# Creating an S3 object for computing the product moment `mtrunmnt` creates an S3 object designed to compute the product moment for a truncated multivariate normal distribution, utilizing the algorithm by Lee (2020). Key attributes of this object are the nodes and weights of the multivariate Gaussian quadrature and the probability of the truncation interval ```r mtrunmnt( mu, lower = -Inf, upper = Inf, Sigma = 1, Sigmae = 1, Z = matrix(rep(1, length(mu)), ncol = 1), D = matrix(1, ncol = 1, nrow = 1), nGH = 35 ) ``` ## Arguments - `mu`: **(Required)** Mean vector of the parent multivariate normal distribution. - `lower`: vector of lower limits. If the lower limits are the same, a scalar value can be given. Defaults to -Inf. - `upper`: Vector of upper limits. If the upper limits are the same, a scalar value can be given. Defaults to Inf. - `Sigma`: The variance-covariance matrix of the parent multivariate normal distribution. It must be given a symmetric positive definite matrix, if Sigmae, D and Z are not specified. - `Sigmae`: Vector of variances of error terms. If the variances are the same, a scalar value can be given. Defaults to 1. - `Z`: Design matrix for the random components. Defaults to $n \times 1$ matrix of 1's where $n$ is the dimension of mu. It must be specified carefully with the argument D. `ncol(Z)` determines the dimension of D. - `D`: Variance-covariance matrix of $u$. See Details. If the random components are independent, you can specify either a vector of variances or a scalar value. A scalar value means that the random components have the same variance. Defaults to 1. - `nGH`: Number of quadrature points. Defaults to 35. ## Returns A mtrunmnt object ## Details Assume the parent multivariate normal distribution comes from a mixed-effects linear model: $$ \mathbf{Y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Zu} +\boldsymbol{\epsilon} $$ where $\mathbf{X}$ and $\mathbf{Z}$ are design matrices corresponding to $\boldsymbol{\beta}$ and $\mathbf{u}$ representing fixed and random effects, respectively, and $\boldsymbol{\epsilon}$ is the vector of errors. It is assumed that the random effects $\mathbf{u}$ follows a multivariate normal distribution with mean $\mathbf{0}$, and symmetric positive definite variance-covariance matrix $\mathbf{D}$. As usual, the distribution of $\boldsymbol{\epsilon}$ is assumed to be a multivariate normal with mean $\mathbf{0}$ and variance-covariance matrix $\sigma^2_{\epsilon} \mathbf{I}$, but for more flexibility, it can be assumed that the error terms are independent, but do not have equal variance. That is, **we assume** c("$\\boldsymbol{\\epsilon} \\sim \n$", "$ N(\\mathbf{0}, \\mathbf{E})$") where $\mathbf{E}$ is a diagonal matrix. Then, $$ \mathbf{Y} \sim N(\boldsymbol{\mu}, \mathbf{\Sigma}) $$ where $\mathbf{\Sigma} = \mathbf{Z}\mathbf{D}\mathbf{Z}' + \mathbf{E}$. The variance-covariance structure in `mtrunmnt` can thus be specified either by providing the individual components $\mathbf{D}, \mathbf{Z}$, and $\mathbf{E}$, or by directly supplying the resulting overall variance-covariance matrix $boldsymbol{\Sigma}$. ## Examples ```r ### Create a mtrunmnt objective ### set.seed(123) sigma2e <- 1 sigma2a <- 2 n <- 5 mu <- seq(-1,1, length.out = n) y <- mu + rnorm(1, sd = sqrt(sigma2a)) + rnorm(n, sd = sqrt(sigma2e)) S <- matrix(sigma2a, ncol = n, nrow = n) + diag(sigma2e, n) a <- rep(-Inf, n) b <- rep(Inf, n) a[y >= 0] <- 0 b[y < 0] <- 0 obj1 <- mtrunmnt(mu, lower = a, upper = b, Sigmae = sigma2e, D = sigma2a) obj2 <- mtrunmnt(mu, lower = a, upper = b, Sigma = S) probntrun(obj1) probntrun(obj2) prodmnt(obj1, c(2,2,0,0,0)) prodmnt(obj2, c(2,2,0,0,0)) meanvar(obj1) meanvar(obj2) ``` ## References Lee, S.-C. (2020). Moments calculation for truncated multivariate normal in nonlinear generalized mixed models. **Communications for Statistical Applications and Methods**, Vol. 27, No. 3, 377–383.

mtrunmnt function