loglqr: Robust Logistic Linear Quantile Regression

Description Usage Arguments Details Value Note Author(s) References See Also Examples

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 lqr function.

Usage

1
2
Log.lqr(y,x,p=0.5,a=0,b=1,dist = "normal",nu="",gama="",precision = 10^-6,
epsilon = 0.001,CI=0.95)

Arguments

We will detail first the only three arguments that differ from lqr function.

a

lower bound for the response (default = 0)

b

upper bound for the response (default = 1)

epsilon

a small quantity ε>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.

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 ν>0. For the Contaminated Normal distribution, ν is the parameter that represents the percentage of outliers. When is not provided, we use the MLE.

gama

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.

CI

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

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(Xβ) + a}{1 + exp(Xβ)}

where Q_{y}(p) represents the conditional quantile of the response. Once estimates for the regression coefficients β_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(β_{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 β coeficient can be interpretated as the increment (or decrement) over the log(odd ratio) when the covariate increases one unit.

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

When a grid of quantiles is provided, a graphical summary with point estimates and Confidence Intervals for model parameters is shown. Also, 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 <cgalarza88@gmail.com>, Luis Benites <lsanchez@ime.usp.br> and Victor H. Lachos <hlachos@ime.unicamp.br>

Maintainer: Christian E. Galarza <cgalarza88@gmail.com>

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. doi: 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.best.lqr,best.lqr,dSKD,QRLMM, QRNLMM

Examples

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## 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,ylab="score",xlab="type")

#Student't median logistic quantile regression
res = Log.lqr(y = score,x = cbind(1,type),a=0,b=4,dist="t")

# 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)


#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)

#Slash (Non logistic) quantile regression for quantiles 0.05, 0.50 and 0.95
xx1  = dose
xx2  = dose^2
xx3  = dose^3
res3 = lqr(y = score,x = cbind(1,xx1,xx2,xx3),p = c(0.05,0.50,0.95),dist="slash")
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

res4 = Log.lqr(y = score,x = cbind(1,xx1,xx2,xx3),a = 0,b = 4,p = c(0.05,0.50,0.95),dist="slash")
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)

#EXAMPLE 1.3
############

#A full model using dose and type for a grid of quantiles

typeB = 1*(type=="B")
res5 = Log.lqr(y = score,x = cbind(1,xx1,xx2,xx3,typeB,typeB*xx1),a = 0,b = 4,
               p = seq(from = 0.05,to = 0.95,by = 0.05),dist = "t")
ddA = ddB = matrix(data = NA,nrow = 1000,ncol = 5)
for(i in 1:5)
{
  k = c(2,5,10,15,18)[i]
  ddA[,i] = rep(res5[[k]]$beta[1],1000) + res5[[k]]$beta[2]*seqq + res5[[k]]$beta[3]*
    seqq^2 + res5[[k]]$beta[4]*seqq^3
  ddB[,i] = rep(res5[[k]]$beta[1],1000) + (res5[[k]]$beta[2] + res5[[k]]$beta[6])*
    seqq + res5[[k]]$beta[3]*seqq^2 + res5[[k]]$beta[4]*seqq^3  + res5[[k]]$beta[5]
}

#Computing quantiles for the original response (Inverse transformation)

for(i in 1:5)
{
  ddA[,i] = pred(ddA[,i],a=0,b=4)
  ddB[,i] = pred(ddB[,i],a=0,b=4)
} 

#Such a beautiful plot
par(mfrow=c(1,2))
plot(dose,score,ylim=c(0,4),col=c((type == "B")*8+(type == "A")*1),main="Type A")
abline(h=c(0,4),lty=2)
lines(seqq,ddA[,1],lwd=2,col=2)
lines(seqq,ddA[,2],lwd=1,col=4)
lines(seqq,ddA[,3],lwd=2,col=1)
lines(seqq,ddA[,4],lwd=1,col=4)
lines(seqq,ddA[,5],lwd=2,col=2)

legend(x = 0,y=4,legend = c("p=0.10","p=0.25","p=0.50","p=0.75","p=0.90")
       ,col=c(2,4,1,4,2),lwd=c(2,1,2,1,2),bty = "n",cex=0.65)

plot(dose,score,ylim=c(0,4),col=c((type == "B")*1 + (type == "A")*8),
     main="Type B");abline(h=c(0,4),lty=2)
lines(seqq,ddB[,1],lwd=2,col=2)
lines(seqq,ddB[,2],lwd=1,col=4)
lines(seqq,ddB[,3],lwd=2,col=1)
lines(seqq,ddB[,4],lwd=1,col=4)
lines(seqq,ddB[,5],lwd=2,col=2)

## End(Not run)

Example output

sh: 1: cannot create /dev/null: Permission denied

Call:
Log.lqr(y = score, x = cbind(1, type), a = 0, b = 4, dist = "t")


Call:
lqr(y = ynovo, x = x, p = p, dist = dist, nu = nu, gama = gama, 
    precision = precision, envelope = FALSE, CI = CI)


--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.5 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -1.63802    0.30818 -5.31508   0.0000 ***
beta 2  0.63249    0.21659  2.92023   0.0035  **
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.81134 
nu    = 2.000045 

------------------------
Model selection criteria
------------------------

        Loglik      AIC      BIC       HQ
Value -755.785 1517.569 1529.726 1522.372


***
The function below converts the logistic quantile prediction curve (predlog) to the original quantile predicted curve for the bounded response. See example.

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

The interpretation of the regression coefficients is analogous to the interpretation of the coefficients of a logistic regression for binary outcomes. For references, please check Bottai et.al. (2009) Logistic quantile regression for bounded outcomes.

[1] 1.882292
[1] 1.882 1.888

Call:
lqr(y = score, x = cbind(1, xx1, xx2, xx3), p = c(0.05, 0.5, 
    0.95), dist = "slash")


--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = slash 
Quantile = 0.05 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -0.41557    0.26643 -1.55976  0.11882    
beta 2  1.63165    0.49207  3.31591  0.00091 ***
beta 3 -1.27379    0.25631 -4.96965  0.00000 ***
beta 4  0.30924    0.04034  7.66557  0.00000 ***
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.08442 
nu    = 9.999948 

------------------------
Model selection criteria
------------------------

        Loglik     AIC     BIC      HQ
Value -279.946 569.891 590.152 577.895

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = slash 
Quantile = 0.5 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1  0.91257    0.40763  2.23870  0.02518   *
beta 2  0.48232    0.72076  0.66918  0.50338    
beta 3 -0.64426    0.38189 -1.68705  0.09159   .
beta 4  0.20654    0.06148  3.35970  0.00078 ***
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.41593 
nu    = 9.999946 

------------------------
Model selection criteria
------------------------

        Loglik     AIC     BIC      HQ
Value -252.009 514.018 534.279 522.022

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = slash 
Quantile = 0.95 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1  1.56600    0.29910  5.23566  0.00000 ***
beta 2  0.88961    0.52408  1.69745  0.08961   .
beta 3 -0.87353    0.27003 -3.23498  0.00122  **
beta 4  0.24088    0.04225  5.70080  0.00000 ***
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.08953 
nu    = 9.999949 

------------------------
Model selection criteria
------------------------

        Loglik     AIC     BIC      HQ
Value -304.797 619.594 639.854 627.598

Call:
Log.lqr(y = score, x = cbind(1, xx1, xx2, xx3), p = c(0.05, 0.5, 
    0.95), a = 0, b = 4, dist = "slash")


Call:
lqr(y = ynovo, x = x, p = p, dist = dist, nu = nu, gama = gama, 
    precision = precision, envelope = FALSE, CI = CI)


--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = slash 
Quantile = 0.05 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -2.40676    0.62478 -3.85217  0.00012 ***
beta 2 -0.15111    1.04926 -0.14401  0.88549    
beta 3 -0.33464    0.53468 -0.62588  0.53140    
beta 4  0.19119    0.08353  2.28891  0.02208   *
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.10803 
nu    = 1.21592 

------------------------
Model selection criteria
------------------------

        Loglik      AIC      BIC       HQ
Value -597.583 1205.167 1225.427 1213.171

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = slash 
Quantile = 0.5 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -1.53485    0.55362 -2.77240  0.00556  **
beta 2  1.27856    1.00306  1.27467  0.20243    
beta 3 -1.21331    0.54550 -2.22422  0.02613   *
beta 4  0.32659    0.08924  3.65980  0.00025 ***
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.40656 
nu    = 0.9580516 

------------------------
Model selection criteria
------------------------

        Loglik      AIC      BIC       HQ
Value -523.549 1057.099 1077.359 1065.103

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = slash 
Quantile = 0.95 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -1.26588    0.38626 -3.27729  0.00105  **
beta 2  2.80266    0.72589  3.86102  0.00011 ***
beta 3 -2.14421    0.39094 -5.48473  0.00000 ***
beta 4  0.49307    0.06259  7.87741  0.00000 ***
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.0798 
nu    = 0.7442275 

------------------------
Model selection criteria
------------------------

        Loglik      AIC      BIC       HQ
Value -633.676 1277.351 1297.612 1285.355


***
The function below converts the logistic quantile prediction curve (predlog) to the original quantile predicted curve for the bounded response. See example.

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

The interpretation of the regression coefficients is analogous to the interpretation of the coefficients of a logistic regression for binary outcomes. For references, please check Bottai et.al. (2009) Logistic quantile regression for bounded outcomes.


Call:
Log.lqr(y = score, x = cbind(1, xx1, xx2, xx3, typeB, typeB * 
    xx1), p = seq(from = 0.05, to = 0.95, by = 0.05), a = 0, 
    b = 4, dist = "t")


Call:
lqr(y = ynovo, x = x, p = p, dist = dist, nu = nu, gama = gama, 
    precision = precision, envelope = FALSE, CI = CI)


--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.05 

-----------
Estimates
-----------

       Estimate Std. Error   z value Pr(>|z|)    
beta 1 -4.23723    0.26409 -16.04457    0e+00 ***
beta 2  2.44640    0.46870   5.21948    0e+00 ***
beta 3 -1.53407    0.26546  -5.77898    0e+00 ***
beta 4  0.36457    0.04410   8.26687    0e+00 ***
beta 5  1.82468    0.21603   8.44632    0e+00 ***
beta 6 -0.38995    0.09049  -4.30928    2e-05 ***
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.11003 
nu    = 3.238599 

------------------------
Model selection criteria
------------------------

        Loglik      AIC      BIC       HQ
Value -510.001 1034.003 1062.367 1045.208

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.1 

-----------
Estimates
-----------

       Estimate Std. Error   z value Pr(>|z|)    
beta 1 -3.91630    0.30013 -13.04873  0.00000 ***
beta 2  2.47281    0.50080   4.93773  0.00000 ***
beta 3 -1.60513    0.28461  -5.63967  0.00000 ***
beta 4  0.37730    0.04792   7.87419  0.00000 ***
beta 5  1.60269    0.24951   6.42329  0.00000 ***
beta 6 -0.31407    0.10420  -3.01412  0.00258  **
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.19923 
nu    = 3.18769 

------------------------
Model selection criteria
------------------------

        Loglik      AIC      BIC      HQ
Value -493.052 1000.104 1028.469 1011.31

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.15 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -3.07449    0.48578 -6.32895  0.00000 ***
beta 2  1.35128    0.78651  1.71808  0.08578   .
beta 3 -1.07554    0.40836 -2.63380  0.00844  **
beta 4  0.29979    0.06501  4.61122  0.00000 ***
beta 5  1.73754    0.30506  5.69576  0.00000 ***
beta 6 -0.38181    0.12895 -2.96083  0.00307  **
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.26718 
nu    = 2.961685 

------------------------
Model selection criteria
------------------------

        Loglik     AIC      BIC      HQ
Value -480.999 975.999 1004.363 987.204

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.2 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -3.12368    0.51912 -6.01728  0.00000 ***
beta 2  1.65617    0.83132  1.99222  0.04635   *
beta 3 -1.21862    0.43059 -2.83009  0.00465  **
beta 4  0.31849    0.06886  4.62531  0.00000 ***
beta 5  1.72579    0.32248  5.35162  0.00000 ***
beta 6 -0.38446    0.13704 -2.80550  0.00502  **
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.32757 
nu    = 2.907099 

------------------------
Model selection criteria
------------------------

        Loglik    AIC     BIC      HQ
Value -474.095 962.19 990.555 973.396

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.25 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -3.17616    0.52248 -6.07896  0.00000 ***
beta 2  1.99358    0.83221  2.39552  0.01660   *
beta 3 -1.39236    0.43316 -3.21446  0.00131  **
beta 4  0.34401    0.06980  4.92824  0.00000 ***
beta 5  1.63717    0.33188  4.93302  0.00000 ***
beta 6 -0.35991    0.13987 -2.57319  0.01008   *
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.37664 
nu    = 2.853568 

------------------------
Model selection criteria
------------------------

        Loglik     AIC    BIC      HQ
Value -469.062 952.125 980.49 963.331

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.3 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -3.07158    0.55468 -5.53757  0.00000 ***
beta 2  2.10041    0.88083  2.38458  0.01710   *
beta 3 -1.47634    0.45668 -3.23274  0.00123  **
beta 4  0.35951    0.07345  4.89467  0.00000 ***
beta 5  1.54738    0.33914  4.56266  0.00001 ***
beta 6 -0.33233    0.14217 -2.33760  0.01941   *
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.41479 
nu    = 2.791061 

------------------------
Model selection criteria
------------------------

        Loglik    AIC     BIC      HQ
Value -465.615 945.23 973.594 956.435

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.35 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -2.91645    0.59210 -4.92559  0.00000 ***
beta 2  2.10606    0.93482  2.25290  0.02427   *
beta 3 -1.49996    0.48124 -3.11686  0.00183  **
beta 4  0.36430    0.07700  4.73103  0.00000 ***
beta 5  1.44849    0.34536  4.19421  0.00003 ***
beta 6 -0.30345    0.14448 -2.10028  0.03570   *
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.43928 
nu    = 2.681429 

------------------------
Model selection criteria
------------------------

       Loglik    AIC     BIC      HQ
Value -462.92 939.84 968.205 951.046

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.4 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -2.81100    0.61678 -4.55756  0.00001 ***
beta 2  2.16727    0.96899  2.23664  0.02531   *
beta 3 -1.54319    0.49604 -3.11101  0.00186  **
beta 4  0.37140    0.07900  4.70114  0.00000 ***
beta 5  1.37463    0.34725  3.95859  0.00008 ***
beta 6 -0.28901    0.14510 -1.99179  0.04639   *
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.45288 
nu    = 2.566255 

------------------------
Model selection criteria
------------------------

        Loglik     AIC     BIC      HQ
Value -461.087 936.174 964.538 947.379

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.45 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -2.74611    0.62923 -4.36425  0.00001 ***
beta 2  2.25736    0.98578  2.28993  0.02203   *
beta 3 -1.59483    0.50306 -3.17024  0.00152  **
beta 4  0.37954    0.07987  4.75166  0.00000 ***
beta 5  1.32746    0.34579  3.83891  0.00012 ***
beta 6 -0.28310    0.14408 -1.96491  0.04943   *
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.4569 
nu    = 2.457635 

------------------------
Model selection criteria
------------------------

        Loglik     AIC     BIC      HQ
Value -459.951 933.902 962.267 945.108

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.5 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -2.70213    0.63058 -4.28515  0.00002 ***
beta 2  2.35012    0.98855  2.37734  0.01744   *
beta 3 -1.64302    0.50395 -3.26027  0.00111  **
beta 4  0.38684    0.07987  4.84321  0.00000 ***
beta 5  1.29570    0.34120  3.79746  0.00015 ***
beta 6 -0.27894    0.14176 -1.96767  0.04911   *
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.45218 
nu    = 2.356639 

------------------------
Model selection criteria
------------------------

        Loglik     AIC     BIC      HQ
Value -459.576 933.152 961.516 944.357

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.55 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -2.66270    0.62406 -4.26676  0.00002 ***
beta 2  2.43469    0.98137  2.48092  0.01310   *
beta 3 -1.68494    0.50055 -3.36617  0.00076 ***
beta 4  0.39310    0.07929  4.95772  0.00000 ***
beta 5  1.26788    0.33405  3.79544  0.00015 ***
beta 6 -0.27499    0.13854 -1.98494  0.04715   *
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.4384 
nu    = 2.254211 

------------------------
Model selection criteria
------------------------

        Loglik     AIC     BIC     HQ
Value -459.907 933.814 962.179 945.02

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.6 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -2.61944    0.61102 -4.28699  0.00002 ***
beta 2  2.48950    0.96533  2.57889  0.00991  **
beta 3 -1.70764    0.49340 -3.46093  0.00054 ***
beta 4  0.39602    0.07824  5.06153  0.00000 ***
beta 5  1.24689    0.32450  3.84247  0.00012 ***
beta 6 -0.27277    0.13462 -2.02612  0.04275   *
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.41578 
nu    = 2.147714 

------------------------
Model selection criteria
------------------------

        Loglik     AIC     BIC      HQ
Value -460.988 935.977 964.341 947.182

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.65 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -2.57527    0.59011 -4.36409  0.00001 ***
beta 2  2.50985    0.93701  2.67858  0.00739  **
beta 3 -1.70214    0.48055 -3.54209  0.00040 ***
beta 4  0.39319    0.07637  5.14844  0.00000 ***
beta 5  1.23583    0.31213  3.95929  0.00008 ***
beta 6 -0.27361    0.12974 -2.10895  0.03495   *
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.38451 
nu    = 2.033051 

------------------------
Model selection criteria
------------------------

        Loglik     AIC     BIC      HQ
Value -463.084 940.169 968.534 951.375

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.7 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -2.54042    0.56245 -4.51668  0.00001 ***
beta 2  2.50949    0.89910  2.79111  0.00525  **
beta 3 -1.67924    0.46351 -3.62289  0.00029 ***
beta 4  0.38703    0.07388  5.23844  0.00000 ***
beta 5  1.23861    0.29838  4.15107  0.00003 ***
beta 6 -0.27738    0.12428 -2.23192  0.02562   *
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.34969 
nu    = 2.000045 

------------------------
Model selection criteria
------------------------

       Loglik    AIC     BIC      HQ
Value -466.69 947.38 975.745 958.586

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.75 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -2.52633    0.52489 -4.81304  0.00000 ***
beta 2  2.53931    0.84800  2.99448  0.00275  **
beta 3 -1.67500    0.44043 -3.80308  0.00014 ***
beta 4  0.38431    0.07047  5.45358  0.00000 ***
beta 5  1.25352    0.28174  4.44921  0.00001 ***
beta 6 -0.28454    0.11734 -2.42495  0.01531   *
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.30944 
nu    = 2.000045 

------------------------
Model selection criteria
------------------------

        Loglik     AIC     BIC      HQ
Value -472.697 959.393 987.758 970.599

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.8 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -2.47172    0.46895 -5.27072  0.00000 ***
beta 2  2.58079    0.77698  3.32159  0.00090 ***
beta 3 -1.71023    0.40883 -4.18318  0.00003 ***
beta 4  0.39200    0.06563  5.97257  0.00000 ***
beta 5  1.20808    0.25700  4.70077  0.00000 ***
beta 6 -0.25951    0.10600 -2.44830  0.01435   *
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.26464 
nu    = 2.000045 

------------------------
Model selection criteria
------------------------

        Loglik     AIC     BIC      HQ
Value -482.293 978.585 1006.95 989.791

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.85 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -2.37942    0.39010 -6.09955  0.00000 ***
beta 2  2.66708    0.68620  3.88671  0.00010 ***
beta 3 -1.80570    0.37104 -4.86656  0.00000 ***
beta 4  0.41380    0.06008  6.88755  0.00000 ***
beta 5  1.11291    0.22493  4.94788  0.00000 ***
beta 6 -0.22048    0.09383 -2.34993  0.01878   *
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.21307 
nu    = 2.000045 

------------------------
Model selection criteria
------------------------

        Loglik      AIC      BIC       HQ
Value -496.011 1006.022 1034.387 1017.228

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.9 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -2.27970    0.33093 -6.88879  0.00000 ***
beta 2  2.71930    0.60104  4.52431  0.00001 ***
beta 3 -1.86117    0.32923 -5.65319  0.00000 ***
beta 4  0.42672    0.05343  7.98593  0.00000 ***
beta 5  1.07583    0.19013  5.65838  0.00000 ***
beta 6 -0.22391    0.08128 -2.75473  0.00587  **
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.15201 
nu    = 2.000045 

------------------------
Model selection criteria
------------------------

       Loglik     AIC      BIC       HQ
Value -517.46 1048.92 1077.284 1060.125

--------------------------------------------------------------
        Quantile Linear Regression using SKD family
--------------------------------------------------------------

Distribution = t 
Quantile = 0.95 

-----------
Estimates
-----------

       Estimate Std. Error  z value Pr(>|z|)    
beta 1 -2.29961    0.41152 -5.58814  0.00000 ***
beta 2  2.94831    0.64417  4.57695  0.00000 ***
beta 3 -1.98837    0.32712 -6.07846  0.00000 ***
beta 4  0.44615    0.05142  8.67678  0.00000 ***
beta 5  1.06071    0.19355  5.48020  0.00000 ***
beta 6 -0.22090    0.08211 -2.69029  0.00714  **
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

sigma = 0.08028 
nu    = 2.000045 

------------------------
Model selection criteria
------------------------

        Loglik      AIC      BIC       HQ
Value -556.264 1126.528 1154.893 1137.734


***
The function below converts the logistic quantile prediction curve (predlog) to the original quantile predicted curve for the bounded response. See example.

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

The interpretation of the regression coefficients is analogous to the interpretation of the coefficients of a logistic regression for binary outcomes. For references, please check Bottai et.al. (2009) Logistic quantile regression for bounded outcomes.

lqr documentation built on Oct. 6, 2021, 1:10 a.m.