Qtools (version 1.3)

qlss: Quantile-based Summary Statistics for Location, Scale and Shape

Description

This function calculates quantile-based summary statistics for location, scale and shape of a distribution, unconditional or conditional.

Usage

qlss(...)
# S3 method for default
qlss(fun = "qnorm", probs = 0.1, ...)
# S3 method for numeric
qlss(x, probs = 0.1, ...)
# S3 method for formula
qlss(formula, probs = 0.1, data = sys.frame(sys.parent()), subset, weights,
	na.action, contrasts = NULL, method = "fn", type = "rq", tsf = "mcjI",
	symm = TRUE, dbounded = FALSE, lambda = NULL, conditional = FALSE, ...)

Arguments

fun

quantile function.

x

a numeric vector.

formula

an object of class formula: a symbolic description of the model to be fitted. The details of model specification are given under "Details".

probs

a vector of probabilities.

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. By default the variables are taken from the environment from which the call is made.

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. Should be NULL or a numeric vector.

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.

method

the algorithm used to solve the linear program. See rq for further details. The Frisch-Newton interior point method is the default.

type

possible options are rq for linear quantile regression (default) or rqt for transformation-based quantile regression.

tsf

transformation to be used. Possible options are mcjI for Proposal I transformation models (default), bc for Box-Cox and ao for Aranda-Ordaz transformation models. See tsrq for further details.

symm

logical flag. If TRUE (default) a symmetric transformation is used.

dbounded

logical flag. If TRUE the response is assumed to be doubly bounded on [a,b]. If FALSE the response is assumed to be singly bounded (ie, strictly positive).

lambda

values of transformation parameters for grid search.

conditional

logical flag. If TRUE, the transformation parameter is assumed to be known and this must be provided via the arguments lambda using a vector of length 3 + 2 x length(probs) (see details).

…

other arguments for fun, rq or tsrq.

Value

qlss returns an object of class qlss. This is a list that contains at least three elements:

location

summary statistic(s) for location.

scale

summary statistic(s) for scale.

shape

summary statistic(s) for shape.

Details

This function computes a number of quantile-based summary statistics for location (median), scale (inter-quartile range and inter-quantile range), and shape (Bowley skewness and shape index) of a distribution. These statistics can be computed for unconditional and conditional distributions.

Let \(Y\) be a continuous random variable and let \(Q(p)\) be its pth quantile. The function qlss computes the median \(Q(0.5)\), the inter-quartile range \(IQR = Q(0.75) - Q(0.25)\), the inter-quantile range \(IPR(p) = Q(1-p) - Q(p)\), the Bowley skewness index \(A(p) = (Q(1-p) + Q(p) - 2Q(0.5))/IPR(p)\), and the shape index \(T(p) = IPR(p)/IQR(p)\), for \(0 < p < 0.25\).

The default qlss function computes the summary statistics of a standard normal distribution or any other theoretical distribution via the argument fun. The latter must be a function with p as its probability argument (see for example qnorm, qt, qchisq, qgamma, etc.). When a variable x is provided, LSS measures are computed using empirical (sample) quantiles.

The argument formula specifies a quantile function for \(Y\) conditional on predictors \(X\). Linear models are fitted via standard quantile regression with type = "rq". Nonlinear models are fitted via transformation-based quantile regression with type = "rqt" (proposal II transformation models are not available.). When conditional = TRUE, lambda is a vector of transformation parameters of length 3 + 2 x np, where np = length(probs) (3 quartiles, np quantiles at level \(p\), np quantiles at level \(1 - p\)).

References

Geraci M and Jones MC. Improved transformation-based quantile regression. Canadian Journal of Statistics 2015;43(1):118-132.

Gilchrist W. Statistical modelling with quantile functions. Chapman and Hall/CRC; 2000.

See Also

predict.qlss, plot.qlss

Examples

# NOT RUN {
# Compute summary statistics of a normal distribution
qlss()

# Compute summary statistics of a t distribution with 3 df
qlss(fun = "qt", df = 3, probs = 0.05)

# Compute summary statistics for a sample using a sequence of probabilities
x <- rnorm(1000)
qlss(x, probs = c(0.1, 0.2, 0.3, 0.4))

# Compute summary statistics for Volume conditional on Height
trees2 <- trees[order(trees$Height),]
fit <- qlss(Volume ~ Height, data = trees2)
plot(fit, z = trees2$Height, xlab = "height")

# Use a quadratic model for Height
fit <- qlss(Volume ~ poly(Height,2), data = trees2)
plot(fit, z = trees2$Height, xlab = "height")


# }