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Beta-Gate Model

Source: R/model_betagate.R
rcogmod_betagate.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, doi:10.1017/pan.2022.20 ). 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

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

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

cogmod_betagate_lpdf_expose()

cogmod_betagate_stanvars()

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

log_lik_cogmod_betagate(i, prep)

posterior_predict_cogmod_betagate(i, prep, ...)

posterior_epred_cogmod_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 (must be strictly positive). Can be conceptualized as an "agreement" indicator: higher phi means less dispersion (more agreement) among ratings, holding mu fixed. Note: In many implementations, phi is parametrized differently, and correspond to the double of our phi argument (cogmod's phi = standard's phi * 2). Our parametrization Makes it phi = 1 corresponds to uniform when mu = 0.5, which makes setting priors more convenient (e.g., on the logit scale)

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

rcogmod_betagate() returns a numeric vector of n simulated ratings on the unit interval [0, 1], including the exact 0s and 1s produced by the gates. dcogmod_betagate() returns the density at each element of x - the log density if log = TRUE - recycled to the length of the longest argument; at 0 and 1 it is the probability mass rather than a density. cogmod_betagate() returns a brms::custom_family object, to put on a brms::bf() formula. cogmod_betagate_stanvars() returns a brms::stanvars object holding the family's Stan functions block, to pass to brms::brm(), and cogmod_betagate_lpdf_expose() compiles that Stan code and returns it as an R function, for checking the density outside of a model. The remaining functions are brms post-processing methods, called by brms rather than directly: log_lik_cogmod_betagate() returns a numeric vector holding one log-likelihood value per posterior draw for observation i, posterior_predict_cogmod_betagate() a draws x 1 matrix of ratings simulated for observation i, and posterior_epred_cogmod_betagate() a draws x observations matrix of expected ratings.

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 <- rcogmod_betagate(10000, mu = 0.5, phi = 3, pex = 0.1, bex = 0.5)
hist(x1, breaks=50, main="rcogmod_betagate: Symmetric Cutpoints (pex=0.1)")


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


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


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

if (FALSE) { # \dontrun{
# Needs cmdstanr and a CmdStan toolchain, which live outside CRAN - see the
# package website to install them. Not run under R CMD check, which executes
# every example in one R session: once brms has fitted a model there (the
# cogmod_inits() and p_outlier() examples do), rstan is live in the process
# and loading an exposed Stan function next to it segfaults on Linux.
lpdf <- cogmod_betagate_lpdf_expose()
lpdf(y = 0.5, mu = 0.6, phi = 10, pex = 0.2, bex = 0.5)
} # }