lqmm: Fitting Linear Quantile Mixed Models

In lqmm: Linear Quantile Mixed Models

Fitting Linear Quantile Mixed Models

Description

lqmm is used to fit linear quantile mixed models based on the asymmetric Laplace distribution.

Usage

lqmm(fixed, random, group, covariance = "pdDiag", tau = 0.5,
	nK = 7, type = "normal", rule = 1, data = sys.frame(sys.parent()),
	subset, weights, na.action = na.fail, control = list(),
	contrasts = NULL, fit = TRUE) 

Arguments

fixed	an object of class formula for fixed effects: a symbolic description of the model to be fitted.
random	a one-sided formula of the form ~x1 + x2 + ... + xn for random effects: a symbolic description of the model to be fitted.
group	grouping factor.
covariance	variance–covariance matrix of the random effects. Default is pdDiag (see details).
tau	the quantile(s) to be estimated.
nK	number of quadrature knots.
type	type of quadrature "c("normal","robust")" (see details).
rule	quadrature rule (see details).
data	an optional data frame containing the variables named in fixed, random and group. By default the variables are taken from the environment from which lqmm is called.
subset	an optional vector specifying a subset of observations to be used in the fitting process.
weights	an optional vector of weights to be used in the fitting process of the same length as the number of rows of data. Weights are given to clusters, therefore units within the same cluster receive the same weight (see details).
na.action	a function that indicates what should happen when the data contain NAs. The default action (na.fail) causes lqmm to print an error message and terminate if there are any incomplete observations.
control	list of control parameters of the fitting process. See lqmmControl.
contrasts	not yet implemented.
fit	logical flag. If FALSE the function returns a list of arguments to be passed to lqmm.fit.

Details

The function computes an estimate on the tau-th quantile function of the response, conditional on the covariates, as specified by the formula argument, and on random effects, as specified by the random argument. The quantile predictor is assumed to be linear. The function maximizes the (log)likelihood of the Laplace regression proposed by Geraci and Bottai (2014). The likelihood is numerically integrated via Gaussian quadrature techniques. The optimization algorithm is based on the gradient of the Laplace log–likelihood (control = list(method = "gs")). An alternative optimization algorithm is based on a Nelder-Mead algorithm (control = list(method = "df")) via optim. The scale parameter is optimized in a refinement step via optimize.

Quadrature approaches include Gauss-Hermite (type = "normal") and Gauss-Laguerre (type = "robust") quadrature. The argument rule takes one of the following: 1 (product rule quadrature), 2 (sparse grid quadrature), 3 (nested quadrature rule - only for type = "normal"), 4 (quadrature rule with the smallest number of nodes between rules 1 or 2). Rules 2 and 3 have not yet been tested extensively.

Different standard types of positive–definite matrices for the random effects can be specified: pdIdent multiple of an identity; pdCompSymm compound symmetry structure (constant diagonal and constant off–diagonal elements); pdDiag diagonal; pdSymm general positive–definite matrix, with no additional structure.

Weights are given to clusters, therefore it is expected that these are constant within cluster. When the weights are specified in the main call, then the first value by group in the vector weights will be replicated for the same length of each group. Alternatively, different weights within the same cluster can be introduced with a direct call to lqmm.fit.gs or lqmm.fit.df.

The lqmm vignette can be accessed by typing help(package = "lqmm") and then following the link 'User guides, package vignettes and other documentation'.

Value

lqmm returns an object of class lqmm.

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

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

theta	a vector containing fixed regression coefficients and parameters of the variance–covariance matrix of the random effects. See VarCorr.lqmm to extract the variance–covariance of the random effects from an "lqmm" object.
theta_x,theta_z	partition of theta: fixed regression coefficients (theta_x) and unique variance–covariance parameters (theta_z).
scale	the scale parameter.
gradient	the gradient (control = list(method = "gs")).
logLik	the log–likelihood.
opt	details on optimization (see lqmm.fit.gs and lqmm.fit.df).
call	the matched call.
nn	column names of mmf.
mm	column names of mmr.
nobs	the number of observations.
dim_theta	the number of columns in mmf and mmr.
dim_theta_z	the length of theta_z.
edf	length of theta.
rdf	the number of residual degrees of freedom.
df	edf + 1 (scale parameter).
tau	the estimated quantile(s).
mmf	the model matrix – fixed effects.
mmr	the model matrix – random effects.
y	the model response.
revOrder	original order of observations (now ordered according to group).
weights	the likelihood weights used in the fitting process (a vector of 1's if weights is missing or NULL).
group	the grouping factor.
ngroups	the number of groups.
QUAD	quadrature nodes and weights.
type	the type of quadrature.
rule	quadrature rule.
InitialPar	starting values for theta.
control	list of control parameters used for optimization (see lqmmControl).
cov_name	class of variance-covariance matrix for the random effects.
mfArgs	arguments for model.frame to return the full data frame.

Note

Updates/FAQ/news are published here https://marcogeraci.wordpress.com/. New versions are usually published here https://github.com/marco-geraci/lqmm/ before going on CRAN.

Author(s)

Marco Geraci

References

Genz A, and Keister BD (1996). Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight. Journal of Computational and Applied Mathematics, 71(2), 299–309. <doi:10.1016/0377-0427(95)00232-4>

Geraci M (2014). Linear quantile mixed models: The lqmm package for Laplace quantile regression. Journal of Statistical Software, 57(13), 1–29. <doi:10.18637/jss.v057.i13>

Geraci M and Bottai M (2007). Quantile regression for longitudinal data using the asymmetric Laplace distribution. Biostatistics 8(1), 140–154. <doi:10.1093/biostatistics/kxj039>

Geraci M and Bottai M (2014). Linear quantile mixed models. Statistics and Computing, 24(3), 461–479. <doi:10.1007/s11222-013-9381-9>.

Heiss F, and Winschel V (2008). Likelihood approximation by numerical integration on sparse grids. Journal of Econometrics, 144(1), 62–80. <doi:10.1016/j.jeconom.2007.12.004>

See Also

lqm, summary.lqmm, coef.lqmm, VarCorr.lqmm, predict.lqmm, residuals.lqmm

Examples

# Test example
set.seed(123)

M <- 50
n <- 10
test <- data.frame(x = runif(n*M,0,1), group = rep(1:M,each=n))
test$y <- 10*test$x + rep(rnorm(M, 0, 2), each = n) + rchisq(n*M, 3)
fit.lqmm <- lqmm(fixed = y ~ x, random = ~ 1, group = group,	
	data = test, tau = 0.5, nK = 11, type = "normal")
fit.lqmm

#Call: lqmm(fixed = y ~ x, random = ~1, group = group, tau = 0.5, nK = 11, 
#    type = "normal", data = test)
#Quantile 0.5 

#Fixed effects:
#(Intercept)            x  
#      3.443        9.258  

#Covariance matrix of the random effects:
#(Intercept) 
#      3.426 

#Residual scale parameter: 0.8697 (standard deviation 2.46)
#Log-likelihood: -1178 

#Number of observations: 500 
#Number of groups: 50 


## Orthodont data
data(Orthodont)

# Random intercept model
fitOi.lqmm <- lqmm(distance ~ age, random = ~ 1, group = Subject,
	tau = c(0.1,0.5,0.9), data = Orthodont)
coef(fitOi.lqmm)

# Random slope model
fitOs.lqmm <- lqmm(distance ~ age, random = ~ age, group = Subject,
	tau = c(0.1,0.5,0.9), cov = "pdDiag", data = Orthodont)

# Extract estimates
VarCorr(fitOs.lqmm)
coef(fitOs.lqmm)
ranef(fitOs.lqmm)

# AIC
AIC(fitOi.lqmm)
AIC(fitOs.lqmm)

lqmm documentation built on April 6, 2022, 5:09 p.m.