Package 'Qest'

Baseline Cumulative Hazard of a Qcoxph Model

Description

Baseline Cumulative Hazard of a Qcoxph Model

Usage

basehaz.Qcoxph(fit, centered = TRUE, se.fit = FALSE, times)

Arguments

fit	an object of class "Qcoxph".
centered	logical: if TRUE, the cumulative hazard at the mean of the covariates; otherwise at $x = 0$.
se.fit	logical: compute the variance of the estimates (delta method)?
times	the times at which $H_0$ is computed (by default, the observed times).

Details

The baseline cumulative hazard of the fitted model is piecewise linear, $H_0(t) = \sum_l \exp(\gamma_l) b_l(t)$. Unlike survival::basehaz(), this is not a step function.

Value

A data frame with columns hazard (the cumulative hazard), time and, if se.fit = TRUE, varhaz.

See Also

Qcoxph(), survfit.Qcoxph()

Examples

library(survival)
set.seed(1)
x <- runif(200)
t <- rexp(200, exp(x))
cens <- runif(200, 0, 3)
fit <- Qcoxph(Surv(pmin(t, cens), t <= cens) ~ x)
head(basehaz.Qcoxph(fit, se.fit = TRUE))

---

Inverse of a Quantile Function

Description

Given a quantile function $Q(\tau \mid x, \theta)$ and a value of $\theta$, computes $F(y_i \mid x_i, \theta)$ by bisection.

Usage

invQ(Q, theta, y, data, n.it = 17)

Arguments

Q	a quantile function Q(theta, tau, data) (see Qest()).
theta	the parameters.
y	the response: one value for each row of data.
data	a data frame.
n.it	the number of iterations of the bisection algorithm (the accuracy is $2^{-n.it}$).

Value

A vector with $F(y_i \mid x_i, \theta)$.

See Also

Qest()

Examples

Qlinmod <- function(theta, tau, data) {
  qnorm(tau, theta[2] + theta[3] * data$x, exp(theta[1]))
}
set.seed(1)
dat <- data.frame(x = runif(10))
U <- runif(10)
y <- Qlinmod(c(0, 1, 2), U, dat)
cbind(U, invQ(Qlinmod, c(0, 1, 2), y, dat))

---

Diagnostic Plot of Q-Estimators

Description

If the model is correctly specified, $F(y_i \mid x_i)$ are a sample from a $U(0, 1)$ distribution. The plot compares their empirical distribution (the Kaplan-Meier estimator, with censored or truncated data) with the uniform distribution function.

Usage

## S3 method for class 'Qest'
plot(x, conf.int = TRUE, ...)

Arguments

x	an object of class "Qest" (or "Qlm").
conf.int	logical: add pointwise 95% confidence bands (Greenwood)?
...	additional graphical parameters passed to graphics::plot().

Value

Invisibly, a data frame with the Kaplan-Meier estimate of the distribution of $F(y_i \mid x_i)$.

See Also

Qest()

Examples

set.seed(1)
y <- rgamma(300, shape = 2, scale = 3)
plot(Qest(y ~ 1, Q = Qgamma))   # good fit
plot(Qest(y ~ 1, Q = Qnorm))    # poor fit

---

Predictions from a Qcoxph Model

Description

Predictions from a Qcoxph Model

Usage

## S3 method for class 'Qcoxph'
predict(
  object,
  newdata,
  type = c("lp", "risk", "expected", "survival", "quantile"),
  se.fit = FALSE,
  p = c(0.25, 0.5, 0.75),
  ...
)

Arguments

object	an object of class "Qcoxph".
newdata	an optional data frame. For type = "expected" and type = "survival" it must contain the response variables.
type	the type of prediction: the linear predictor ("lp", centered at the mean of the covariates, as in survival::coxph()), the relative risk ("risk", $\exp(lp)$), the expected number of events $H(y \mid x)$ ("expected", minus $H(z \mid x)$ with truncation), the survival probability $\exp(-expected)$ ("survival"), or the quantiles of the time-to-event ("quantile").
se.fit	logical: compute standard errors (delta method)? Not available for type = "quantile".
p	the quantile orders for type = "quantile".
...	not used.

Value

A vector, or a list with items fit and se.fit; for type = "quantile", a matrix with one column for each value of p.

See Also

Qcoxph(), survfit.Qcoxph()

Examples

library(survival)
set.seed(1)
dat <- data.frame(x = runif(200))
t <- rexp(200, exp(dat$x)); cens <- runif(200, 0, 3)
dat$y <- pmin(t, cens); dat$d <- as.numeric(t <= cens)
fit <- Qcoxph(Surv(y, d) ~ x, data = dat)
predict(fit, newdata = data.frame(x = c(0, 0.5, 1)), type = "risk", se.fit = TRUE)
predict(fit, newdata = data.frame(x = c(0, 1)), type = "quantile", p = c(0.25, 0.5))

---

Predictions from Q-Estimators

Description

Computes the fitted quantiles $Q(p \mid x)$, or the fitted distribution function $F(y \mid x)$ at the observed (or new) values of the response.

Usage

## S3 method for class 'Qest'
predict(
  object,
  newdata,
  type = c("quantile", "CDF"),
  p = c(0.1, 0.25, 0.5, 0.75, 0.9),
  ...
)

Arguments

object	an object of class "Qest".
newdata	an optional data frame. For type = "CDF", it must contain the response.
type	"quantile" (default) or "CDF".
p	the quantile orders for type = "quantile".
...	additional arguments of a user-defined Q, if they differ from those used to fit the model.

Details

For a user-defined Q, newdata is passed as the argument data of Q. For type = "CDF", $F(y \mid x)$ is computed by inverting $Q$ (see invQ()). For censored data, the CDF is evaluated at the observed times.

Value

For type = "quantile", a matrix with one row for each observation and one column for each value of p; for type = "CDF", a vector.

See Also

Qest(), invQ()

Examples

set.seed(1)
dat <- data.frame(x = runif(200))
dat$y <- rgamma(200, shape = 2, scale = exp(1 + dat$x))
fit <- Qest(y ~ x, Q = Qgamma, data = dat)
predict(fit, newdata = data.frame(x = c(0, 1)), p = c(0.1, 0.5, 0.9))
head(predict(fit, type = "CDF"))

---

Predictions from Qlm Models

Description

Predictions from Qlm Models

Usage

## S3 method for class 'Qlm'
predict(
  object,
  newdata,
  type = c("response", "quantile"),
  se.fit = FALSE,
  p = c(0.1, 0.25, 0.5, 0.75, 0.9),
  na.action = stats::na.pass,
  ...
)

Arguments

object	an object of class "Qlm".
newdata	an optional data frame.
type	"response" (the fitted mean, i.e. median, $x^\top\beta$) or "quantile" ($x^\top\beta + \sigma\Phi^{-1}(p)$).
se.fit	logical: return the standard errors of the fitted mean?
p	the quantile orders for type = "quantile".
na.action	how to treat missing values in newdata.
...	not used.

Value

A vector, a list with items fit and se.fit, or (for type = "quantile") a matrix with one column for each value of p.

See Also

Qlm()

Examples

set.seed(1)
x <- runif(100)
y <- 1 + 2 * x + rnorm(100)
fit <- Qlm(y ~ x)
predict(fit, newdata = data.frame(x = c(0, 0.5)), se.fit = TRUE)
predict(fit, newdata = data.frame(x = c(0, 0.5)), type = "quantile", p = c(0.1, 0.9))

---

Q-Estimation of Proportional Hazards Regression Models

Description

Fits a proportional hazards model with the Q-estimator of Sottile and Frumento (2022). The baseline cumulative hazard $H_0(t)$ is piecewise linear, so that the model is fully parametric, and $Q(\tau \mid x) = H_0^{-1}(-\exp(-x^\top\beta)\log(1 - \tau))$.

Usage

Qcoxph(
  formula,
  weights,
  start,
  data,
  subset,
  na.action,
  knots,
  wtau = NULL,
  control = Qcoxph.control(),
  ...
)

Arguments

formula	a formula with a survival::Surv() response: Surv(time, event) for right-censored data, or Surv(start, stop, event) for left-truncated and right-censored data. Times must be positive.
weights	an optional vector of non-negative (sampling) weights.
start	optional starting values for the regression coefficients $\beta$.
data	an optional data frame.
subset, na.action	as in stats::lm().
knots	the knots of $H_0(t)$: a number (the number of internal knots, placed at the quantiles of the observed event times) or a vector with the positions of all knots, including the boundaries (the first knot must not be larger than the smallest time, the last not smaller than the largest time). By default, max(1, min(floor(n.events / 30), 3)) internal knots are used.
wtau	an optional weighting function $w(\tau)$, e.g. wtrunc().
control	a list created by Qcoxph.control().
...	additional arguments of wtau.

Details

The model is $h(t \mid x) = h_0(t)\exp(x^\top\beta)$, with $H_0(t) = \sum_l \exp(\gamma_l)\,b_l(t)$, where $b_l$ is a piecewise linear basis. All quantities of the estimating equations have closed forms (Appendix D2 of the paper). The coefficients have the same interpretation as in survival::coxph(); the estimates are typically very close, but more robust. Starting points are obtained from the partial likelihood and the Breslow estimator of $H_0$.

Methods are available to compute the fitted survival curves (survfit.Qcoxph(), which returns an object of class "survfit"), predictions (predict.Qcoxph()), residuals (residuals.Qcoxph()) and the baseline cumulative hazard (basehaz.Qcoxph()). All of them are based on the fitted parametric model.

Value

An object of class c("Qcoxph", "Qest"), a list with items

coefficients, var	the estimated $\beta$ and their covariance matrix.
obj.function	the loss at the solution.
iter, converged	number of iterations and convergence.
linear.predictors	$x_i^\top\beta - \bar x^\top\beta$.
residuals	martingale residuals.
means, n, nevent	means of the covariates, number of observations and of events.
concordance	Harrell's concordance index (from survival::concordancefit()).
internal	the knots, the coefficients $\gamma$ of $H_0$, the full covariance matrix (covar), $F(y_i \mid x_i)$ (CDF), $f(y_i \mid x_i)$ (PDF), and other quantities used by the methods.

and terms, assign, formula, call, y, x, model, xlevels, na.action.

References

Sottile, G. and Frumento, P. (2022). Robust estimation and regression with parametric quantile functions. Computational Statistics and Data Analysis, 171, 107471. doi:10.1016/j.csda.2022.107471

See Also

survfit.Qcoxph(), predict.Qcoxph(), residuals.Qcoxph(), basehaz.Qcoxph(), Qest()

Examples

library(survival)
set.seed(1)
n <- 200
x <- runif(n, 0, 3)
t <- rweibull(n, shape = 2, scale = (1 / exp(2 + 2 * x))^(1 / 2))
cens <- runif(n, 0, 1)
dat <- data.frame(y = pmin(t, cens), d = as.numeric(t <= cens), x = x)

m1 <- coxph(Surv(y, d) ~ x, data = dat)
m2 <- Qcoxph(Surv(y, d) ~ x, data = dat)
summary(m2)
cbind(coxph = coef(m1), Qcoxph = coef(m2))
sf <- survfit(m2, newdata = data.frame(x = c(0.5, 1.5)))
plot(sf, col = 1:2)

---

Quantile-Based Estimators (Q-Estimators)

Description

Fits a parametric model by minimizing the integrated loss of quantile regression (Sottile and Frumento, 2022). The model is specified through its quantile function $Q(\tau \mid x, \theta)$, either written by the user or chosen among the built-in families (see Qfamily). Censored and truncated data are allowed.

Usage

Qest(
  formula,
  Q,
  weights,
  start,
  data,
  subset,
  na.action,
  ntau = 199,
  wtau = NULL,
  control = Qest.control(),
  ...
)

Arguments

formula	a two-sided formula of the form y ~ x. The response can be a numeric vector, a right-censored survival::Surv(time, event) object, or a left-truncated and right-censored survival::Surv(start, stop, event) object. With a user-defined Q, the right-hand side of the formula is only used to define the covariates of the starting points: write y ~ 1 if no covariates are used by Q.
Q	a quantile function Q(theta, tau, data) (see "Details"), or a Qfamily (Qnorm, Qgamma, Qpois, Qunif).
weights	an optional vector of non-negative weights. They are treated as sampling weights: the (sandwich) standard errors do not depend on their scale.
start	the starting points. With a user-defined Q, start is required, because its length defines the number of parameters: use NA for the parameters without a good starting value (with any NA, the starting points are estimated, see "Details"). With a Qfamily, start is optional.
data	an optional data frame containing the variables in the model. It is also the argument data of Q.
subset, na.action	as in stats::lm().
ntau	the number of points of the grid of quantile orders used for numerical integration (default 199).
wtau	an optional weighting function $w(\tau)$, e.g. wtrunc(). Its first argument must be tau; other arguments are passed through ....
control	a list of operational parameters created by Qest.control().
...	additional named arguments of wtau or Q.

Details

The Q-estimator minimizes

$L(\theta) = \sum_i \int_0^1 w(\tau)\rho_\tau(y_i - Q(\tau \mid x_i, \theta))\,d\tau,$

where $\rho_\tau(u) = u(\tau - I(u < 0))$. It is consistent and asymptotically normal, and its robustness can be tuned by choosing $w(\tau)$ (Sottile and Frumento, 2022). For censored and truncated data, the estimating equations of Appendix A of the paper are solved.

User-defined quantile functions. Q(theta, tau, data) must have these three arguments, in this order (other arguments can follow and are passed through ...), and must return, for each row of data, $Q(\tau \mid x_i, \theta)$. The argument tau is either a matrix with one row for each row of data (then Q returns a matrix of the same size), or a vector with one value for each row of data (then Q returns a vector). data is the data frame used to fit the model (after removing missing values, subset and observations with zero weight). Q must be increasing in tau and twice differentiable in theta; it is evaluated on a grid of ntau quantile orders in $[5 \cdot 10^{-4}, 1 - 5 \cdot 10^{-4}]$, and its derivatives with respect to theta are computed by central differences.

Starting points. If start contains NA (or control$restart = TRUE), a nonparametric estimate $\hat F(y \mid x)$ is obtained from a piecewise-constant hazard model that uses the covariates in formula, and the starting points minimize $\sum_i (y_i - Q(\hat F(y_i \mid x_i) \mid x_i, \theta))^2$ (Sottile and Frumento, 2022, Algorithm 1), searched from the values in start (with NA replaced by zero): with poorly scaled data, rough starting values for all parameters are recommended.

Algorithm. The estimating equations are solved by Newton-Raphson with backtracking, switching to Levenberg-Marquardt steps when the Newton step fails. The Jacobian is the exact derivative of the discretized estimating equations. Standard errors are computed with the sandwich formula.

Value

An object of class "Qest", a list with items

coefficients	the estimated parameters.
std.errs	their standard errors.
covar	the estimated covariance matrix.
obj.function	the value of the loss at the solution (for censored or truncated data, the loss of Appendix A, which is not the function whose gradient is solved).
ee, jacobian	the estimating equations (divided by the sample size) and their Jacobian at the solution.
CDF, PDF	the fitted $F(y_i \mid x_i)$ and $f(y_i \mid x_i)$.
converged, n.it	convergence of the algorithm and number of iterations.
internal	quantities used by the methods.

and the usual call, formula, terms, data, na.action, xlevels, contrasts, control.

References

Sottile, G. and Frumento, P. (2022). Robust estimation and regression with parametric quantile functions. Computational Statistics and Data Analysis, 171, 107471. doi:10.1016/j.csda.2022.107471

See Also

Qfamily, Qlm(), Qcoxph(), summary.Qest(), wtrunc(), invQ().

Examples

# Normal linear model
Qlinmod <- function(theta, tau, data) {
  X <- model.matrix(~ x1 + x2, data = data)
  qnorm(tau, c(X %*% theta[-1]), exp(theta[1]))
}
set.seed(1)
n <- 100
dat <- data.frame(x1 = rnorm(n), x2 = runif(n, 0, 3))
dat$y <- 4 + dat$x1 + 2 * dat$x2 + rnorm(n)
m1 <- Qest(y ~ x1 + x2, Q = Qlinmod, start = rep(NA, 4), data = dat)
m2 <- Qest(y ~ x1 + x2, Q = Qnorm, data = dat)    # the same model, as a Qfamily
m3 <- Qlm(y ~ x1 + x2, data = dat)                # ... or with Qlm (fastest)
summary(m1)
cbind(user = coef(m1), family = coef(m2))


# A Weibull proportional hazards model with censored data
QWeibPH <- function(theta, tau, data) {
  shape <- exp(theta[1])
  X <- model.matrix(~ x, data = data)
  qweibull(tau, shape = shape, scale = exp(-c(X %*% theta[-1]) / shape))
}
dat <- data.frame(x = runif(n, 0, 3))
t <- QWeibPH(c(log(2), -0.5, 1), runif(n), dat)
cens <- runif(n, 0, 2)
dat$y <- pmin(t, cens); dat$d <- as.numeric(t <= cens)
m4 <- Qest(survival::Surv(y, d) ~ x, Q = QWeibPH, start = rep(NA, 3), data = dat)
summary(m4)

# Gamma model with more weight around the median
y <- rgamma(n, shape = 3, scale = 2)
m5 <- Qest(y ~ 1, Q = Qgamma, wtau = function(tau, h) dnorm((tau - 0.5) / h), h = 0.2)
coef(m5)

---

Control Parameters of Qest, Qlm and Qcoxph

Description

Control Parameters of Qest, Qlm and Qcoxph

Usage

Qest.control(
  tol = 1e-08,
  maxit = NULL,
  display = FALSE,
  restart = FALSE,
  deriv.step = 1e-04,
  ...
)

Qcoxph.control(tol = 1e-08, maxit = NULL, display = FALSE, ...)

Arguments

tol	convergence tolerance on the norm of the estimating equations (divided by the sample size).
maxit	maximum number of iterations (default 50 + 25 * npar).
display	logical: print the iterations?
restart	logical: if TRUE, the starting points of Qest() with a user-defined Q are re-estimated even if start has no missing values.
deriv.step	relative step of the central differences used for the derivatives of a user-defined Q with respect to theta: the step for $\theta_j$ is deriv.step * max(1, abs(theta_j)).
...	arguments of 'Qest' 1.0.x that were removed (safeit, alpha0): they give an error.

Value

A list with the control parameters.

See Also

Qest(), Qlm(), Qcoxph()

Examples

Qest.control(tol = 1e-10, display = TRUE)

---

Family Objects for Qest

Description

Built-in parametric models for Qest(), with analytical (or nearly analytical) estimating equations: much faster and more accurate than the equivalent user-defined quantile functions.

Usage

Qgamma()

Qnorm()

Qpois(offset = NULL)

Qunif(min = TRUE)

Arguments

offset	an optional vector of offsets of the log-rate (Qpois only). An offset() term in the formula can be used instead.
min	logical (Qunif only): if TRUE, fit a $U(a, b)$ model; if FALSE, a $U(0, b)$ model.

Details

A family can be passed to Qest() by name ("Qpois"), as a function (Qpois) or as a call (Qpois()); the latter allows to pass arguments.

• Qnorm: normal homoskedastic model, $Q(\tau \mid x) = x^\top\beta + \sigma\Phi^{-1}(\tau)$; parameters log(sigma), beta. With complete data, Qlm() fits the same model with a richer output.

• Qgamma: Gamma model with shape $a$ and log-scale $x^\top\beta$; parameters log(shape), beta.

• Qpois: Poisson model with log-rate $x^\top\beta$. To obtain a continuous quantile function, qpois is replaced by the inverse with respect to $y$ of the regularized upper incomplete gamma function $\Gamma(y, \lambda)/\Gamma(y)$ (Ilienko, 2013). The model should be applied to a jittered response, y + runif(n).

• Qunif: Uniform model $U(a(x), b(x))$ with $a(x) = x^\top\alpha$ and $b(x) = x^\top\beta$, or $U(0, b(x))$ if min = FALSE.

Qnorm and Qgamma can be used with censored and truncated data; Qpois and Qunif cannot. Qgamma and Qpois require an intercept.

Value

An object of class "Qfamily".

References

Ilienko, A. (2013). Continuous counterparts of Poisson and binomial distributions and their properties. Annales Univ. Sci. Budapest., Sect. Comp., 39, 137-147.

Sottile, G. and Frumento, P. (2022). Robust estimation and regression with parametric quantile functions. Computational Statistics and Data Analysis, 171, 107471. doi:10.1016/j.csda.2022.107471

See Also

Qest()

Examples

set.seed(1)
n <- 250
x <- runif(n)
eta <- 1 + 2 * x

m1 <- Qest(y ~ x, Qnorm, data = data.frame(y = rnorm(n, eta, exp(1)), x = x))
m2 <- Qest(y ~ x, Qgamma, data = data.frame(y = rgamma(n, shape = exp(1), scale = exp(eta)), x = x))
m3 <- Qest(y ~ x, Qpois, data = data.frame(y = rpois(n, exp(eta)) + runif(n), x = x))
m4 <- Qest(y ~ x, Qunif(min = TRUE), data = data.frame(y = runif(n, 0, eta), x = x))
m5 <- Qest(y ~ x, Qunif(min = FALSE), data = data.frame(y = runif(n, 0, eta), x = x))
coef(m2)

---

Q-Estimation of Linear Regression Models

Description

Fits a linear model with normal homoskedastic errors, $Q(\tau \mid x) = x^\top\beta + \sigma\Phi^{-1}(\tau)$, with the Q-estimator of Sottile and Frumento (2022). The estimating equations and the Hessian have closed forms (Appendix D1 of the paper), and the estimator is fast and robust to outliers in the response.

Usage

Qlm(
  formula,
  data,
  subset,
  weights,
  na.action,
  start = NULL,
  contrasts = NULL,
  wtau = NULL,
  control = Qest.control(),
  ...
)

Qlm.fit(
  y,
  X,
  w = rep(1, nobs),
  start = NULL,
  wtau = NULL,
  control = Qest.control(),
  ...
)

Arguments

formula, data, subset, na.action, contrasts	as in stats::lm().
weights	an optional vector of non-negative (sampling) weights; the standard errors do not depend on their scale.
start	optional starting values for the regression coefficients (excluding $\sigma$).
wtau	an optional weighting function $w(\tau)$, e.g. wtrunc().
control	a list created by Qest.control().
...	additional arguments of wtau.
y, X, w	for Qlm.fit: the response, the design matrix and the weights.

Details

Qlm(formula) estimates the same model as Qest(formula, Q = Qnorm), with a much richer output. Censored and truncated data are not allowed (use Qest() with Q = Qnorm). Standard errors are computed with the sandwich formula, and the tests in summary.Qlm() are based on the asymptotic normal distribution.

Value

Qlm returns an object of class c("Qlm", "Qest") with items

coefficients, std.errs, covar	the estimated regression coefficients, their standard errors and covariance matrix.
dispersion	the estimated $\sigma^2$.
sigma	the estimated $\sigma$, with attribute "std.err" (the standard error of $\log\sigma$).
residuals, fitted.values	residuals and fitted values.
CDF	the fitted $F(y_i \mid x_i)$.
obj.function	the minimized loss.
gradient, hessian	the estimating equations and the Hessian of the loss at the solution (including $\log\sigma$).
covar.full	the covariance matrix of $(\log\sigma, \beta)$.

and rank, df.residual, weights, converged, n.it, call, terms, model, xlevels, contrasts, na.action. Qlm.fit returns a similar list.

References

Sottile, G. and Frumento, P. (2022). Robust estimation and regression with parametric quantile functions. Computational Statistics and Data Analysis, 171, 107471. doi:10.1016/j.csda.2022.107471

See Also

summary.Qlm(), predict.Qlm(), Qest()

Examples

set.seed(1234)
n <- 100
x1 <- rnorm(n)
x2 <- runif(n, 0, 3)
y <- rnorm(n, 4 + x1 + 2 * x2, 1)
y[1:5] <- y[1:5] + 20      # outliers

m1 <- Qlm(y ~ x1 + x2)
summary(m1)
m2 <- Qlm(y ~ x1 + x2, wtau = wtrunc, delta.left = 0.05, delta.right = 0.05)
cbind(lm = coef(lm(y ~ x1 + x2)), Qlm = coef(m1), Qlm.trunc = coef(m2))

---

Residuals of a Qcoxph Model

Description

Residuals of a Qcoxph Model

Usage

## S3 method for class 'Qcoxph'
residuals(object, type = c("martingale", "deviance"), ...)

Arguments

object	an object of class "Qcoxph".
type	"martingale" ($d_i - H(y_i \mid x_i)$) or "deviance".
...	not used.

Value

A vector of residuals.

See Also

Qcoxph()

---

Summarizing Qcoxph Fits

Description

Summarizing Qcoxph Fits

Usage

## S3 method for class 'Qcoxph'
summary(object, conf.int = 0.95, scale = 1, ...)

## S3 method for class 'summary.Qcoxph'
print(
  x,
  digits = max(getOption("digits") - 3, 3),
  signif.stars = getOption("show.signif.stars"),
  ...
)

Arguments

object	an object of class "Qcoxph".
conf.int	the level of the confidence intervals for the hazard ratios.
scale	a vector of scale factors for the coefficients.
...	not used.
x	an object of class "summary.Qcoxph".
digits, signif.stars	printing options.

Value

An object of class "summary.Qcoxph", with the table of coefficients (coefficients), the confidence intervals of the hazard ratios (conf.int), the concordance index and its standard error (concordance), and the loss (obj.function).

See Also

Qcoxph()

---

Summarizing Q-Estimators

Description

Summary methods for objects of class "Qest" (fitted by Qest()) and "Qlm" (fitted by Qlm()).

Usage

## S3 method for class 'Qest'
summary(object, covar = FALSE, ...)

## S3 method for class 'summary.Qest'
print(
  x,
  digits = max(3L, getOption("digits") - 3L),
  signif.stars = getOption("show.signif.stars"),
  ...
)

## S3 method for class 'Qlm'
summary(object, correlation = FALSE, ...)

## S3 method for class 'summary.Qlm'
print(
  x,
  digits = max(3L, getOption("digits") - 3L),
  signif.stars = getOption("show.signif.stars"),
  ...
)

Arguments

object	an object of class "Qest" or "Qlm".
covar	logical: include the covariance matrix?
...	not used.
x	a summary object.
digits, signif.stars	printing options.
correlation	logical: include the correlation matrix of the coefficients (summary.Qlm)?

Details

The tests are based on the asymptotic normal distribution of the estimators, with standard errors from the sandwich formula (Sottile and Frumento, 2022).

Value

summary.Qest returns an object of class "summary.Qest" with the table of coefficients (coefficients), the loss (obj.function), the numbers of observations and parameters, the convergence status and, optionally, the covariance matrix. summary.Qlm returns an object of class "summary.Qlm" with the table of coefficients, the estimate of $\sigma$ and its standard error, the residuals and, optionally, the correlation matrix.

See Also

Qest(), Qlm()

Examples

set.seed(1)
y <- rgamma(200, shape = 3, scale = 2)
fit <- Qest(y ~ 1, Q = Qgamma)
summary(fit, covar = TRUE)

---

Survival Curves from a Qcoxph Model

Description

Computes the survival curves $S(t \mid x) = \exp\{-H_0(t)\exp(x^\top\beta)\}$ of the fitted model, with pointwise confidence intervals based on the delta method.

Usage

## S3 method for class 'Qcoxph'
survfit(
  formula,
  newdata,
  se.fit = TRUE,
  conf.int = 0.95,
  conf.type = c("log", "log-log", "plain", "none"),
  times,
  ...
)

Arguments

formula	an object of class "Qcoxph".
newdata	a data frame with the covariates of the curves. By default, one curve at the mean of the covariates.
se.fit	logical: compute the standard errors?
conf.int	the level of the confidence intervals.
conf.type	the transformation used for the confidence intervals: "log" (default, symmetric on the scale of $H$), "log-log", "plain" or "none".
times	the times (by default, the observed times).
...	not used.

Value

An object of class "survfit" (see survival::survfit.object): the methods print, plot, lines and summary of 'survival' can be used.

See Also

Qcoxph(), predict.Qcoxph()

Examples

library(survival)
set.seed(1)
dat <- data.frame(x = runif(200))
t <- rexp(200, exp(dat$x)); cens <- runif(200, 0, 3)
dat$y <- pmin(t, cens); dat$d <- as.numeric(t <= cens)
fit <- Qcoxph(Surv(y, d) ~ x, data = dat)
sf <- survfit(fit, newdata = data.frame(x = c(0, 1)))
plot(sf, col = 1:2, conf.int = TRUE)

---

Weights for Truncated Quantile Integration

Description

A weighting function $w(\tau)$ that assigns zero weight to the quantiles below delta.left and above 1 - delta.right, to be used as the argument wtau of Qest(), Qlm() and Qcoxph(). Truncated weights make the estimators more robust to outliers (Sottile and Frumento, 2022, Section 3).

Usage

wtrunc(
  tau,
  delta.left = 0.05,
  delta.right = 0.05,
  smooth = FALSE,
  sigma = 0.01
)

Arguments

tau	a vector of quantile orders in $[0, 1]$.
delta.left, delta.right	the proportions of quantiles with zero weight in the left and right tail.
smooth	logical: if TRUE, a smooth version based on the normal distribution function is used.
sigma	the scale of the smooth version (only used if smooth = TRUE).

Details

With smooth = FALSE, $w(\tau) = I(\delta_L \le \tau \le 1 - \delta_R)$. With smooth = TRUE, the indicators are replaced by $\Phi((\tau - \delta_L)/\sigma)$ and $1 - \Phi((\tau - 1 + \delta_R)/\sigma)$. A delta equal to zero means no truncation on that side. The arguments delta.left, delta.right, smooth and sigma are passed through the ... of the fitting functions.

Value

A vector of weights of the same length as tau.

References

Sottile, G. and Frumento, P. (2022). Robust estimation and regression with parametric quantile functions. Computational Statistics and Data Analysis, 171, 107471. doi:10.1016/j.csda.2022.107471

Examples

taus <- seq(0, 1, length.out = 1000)
plot(taus, wtrunc(taus, delta.left = 0.05, delta.right = 0.1), type = "l")
lines(taus, wtrunc(taus, 0.05, 0.1, smooth = TRUE, sigma = 0.01), col = 2)

# in a model
set.seed(1)
y <- c(rnorm(95), rnorm(5, 10))   # 5% of outliers
fit <- Qlm(y ~ 1, wtau = wtrunc, delta.left = 0.05, delta.right = 0.05)
coef(fit)

---