knitr::opts_chunk$set( collapse = TRUE, comment = "#>", fig.align = "center", fig.width = 12, fig.height = 6)
The unitquantreg
R package provide efficient tools for estimation and inference
in parametric quantile regression models for bounded data.
The current version of unitquantreg
has 11 probability distributions available
for user choice. The Table above lists the families of distributions,
their abbreviations and the paper reference.
fam_name <- c("unit-Weibull", "Kumaraswamy", "unit-Logistic", "unit-Chen", "unit-Birnbaum-Saunders", "log-extended Exponential-Geometric", "unit-Generalized Half-Normal-E", "unit-Generalized Half-Normal-X", "unit-Gompertz", "unit-Burr-XII", "Johnson-SB", "arc-secant hyperbolic Weibull", "unit-Gumbel") abbrev_name <- c("uweibull", "kum", "ulogistic", "uchen", "ubs", "leeg", "ughne", "ughnx", "ugompertz", "uburrxii", "johnsonsb", "ashw", "ugumbel") refs <- c("[Mazucheli, et al. (2018)](http://japs.isoss.net/13(2)1%2011046.pdf)", "[Kumaraswamy, (1980)](https://www.sciencedirect.com/science/article/abs/pii/0022169480900360)", "[Tadikamalla and Johnson (1982)](https://doi.org/10.2307/2335422)", "[Korkmaz, et al. (2020)](https://doi.org/10.1515/ms-2022-0052)", "[Mazucheli, et al. (2021)](https://www.mdpi.com/2073-8994/13/4/682)", "[Jodrá and Jiménez-Gamero (2020)](https://doi.org/10.57805/revstat.v18i4.309)", "[Korkmaz MÇ (2020)](https://www.scientificbulletin.upb.ro/rev_docs_arhiva/full6b9_464742.pdf)", "New", "[Mazucheli et al. (2019)](https://rivista-statistica.unibo.it/article/view/8497)", "[Korkmaz and Chesneau (2021)](https://link.springer.com/article/10.1007/s40314-021-01418-5)", "[Johnson (1949)](https://doi.org/10.2307/2332539)", "[Korkmaz et al. (2021)](https://www.tandfonline.com/doi/full/10.1080/02664763.2021.1981834)", "New") tab <- data.frame(fam_name, abbrev_name, refs) knitr::kable(tab[order(tab$abbrev_name), ], col.names = c("Family", "Abbreviation", "Reference"), caption = "Available families of distributions their abbreviations and reference.", label = "distributions", row.names = FALSE)
The [dpqr
]'s functions of the distributions are vectorized and implemented in C++
.
The log likelihood, score and hessian functions are also implemented in C++
in order
to guarantee more computational efficiency.
The parameter estimation and inference are performed under the frequentist paradigm.
Maximization of the log-likelihood function is done by optimization techniques available
in the R
through the optimx
package, which is a general-purpose optimization wrapper function that allows the use of
several R
tools for optimization, including the existing stats::optim()
function.
To achieve quick and better convergence the analytical score function is use during the
maximization. Also, standard errors of parameter estimates are computed using
the analytical hessian matrix.
The unitquantreg
package is built around the unitquantreg()
function which
perform the fit of parametric quantile regression models via likelihood method.
The unitquantreg()
function has standard arguments as stats::glm()
function,
and they are as follows:
library(unitquantreg) args(unitquantreg)
The formula
argument use the concept of Formula
package allows multiple parts on the right-hand side, which indicates regression structure
for the quantile and shape parameter of the distribution. For instance,
formula = y ~ x1 | z1
means the following regression structure
$$ g_1(\mu) = \beta_0 + \beta_1\,x_1 \quad \textrm{and} \quad g_2(\theta) = \gamma_0 + \gamma_1\,z_1 $$ where $\mu$ indicates the quantile of order $\tau$ and $\theta$ is the shape parameter.
The tau
argument indicates the quantile(s) to be estimated, being possible to specify a
vector of quantiles. family
argument specify the distribution family using the
abbreviation of dpqr
functions, listed in Table above.
The control
argument controls the fitting process through the unitquantreg.control()
function which returned a list
and the default values are:
unlist(unitquantreg.control())
The two most important arguments are hessian
and gradient
which tell the optimx::optimx()
whether it should use the numerical hessian matrix and the analytical score vector,
respectively. That is, if hessian = TRUE
, then the standard errors are computed using
the numerical hessian matrix. For detailed description of other arguments see the package
documentation.
The unitquanreg()
function returns an object of class unitquanreg
if the
argument tau
is scalar or unitquanregs
if tau
is a vector.
The currently methods implemented for unitquantreg
objects are:
methods(class = "unitquantreg")
And for the unitquantregs
objects are
methods(class = "unitquantregs")
It is important to mention that the unitquantregs
objects consists of a list
with unitquantreg
objects for according to the vector of tau
.
Furthermore, the package provide functions designated for model comparison
between uniquantreg
objects. Particularly,
likelihood_stats()
function computes likelihood-based statistics (Neg2LogLike, AIC, BIC and HQIC),
vuong.test()
function performs Vuong test between two fitted non nested models,
pairwise.vuong.test()
function performs pairwise Vuong test with adjusted p-value according to stats::p.adjust.methods
between fitted models
Finally, uniquantreg
objects also permits use the inference methods functions
lmtest::coeftest()
, lmtest::coefci
, lmtest::coefci
, lmtest::waldtest
and lmtest::lrtest
implemented in lmtest
to
perform hypothesis test, confidence intervals for nested models.
Next, a detailed account of the usage of all these functions is provided.
As in Mazucheli et al. (2020)
consider the data set related to the access of people in
households with piped water supply in the cities of Brazil from the Southeast and Northeast
regions. The response variable phpws
is the proportion of households with piped water
supply. The covariates are:
mhdi
: human development index.
incpc
: per capita income.
region
: 0 for southeast, 1 for northeast.
pop
: population.
data(water) head(water)
Assuming the following regression structure for the parameters: $$ \textrm{logit}(\mu_i) = \beta_0 + \beta_1 \texttt{mhdi}{i1} + \beta_2 \texttt{incp}{i2} + \beta_3 \texttt{region}{i3} + \beta_4 \log\left(\texttt{pop}{i4}\right), $$ and $$ \log(\theta_i) = \gamma_0. $$ for $i = 1, \ldots, 3457$.
For $\tau = 0.5$, that is, the median regression model we fitted for all families of distributions as follows:
lt_families <- list("unit-Weibull" = "uweibull", "Kumaraswamy" = "kum", "unit-Logistic" = "ulogistic", "unit-Birnbaum-Saunders" = "ubs", "log-extended Exponential-Geometric" = "leeg", "unit-Chen" = "uchen", "unit-Generalized Half-Normal-E" = "ughne", "unit-Generalized Half-Normal-X" = "ughnx", "unit-Gompertz" = "ugompertz", "Johnson-SB" = "johnsonsb", "unit-Burr-XII" = "uburrxii", "arc-secant hyperbolic Weibull" = "ashw", "unit-Gumbel" = "ugumbel") lt_fits <- lapply(lt_families, function(fam) { unitquantreg(formula = phpws ~ mhdi + incpc + region + log(pop), data = water, tau = 0.5, family = fam, link = "logit", link.theta = "log") }) t(sapply(lt_fits, coef))
Let's check the likelihood-based statistics of fit
likelihood_stats(lt = lt_fits)
According to the statistics the unit-Logistic, Johnson-SB, unit-Burr-XII and unit-Weibull were the best models. Now, let's perform the pairwise vuong test to check if there is statistical significant difference between the four models.
lt_chosen <- lt_fits[c("unit-Logistic", "Johnson-SB", "unit-Burr-XII", "unit-Weibull")] pairwise.vuong.test(lt = lt_chosen)
The adjusted p-values of pairwise Vuong tests shows that there is a large statistical significance difference between the models. In particular, the pairwise comparison between unit-Logistic and the other models provide a smaller p-values, indicating that the unit-Logistic median regression model is the most suitable model for this data set comparing to the others families of distributions.
It is possible to check model assumptions from diagnostic plots using the plot()
function method for unitquantreg
objects.
The residuals()
method provides quantile
, cox-snell
, working
and partial
residuals type. The randomize quantile residuals is the default choice of plot()
method.
oldpar <- par(no.readonly = TRUE) par(mfrow = c(2, 2)) plot(lt_fits[["unit-Logistic"]]) par(oldpar)
Plots of the residuals against the fitted linear predictor and the residuals against indices of observations are the tools for diagnostic analysis to check the structural form of the model. Two features of the plots are important:
Trends: Any trends appearing in these plots indicate that the systematic component can be improved. This could mean changing the link function, adding extra explanatory variables, or transforming the explanatory variables.
Constant variation: If the random component is correct then the variance of the points is approximately constant.
Working residuals versus linear predictor is used to check possible misspecification of link function and Half-normal plot of residuals to check distribution assumption.
Another best practice in diagnostic analysis is to inspect the (Half)-Normal plots with simulated envelope for several quantile value. This is done to obtain a more robust evaluation of the model assumptions. Thus, let's fit the unit-Logistic quantile regression model for various quantiles.
system.time( fits_ulogistic <- unitquantreg(formula = phpws ~ mhdi + incpc + region + log(pop), data = water, tau = 1:49/50, family = "ulogistic", link = "logit", link.theta = "log"))
Now, we can check the (Half)-Normal plots using the output of hnp()
method.
library(ggplot2) get_data <- function(obj) { tmp <- hnp(obj, halfnormal = FALSE, plot = FALSE, nsim = 10) tmp <- as.data.frame(do.call("cbind", tmp)) tmp$tau <- as.character(obj$tau) tmp } chosen_taus <- c("0.02", "0.5", "0.98") df_plot <- do.call("rbind", lapply(fits_ulogistic[chosen_taus], get_data)) df_plot$tau <- paste0(expression(tau), " == ", df_plot$tau) ggplot(df_plot, aes(x = teo, y = obs)) + facet_wrap(~tau, labeller = label_parsed) + geom_point(shape = 3, size = 1.4) + geom_line(aes(y = median), linetype = "dashed") + geom_line(aes(y = lower), col = "#0080ff") + geom_line(aes(y = upper), col = "#0080ff") + theme_bw() + labs(x = "Theoretical quantiles", y = "Randomized quantile residuals") + scale_x_continuous(breaks = seq(-3, 3, by = 1)) + scale_y_continuous(breaks = seq(-3, 3, by = 1)) + theme_bw() + theme(text = element_text(size = 16, family = "Palatino"), panel.grid.minor = element_blank())
Inference results about the parameter estimates can be accessed through the
summary
method. For instance,
summary(lt_fits[["unit-Logistic"]])
For unitquantregs
objects the plot
method provide a convenience to check the
significance as well as the effect of estimate along the specify quantile value.
plot(fits_ulogistic, which = "coef")
Curiously, the unit-Logistic quantile regression models capture constant effect for all
covariates along the different quantiles. In contrast, the unit-Weibull model
(the fourth best model) found a decrease effect of mhdi
covaraite on the response as
the quantile increases and increase effects of incpc
and region
on the response
variable as the quantile increases.
system.time( fits_uweibull <- unitquantreg(formula = phpws ~ mhdi + incpc + region + log(pop), data = water, tau = 1:49/50, family = "uweibull", link = "logit", link.theta = "log")) plot(fits_uweibull, which = "coef")
Using the plot()
method with argument which = "conddist"
for unitquantregs
objects
it is possible to
estimate and visualize the conditional distribution of a response variable at
different values of covariates. For instance,
lt_data <- list(mhdi = c(0.5, 0.7), incpc = round(mean(water$incpc)), region = c(1, 0), pop = round(mean(water$pop))) plot(fits_ulogistic, which = "conddist", at_obs = lt_data, at_avg = FALSE, dist_type = "density") plot(fits_ulogistic, which = "conddist", at_obs = lt_data, at_avg = FALSE, dist_type = "cdf")
sessionInfo()
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