midrq {Qtools}

Mid-Quantile Regression for Discrete Responses

Description

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.

Author(s)

Marco Geraci with contributions from Alessio Farcomeni

References

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

See Also

residuals.midrq, predict.midrq, coef.midrq

Examples

## Not run: 
# 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"))


## End(Not run)