lqr (version 4.1)

Log.best.lqr: Best Fit in Robust Logistic Linear Quantile Regression

Description

It performs the logistic transformation in Galarza et.al.(2020) (see references) for estimating quantiles for a bounded response. Once the response is transformed, it uses the best.lqr function.

Usage

Log.best.lqr(y,x,p=0.5,a=0,b=1,epsilon = 0.001,
precision = 10^-6,criterion = "AIC")

Arguments

a

lower bound for the response (default = 0)

b

upper bound for the response (default = 1)

epsilon

a small quantity \(\epsilon>0\) that ensures that the logistic transform is defined for all values of y

y

the response vector of dimension \(n\) where \(n\) is the total of observations.

x

design matrix for the fixed effects of dimension \(N x d\) where \(d\) represents the number of fixed effects including the intercept, if considered.

p

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

precision

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

criterion

Likelihood-based criterion to be used for choosen the best model. It could be AIC, BIC, HQ or loglik (log-likelihood). By default AIC criterion will be used.

Value

For the best model:

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.

Details

We follow the transformation in Bottai et.al. (2009) defined as

$$h(y)=logit(y)=log(\frac{y-a}{b-y})$$

that implies

$$Q_{y}(p)=\frac{b\,exp(X\beta) + a}{1 + exp(X\beta)}$$

where \(Q_{y}(p)\) represents the conditional quantile of the response. Once estimates for the regression coefficients \(\beta_p\) are obtained, inference on \(Q_{y}(p)\) can then be made through the inverse transform above. This equation (as function) is provided in the output. See example.

The interpretation of the regression coefficients is analogous to the interpretation of the coefficients of a logistic regression for binary outcomes.

For example, let \(x_1\) be the gender (male = 0, female=1). Then \(exp(\beta_{0.5,1})\) represents the odds ratio of median score in males vs females, where the odds are defined using the score instead of a probability, \((y-a)/(b-y)\). When the covariate is continous, the respective \(\beta\) coeficient can be interpretated as the increment (or decrement) over the log(odd ratio) when the covariate increases one unit.

References

Galarza, C.M., Zhang P. and Lachos, V.H. (2020). Logistic Quantile Regression for Bounded Outcomes Using a Family of Heavy-Tailed Distributions. Sankhya B: The Indian Journal of Statistics. 10.1007/s13571-020-00231-0

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.

See Also

Log.lqr,best.lqr,dSKD,QRLMM, QRNLMM

Examples

# NOT RUN {
##Load the data
data(resistance)
attach(resistance)

#EXAMPLE 1.1

#Comparing the resistence to death of two types of tumor-cells.
#The response is a score in [0,4].

boxplot(score~type)

#Median logistic quantile regression (Best fit distribution)
res = Log.best.lqr(y = score,x = cbind(1,type),a=0,b=4)

# The odds ratio of median score in type B vs type A
exp(res$beta[2])

#Proving that exp(res$beta[2])  is approx median odd ratio
medA  = median(score[type=="A"])
medB  = median(score[type=="B"])
rateA = (medA - 0)/(4 - medA)
rateB = (medB - 0)/(4 - medB)
odd   = rateB/rateA

round(c(exp(res$beta[2]),odd),3) #better fitted

#EXAMPLE 1.2
############

#Comparing the resistence to death depending of dose.

#descriptive
plot(dose,score,ylim=c(0,4),col="dark gray");abline(h=c(0,4),lty=2)
dosecat<-cut(dose, 6, ordered = TRUE)
boxplot(score~dosecat,ylim=c(0,4))
abline(h=c(0,4),lty=2)

#(Non logistic) Best quantile regression for quantiles
# 0.05, 0.50 and 0.95
xx1  = dose
xx2  = dose^2
xx3  = dose^3
res31 = best.lqr(y = score,x = cbind(1,xx1,xx2,xx3),p = 0.05)
res32 = best.lqr(y = score,x = cbind(1,xx1,xx2,xx3),p = 0.50)
res33 = best.lqr(y = score,x = cbind(1,xx1,xx2,xx3),p = 0.95)
res3  = list(res31,res32,res33)
seqq=seq(min(dose),max(dose),length.out = 1000)
dd = matrix(data = NA,nrow = 1000,ncol =3)
for(i in 1:3)
{
  dd[,i] = rep(res3[[i]]$beta[1],1000) + res3[[i]]$beta[2]*seqq +
    res3[[i]]$beta[3]*seqq^2 + res3[[i]]$beta[4]*seqq^3
}

plot(dose,score,ylim=c(-1,5),col="gray");abline(h=c(0,4),lty=2)
lines(seqq,dd[,1],lwd=1,col=2)
lines(seqq,dd[,2],lwd=1,col=1)
lines(seqq,dd[,3],lwd=1,col=2)

#Using logistic quantile regression for obtaining predictions inside bounds

res41 = Log.best.lqr(y = score,x = cbind(1,xx1,xx2,xx3),a=0,b=4,p = 0.05)
res42 = Log.best.lqr(y = score,x = cbind(1,xx1,xx2,xx3),a=0,b=4,p = 0.50)
res43 = Log.best.lqr(y = score,x = cbind(1,xx1,xx2,xx3),a=0,b=4,p = 0.95)
res4  = list(res41,res42,res43)
dd = matrix(data = NA,nrow = 1000,ncol =3)
for(i in 1:3)
{
  dd[,i] = rep(res4[[i]]$beta[1],1000) + res4[[i]]$beta[2]*seqq +
    res4[[i]]$beta[3]*seqq^2 + res4[[i]]$beta[4]*seqq^3
}

#Computing quantiles for the original response (Inverse trnasformation)

pred = function(predlog,a,b)
{
  return((b*exp(predlog)+a)/(1+exp(predlog)))
}

for(i in 1:3)
{
  dd[,i] = pred(dd[,i],a=0,b=4)
}           

#No more prediction curves outof bounds
plot(dose,score,ylim=c(0,4),col="gray");abline(h=c(0,4),lty=2)
lines(seqq,dd[,1],lwd=1,col=2)
lines(seqq,dd[,2],lwd=1,col=1)
lines(seqq,dd[,3],lwd=1,col=2)

# }