piqr: Penalized Quantile Regression Coefficients Modeling

In gianluca-sottile/qrcmPen: Penalized Quantile Regression Coefficients Modeling

Description

This package implements a penalized Frumento and Bottai's (2015) 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 coordinate descent.

Usage

piqr(formula, formula.p = ~ slp(p, 1), weights, data, s, nl=70,
     display=TRUE, tol=1e-6, maxit=100)

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’).
nl	the number of lambda values - default is 70.
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.

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) = β0(p) + β1(p)*x1 + β2(p)*x2 + ….

The model coefficients β(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:

β(p | θ) = θ0 + θ1*b1(p) + θ2*b2(p) + …

where b1(p), b2(p), … are known functions of p. If q is the dimension of x = (1, x1, x2, …) and k is that of b(p) = (1, b1(p), b2(p), …), the entire conditional quantile function is described by a q*k matrix θ 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, …) can be used, see ‘Examples’.

Estimation of penalized θ 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

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.

Note

By expressing quantile regression coefficients as functions of p, a parametric model for the conditional quantile function is specified. The induced PDF and CDF can be used as diagnostic tools. Negative values of PDF indicate quantile crossing, i.e., the conditional quantile function is not monotonically increasing. Null values of PDF indicate observations that lie outside the estimated support of the data, defined by quantiles of order 0 and 1. If null or negative PDF values occur for a relatively large proportion of data, the model is probably misspecified or ill-defined. If the model is correct, the fitted CDF should approximately follow a Uniform(0,1) distribution. This idea is used to implement a goodness-of-fit test, see summary.iqr and test.fit.

The intercept can be excluded from formula, e.g., iqr(y ~ -1 + x). This, however, implies that when x = 0, y is always 0. See example 5 in ‘Examples’. The intercept can also be removed from formula.p. This is recommended if the data are bounded. For example, for strictly positive data, use iqr(y ~ 1, formula.p = -1 + slp(p,3)) to force the smallest quantile to be zero. See example 6 in ‘Examples’.

Author(s)

Gianluca Sottile [e-mail]

References

Frumento, P., and Bottai, M. (2015). Parametric modeling of quantile regression coefficient functions. Biometrics, doi: 10.1111/biom.12410.

See Also

summary.piqr, plot.piqr, predict.piqr, for summary, plotting, and prediction. gof.piqr to select the best value of the tuning parameter though AIC, BIC, GIC, GCV criteria.

Examples

  ##### Using simulated data in all examples


  ##### Example 1

  n <- 1000
  x <- runif(n)
  y <- rnorm(n, 1 + x, 1 + x)
  # true quantile function: Q(p | x) = beta0(p) + beta1(p)*x, with
    # beta0(p) = beta1(p) = 1 + qnorm(p)

  # fit the true model: b(p) = (1 , qnorm(p))
  m1 <- piqr(y ~ x, formula.p = ~ I(qnorm(p)))
  best1 <- gof.piqr(m1, method="AIC", plot=FALSE)
  # the fitted quantile regression coefficient functions are
    # beta0(p) = m1$coef[1,1] + m1$coef[1,2]*qnorm(p)
    # beta1(p) = m1$coef[2,1] + m1$coef[2,2]*qnorm(p)

  # a basis b(p) = (1, p), i.e., beta(p) is assumed to be a linear function of p
  m2 <- piqr(y ~ x, formula.p = ~ p)
  best2 <- gof.piqr(m2, method="AIC", plot=FALSE)

  # a 'rich' basis b(p) = (1, p, p^2, log(p), log(1 - p))
  m3 <- piqr(y ~ x, formula.p = ~ p + I(p^2) + I(log(p)) + I(log(1 - p)))
  best3 <- gof.piqr(m3, method="AIC", plot=FALSE)

  # 'slp' creates an orthogonal spline basis using shifted Legendre polynomials
  m4 <- piqr(y ~ x, formula.p = ~ slp(p, k = 3)) # note that this is the default
  best4 <- gof.piqr(m4, method="AIC", plot=FALSE)

  # 'plf' creates the basis of a piecewise linear function
  m5 <- piqr(y ~ x, formula.p = ~ plf(p, knots = c(0.1,0.9)))
  best5 <- gof.piqr(m5, method="AIC", plot=FALSE)


  summary(m4, best4$minLambda)
  par(mfrow = c(2,2))
  plot(m4, xvar="norm")
  plot(m4, xvar="lambda")
  plot(m4, xvar="objective")
  plot(m4, xvar="grad")
  # see the documentation for 'summary.piqr', and 'plot.piqr'

gianluca-sottile/qrcmPen documentation built on May 8, 2019, 9:23 a.m.