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Compute (Negative) Log-Likelihood for Beta Error Model

Source: R/loglik_err.R
loglik_err.Rd

Calculates the log-likelihood (or negative log-likelihood) of a set of observed values under a Beta-distributed error model.

Usage

loglik_err(observed, predicted, phi, neg = TRUE, epsilon = 0.001)

Arguments

observed

A numeric vector of observed values (e.g., proportions), each in the range [0, 1].

predicted

A numeric vector of predicted values from the model, also in [0, 1].

phi

A scalar or numeric vector specifying the precision (concentration) parameter(s) of the Beta distribution. If a vector, it must be the same length as observed and predicted.

neg

Logical. If TRUE (default), returns the negative log-likelihood (useful for optimisation). If FALSE, returns the log-likelihood.

epsilon

A small numeric value used to constrain observed and predicted away from 0 and 1, ensuring numerical stability (default is 1e-3). See Details.

Value

A single numeric value: the negative log-likelihood or log-likelihood of the data under the Beta error model.

Details

This function calculates the (log) likelihood of observed choice proportions under a Beta-distributed error model. Since the Beta distribution is only defined on the open interval (0, 1), a small continuity correction (epsilon) is applied to both the observed and predicted values to avoid zero or one values that would result in undefined or infinite log-likelihoods.

The choice of epsilon can substantially affect the resulting log-likelihood, particularly when any values are near 0 or 1. This is due to the steep curvature of the Beta distribution near the boundaries, especially when the shape parameters are less than 1. Smaller values of epsilon reduce the size of the correction, but can introduce large penalties to the likelihood if the data approach the boundaries.

For this reason, it is important to choose an epsilon value that balances numerical stability and minimal distortion of the data. A default of epsilon = 0.001 is typically sufficient for behavioural data, but users are encouraged to check the sensitivity of their results when values are close to 0 or 1.

Note

If the observed data include values near 0 or 1, the log-likelihood may be highly sensitive to the choice of epsilon. See Details.

Examples

# Scalar phi
loglik_err(c(0.3, 0.6), c(0.25, 0.55), phi = 20)
#> [1] -2.353555

# Vectorised phi
loglik_err(c(0.3, 0.6), c(0.25, 0.55), phi = c(15, 25))
#> [1] -2.32292

# Return log-likelihood instead of negative log-likelihood
loglik_err(c(0.3, 0.6), c(0.25, 0.55), phi = 20, neg = FALSE)
#> [1] 2.353555