Quantile-based Summary Statistics for Location, Scale and Shape

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

Usage

qlss(...)
# Default S3 method
qlss(fun = "qnorm", probs = 0.1, ...)
# S3 method for class 'numeric'
qlss(x, probs = 0.1, ...)
# S3 method for class '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.

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 \(T(p) = IPR(p)/IQR\), 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 \(1 - p\)).

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.

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.

Author

Marco Geraci

Examples


# Compute summary statistics of a normal distribution
qlss()
#> call:
#> qlss.default()
#> 
#> Unconditional Quantile-Based Location, Scale, and Shape 
#> 
#> ** Location ** 
#> Median 
#> [1] 0
#> ** Scale ** 
#> Inter-quartile range (IQR) 
#> [1] 1.34898
#> Inter-quantile range (IPR) 
#>      0.1 
#> 2.563103 
#> ** Shape ** 
#> Skewness index 
#> 0.1 
#>   0 
#> Shape index 
#>      0.1 
#> 1.900031 

# Compute summary statistics of a t distribution with 3 df
qlss(fun = "qt", df = 3, probs = 0.05)
#> call:
#> qlss.default(fun = "qt", probs = 0.05, df = 3)
#> 
#> Unconditional Quantile-Based Location, Scale, and Shape 
#> 
#> ** Location ** 
#> Median 
#> [1] 0
#> ** Scale ** 
#> Inter-quartile range (IQR) 
#> [1] 1.529785
#> Inter-quantile range (IPR) 
#>     0.05 
#> 4.706727 
#> ** Shape ** 
#> Skewness index 
#>         0.05 
#> -1.88704e-16 
#> Shape index 
#>     0.05 
#> 3.076725 

# 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))
#> call:
#> qlss.numeric(x = x, probs = c(0.1, 0.2, 0.3, 0.4))
#> 
#> Unconditional Quantile-Based Location, Scale, and Shape 
#> 
#> ** Location ** 
#> Median 
#> [1] -0.03489193
#> ** Scale ** 
#> Inter-quartile range (IQR) 
#> [1] 1.282709
#> Inter-quantile range (IPR) 
#>       0.1       0.2       0.3       0.4 
#> 2.4557636 1.6380151 0.9847508 0.4321300 
#> ** Shape ** 
#> Skewness index 
#>         0.1         0.2         0.3         0.4 
#> -0.01905402  0.00627749  0.03288934  0.11093977 
#> Shape index 
#>       0.1       0.2       0.3       0.4 
#> 1.9145141 1.2769971 0.7677121 0.3368887 

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


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