lqr {lqr}

Robust Linear Quantile Regression

Description

It fits a robust linear quantile regression model using a new family of zero-quantile distributions for the error term. This family of distribution includes skewed versions of the Normal, Student's t, Laplace, Slash and Contaminated Normal distribution. It provides estimates and full inference. It also provides envelopes plots for assessing the fit and confidences bands when several quantiles are provided simultaneously.

Usage

lqr(formula,data = NULL,subset = NULL,
               p=0.5,dist = "normal",
               nu=NULL,gamma=NULL,
               precision = 10^-6,envelope=FALSE,
               CI=0.95,silent = FALSE
)

#lqr(y~x, data, p = 0.5, dist = "normal")
#lqr(y~x, data, p = 0.5, dist = "t")
#lqr(y~x, data, p = 0.5, dist = "laplace")
#lqr(y~x, data, p = 0.5, dist = "slash")
#lqr(y~x, data, p = 0.5, dist = "cont")

#lqr(y~x, p = c(0.25,0.50,0.75), dist = "normal")

Arguments

formula

an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted.

data

an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula).

subset

an optional string specifying a subset of observations to be used in the fitting process. Be aware of the use of double quotes in a proper way when necessary, e.g., in "(sex=='F')".

p

An unique quantile or a set of quantiles related to the quantile regression.

dist

represents the distribution to be used for the error term. The values are normal for Normal distribution, t for Student's t distribution, laplace for Laplace distribution, slash for Slash distribution and cont for the Contaminated normal distribution.

nu

It represents the degrees of freedom when dist = t. For the Slash distribution (dist = slash) it is a shape parameter \nu>0. For the Contaminated Normal distribution, \nu is the parameter that represents the percentage of outliers. When is not provided, we use the MLE.

gamma

It represents a scale factor for the contaminated normal distribution. When is not provided, we use the MLE.

precision

The convergence maximum error permitted. By default is 10^-6.

envelope

if TRUE, it will show a confidence envelope for a curve based on bootstrap replicates. By default it is TRUE when just one quantile is provided. If a grid of quantiles is provided it will be FALSE by default.

CI

Confidence to be used for the Confidence Interval when a grid of quantiles is provided. Default = 0.95.

silent

if FALSE (by default), the function prints some output.

Details

When a grid of quantiles is provided, a graphical summary with point estimates and Confidence Intervals for model parameters is shown.

Value

iter

number of iterations.

criteria

attained criteria value.

beta

fixed effects estimates.

sigma

scale parameter estimate for the error term.

nu

Estimate of nu parameter detailed above.

gamma

Estimate of gamma parameter detailed above.

SE

Standard Error estimates.

table

Table containing the inference for the fixed effects parameters.

loglik

Log-likelihood value.

AIC

Akaike information criterion.

BIC

Bayesian information criterion.

HQ

Hannan-Quinn information criterion.

fitted.values

vector containing the fitted values.

residuals

vector containing the residuals.

Note

If a grid of quantiles is provided, the result will be a list of the same dimension where each element corresponds to each quantile as detailed above.

Author(s)

Christian E. Galarza <[e-mail]>, Luis Benites <[e-mail]> and Victor H. Lachos <[e-mail]>

Maintainer: Christian E. Galarza <[e-mail]>

References

Galarza, C., Lachos, V. H., Cabral, C. R. B., & Castro, C. L. (2017). Robust quantile regression using a generalized class of skewed distributions. Stat, 6(1), 113-130.

Wichitaksorn, N., Choy, S. T., & Gerlach, R. (2014). A generalized class of skew distributions and associated robust quantile regression models. Canadian Journal of Statistics, 42(4), 579-596.

See Also

cens.lqr,best.lqr,Log.lqr, Log.best.lqr,dSKD

Examples

#Example 1

##Load the data
data(ais)
attach(ais)

## Fitting a median regression with Normal errors (by default)

modelF = lqr(BMI~LBM,data = ais,subset = "(Sex==1)")
modelM = lqr(BMI~LBM,data = ais,subset = "(Sex==0)")

plot(LBM,BMI,col=Sex*2+1,
     xlab="Lean Body Mass",
     ylab="Body4 Mass Index",
     main="Quantile Regression")

abline(a = modelF$beta[1],b = modelF$beta[2],lwd=2,col=3)
abline(a = modelM$beta[1],b = modelM$beta[2],lwd=2,col=1)
legend(x = "topleft",legend = c("Male","Female"),lwd = 2,col = c(1,3))

#COMPARING SOME MODELS for median regression
modelN  = lqr(BMI~LBM,dist = "normal")
modelT  = lqr(BMI~LBM,dist = "t")
modelL  = lqr(BMI~LBM,dist = "laplace")

#Comparing AIC criteria
modelN$AIC;modelT$AIC;modelL$AIC

#This could be automatically done using best.lqr()
best.model = best.lqr(BMI~LBM,data = ais,
                      p = 0.75, #third quartile
                      criterion = "AIC")

#Let's use a grid of quantiles (no output)
modelfull = lqr(BMI~LBM,data = ais,
                p = seq(from = 0.10,to = 0.90,by = 0.05),
                dist = "normal",silent = TRUE)

#Plotting quantiles 0.10,0.25,0.50,0.75 and 0.90

if(TRUE){
  plot(LBM,BMI,xlab = "Lean Body Mass"
       ,ylab = "Body Mass Index", main = "Quantile Regression",pch=16)
  
  colvec = c(2,2,3,3,4)
  imodel = c(1,17,4,14,9)
  for(i in 1:5){
    abline(a = modelfull[[imodel[i]]]$beta[1],
           b = modelfull[[imodel[i]]]$beta[2],
           lwd=2,col=colvec[i])  
  }
  legend(x = "topleft",
         legend = rev(c("0.10","0.25","0.50","0.75","0.90")),
         lwd = 2,col = c(2,3,4,3,2))
}

#Example 2
##Load the data

data(crabs,package = "MASS")
attach(crabs)

## Fitting a median regression with Normal errors (by default) #Note the double quotes
crabsF = lqr(BD~FL,data = crabs,subset = "(sex=='F')")
crabsM = lqr(BD~FL,data = crabs,subset = "(sex=='M')")

if(TRUE){
  plot(FL,BD,col=as.numeric(sex)+1,
       xlab="Frontal lobe size",ylab="Body depth",main="Quantile Regression")
  abline(a = crabsF$beta[1],b = crabsF$beta[2],lwd=2,col=2)
  abline(a = crabsM$beta[1],b = crabsM$beta[2],lwd=2,col=3)
  legend(x = "topleft",legend = c("Male","Female"),
         lwd = 2,col = c(3,2))
}

#Median regression for different distributions

modelN  = lqr(BD~FL,dist = "normal")
modelT  = lqr(BD~FL,dist = "t")
modelL  = lqr(BD~FL,dist = "laplace")
modelS  = lqr(BD~FL,dist = "slash")
modelC  = lqr(BD~FL,dist = "cont" )

#Comparing AIC criterias
modelN$AIC;modelT$AIC;modelL$AIC;modelS$AIC;modelC$AIC

# best model based on BIC
best.lqr(BD~FL,criterion = "BIC")

#Let's use a grid of quantiles for the Student's t distribution
modelfull = lqr(BD~FL,data = crabs,
                p = seq(from = 0.10,to = 0.90,by = 0.05),
                dist = "t") # silent = FALSE

#Plotting quantiles 0.10,0.25,0.50,0.75 and 0.90
if(TRUE){
  plot(FL,BD,xlab = "Frontal lobe size"
       ,ylab = "Body depth", main = "Quantile Regression",pch=16)
  colvec = c(2,2,3,3,4)
  imodel = c(1,17,4,14,9)
  for(i in 1:5){
    abline(a = modelfull[[imodel[i]]]$beta[1],
           b = modelfull[[imodel[i]]]$beta[2],
           lwd=2,col=colvec[i])  
  }
  legend(x = "topleft",
         legend = rev(c("0.10","0.25","0.50","0.75","0.90")),
         lwd = 2,col = c(2,3,4,3,2))
}