nlmm (version 1.1.1)

generate.data: Simulate Data from Mixed-Effects Models

Description

This function generates data from a 2-level hierarchical design.

Usage

generate.data(R, n, M, sigma_1 = NULL, sigma_2 = NULL,
	shape_1 = NULL, shape_2 = NULL, dist.u, dist.e,
	beta, gamma, fixed = FALSE, seed = round(runif(1,1,1000)))

Value

nlmm returns an object of class

nlmm.

The function summary is used to obtain and print a summary of the results.

An object of class nlmm is a list containing the following components:

Y

a matrix \(R x N\), where \(N = n x M\), with responses

X

an array \(N x 3 x R\) with fixed design matrix

group

vector of length \(N\) with cluster labels

u

an array \(M x 2 x R\) with random effects

e

a matrix \(R x N\) with errors

Arguments

R

number of replications.

n

number of observations within cluster.

M

number of clusters.

sigma_1

scale parameter for the random effects.

sigma_2

scale parameter for the errors.

shape_1

shape parameter for the random effects.

shape_2

shape parameter for the errors.

dist.u

distribution of the random effects.

dist.e

distribution of the errors.

beta

vector of coefficients for fixed effects.

gamma

vector of coefficients for heteroscedasticity.

fixed

logical flag. See details.

seed

seed for random number generation.

Author

Marco Geraci

Details

This function generates data as in the simulation study by Geraci and Farcomeni (2020). The data-generating model is $$ y[ij] = \beta[0] + \beta[1]x[ij] + \beta[2]z[ij] + u[i] + v[i]x[ij] + (\gamma[0] + \gamma[1]x[ij])e[ij] $$ where \((u[i],v[i])\) follows a distribution with scale sigma_1 and shape shape_1, and \(e\) follows a distribution with scale sigma_2 and shape shape_2.

The scale parameter sigma_1 must be a 1 by 1 or a 2 by 2 matrix. In the former case, the model will include only random intercepts. In the latter case, then both random intercepts and slopes will be included. Currently, no more than 2 random effects can be specified. The scale parameter sigma_2 must be a matrix n by n.

The options for dist.u and dist.e are: multivariate normal ("norm") (rmvnorm), multivariate symmetric Laplace ("laplace") (rmal), multivariate symmetric generalized Laplace ("genlaplace") rmgl, and multivariate Student's t ("t") (rmvt).

The shape parameter specifies the degrees of freedom for Student's t and chi-squared, and the kurtosis of the generalized Laplace.

The values \(x[ij]\) are generated as \(x[ij] = \delta[i] + \zeta[ij]\), where \(\delta[i]\) and \(\zeta[ij]\) are independent standard normal. If the argument fixed = TRUE, then \(x[ij] = j\). The values \(z[ij]\) are generated from Bernoullis with probability 0.5.

References

Geraci, M. and Farcomeni A. (2020). A family of linear mixed-effects models using the generalized Laplace distribution. Statistical Methods in Medical Research, 29(9), 2665-2682.

See Also

nlmm

Examples

# Simulate 10 replications from a homoscedastic normal mixed model.
generate.data(R = 10, n = 3, M = 5, sigma_1 = diag(2), sigma_2 = diag(3),
shape_1 = NULL, shape_2 = NULL, dist.u = "norm", dist.e = "norm",
beta = c(1,2,1), gamma = c(1,0))

# Simulate 10 replications from a generalized Laplace. Note: the shape
# parameter that is passed to rmgl corresponds to the reciprocal of the
# parameter alpha in Geraci and Farcomeni (2020)
generate.data(R = 10, n = 3, M = 5, sigma_1 = diag(2), sigma_2 = diag(3),
shape_1 = 1/0.5, shape_2 = 1/0.5, dist.u = "genlaplace", dist.e = "genlaplace",
beta = c(1,2,1), gamma = c(1,0))