iqr: Quantile Regression Coefficients Modeling
In qrcm: Quantile Regression Coefficients Modeling

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

This function implements Frumento and Bottai's (2016, 2017) methods for quantile regression coefficients modeling (qrcm). Quantile regression coefficients are described by (flexible) parametric functions of the order of the quantile. Quantile crossing can be eliminated using the method described in Sottile and Frumento (2021).

1 2	iqr(formula, formula.p = ~ slp(p,3), weights, data, s, tol = 1e-6, maxit, remove.qc = FALSE)

`formula`	a two-sided formula of the form `y ~ x1 + x2 + ...`: a symbolic description of the quantile regression model. The left side of the formula is `Surv(time,event)` if the data are right-censored, and `Surv(time,time2,event)` if the data are right-censored and left-truncated (`time < time2`, `time` can be `-Inf`).
`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. The weights will always be normalized to sum to the sample size. This implies that, for example, using double weights will not halve the standard errors.
`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’).
`tol`	convergence criterion for numerical optimization.
`maxit`	maximum number of iterations.
`remove.qc`	either a logical value, or a list created with `qc.control`. See ‘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’.

If no censoring and truncation are present, estimation of θ is carried out by minimizing an objective function that corresponds to the integral, with respect to p, of the loss function of standard quantile regression. Details are in Frumento and Bottai (2016). If the data are censored or truncated, instead, θ is estimated by solving the estimating equations described in Frumento and Bottai (2017).

The option remove.qc applies the method described by Sottile and Frumento (2021) to remove quantile crossing. You can either choose remove.qc = TRUE, or use remove.qc = qc.control(...), which allows to specify the operational parameters of the algorithm. Please read qc.control for more details on the method, and use diagnose.qc to diagnose quantile crossing.

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

`coefficients`	a matrix of estimated model parameters describing the fitted quantile function.
`converged`	logical. The convergence status.
`n.it`	the number of iterations.
`call`	the matched call.
`obj.function`	if the data are neither censored nor truncated, the value of the minimized loss function; otherwise, a meaningful loss function which, however, is not the objective function of the model (see note 3). The number of model parameter is returned as an attribute.
`mf`	the model frame used.
`PDF, CDF`	the fitted values of the conditional probability density function (PDF) and cumulative distribution function (CDF). See note 1 for details.
`covar`	the estimated covariance matrix.
`s`	the used ‘`s`’ matrix.

Use summary.iqr, plot.iqr, and predict.iqr for summary information, plotting, and predictions from the fitted model. The function test.fit can be used for goodness-of-fit assessment. The generic accessory functions coefficients, formula, terms, model.matrix, vcov are available to extract information from the fitted model. The special function diagnose.qc can be used to diagnose quantile crossing.

NOTE 1 (PDF, CDF, quantile crossing, and goodness-of-fit). By expressing quantile regression coefficients as functions of p, you practically specify a parametric model for the entire conditional distribution. The induced CDF is the value p* such that y = Q(p* | x). The corresponding PDF is given by 1/Q'(p* | x). Negative values of PDF indicate quantile crossing, occurring when the estimated quantile function is not monotonically increasing. If 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 test.fit.

NOTE 2 (model intercept). The intercept can be excluded from formula, e.g., iqr(y ~ -1 + x). This, however, implies that when x = 0, y is zero at all quantiles. 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’.

NOTE 3 (censoring, truncation, and loss function). Data are right-censored when, instead of a response variable T, one can only observe Y = min(T,C) and d = I(T ≤ C). Here, C is a censoring variable that is assumed to be conditionally independent of T. Additionally, left truncation occurs if Y can only be observed when it exceeds another random variable Z. For example, in the prevalent sampling design, subjects with a disease are enrolled; those who died before enrollment are not observed.

Ordinary quantile regression minimizes L(β(p)) = ∑ (p - ω)(t - x'β(p)) where ω = I(t ≤ x'β(p)). Equivalently, it solves its first derivative, S(β(p)) = ∑ x(ω - p). The objective function of iqr is simply the integral of L(β(p | θ)) with respect to p.

If the data are censored and truncated, ω is replaced by

ω* = ω.y + (1 - d)ω.y(p - 1)/S.y - ω.z - ω.z(p - 1)/S.z + p

where ω.y = I(y ≤ x'β(p)), ω.z = I(z ≤ x'β(p)), S.y = P(T > y), and S.z = P(T > z). The above formula can be obtained from equation (7) of Frumento and Bottai, 2017. Replacing ω with ω* in L(β(p)) is NOT equivalent to replacing ω with ω* in S(β(p)). The latter option leads to a much simpler computation, and generates the estimating equation used by iqr. This means that, if the data are censored or truncated, the obj.function returned by iqr is NOT the objective function being minimized, and should not be used to compare models. However, if one of two models has a much larger value of the obj.function, this may be a sign of severe misspecification or poor convergence.

Paolo Frumento paolo.frumento@unipi.it

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

Frumento, P., and Bottai, M. (2017). Parametric modeling of quantile regression coefficient functions with censored and truncated data. Biometrics, doi: 10.1111/biom.12675.

Sottile, G., and Frumento, P. (2021). Parametric estimation of non-crossing quantile functions. Statistical Modelling [forthcoming].

summary.iqr, plot.iqr, predict.iqr, for summary, plotting, and prediction, and test.fit.iqr for goodness-of-fit assessment; plf and slp to define b(p) to be a piecewise linear function and a shifted Legendre polynomial basis, respectively; diagnose.qc to diagnose quantile crossing.

  ##### 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 <- iqr(y ~ x, formula.p = ~ I(qnorm(p)))
  # 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 <- iqr(y ~ x, formula.p = ~ p)

  # a 'rich' basis b(p) = (1, p, p^2, log(p), log(1 - p))
  m3 <- iqr(y ~ x, formula.p = ~ p + I(p^2) + I(log(p)) + I(log(1 - p)))

  # 'slp' creates an orthogonal spline basis using shifted Legendre polynomials
  m4 <- iqr(y ~ x, formula.p = ~ slp(p, k = 3)) # note that this is the default
  
  # 'plf' creates the basis of a piecewise linear function
  m5 <- iqr(y ~ x, formula.p = ~ plf(p, knots = c(0.1,0.9)))

  
  summary(m1)
  summary(m1, p = c(0.25,0.5,0.75))
  test.fit(m1)
  par(mfrow = c(1,2)); plot(m1, ask = FALSE)
  # see the documentation for 'summary.iqr', 'test.fit.iqr', and 'plot.iqr'
  



  ##### Example 2 ### excluding coefficients
  
  n <- 1000
  x <- runif(n)
  qy <- function(p,x){(1 + qnorm(p)) + (1 + log(p))*x}
  # true quantile function: Q(p | x) = beta0(p) + beta1(p)*x, with
    # beta0(p) = 1 + qnorm(p) 
    # beta1(p) = 1 + log(p)
  
  y <- qy(runif(n), x) # to generate y, plug uniform p in qy(p,x) 
  iqr(y ~ x, formula.p = ~ I(qnorm(p)) + I(log(p)))

  # I would like to exclude log(p) from beta0(p), and qnorm(p) from beta1(p)
  # I set to 0 the corresponding entries of 's'

  s <- matrix(1,2,3); s[1,3] <- s[2,2] <- 0
  iqr(y ~ x, formula.p = ~ I(qnorm(p)) + I(log(p)), s = s)


  

  ##### Example 3 ### excluding coefficients when b(p) is singular
  
  n <- 1000
  x <- runif(n)
  qy <- function(p,x){(1 + log(p) - 2*log(1 - p)) + (1 + log(p/(1 - p)))*x} 
  # true quantile function: Q(p | x) = beta0(p) + beta1(p)*x, with
    # beta0(p) = 1 + log(p) - 2*log(1 - p)
    # beta1(p) = 1 + log(p/(1 - p))

  y <- qy(runif(n), x) # to generate y, plug uniform p in qy(p,x)

  iqr(y ~ x, formula.p = ~ I(log(p)) + I(log(1 - p)) + I(log(p/(1 - p))))
  # log(p/(1 - p)) is dropped due to singularity
  
  # I want beta0(p) to be a function of log(p) and log(1 - p),
  # and beta1(p) to depend on log(p/(1 - p)) alone

  s <- matrix(1,2,4); s[2,2:3] <- 0
  iqr(y ~ x, formula.p = ~ I(log(p)) + I(log(1 - p)) + I(log(p/(1 - p))), s = s)
  # log(p/(1 - p)) is not dropped




  ##### Example 4 ### using slp to test deviations from normality
  
  n <- 1000
  x <- runif(n)
  y <- rnorm(n, 2 + x) 
  # the true model is normal, i.e., b(p) = (1, qnorm(p))
  
  summary(iqr(y ~ x, formula.p = ~ I(qnorm(p)) + slp(p,3))) 
  # if slp(p,3) is not significant, no deviation from normality




  ##### Example 5 ### formula without intercept
  
  n <- 1000
  x <- runif(n)
  y <- runif(n, 0,x) 

  # True quantile function: Q(p | x) = p*x, i.e., beta0(p) = 0, beta1(p) = p
  # When x = 0, all quantiles of y are 0, i.e., the distribution is degenerated
  # To explicitly model this, remove the intercept from 'formula'
  
  iqr(y ~ -1 + x, formula.p = ~ p)
  
  # the true model does not have intercept in b(p) either:

  iqr(y ~ -1 + x, formula.p = ~ -1 + p)




  ##### Example 6 ### no covariates, strictly positive outcome
  
  n <- 1000
  y <- rgamma(n, 3,1) 

  # you know that Q(0) = 0
  # remove intercept from 'formula.p', and use b(p) such that b(0) = 0
  
  summary(iqr(y ~ 1, formula.p = ~ -1 + slp(p,5))) # shifted Legendre polynomials
  summary(iqr(y ~ 1, formula.p = ~ -1 + sin(p*pi/2) + I(qbeta(p,2,4)))) # unusual basis
  summary(iqr(y ~ 1, formula.p = ~ -1 + I(sqrt(p))*I(log(1 - p)))) # you can include interactions




  ##### Example 7 ### revisiting the classical linear model
  
  n <- 1000
  x <- runif(n)
  y <- 2 + 3*x + rnorm(n,0,1) # beta0 = 2, beta1 = 3
  
  iqr(y ~ x, formula.p = ~ I(qnorm(p)), s = matrix(c(1,1,1,0),2))
  # first column of coefficients: (beta0, beta1)
  # top-right coefficient: residual standard deviation
  
  


  ##### Example 8 ### censored data
  
  n <- 1000
  x <- runif(n,0,5)
	
  u <- runif(n)
  beta0 <- -log(1 - u)
  beta1 <- 0.2*log(1 - u)
  t <- beta0 + beta1*x  # time variable
  c <- rexp(n,2)        # censoring variable
  y <- pmin(t,c)        # observed events
  d <- (t <= c)         # 1 = event, 0 = censored
  
  iqr(Surv(y,d) ~ x, formula.p = ~ I(log(1 - p)))
  
  ##### Example 8 (cont.) ### censored and truncated data

  z <- rexp(n,10)   # truncation variable
  w <- which(y > z) # only observe z,y,d,x when y > z
  z <- z[w]; y <- y[w]; d <- d[w]; x <- x[w]

  iqr(Surv(z,y,d) ~ x, formula.p = ~ I(log(1 - p)))