piqr: Penalized Quantile Regression Coefficients Modeling

Description

This package implements a penalized Frumento and Bottai's (2016) method for quantile regression coefficient modeling (qrcm), in which quantile regression coefficients are described by (flexible) parametric functions of the order of the quantile. This package fits lasso qrcm using pathwise coordinate descent algorithm.

Usage

piqr(formula, formula.p = ~ slp(p, 3), weights, data, s, nlambda=100,
     lambda.min.ratio=ifelse(nobs

Value

An object of class “piqr”, a list containing the following items:

call: the matched call.
lambda: The actual sequence of lambda values used.
coefficients: a list of estimated model parameters describing the fitted quantile function along the path.
minimum: the value of the minimized integrated loss function for each value of lambda.
dl: a matrix of gradient values along the path.
df: The number of nonzero coefficients for each value of lambda.
seqS: a list containg each matrix s for each value of lambda.
internal: a list containing some initial object.

Arguments

formula: a two-sided formula of the form y ~ x1 + x2 + ...: a symbolic description of the quantile regression model.
formula.p: a one-sided formula of the form ~ b1(p, ...) + b2(p, ...) + ..., describing how quantile regression coefficients depend on p, the order of the quantile.
weights: an optional vector of weights to be used in the fitting process.
data: an optional data frame, list or environment containing the variables in formula.
s: an optional 0/1 matrix that permits excluding some model coefficients (see ‘Examples’).
nlambda: the number of lambda values - default is 100.
lambda.min.ratio: Smallest value for lambda, as a fraction of lambda.max. The default depends on the sample size nobs relative to the number of variables nvars. If nobs > nvars, the default is 0.0001, close to zero. If nobs < nvars, the default is 0.01.
lambda: A user supplied lambda sequence.
display: if TRUE something is printed - default is TRUE.
tol: convergence criterion for numerical optimization - default is 1e-6.
maxit: maximum number of iterations - default is 100.

Author

Gianluca Sottile gianluca.sottile@unipa.it

Details

Quantile regression permits modeling conditional quantiles of a response variabile, given a set of covariates. A linear model is used to describe the conditional quantile function: $$Q(p | x) = \beta_0(p) + \beta_1(p)x_1 + \beta_2(p)x_2 + \ldots.$$ The model coefficients $\beta(p)$ describe the effect of covariates on the $p$-th quantile of the response variable. Usually, one or more quantiles are estimated, corresponding to different values of $p$.

Assume that each coefficient can be expressed as a parametric function of $p$ of the form: $$\beta(p | \theta) = \theta_{0} + \theta_1 b_1(p) + \theta_2 b_2(p) + \ldots$$ where $b_1(p), b_2(p, \ldots)$ are known functions of $p$. If $q$ is the dimension of $x = (1, x_1, x_2, \ldots)$ and $k$ is that of $b(p) = (1, b_1(p), b_2(p), \ldots)$, the entire conditional quantile function is described by a $q \times k$ matrix $\theta$ of model parameters.

Users are required to specify two formulas: formula describes the regression model, while formula.p identifies the 'basis' $b(p)$. By default, formula.p = ~ slp(p, k = 3), a 3rd-degree shifted Legendre polynomial (see slp). Any user-defined function $b(p, \ldots)$ can be used, see ‘Examples’.

Estimation of penalized $\theta$ is carried out by minimizing a penalized integrated loss function, corresponding to the integral, over $p$, of the penalized loss function of standard quantile regression. This motivates the acronym piqr (penalized integrated quantile regression).

See details in iqr

References

Sottile G, Frumento P, Chiodi M, Bottai M. (2020). A penalized approach to covariate selection through quantile regression coefficient models. Statistical Modelling, 20(4), pp 369-385. doi:10.1177/1471082X19825523.

Frumento, P., and Bottai, M. (2016). Parametric modeling of quantile regression coefficient functions. Biometrics, 72(1), pp 74-84, doi:10.1111/biom.12410.

Friedman, J., Hastie, T. and Tibshirani, R. (2008). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, Vol. 33(1), pp 1-22 Feb 2010.

Examples

Run this code


  ##### Using simulated data in all examples

  ##### Example 1
  set.seed(1234)
  n <- 300
  x1 <- rexp(n)
  x2 <- runif(n, 0, 5)
  x <- cbind(x1,x2)

  b <- function(p){matrix(cbind(1, qnorm(p), slp(p, 2)), nrow=4, byrow=TRUE)}
  theta <- matrix(0, nrow=3, ncol=4); theta[, 1] <- 1; theta[1,2] <- 1; theta[2:3,3] <- 2
  qy <- function(p, theta, b, x){rowSums(x * t(theta %*% b(p)))}

  y <- qy(runif(n), theta, b, cbind(1, x))

  s <- matrix(1, nrow=3, ncol=4); s[1,3:4] <- 0; s[2:3, 2] <- 0
  obj <- piqr(y ~ x1 + x2, formula.p = ~ I(qnorm(p)) + slp(p, 2), s=s, nlambda=50)

  best <- gof.piqr(obj, method="AIC", plot=FALSE)
  best2 <- gof.piqr(obj, method="BIC", plot=FALSE)

  summary(obj, best$posMinLambda)
  summary(obj, best2$posMinLambda)

  if (FALSE) {
  ##### other examples
  set.seed(1234)
  n <- 1000
  q <- 5
  k <- 3
  X <- matrix(abs(rnorm(n*q)), n, q)
  rownames(X) <- 1:n
  colnames(X) <- paste0("X", 1:q)
  theta <- matrix(c(3, 1.5, 1, 1,
                    2, 1, 1, 1,
                    0, 0, 0, 0,
                    0, 0, 0, 0,
                    1.5, 1, 1, 1,
                    0, 0, 0, 0),
                  ncol=(k+1), byrow=TRUE)
  rownames(theta) <- c("(intercept)", paste0("X", 1:q))
  colnames(theta) <- c("(intercept)", "slp(p,1)", "slp(p,2)", "slp(p,3)")
  B <- function(p, k){matrix(cbind(1, slp(p, k)), nrow=(k+1), byrow=TRUE)}
  Q <- function(p, theta, B, k, X){rowSums(X * t(theta %*% B(p, k)))}

  pp <- runif(n)
  y <- Q(p=pp, theta=theta, B=B, k=k, X=cbind(1, X))
  m1 <- piqr(y ~ X, formula.p = ~ slp(p, k))
  best1 <- gof.piqr(m1, method="AIC", plot=FALSE)
  best2 <- gof.piqr(m1, method="BIC", plot=FALSE)
  summary(m1, best1$posMinLambda)
  summary(m1, best2$posMinLambda)
  par(mfrow = c(1,3)); plot(m1, xvar="lambda");
                       plot(m1, xvar="objective"); plot(m1, xvar="grad")

  set.seed(1234)
  n <- 1000
  q <- 6
  k <- 4
  # x <- runif(n)
  X <- matrix(abs(rnorm(n*q)), n, q)
  rownames(X) <- 1:n
  colnames(X) <- paste0("X", 1:q)
  theta <- matrix(c(1, 2, 0, 0, 0,
                    2, 0, 1, 0, 0,
                    0, 0, 0, 0, 0,
                    1, 0, 0, 1, -1.2,
                    0, 0, 0, 0, 0,
                    1.5, 0, .5, 0, 0,
                    0, 0, 0, 0, 0),
                  ncol=(k+1), byrow=TRUE)
  rownames(theta) <- c("(intercept)", paste0("X", 1:q))
  colnames(theta) <- c("(intercept)", "qnorm(p)", "p", "log(p)", "log(1-p)")
  B <- function(p, k){matrix(cbind(1, qnorm(p), p, log(p), log(1-p)), nrow=(k+1), byrow=TRUE)}
  Q <- function(p, theta, B, k, X){rowSums(X * t(theta %*% B(p, k)))}

  pp <- runif(n)
  y <- Q(p=pp, theta=theta, B=B, k=k, X=cbind(1, X))
  s <- matrix(1, q+1, k+1); s[2:(q+1), 2] <- 0; s[1, 3:(k+1)] <- 0; s[2:3, 4:5] <- 0
  s[4:5, 3] <- 0; s[6:7, 4:5] <- 0
  m2 <- piqr(y ~ X, formula.p = ~ qnorm(p) + p + I(log(p)) + I(log(1-p)), s=s)
  best1 <- gof.piqr(m2, method="AIC", plot=FALSE)
  best2 <- gof.piqr(m2, method="BIC", plot=FALSE)
  summary(m2, best1$posMinLambda)
  summary(m2, best2$posMinLambda)
  par(mfrow = c(1,3)); plot(m2, xvar="lambda");
                       plot(m2, xvar="objective"); plot(m2, xvar="grad")
  }
  # see the documentation for 'summary.piqr', and 'plot.piqr'

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