ObtainMultBinaryDist: Generating a multivariate Bernoulli joint-distribution

Description

This function applies the IPFP procedure to obtain a joint distribution of \(K\) multivariate binary (Bernoulli) variables \(X_1\), ..., \(X_K\).

It requires as input the odds ratio or alternatively the correlation as a measure of association between all the binary variables and a vector of marginal probabilities.

This function is useful when one wants to simulate and draw from a multivariate binary distribution when only first order (marginal probabilities) and second order moments (correlation or odds ratio) are available.

Usage

ObtainMultBinaryDist(odds = NULL, corr = NULL, marg.probs, ...)

Arguments

odds

A \(K \times K\) matrix where the \(i\)-th row and the \(j\)-th column represents the Odds ratio between variables \(i\) and \(j\). Must be provided if corr is not.

corr

A \(K \times K\) matrix where the \(i\)-th row and the \(j\)-th column represents the correlation between variables \(i\) and \(j\). Must be provided if odds is not.

marg.probs

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

...

Additional arguments that can be passed to the Ipfp function such as tol, iter, print and compute.cov.

Value

A list whose elements are mainly determined by the Ipfp function.

joint.proba

The resulting multivariate joint-probabilities (from Ipfp).

stp.crit

The final value of the Ipfp stopping criterion.

conv

Boolean indicating whether the Ipfp algorithm converged to a solution.

check.margins

A list returning, for each margin, the absolute maximum deviation between the desired and generated margin. Ideally the elements should approximate 0 (from Ipfp).

label

The names of the variables.

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

Ipfp for the function used to estimate the distribution; RMultBinary to simulate the estimated joint-distribution; Corr2Odds and Odds2Corr to convert odds ratio to correlation and conversely.

Examples

# NOT RUN {
# initial odds ratios 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)

# estimating the joint-distribution
p.joint <- ObtainMultBinaryDist(odds = or, corr = NULL, marg.probs = p)
print(p.joint$joint.proba)

# obtain identical solution when providing correlation
corr <- Odds2Corr(odds = or, marg.probs = p)$corr
p.joint.alt <- ObtainMultBinaryDist(corr = corr, marg.probs = p)

# checking if the results are truly identicals
diff <- sum(abs(p.joint.alt$joint.proba - p.joint$joint.proba))
cat('Sum of the absolute deviations: ', diff, '\n')
# }