# Computing the "intermediate" correlation matrix for the multivariate standard normal in order to achieve the "target" correlation matrix for the multivariate discrete variable The function computes the correlation matrix of the $k$-dimensional standard normal r.v. yielding the desired correlation matrix `Sigma` for the $k$-dimensional r.v. with desired marginal distributions `marginal` ```r ordcont(marginal, Sigma, support = list(), Spearman = FALSE, epsilon = 1e-06, maxit = 100, df=Inf) ``` ## 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 target correlation matrix of the discrete variables - `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 - `epsilon`: the maximum tolerated error between target and actual correlations - `maxit`: the maximum number of iterations allowed for the algorithm - `df`: the degrees of freedom of the multivariate Student's t ## Returns a list of five elements - **SigmaC**: the correlation matrix of the multivariate standard normal variable - **SigmaO**: the actual correlation matrix of the discretized variables (it should approximately coincide with the target correlation matrix `Sigma`) - **Sigma**: the target correlation matrix of the discrete variables - **niter**: a matrix containing the number of iterations performed by the algorithm, one for each pair of variables - **maxerr**: the actual maximum error (the maximum absolute deviation between actual and target correlations of the discrete variables) ## Author(s) Alessandro Barbiero, Pier Alda Ferrari ## Note For some choices of `marginal` and `Sigma`, there may not exist a feasible $k$-variate probability mass function or the algorithm may not provide a feasible correlation matrix `SigmaC`. In this case, the procedure stops and exits with an error. The value of the maximum tolerated absolute error `epsilon` on the elements of the correlation matrix for the target r.v. can be set by the user: a value between 1e-6 and 1e-2 seems to be an acceptable compromise assuring both the precision of the results and the convergence of the algorithm; moreover, a maximum number of iterations can be chosen (`maxit`), in order to avoid possible endless loops ## See Also `contord`, `ordsample`, `corrcheck` ## Examples ```r # consider a 4-dimensional ordinal variable k <- 4 # with different number of categories kj <- 2:5 # and uniform marginal distributions with variable number of categories marginal <- list(0.5, (1:2)/3, (1:3)/4, (1:4)/5) corrcheck(marginal) # and the following correlation matrix Sigma <- matrix(c(1,0.5,0.4,0.3,0.5,1,0.5,0.4,0.4,0.5,1,0.5,0.3,0.4,0.5,1), 4, 4, byrow=TRUE) Sigma # the correlation matrix of the standard 4-dimensional standard normal # ensuring Sigma is res <- ordcont(marginal, Sigma) res[[1]] # change some marginal distributions marginal <- list(0.3, c(1/3, 2/3), c(1/5, 2/5, 3/5), c(0.1, 0.2, 0.4, 0.6)) corrcheck(marginal) # and notice how the correlation matrix of the multivariate normal changes... res <- ordcont(marginal, Sigma) res[[1]] # change Sigma, adding a negative correlation Sigma[1,2] <- -0.2 Sigma[2,1] <- Sigma[1,2] Sigma # checking whether Sigma is still positive definite eigen(Sigma)$values # all >0, OK res <- ordcont(marginal, Sigma) res[[1]] # consider now a multivariate Student's t with 10 dof res.t <- ordcont(marginal, Sigma, df=10) res.t$SigmaC res.t$SigmaO ```

ordcont function