# Correlations of discretized variables The function computes the correlation matrix of the $k$ variables, with given marginal distributions, derived discretizing a $k$-variate standard normal or Student's $t$ variable with given correlation matrix ```r contord(marginal, Sigma, support = list(), Spearman = FALSE, df=Inf, integerdf=TRUE, prob=FALSE) ``` ## Arguments - `marginal`: a list of $k$ elements, where $k$ is the number of variables. The $i$-th element of `marginal` is the vector of the cumulative probabilities defining the marginal distribution of the $i$-th component of the multivariate variable. If the $i$-th component can take $k_i$ values, the $i$-th element of `marginal` will contain $k_i-1$ probabilities (the $k_i$-th is obviously 1 and shall not be included). - `Sigma`: the correlation matrix of the standard multivariate normal variable - `support`: a list of $k$ elements, where $k$ is the number of variables. The $i$-th element of `support` is the vector containing the ordered values of the support of the $i$-th variable. By default, the support of the $i$-th variable is $1,2,...,k_i$ - `Spearman`: if `TRUE`, the function finds Spearman's correlations (and it is not necessary to provide `support`), if `FALSE` (default) Pearson's correlations - `df`: the degrees of freedom of the multivariate Student's $t$ - `integerdf`: if `TRUE` uses `pmvt`, which requires an integer value for `df`; otherwise, uses `pmvt.alt`, which integrates `dmvt` with possibly non-integer `df` - `prob`: if `TRUE`, it return also the probability table of the bivariate discrete random variable ## Returns the correlation matrix of the discretized variables; if `prob=TRUE`, it returns a list containing the correlation along with the probability table, in case of a bivariate random variable ## Author(s) Alessandro Barbiero, Pier Alda Ferrari ## See Also `ordcont`, `ordsample`, `corrcheck` ## Examples ```r # Example 1 # consider a bivariate discrete random vector k <- 2 # with these cumulative margins marginal <- list(c(1/3, 2/3), c(0.1, 0.3, 0.6)) # generated discretizing a multivariate standard normal variable # with correlation matrix Sigma <- matrix(c(1, .75, .75, 1), 2, 2) # the resulting joint distribution and correlation matrix # for the bivariate discrete random vector are res <- contord(marginal, Sigma, prob=TRUE) res$pij res$SigmaO # let's check the margins are those assigned cumsum(margin.table(res$pij,1)) cumsum(margin.table(res$pij,2)) # -> OK # Example 2 # consider 4 discrete variables k <- 4 # with these marginal distributions marginal <- list(0.4,c(0.3,0.6), c(0.25,0.5,0.75), c(0.1,0.2,0.8,0.9)) # generated discretizing a multivariate standard normal variable # with correlation matrix Sigma <- matrix(0.5,4,4) diag(Sigma) <- 1 # the resulting correlation matrix for the discrete variables is contord(marginal, Sigma) # note all the correlations are smaller than the original 0.6 # change Sigma, adding a negative correlation Sigma[1,2] <- -0.15 Sigma[2,1] <- Sigma[1,2] Sigma # checking whether Sigma is still positive definite eigen(Sigma)$values # all >0, OK contord(marginal, Sigma) # Example 2 # the same margins and the same correlation matrix # but now we consider a 4-variate Students's t with df=3 contord(marginal, Sigma, df=3) # -> a slight reduction in magnitude for all the correlations ```

contord function