Skip to contents

Beta-Gate Model

Source: R/betagate.R, R/betagate_brms.R
rbetagate.Rd

The Beta-Gate model represents subjective ratings as a mixture of a continuous Beta distribution with additional point masses at the extremes (0 and 1). This structure effectively captures common patterns in subjective rating data where respondents often select extreme values at higher rates than would be expected from a Beta distribution alone.

The Beta-Gate model corresponds to a reparametrized ordered beta model (Kubinec, 2023). In the ordered Beta model, the extreme values (0 and 1) arise from censoring an underlying latent process based on cutpoints ("gates"). Values falling past the gates are considered extremes (zeros and ones). The difference from the Ordered Beta is the way the cutpoints are defined, as well as the scale of the precision parameter phi.

It differs from the Zero-One-Inflated Beta (ZOIB) model in that the ZOIB model has zoi and coi parameters, directly controlling the likelihood of extreme values. Instead, Beta-Gate uses pex and bex to define "cutpoints" after which extreme values become likely. In an ordered beta framework, the boundary probabilities arise through a single underlying ordering process (the location of the cutpoints on the latent scale). In a ZOIB framework, the boundaries are more like additional mass points inserted into a beta distribution. In Beta-gate models, extreme values arise naturally from thresholding a single latent process.

Usage

rbetagate(n, mu = 0.5, phi = 3, pex = 0.1, bex = 0.5)

dbetagate(x, mu = 0.5, phi = 3, pex = 0.1, bex = 0.5, log = FALSE)

betagate_lpdf_expose()

betagate_stanvars()

betagate(
  link_mu = "logit",
  link_phi = "softplus",
  link_pex = "logit",
  link_bex = "logit"
)

log_lik_betagate(i, prep)

posterior_predict_betagate(i, prep, ...)

posterior_epred_betagate(prep)

Arguments

n

Number of simulated values.

mu

Mean of the underlying Beta distribution (0 < mu < 1).

phi

Precision parameter of the underlying Beta distribution (phi > 0). Note: precision = phi * 2. phi = 1 corresponds to uniform when mu = 0.5.

pex

Controls the location of the lower and upper boundary gates (0 <= pex <= 1). It defines the total probability mass allocated to the extremes (0 or 1). Higher pex increases the probability of extreme values (0 or 1).

bex

Balances the extreme probability mass pex between 0 and 1 (0 <= bex <= 1). A balance of 0.5 means that the 'gates' are symmetrically placed around the center of the distribution, and values higher or lower than 0.5 will shift the relative "ease" of crossing the gates towards 1 or 0, respectively.

x

Vector of quantiles (values at which to evaluate the density). Must be between 0 and 1, inclusive.

log

Logical; if TRUE, returns the log-density.

Link functions for the parameters.

i, prep

For brms' functions to run: index of the observation and a brms preparation object.

...

Additional arguments.

Value

A vector of simulated outcomes in the range 0-1.

Details

Special cases:

  • When pex = 0: Pure Beta distribution with mean mu and precision phi * 2.

  • When pex = 1: Pure Bernoulli distribution with P(1) = bex, P(0) = 1-bex.

  • When bex = 0 and pex = 1: All mass at 0.

  • When bex = 1 and pex = 1: All mass at 1.

Psychological Interpretation:

  • mu: Can be interpreted as the underlying average tendency or preference strength, disregarding extreme "all-or-nothing" responses.

  • phi: Reflects the certainty or consistency of the non-extreme responses. Higher phi indicates responses tightly clustered around mu (more certainty), while lower phi (especially phi = 1) suggests more uniform or uncertain responses.

  • pex: Represents the overall tendency towards extreme responding (choosing 0 or 1). This could reflect individual response styles (e.g., acquiescence, yea-saying/nay-saying) or properties of the item itself (e.g., polarizing questions).

  • bex: Indicates the direction of the extreme response bias. bex > 0.5 suggests a bias for producing ones more easily, while bex < 0.5 suggests a bias towards zero.

References

  • Kubinec, R. (2023). Ordered beta regression: a parsimonious, well-fitting model for continuous data with lower and upper bounds. Political Analysis, 31(4), 519-536.

Examples

# Symmetric gates (c0=0.05, c1=0.95), pex=0.1, bex=0.5
x1 <- rbetagate(10000, mu = 0.5, phi = 3, pex = 0.1, bex = 0.5)
# hist(x1, breaks=50, main="rbetagate: Symmetric Cutpoints (pex=0.1)")

# Asymmetric gates (c0=0.15, c1=0.95), pex=0.2, bex=0.25
x2 <- rbetagate(10000, mu = 0.5, phi = 3, pex = 0.2, bex = 0.25)
# hist(x2, breaks=50, main="rbetagate: Asymmetric Cutpoints (pex=0.2, bex=0.25)")

# No gating (pure Beta)
x3 <- rbetagate(10000, mu = 0.7, phi = 5, pex = 0, bex = 0.5)
# hist(x3, breaks=50, main="rbetagate: No Extreme Values (pex=0)")

x <- seq(0, 1, length.out = 1001)
densities <- dbetagate(x, mu = 0.5, phi = 5, pex = 0.2, bex = 0.5)
plot(x, densities, type = "l", main = "Density Function", xlab = "y", ylab = "Density")

# You can expose the lpdf function as follows:
# betagate_lpdf <- betagate_lpdf_expose()
# betagate_lpdf(y = 0.5, mu = 0.6, phi = 10, pex = 0.2, bex = 0.5)