Nonlinear and Penalized Quantile Regression Coefficients Modeling

Penalized and nonlinear parametric modeling of quantile regression coefficient functions. The penalized integrated loss minimization estimator (lasso and group lasso penalties) of Sottile, Frumento, Chiodi and Bottai (2020) performs covariate selection using the information on all quantiles simultaneously. Nonlinear parametric quantile models, in which the conditional quantile function can be a nonlinear function of the parameters, of the covariates or both, are estimated by integrated loss minimization as in Bottai and Cilluffo (2020) . Both approaches extend the quantile regression coefficients modeling framework of Frumento and Bottai (2016) implemented in 'qrcm'.

qrcmNP

‘qrcmNP’ extends quantile regression coefficients modeling (QRCM, Frumento and Bottai, 2016; package ‘qrcm’) in two directions:

piqr(): penalized QRCM for covariate selection using the information on all quantiles simultaneously (Sottile, Frumento, Chiodi and Bottai, 2020), with lasso and group lasso penalties;
niqr(): nonlinear parametric quantile models, nonlinear in the parameters, in the covariates, or both (Bottai and Cilluffo, 2020).

Installation

# CRAN version
install.packages("qrcmNP")
# development version
# install.packages("remotes")
remotes::install_github("gianluca-sottile/qrcmNP")

Penalized QRCM

library(qrcmNP)
set.seed(1234)
n <- 300
d <- data.frame(x1 = rexp(n), x2 = runif(n, 0, 5), x3 = rnorm(n), x4 = runif(n))
u <- runif(n)
d$y <- 1 + qnorm(u) + (1 + 2 * u) * d$x1 + d$x2      # x3 and x4 are irrelevant

fit <- piqr(y ~ x1 + x2 + x3 + x4, formula.p = ~ slp(p, 2), data = d, nlambda = 30)
best <- gof.piqr(fit, method = "BIC", plot = FALSE)
best
#> 
#> Selection of lambda by BIC 
#> 
#> Selected lambda: 9.051  (position 7 )
#> Degrees of freedom: 8 
#> 
#> Coefficients:
#>              (Intercept)  slp1      slp2    
#> (Intercept)   0.17508      1.57385   0.42889
#> x1            0.32843      0.93906  -0.38235
#> x2            0.89827           NA  -0.07583
#> x3                 NA           NA        NA
#> x4                 NA           NA        NA

# unpenalized re-estimation of the selected model, with standard errors
m <- refit_piqr(fit, pos.lambda = best$posMinLambda)
summary(m)
#> 
#> Call: qrcm::iqr(formula = y ~ x1 + x2 + x3 + x4, formula.p = ~slp(p, 
#>     2), data = d, s = structure(c(1, 1, 1, 0, 0, 1, 1, 1, 0, 
#> 0, 1, 1, 1, 0, 0), dim = c(5L, 3L)))
#> 
#> converged: TRUE 
#> n. of iterations: 8 
#> n. of free parameters: 9 
#> n. of observations: 300 
#> 
#> ###################### 
#> ###################### 
#> 
#> Coefficients:
#>              (Intercept)  slp1      slp2    
#> (Intercept)  -0.63231      1.67534   0.18250
#> x1            0.94257      0.81220  -0.16305
#> x2            1.03363     -0.01867  -0.05563
#> x3                 NA           NA        NA
#> x4                 NA           NA        NA
#> 
#> Standard errors:
#>              (Intercept)  slp1     slp2   
#> (Intercept)  0.44946      0.24757  0.24323
#> x1           0.27067      0.15583  0.14521
#> x2           0.10498      0.07303  0.06567
#> x3                NA           NA       NA
#> x4                NA           NA       NA
#> 
#> ###################### 
#> ###################### 
#> 
#> Wald test for x:
#>             chi-square df P(> chi)    
#> (Intercept)      75.11  3 3.43e-16 ***
#> x1              442.62  3  < 2e-16 ***
#> x2              343.33  3  < 2e-16 ***
#> x3                  NA  0       NA    
#> x4                  NA  0       NA    
#> 
#> 
#> Wald test for b(p):
#>             chi-square df P(> chi)    
#> (Intercept)    555.687  3   <2e-16 ***
#> slp1           528.039  3   <2e-16 ***
#> slp2             2.719  3    0.437    
#> 
#> Minimized loss function: 128.1989

Nonlinear parametric quantile models

set.seed(1234)
x <- runif(n)
fun <- function(theta, p) cbind(theta[1] + exp(theta[2] * p), theta[3] + theta[4] * p)
b <- fun(c(1, 1, 1, 1), runif(n))
dn <- data.frame(y = b[, 1] + b[, 2] * x, x = x)

m2 <- niqr(y ~ x, data = dn, fun = fun, x0 = rep(0.5, 4))
summary(m2)
#> 
#> Call: niqr(formula = y ~ x, data = dn, fun = fun, x0 = rep(0.5, 4))
#> 
#> n. of observations: 300    n. of iterations: 5 
#> Standard errors: sandwich 
#> 
#>        Estimate Std. Error z value Pr(>|z|)    
#> theta0  1.05523    0.06841  15.426  < 2e-16 ***
#> theta1  1.00759    0.03903  25.816  < 2e-16 ***
#> theta2  1.02334    0.14501   7.057 1.70e-12 ***
#> theta3  1.04106    0.20787   5.008 5.49e-07 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Minimized loss function: 0.1852556 
#> AIC = 63.5767   BIC = 78.3918   AIC* = -1.67269   BIC* = -1.64799

References

Frumento, P. and Bottai, M. (2016). Parametric modeling of quantile regression coefficient functions. Biometrics, 72(1), 74-84. https://doi.org/10.1111/biom.12410
Sottile, G., Frumento, P., Chiodi, M. and Bottai, M. (2020). A penalized approach to covariate selection through quantile regression coefficient models. Statistical Modelling, 20(4), 369-385. https://doi.org/10.1177/1471082X19825523
Bottai, M. and Cilluffo, G. (2020). Nonlinear parametric quantile models. Statistical Methods in Medical Research, 29(12), 3757-3769. https://doi.org/10.1177/0962280220941159