Linear Quantile Models and Linear Quantile Mixed Models

Description

Fit quantile regression models for independent and hierarchical data

Details

Author(s)

Marco Geraci

Maintainer: Marco Geraci <geraci@mailbox.sc.edu>

References

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>.

Growth curve data on an orthdontic measurement

Description

The Orthodont data frame has 108 rows and 4 columns of the change in an orthdontic measurement over time for several young subjects.

Format

This data frame contains the following columns:

distance

a numeric vector of distances from the pituitary to the pterygomaxillary fissure (mm). These distances are measured on x-ray images of the skull.

age

a numeric vector of ages of the subject (yr).

Subject

an ordered factor indicating the subject on which the measurement was made. The levels are labelled M01 to M16 for the males and F01 to F13 for the females. The ordering is by increasing average distance within sex.

Sex

a factor with levels Male and Female

Details

Investigators at the University of North Carolina Dental School followed the growth of 27 children (16 males, 11 females) from age 8 until age 14. Every two years they measured the distance between the pituitary and the pterygomaxillary fissure, two points that are easily identified on x-ray exposures of the side of the head.

Source

Pinheiro, J. C. and Bates, D. M. (2000), Mixed-Effects Models in S and S-PLUS, Springer, New York. (Appendix A.17)

Potthoff, R. F. and Roy, S. N. (1964), “A generalized multivariate analysis of variance model useful especially for growth curve problems”, Biometrika, 51, 313–326.

Jose Pinheiro, Douglas Bates, Saikat DebRoy, Deepayan Sarkar and the R Development Core Team (2011). nlme: Linear and Nonlinear Mixed Effects Models. R package version 3.1-100. https://CRAN.R-project.org/package=nlme

Extract Variance-Covariance Matrix

Description

This function extracts the variance-covariance matrix of the random effects from a fitted lqmm object.

Usage

## S3 method for class 'lqmm'
VarCorr(x, sigma = NULL, ...)

Arguments

Details

This function returns the variance or the variance-covariance matrix of the random effects. It calls covHandling to manage the output of lqmm.fit.gs or lqmm.fit.df. A post-fitting approximation to the nearest positive (semi)definite matrix (Higham, 2002) is applied if necessary. The generic function VarCorr is imported from the nlme package (Pinheiro et al, 2014).

Author(s)

Marco Geraci

References

Higham N (2002). Computing the Nearest Correlation Matrix - A Problem from Finance. IMA Journal of Numerical Analysis, 22, 329-343.

Pinheiro J, Bates D, DebRoy S, Sarkar D and R Core Team (2014). nlme: Linear and Nonlinear Mixed Effects Models. R package version 3.1-117, https://CRAN.R-project.org/package=nlme.

See Also

lqmm coef.lqmm

Bootstrap functions for LQM and LQMM

Description

This function is used to obtain a bootstrap sample of a fitted LQM or LQMM. It is a generic function.

Usage

boot(object, R = 50, seed = round(runif(1, 1, 10000)), startQR = FALSE)
## S3 method for class 'lqm'
boot(object, R = 50, seed = round(runif(1, 1, 10000)), startQR = FALSE)
## S3 method for class 'lqmm'
boot(object, R = 50, seed = round(runif(1, 1, 10000)), startQR = FALSE)

Arguments

Value

An object of class boot.lqm is a data frame with R rows and npars columns containing the bootstrap estimates of theta. If object contains results for multiple quantiles, boot.lqm returns an array of dimension c(R,npars,nt), where nt is the length of tau.

An object of class boot.lqmm is a data frame with R rows and npars columns containing the bootstrap estimates of theta_x, theta_z, and scale. If object contains results for multiple quantiles, boot.lqmm returns an array of dimension c(R,npars,nt), where nt is the length of tau. The elements of theta_z are labelled with reStruct. See function covHandling and the example below on how to derive the variance-covariance matrix of the random effects starting from theta_z.

The following attributes are available:

Author(s)

Marco Geraci

Examples

# boot.lqm
set.seed(123)
n <- 500
test <- data.frame(x = runif(n,0,1))
test$y <- 30 + test$x + rnorm(n)
fit.lqm <- lqm(y ~ x, data = test, tau = 0.5)
fit.boot <- boot(fit.lqm)
str(fit.boot)

# boot.lqmm
data(Orthodont)
fit <- lqmm(distance ~ age, random = ~ 1, group = Subject,
	tau = 0.5, data = Orthodont)
fit.boot <- boot(fit)
str(fit.boot)

Extract LQM Coefficients

Description

coef extracts model coefficients from lqm, lqm.counts objects.

Usage

## S3 method for class 'lqm'
coef(object, ...)

Arguments

Value

a vector for single quantiles or a matrix for multiple quantiles.

Author(s)

Marco Geraci

See Also

lqm summary.lqm lqm.counts

Extract LQMM Coefficients

Description

coef extracts model coefficients from lqmm objects.

Usage

## S3 method for class 'lqmm'
coef(object, ...)

Arguments

Value

a vector for single quantiles or a matrix for multiple quantiles.

Author(s)

Marco Geraci

See Also

lqmm summary.lqmm

Variance-Covariance Matrix

Description

This is an auxiliary function.

Usage

covHandling(theta, n, cov_name, quad_type)

Arguments

Author(s)

Marco Geraci

See Also

VarCorr.lqmm

The Asymmetric Laplace Distribution

Description

Density, distribution function, quantile function and random generation for the asymmetric Laplace distribution.

Usage

dal(x, mu = 0, sigma = 1, tau = 0.5, log = FALSE)
pal(x, mu = 0, sigma = 1, tau = 0.5)
qal(x, mu = 0, sigma = 1, tau = 0.5)
ral(n, mu = 0, sigma = 1, tau = 0.5)

Arguments

Details

The asymmetric Laplace distribution with parameters (mu, sigma, tau) has density

f(x) = \tau(1-\tau)/\sigma e^{-1/(2\sigma) (\theta max(x,0) + (1 - \theta) max(-x,0))}

Author(s)

Marco Geraci

See Also

lqmm, lqm

Extract Fixed and Random Bootstrapped Parameters

Description

This generic function extracts the fixed and random components of bootstrapped estimates of an lqmm object.

Usage

extractBoot(object, which = "fixed")
## S3 method for class 'boot.lqmm'
extractBoot(object, which = "fixed")

Arguments

Details

The "random" parameters refer to the "raw" parameters of the variance-covariance matrix of the random effects as returned by lqmm.fit.gs and lqmm.fit.df.

Value

a matrix of bootstrapped estimates.

Author(s)

Marco Geraci

See Also

boot.lqmm, lqmm.fit.gs, lqmm.fit.df

Examples

## Orthodont data
data(Orthodont)

# Random intercept model
fit <- lqmm(distance ~ age, random = ~ 1, group = Subject,
	tau = 0.5, data = Orthodont)
fit.boot <- boot(fit)

# extract fixed effects
B <- extractBoot(fit.boot, which = "fixed")

# covariance matrix estimated fixed parameters
cov(B)

Gaussian Quadrature

Description

This function calculates nodes and weights for Gaussian quadrature. See help("gauss.quad") from package statmod.

Author(s)

Original version by Gordon Smyth

Source

Gordon Smyth with contributions from Yifang Hu, Peter Dunn and Belinda Phipson. (2011). statmod: Statistical Modeling. R package version 1.4.11. https://CRAN.R-project.org/package=statmod

Gaussian Quadrature

Description

This function calculates nodes and weights for Gaussian quadrature in terms of probability distributions. See help("gauss.quad.prob") from package statmod.

Author(s)

Original version by Gordon Smyth

Source

Gordon Smyth with contributions from Yifang Hu, Peter Dunn and Belinda Phipson. (2011). statmod: Statistical Modeling. R package version 1.4.11. https://CRAN.R-project.org/package=statmod

Test for Positive Definiteness

Description

This function tests whether all eigenvalues of a symmetric matrix are positive. See help("is.positive.definite") from package corpcor.

Author(s)

Original version by Korbinian Strimmer

Source

Juliane Schaefer, Rainer Opgen-Rhein, Verena Zuber, A. Pedro Duarte Silva and Korbinian Strimmer. (2011). corpcor: Efficient Estimation of Covariance and (Partial) Correlation. R package version 1.6.0. https://CRAN.R-project.org/package=corpcor

Labor Pain Data

Description

The labor data frame has 358 rows and 4 columns of the change in pain over time for several 83 women in labor.

Format

This data frame contains the following columns:

subject

an ordered factor indicating the subject on which the measurement was made. The levels are labelled 1 to 83.

pain

a numeric vector of self–reported pain scores on a 100mm line.

treatment

a dummy variable with values 1 for subjects who received a pain medication and 0 for subjects who received a placebo.

time

a numeric vector of times (minutes since randomization) at which pain was measured.

Details

The labor pain data were reported by Davis (1991) and successively analyzed by Jung (1996) and Geraci and Bottai (2007). The data set consists of repeated measurements of self–reported amount of pain on N = 83 women in labor, of which 43 were randomly assigned to a pain medication group and 40 to a placebo group. The response was measured every 30 min on a 100–mm line, where 0 means no pain and 100 means extreme pain. A nearly monotone pattern of missing data was found for the response variable and the maximum number of measurements for each woman was six.

Source

Davis CS (1991). Semi–parametric and non–parametric methods for the analysis of repeated measurements with applications to clinical trials. Statistics in Medicine 10, 1959–80.

References

Geraci M and Bottai M (2007). Quantile regression for longitudinal data using the asymmetric Laplace distribution. Biostatistics 8(1), 140–154.

Jung S (1996). Quasi–likelihood for median regression models. Journal of the American Statistical Association 91, 251–7.

Extract Log-Likelihood

Description

logLik.lqm extracts the log-likelihood of a fitted LQM.

Usage

## S3 method for class 'lqm'
logLik(object, ...)

Arguments

Author(s)

Marco Geraci

See Also

lqm AIC

Extract Log-Likelihood

Description

logLik.lqmm extracts the log-likelihood of a fitted LQMM.

Usage

## S3 method for class 'lqmm'
logLik(object, ...)

Arguments

Author(s)

Marco Geraci

See Also

lqmm AIC

Fitting Linear Quantile Models

Description

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

Usage

lqm(formula, data, subset, na.action, weights = NULL, tau = 0.5,
	contrasts = NULL, control = list(), fit = TRUE)

Arguments

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. The quantile predictor is assumed to be linear. The function maximizes the (log)likelihood of a Laplace regression which is equivalent to the minimization of the weighted sum of absolute residuals (Koenker and Bassett, 1978). The optimization algorithm is based on the gradient of the Laplace log–likelihood (Bottai, Orsini and Geraci, 2013).

Value

lqm returns an object of class lqm.

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

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

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

Bottai M, Orsini N, Geraci M (2015). A Gradient Search Maximization Algorithm for the Asymmetric Laplace Likelihood, Journal of Statistical Computation and Simulation, 85(10), 1919-1925.

Chen C (2007). A finite smoothing algorithm for quantile regression. Journal of Computational and Graphical Statistics, 16(1), 136-164.

Koenker R and Bassett G (1978). Regression Quantiles. Econometrica 46(1), 33–50.

See Also

summary.lqm, coef.lqm, predict.lqm, residuals.lqm

Examples

set.seed(123)
n <- 500
p <- 1:3/4
test <- data.frame(x = runif(n,0,1))
test$y <- 30 + test$x + rnorm(n)
fit.lqm <- lqm(y ~ x, data = test, tau = p,
	control = list(verbose = FALSE, loop_tol_ll = 1e-9), fit = TRUE)
fit.lqm

Quantile Regression for Counts

Description

This function is used to fit a quantile regression model when the response is a count variable.

Usage

lqm.counts(formula, data, weights = NULL, offset = NULL, contrasts = NULL,
	tau = 0.5, M = 50, zeta = 1e-05, B = 0.999, cn = NULL, alpha = 0.05,
	control = list())

Arguments

Details

A linear quantile regression model if fitted to the log–transformed response. Additional tranformation functions will be implemented. The notation used here follows closely that of Machado and Santos Silva (2005).

Value

an object of class "lqm.counts" containing the following components

Author(s)

Marco Geraci

References

Machado JAF and Santos Silva JMC (2005). Quantiles for counts. Journal of the American Statistical Association, 100(472), 1226–1237.

Examples

n <- 100
x <- runif(n)
test <- data.frame(x = x, y = rpois(n, 2*x))
lqm.counts(y ~ x, data = test, M = 50)

Quantile Regression Fitting by Gradient Search

Description

This function controls the arguments to be passed to routines written in C for LQM estimation. The optimization algorithm is based on the gradient of the Laplace log–likelihood (Bottai, Orsini and Geraci, 2013).

Usage

lqm.fit.gs(theta, x, y, weights, tau, control)

Arguments

Details

See argument fit in lqm for generating a list of arguments to be called by this function.

Value

An object of class list containing the following components:

.

Author(s)

Marco Geraci

References

Bottai M, Orsini N, Geraci M (2014). A Gradient Search Maximization Algorithm for the Asymmetric Laplace Likelihood, Journal of Statistical Computation and Simulation, 85, 1919-1925.

See Also

lqm

Examples

set.seed(123)
n <- 500
test <- data.frame(x = runif(n,0,1))
test$y <- 30 + test$x + rnorm(n)
lqm.ls <- lqm(y ~ x, data = test, fit = FALSE)

do.call("lqm.fit.gs", lqm.ls)

Control parameters for lqm estimation

Description

A list of parameters for controlling the fitting process.

Usage

lqmControl(method = "gs1", loop_tol_ll = 1e-5, loop_tol_theta = 1e-3,
	check_theta = FALSE, loop_step = NULL, beta = 0.5, gamma = 1.25,
	reset_step = FALSE, loop_max_iter = 1000, smooth = FALSE,
	omicron = 0.001, verbose = FALSE)

Arguments

Details

The methods "gs1" and "gs2" implement the same algorithm (Bottai et al, 2015). The former is based on C code, the latter on R code. While the C code is faster, the R code seems to be more efficient in handling large datasets. For method "gs2", it is possible to replace the classical non-differentiable loss function with a smooth version (Chen, 2007).

Value

a list of control parameters.

Author(s)

Marco Geraci

References

Bottai M, Orsini N, Geraci M (2015). A Gradient Search Maximization Algorithm for the Asymmetric Laplace Likelihood, Journal of Statistical Computation and Simulation, 85(10), 1919-1925.

Chen C (2007). A finite smoothing algorithm for quantile regression. Journal of Computational and Graphical Statistics, 16(1), 136-164.

See Also

lqm

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

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:

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)

Internal lqmm objects

Description

Internal lqmm objects.

Details

These are not to be called by the user.

Linear Quantile Mixed Models Fitting by Derivative-Free Optimization

Description

This function controls the arguments to be passed to optim and optimize for LQMM estimation.

Usage

lqmm.fit.df(theta_0, x, y, z, weights, cov_name, V, W, sigma_0,	
	tau, group, control)

Arguments

Details

In lqmm, see argument fit for generating a list of arguments to be called by this function; see argument covariance for alternative variance–covariance matrices.

NOTE: the data should be ordered by group when passed to lqmm.fit.df (such ordering is performed by lqmm).

Value

An object of class "list" containing the following components:

.

Author(s)

Marco Geraci

See Also

lqmm

Examples

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)
lqmm.ls <- lqmm(fixed = y ~ x, random = ~ 1, group = group, data = test,
	fit = FALSE)

do.call("lqmm.fit.df", lqmm.ls)

Linear Quantile Mixed Models Fitting by Gradient Search

Description

This function controls the arguments to be passed to routines written in C for LQMM estimation. The optimization algorithm is based on the gradient of the Laplace log–likelihood (Bottai, Orsini and Geraci, 2014; Geraci and Bottai, 2014).

Usage

lqmm.fit.gs(theta_0, x, y, z, weights, cov_name, V, W, sigma_0, tau,
	group, control)

Arguments

Details

In lqmm, see argument fit for generating a list of arguments to be called by this function; see argument covariance for alternative variance–covariance matrices.

NOTE: the data should be ordered by group when passed to lqmm.fit.gs (such ordering is performed by lqmm).

Value

An object of class "list" containing the following components:

.

Author(s)

Marco Geraci

References

Bottai M, Orsini N, Geraci M. (2014). A gradient search maximization algorithm for the asymmetric Laplace likelihood, Journal of Statistical Computation and Simulation (in press).

Geraci M and Bottai M (2014). Linear quantile mixed models. Statistics and Computing, 24(3), 461–479.

See Also

lqmm

Examples

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)
lqmm.ls <- lqmm(fixed = y ~ x, random = ~ 1, group = group,
	data = test, fit = FALSE)

do.call("lqmm.fit.gs", lqmm.ls)

Control parameters for lqmm estimation

Description

A list of parameters for controlling the fitting process.

Usage

lqmmControl(method = "gs", LP_tol_ll = 1e-5, LP_tol_theta = 1e-5,
	check_theta = FALSE, LP_step = NULL, beta = 0.5, gamma = 1,
	reset_step = FALSE, LP_max_iter = 500, UP_tol = 1e-4,
	UP_max_iter = 20, startQR = FALSE, verbose = FALSE)

Arguments

Details

LP (lower loop) refers to the estimation of regression coefficients and variance-covariance parameters. UP (upper loop) refers to the estimation of the scale parameter.

Value

a list of control parameters.

Author(s)

Marco Geraci

See Also

lqmm

Compute Nearest Positive Definite Matrix

Description

This function computes the nearest positive definite of a real symmetric matrix. See help("make.positive.definite") from package corpcor.

Author(s)

Original version by Korbinian Strimmer

Source

Juliane Schaefer, Rainer Opgen-Rhein, Verena Zuber, A. Pedro Duarte Silva and Korbinian Strimmer. (2011). corpcor: Efficient Estimation of Covariance and (Partial) Correlation. R package version 1.6.0. https://CRAN.R-project.org/package=corpcor

Functions for Asymmetric Laplace Distribution Parameters

Description

Accessory functions.

Usage

meanAL(mu, sigma, tau)
varAL(sigma, tau)
invvarAL(x, tau)

Arguments

Details

meanAL computes the mean of an asymmetric Laplace with parameters mu, sigma and tau.

varAL computes the variance of an asymmetric Laplace with parameters sigma and tau.

invvarAL computes the scale parameter of an asymmetric Laplace with parameter tau and variance x.

Author(s)

Marco Geraci

References

Yu K and Zhang J (2005). A three-parameter asymmetric Laplace distribution and its extension. Communications in Statistics-Theory and Methods 34, 1867–1879.

See Also

dal, mleAL

Maximum Likelihood Estimation of Asymmetric Laplace Distribution

Description

This function estimates the parameters of an asymmetric Laplace distribution for a sample.

Usage

mleAL(x)

Arguments

Value

an object of class list containing the following components:

Author(s)

Marco Geraci

References

Yu K and Zhang J (2005). A three-parameter asymmetric Laplace distribution and its extension. Communications in Statistics-Theory and Methods 34, 1867–1879.

See Also

dal, meanAL

Predictions from LQM Objects

Description

This function computes predictions based on fitted linear quantile model.

Usage

## S3 method for class 'lqm'
predict(object, newdata, interval = FALSE,
	level = 0.95, na.action = na.pass, ...)
## S3 method for class 'lqm.counts'
predict(object, newdata,
	na.action = na.pass, ...)

Arguments

Value

a vector or a matrix or an array of predictions.

Author(s)

Marco Geraci

See Also

residuals.lqm, residuals.lqm.counts, lqm, lqm.counts, coef.lqm, boot.lqm

Predictions from an lqmm Object

Description

The predictions at level 0 correspond to predictions based only on the fixed effects estimates. The predictions at level 1 are obtained by adding the best linear predictions of the random effects to the predictions at level 0. See details for interpretation. The function predint will produce 1-alpha confidence intervals based on bootstrap centiles.

Usage

## S3 method for class 'lqmm'
predict(object, newdata, level = 0,
	na.action = na.pass, ...)
## S3 method for class 'lqmm'
predint(object, level = 0, alpha = 0.05,
	R = 50, seed = round(runif(1, 1, 10000)))

Arguments

Details

As discussed by Geraci and Bottai (2014), integrating over the random effects will give "weighted averages" of the cluster-specific quantile effects. These may be interpreted strictly as population regression quantiles for the median (tau=0.5) only. Therefore, predictions at the population level (code=0) should be interpreted analogously.

Value

a vector or a matrix of predictions for predict.lqmm. A data frame or a list of data frames for predint.lqmm containing predictions, lower and upper bounds of prediction intervals, and standard errors.

Author(s)

Marco Geraci

References

Geraci M and Bottai M (2014). Linear quantile mixed models. Statistics and Computing, 24(3), 461–479.

See Also

lqmm, ranef.lqmm, coef.lqmm

Examples

## 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)

# Predict (y - Xb)	
predict(fitOi.lqmm, level = 0)

# Predict (y - Xb - Zu)
predict(fitOi.lqmm, level = 1)

# 95% confidence intervals
predint(fitOi.lqmm, level = 0, alpha = 0.05)

Print LQM Objects

Description

Print an object generated by lqm or lqm.counts.

Usage

## S3 method for class 'lqm'
print(x, digits = max(6, getOption("digits")), ...)

Arguments

Author(s)

Marco Geraci

See Also

lqm, lqm.counts

Print an lqmm Object

Description

Print an object generated by lqmm.

Usage

## S3 method for class 'lqmm'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Author(s)

Marco Geraci

See Also

lqmm

Print an lqm Summary Object

Description

Print summary of an lqm object.

Usage

## S3 method for class 'summary.lqm'
print(x, ...)

Arguments

Author(s)

Marco Geraci

See Also

lqm, summary.lqm

Print an lqmm Summary Object

Description

Print summary of an lqmm object.

Usage

## S3 method for class 'summary.lqmm'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Author(s)

Marco Geraci

See Also

lqmm, summary.lqmm

Extract Random Effects

Description

This function computes random effects for a linear quantile mixed model.

Usage

## S3 method for class 'lqmm'
ranef(object, ...)

Arguments

Details

The prediction of the random effects is done via estimated best linear prediction (Geraci and Bottai, 2014). The generic function ranef is imported from the nlme package (Pinheiro et al, 2014).

Value

a data frame or a list of data frames of predicted random effects.

Author(s)

Marco Geraci

References

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

Pinheiro J, Bates D, DebRoy S, Sarkar D and R Core Team (2014). nlme: Linear and Nonlinear Mixed Effects Models. R package version 3.1-117, https://CRAN.R-project.org/package=nlme.

See Also

lqmm, coef.lqmm

Residuals from an LQM Objects

Description

This function computes the residuals from a fitted linear quantile model.

Usage

## S3 method for class 'lqm'
residuals(object, ...)

Arguments

Value

a vector or matrix of residuals.

Author(s)

Marco Geraci

See Also

lqm, lqm.counts, predict.lqm, coef.lqm

Residuals from an lqmm Object

Description

The residuals at level 0 correspond to population residuals (based only on the fixed effects estimates). The residuals at level 1 are obtained by adding the best linear predictions of the random effects to the predictions at level 0 and the subtracting these from the model response.

Usage

## S3 method for class 'lqmm'
residuals(object, level = 0, ...)

Arguments

Value

a matrix of residuals.

Author(s)

Marco Geraci

References

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

See Also

lqmm, predict.lqmm, coef.lqmm, ranef.lqmm,

Summary for a boot.lqm Object

Description

Summary method for class boot.lqm.

Usage

## S3 method for class 'boot.lqm'
summary(object, alpha = 0.05, digits = max(3, getOption("digits") - 3), ...)

Arguments

Author(s)

Marco Geraci

See Also

boot.lqm, lqm,

Summary for a boot.lqmm Object

Description

This function gives a summary of a botstrapped lqmm object

Usage

## S3 method for class 'boot.lqmm'
summary(object, alpha = 0.05, digits = max(3, getOption("digits") - 3), ...)

Arguments

Author(s)

Marco Geraci

References

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

See Also

boot.lqmm, lqmm,

Summary for an lqm Object

Description

Summary method for class lqm.

Usage

## S3 method for class 'lqm'
summary(object, method = "boot", alpha = 0.05, covariance = FALSE, ...)

Arguments

Details

print.summary.lqm formats the coefficients, standard errors, etc. and additionally gives ‘significance stars’.

Value

an object of class summary.lqm. The function summary.lqm computes and returns a list of summary statistics of the fitted linear quantile mixed model given in object, using the components (list elements) from its argument, plus

Author(s)

Marco Geraci

Source

The code for the "nid" method has been adapted from the function summary.rq in package quantreg. It depends on the function bandwidth.rq.

Roger Koenker (2016). quantreg: Quantile Regression. R package version 5.29. https://CRAN.R-project.org/package=quantreg

See Also

print.summary.lqm lqm

Examples

set.seed(12356)
n <- 200
p <- 1:3/4
test <- data.frame(x = runif(n,0,1))
test$y <- 30 + test$x + rnorm(n)
fit.lqm <- lqm(y ~ x, data = test, tau = p)
summary(fit.lqm, R = 50)

Summary for an lqmm Object

Description

Summary method for class lqmm.

Usage

## S3 method for class 'lqmm'
summary(object, method = "boot", alpha = 0.05, covariance = FALSE, ...)

Arguments

Details

print.summary.lqmm formats the coefficients, standard errors, etc. and additionally gives ‘significance stars’.

Value

an object of class summary.lqmm. The function summary.lqmm computes and returns a list of summary statistics of the fitted linear quantile mixed model given in object, using the components (list elements) from its argument, plus

Author(s)

Marco Geraci

See Also

print.summary.lqmm lqmm

Examples

data(Orthodont)
fitOi.lqmm <- lqmm(distance ~ age, random = ~ 1, group = Subject,
	tau = c(0.1,0.5,0.9), data = Orthodont)
summary(fitOi.lqmm)