# CLME-package: Constrained inference for linear mixed models. In CLME: Constrained Inference for Linear Mixed Effects Models

## Description

Constrained inference on linear fixed and mixed models using residual bootstrap. Covariates and random effects are permitted but not required.

Appropriate credit should be given when publishing results obtained using CLME, or when developing other programs/packages based off of this one. Use `citation(package="CLME")` for Bibtex information.

The work was produced in part with funding from the Intramural Research Program of the NIH, National Institute of Environmental Health Sciences (Z01 ES101744).

## Details

This package was introduced in Jelsema and Peddada (2016). The primary function is `clme`. The other functions in this package may be run separately, but in general are designed for use by `clme`.

The method which is implemented is the constrained linear mixed effects model described in Farnan, Ivanova, and Peddada (2014). See that paper for more details regarding the method. Here we give a brief overview of the assumed model:

Y = X1*theta1 + X2*theta2 + U*xi + e

where

• X1 is a N x p1 design matrix.

• theta1 are the coefficients (often treatment effects).

• X2 is a N x p2 matrix of fixed covariates.

• theta2 are the coefficients for the covariates.

• U is a N x c matrix of random effects.

• xi is a zero-mean random vector with covariance T (see below).

• e is a zero-mean random vector with covariance Sigma (see below).

Neither covariates (X2) nor random effects (U) are required by the model or CLME. The covariance matrix of xi is given by:

T = diag( tau1^2 I_c1, tau2^2 I_c2 , ... , tauq^2 I_cq)

The first c1 random effects will share a common variance, tau1^2, the next c2 random effects will share a common variance, and so on. Note that c = SUM(ci), i=1,...q. Homogeneity of variances in the random effects can be induced by letting q=1 (hence c1=c=ncol(U)).

Similarly, the covariance matrix of e is given by:

Sigma = diag( sigma1^2 I_n1, sigma2^2 I_n2 , ... , sigmak^2 I_nk)

Again, the first n1 observations will share a common variance, sigma1^2, the next n2 will share a common variance, and so on. Note that N = SUM(n_i), i=1,...k. Homogeneity of variances in the residuals can be induced by letting k=1.

The order constraints are defined by the matrix A. This is an r x p matrix where r is the number of constraints, and p = p1 + p2 is the dimension of theta = ( theta1' , theta2')'. Formally the hypothesis being tested is:

Ha: A*theta > 0

For several default orders (simple, umbrella, simple tree) the A matrix can be automatically generated. Alternatively, the user may define a custom A matrix to test other patterns among the elements of theta. See `create.constraints` and `clme` for more details.

For computational reasons, the implementation is not identical to the model expressed. Particularly, the fixed-effects matrix (or matrices) and the random effects matrix are assumed to be columns in a data frame, not passed as matrices. The A matrix is not r x p, but r x 2, where each row gives the indices of the constrained coefficients. See `create.constraints` for further explanation.

The creation of this package CLME, this manual, and the vignette were all supported by the Intramural Research Program of the United States' National Institutes of Health (Z01 ES101744).

## Author(s)

Maintainer: Casey M. Jelsema jelsema.casey@gmail.com

Authors:

## References

Jelsema, C. M. and Peddada, S. D. (2016). CLME: An R Package for Linear Mixed Effects Models under Inequality Constraints. Journal of Statistical Software, 75(1), 1-32. doi:10.18637/jss.v075.i01

Farnan, L., Ivanova, A., and Peddada, S. D. (2014). Linear Mixed Efects Models under Inequality Constraints with Applications. PLOS ONE, 9(1). e84778. doi: 10.1371/journal.pone.0084778