Provides analytic derivatives and information matrices for
    fitted linear mixed effects (lme) models and generalized least squares (gls) models
    estimated using lme() (from package 'nlme') and gls() (from package 'nlme'), respectively.
    The package includes functions for estimating the sampling variance-covariance of variance
    component parameters using the inverse Fisher information. The variance components include
    the parameters of the random effects structure (for lme models), the variance structure,
    and the correlation structure. The expected and average forms of the Fisher information matrix
    are used in the calculations, and models estimated by full maximum likelihood or
    restricted maximum likelihood are supported. The package also includes a function for estimating
    standardized mean difference effect sizes (Pustejovsky, Hedges, and Shadish (2014) <DOI:10.3102/1076998614547577>)
    based on fitted lme or gls models.

The main function of the package is to perform backward selection of fixed effects, forward fitting of the random effects, and post-hoc analysis using parallel capabilities. Other functionality includes the computation of ANOVAs with upper- or lower-bound p-values and R-squared values for each model term, model criticism plots, data trimming on model residuals, and data visualization. The data to run examples is contained in package LCF_data.

Bootstrap routines for nested linear mixed effects models fit using
    either 'lme4' or 'nlme'. The provided 'bootstrap()' function implements the
    parametric, residual, cases, random effect block (REB), and wild bootstrap 
    procedures. An overview of these procedures can be found 
    in Van der Leeden et al. (2008) <doi: 10.1007/978-0-387-73186-5_11>, 
    Carpenter, Goldstein & Rasbash (2003) <doi: 10.1111/1467-9876.00415>,
    and Chambers & Chandra (2013) <doi: 10.1080/10618600.2012.681216>.

We provide a solution for performing permutation tests on linear and mixed linear regression models. It
             allows users to obtain accurate p-values without making distributional assumptions about the data. By 
             generating a null distribution of the test statistics through repeated permutations of the response variable,
             permutation tests provide a powerful alternative to traditional parameter tests (Holt et al. (2023) 
             <doi:10.1007/s10683-023-09799-6>). In this early version, we focus on the permutation tests over observed 
             t values of beta coefficients, i.e.original t values generated by parameter tests. After generating a null 
             distribution of the test statistic through repeated permutations of the response variable, each observed t 
             values would be compared to the null distribution to generate a p-value. To improve the efficiency,a stop 
             criterion (Anscombe (1953) <doi:10.1111/j.2517-6161.1953.tb00121.x>) is adopted to force permutation to stop 
             if the estimated standard deviation of the value falls below a fraction of the estimated p-value. By doing so,
             we avoid the need for massive calculations in exact permutation methods while still generating stable and accurate 
             p-values.