betareg

Family Objects for (Extended-Support) Beta Regression

Description

Family objects for distributional regression with the (extended-support) beta distribution via bamlss or gamlss2.

Usage

betar_family(link = "logit", link.phi = "log", ...)

xbetax_family(link = "logit", link.phi = "log", link.nu = "log",
  quad = 20, tol = .Machine\$double.eps^0.7, ...)

Arguments

link character specification of the link function in the mean model (mu). Currently, “logit”, “probit”, “cloglog”, “cauchit”, “log”, “loglog” are supported.

link.phi character specification of the link function in the precision model (phi). Currently, “log”, “identity”, “sqrt” are supported.

link.nu character specification of the link function in the exceedence model (nu). Currently, “log”, “identity”, “sqrt” are supported.

quad numeric. The number of quadrature points for numeric integration of the continuous mixture in dxbetax. Alternatively, a matrix with nodes and weights for the quadrature points can be specified.

tol numeric. Accuracy (convergence tolerance) for numerically determining quantiles based on uniroot and pxbetax.

… Arguments passed to functions that are called within the family object.

Details

Family objects for bamlss (Umlauf et al. 2019, 2021) and gamlss2 (Umlauf et al. 2024) are essentially lists of functions providing a standardized interface to the d/p/q/r functions of distributions. Hence, betar_family interfaces the classical beta distribution in regression specification, see dbetar. Analogously, xbetax_family interfaces the extended-support beta mixture specification (Kosmidis and Zeileis 2024), see dxbetax.

Value

A list of class family.bamlss.

References

Kosmidis I, Zeileis A (2024). Extended-Support Beta Regression for [0, 1] Responses. 2409.07233, arXiv.org E-Print Archive. doi:10.48550/arXiv.2409.07233

Umlauf N, Klein N, Zeileis A (2019). BAMLSS: Bayesian Additive Models for Location, Scale and Shape (and Beyond). Journal of Computational and Graphical Statistics, 27(3), 612–627. doi:10.1080/10618600.2017.1407325

Umlauf N, Klein N, Simon T, Zeileis A (2021). bamlss: A Lego Toolbox for Flexible Bayesian Regression (and Beyond). Journal of Statistical Software, 100(4), 1–53. doi:10.18637/jss.v100.i04

Umlauf N, Stasinopoulos M, Rigby R, Stauffer R (2024). gamlss2: Infrastructure for Flexible Distributional Regression. R package version 0.1-0. https://gamlss-dev.github.io/gamlss2/

Examples

library("betareg")

## IGNORE_RDIFF_BEGIN

## package and data
library("betareg")
library("bamlss")
data("ReadingSkills", package = "betareg")

## classical beta regression via ML
rs1 <- betareg(accuracy ~ dyslexia * iq | dyslexia + iq, data = ReadingSkills)

## Bayesian additive model (with low number of iterations to speed up the example)
set.seed(0)
rs2 <- bamlss(accuracy ~ s(iq, by = dyslexia) | dyslexia + iq, data = ReadingSkills,
  family = betar_family(), eps = 1e-7, n.iter = 400, burnin = 100)

AICc  -0.8829 logPost -64.6792 logLik  21.6241 edf 13.973 eps 1.0000 iteration   1
AICc -51.8915 logPost -25.8044 logLik  39.7035 edf 10.242 eps 2.8731 iteration   2
AICc -68.1251 logPost  -7.3858 logLik  46.8183 edf 9.6642 eps 0.5561 iteration   3
AICc -73.1290 logPost   4.7269 logLik  48.3399 edf 9.0783 eps 0.0895 iteration   4
AICc -74.7842 logPost   5.5545 logLik  49.1675 edf 9.0783 eps 0.0629 iteration   5
AICc -75.7464 logPost   6.0357 logLik  49.6487 edf 9.0783 eps 0.0559 iteration   6
AICc -76.3072 logPost   6.3160 logLik  49.9291 edf 9.0783 eps 0.2490 iteration   7
AICc -77.3399 logPost   6.8324 logLik  50.4454 edf 9.0783 eps 0.2694 iteration   8
AICc -77.6557 logPost   6.9903 logLik  50.6033 edf 9.0783 eps 0.0207 iteration   9
AICc -77.8176 logPost   7.0713 logLik  50.6843 edf 9.0783 eps 0.0123 iteration  10
AICc -80.2391 logPost   8.7295 logLik  50.7321 edf 8.3559 eps 0.0099 iteration  11
AICc -81.1323 logPost -61.6299 logLik  50.7922 edf 8.1090 eps 0.0333 iteration  12
AICc -81.3164 logPost -654.389 logLik  50.8351 edf 8.0774 eps 0.0089 iteration  13
AICc -81.3726 logPost -4694.96 logLik  50.8579 edf 8.0740 eps 0.0054 iteration  14
AICc -81.4322 logPost -25766.3 logLik  50.8816 edf 8.0700 eps 0.0077 iteration  15
AICc -81.6598 logPost -2043.53 logLik  50.9498 edf 8.0406 eps 0.0179 iteration  16
AICc -82.3830 logPost -8852.97 logLik  51.1594 edf 7.9421 eps 0.0137 iteration  17
AICc -82.7904 logPost -13085.5 logLik  51.2898 edf 7.8944 eps 0.0124 iteration  18
AICc -82.9776 logPost -13740.2 logLik  51.3740 edf 7.8883 eps 0.0431 iteration  19
AICc -83.0956 logPost -13809.0 logLik  51.4431 edf 7.8949 eps 0.0096 iteration  20
AICc -83.2026 logPost -13815.6 logLik  51.4812 edf 7.8848 eps 0.0058 iteration  21
AICc -83.2582 logPost -13816.0 logLik  51.5109 edf 7.8861 eps 0.0574 iteration  22
AICc -83.3488 logPost -13815.9 logLik  51.5729 edf 7.8970 eps 0.0084 iteration  23
AICc -83.4298 logPost -13815.7 logLik  51.5929 edf 7.8836 eps 0.0035 iteration  24
AICc -83.4578 logPost -13815.6 logLik  51.6069 edf 7.8837 eps 0.0029 iteration  25
AICc -83.4771 logPost -13815.4 logLik  51.6166 edf 7.8837 eps 0.0025 iteration  26
AICc -83.4921 logPost -13815.3 logLik  51.6242 edf 7.8837 eps 0.0042 iteration  27
AICc -83.5087 logPost -13815.3 logLik  51.6291 edf 7.8815 eps 0.0021 iteration  28
AICc -83.5136 logPost -13815.2 logLik  51.6323 edf 7.8820 eps 0.0016 iteration  29
AICc -83.5176 logPost -13815.1 logLik  51.6343 edf 7.8820 eps 0.0013 iteration  30
AICc -83.5199 logPost -13815.1 logLik  51.6355 edf 7.8820 eps 0.0011 iteration  31
AICc -83.5219 logPost -13815.1 logLik  51.6365 edf 7.8820 eps 0.0042 iteration  32
AICc -83.5216 logPost -13815.0 logLik  51.6382 edf 7.8832 eps 0.0022 iteration  33
AICc -83.5300 logPost -13815.0 logLik  51.6383 edf 7.8806 eps 0.0009 iteration  34
AICc -83.5294 logPost -13815.0 logLik  51.6384 edf 7.8808 eps 0.0003 iteration  35
AICc -83.5293 logPost -13815.0 logLik  51.6384 edf 7.8809 eps 0.0000 iteration  36
AICc -83.5293 logPost -13815.0 logLik  51.6384 edf 7.8809 eps 0.0000 iteration  37
AICc -83.5293 logPost -13815.0 logLik  51.6384 edf 7.8809 eps 0.0000 iteration  38
AICc -83.5293 logPost -13815.0 logLik  51.6384 edf 7.8809 eps 0.0000 iteration  39
AICc -83.5293 logPost -13815.0 logLik  51.6384 edf 7.8809 eps 0.0000 iteration  39
elapsed time:  3.19sec
Starting the sampler...

|                    |   0%  3.55sec
|*                   |   5%  3.33sec  0.18sec
|**                  |  10%  3.00sec  0.33sec
|***                 |  15%  2.77sec  0.49sec
|****                |  20%  2.58sec  0.65sec
|*****               |  25%  2.45sec  0.82sec
|******              |  30%  2.30sec  0.99sec
|*******             |  35%  2.14sec  1.15sec
|********            |  40%  1.99sec  1.32sec
|*********           |  45%  1.84sec  1.51sec
|**********          |  50%  1.68sec  1.68sec
|***********         |  55%  1.51sec  1.85sec
|************        |  60%  1.34sec  2.01sec
|*************       |  65%  1.18sec  2.20sec
|**************      |  70%  1.02sec  2.37sec
|***************     |  75%  0.85sec  2.54sec
|****************    |  80%  0.68sec  2.72sec
|*****************   |  85%  0.51sec  2.89sec
|******************  |  90%  0.34sec  3.06sec
|******************* |  95%  0.17sec  3.25sec
|********************| 100%  0.00sec  3.43sec

## Bayesian model shrinks the effects compared to ML
plot(accuracy ~ iq, data = ReadingSkills, pch = 19, col = dyslexia)
nd <- data.frame(
  iq = rep(-19:20/10, 2),
  dyslexia = factor(rep(c("no", "yes"), each = 40), levels = c("no", "yes"))
)
nd$betareg <- predict(rs1, newdata = nd, type = "response")
nd$bamlss  <- predict(rs2, newdata = nd, type = "parameter", model = "mu")
lines(betareg ~ iq, data = nd, subset = dyslexia == "no",  col = 1, lwd = 2, lty = 1)
lines(betareg ~ iq, data = nd, subset = dyslexia == "yes", col = 2, lwd = 2, lty = 1)
lines(bamlss  ~ iq, data = nd, subset = dyslexia == "no",  col = 1, lwd = 2, lty = 2)
lines(bamlss  ~ iq, data = nd, subset = dyslexia == "yes", col = 2, lwd = 2, lty = 2)
legend("topleft", c("Dyslexia: no", "Dyslexia: yes", "betareg", "bamlss"),
  lty = c(0, 0, 1, 2), pch = c(19, 19, NA, NA), col = c(1, 2, 1, 1), bty = "n")

## IGNORE_RDIFF_END

## xbetax_family(): requires more time due to Gaussian quadrature
## for gamlss2: install.packages("gamlss2", repos = "https://gamlss-dev.R-universe.dev")