Mid-Quantile Regression for Discrete Responses

This function is used to fit a mid-quantile regression model when the response is discrete.

Usage

midrq(formula, data, tau = 0.5, lambda = NULL, subset, weights, na.action,
  contrasts = NULL, offset, type = 3, midFit = NULL, control = list())
midrq.fit(x, y, offset, lambda, binary, midFit, type, tau, method)

Arguments

formula: an object of class formula: a symbolic description of the model to be fitted.
data: an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which midrq is called.
tau: quantile to be estimated. This can be a vector of quantiles in midrq, but must be one single quantile in midrq.fit.
lambda: a numerical value for the transformation parameter. This is provided by the user or set to NULL. The transformation is always Box-Cox, unless the response is binary (0-1) in which case the trasformation is Aranda-Ordaz. See bc and ao.
subset: an optional vector specifying a subset of observations to be used in the fitting process.
weights: an optional vector of weights to be used in the fitting process.
na.action: a function which indicates what should happen when the data contain NAs.
contrasts: an optional list. See the contrasts.arg of model.matrix.default.
offset: an optional offset to be included in the model frame. This must be provided in midrq.fit (e.g., a vector of zeros).
type: estimation method for the fitting process. See details.
midFit: cmidecdf object used for fitting conditional mid-quantiles. If set to NULL in midrq, it is automatically created. It must be provided in midrq.fit.
control: list of control parameters of the fitting process. See midrqControl.
x: design matrix of dimension \(n * p\).
y: vector of observations of length \(n\).
binary: logical flag. Is the response binary?
method: character vector that specifies the optimization algorithm in optim to fit a conditional mid-quantile model when type = 1 or type = 2. Only "Nelder-Mead" has been tested.

Details

A linear mid-quantile regression model is fitted to the transformed response. The transformation of the response can be changed with the argument lambda. If lambda = NULL, then no transformation is applied (i.e., identity); if lambda is a numeric value, then the Box-Cox transformation is applied (e.g., 0 for log-transformation). However, midrq will automatically detect whether the response is binary, in which case the Aranda-Ordaz transformation is applied. In contrast, the user must declare whether the response is binary in midrq.fit.

There are 3 different estimators. type = 1 is based on a general-purpose estimator (i.e., optim). type = 2 is similar to type = 1, except the loss function is averaged over the space of the predictors (i.e., CUSUM). type = 3 is the least-squares estimator discussed by Geraci and Farcomeni (2019).

The warning `tau is outside mid-probabilities range' indicates that there are observations for which tau is below or above the range of the corresponding estimated conditional mid-probabilities. This affects estimation in a way similar to censoring.

Value

a list of class midrq containing the following components

call: the matched call.
x: the model matrix.
y: the model response.
hy: the tranformed model response.
tau: the order of the estimated quantile(s).
coefficients: regression quantile (on the log–scale).
fitted.values: fitted values (on the response scale).
offset: offset.
terms: the terms object used.
term.labels: names of coefficients.

References

Geraci, M. and A. Farcomeni. Mid-quantile regression for discrete responses. arXiv:1907.01945 [stat.ME]. URL: https://arxiv.org/abs/1907.01945.

Author

Marco Geraci with contributions from Alessio Farcomeni

Examples


if (FALSE) { # \dontrun{
# Esterase data
data(esterase)

# Fit quantiles 0.25 and 0.75
fit <- midrq(Count ~ Esterase, tau = c(0.25, 0.75), data = esterase, type = 3, lambda = 0)
coef(fit)

# Plot
with(esterase, plot(Count ~ Esterase))
lines(esterase$Esterase, fit$fitted.values[,1], col = "blue")
lines(esterase$Esterase, fit$fitted.values[,2], col = "red")
legend(8, 1000, lty = c(1,1), col = c("blue", "red"), legend = c("tau = 0.25","tau = 0.75"))

} # }