Probit response functions of Thurstonian IRT

The probability of a binary outcome as the standard normal distribution function \(\Phi\) of the latent trait \(\eta\), in three forms:

prob_tirt()

a single outcome, with the parameters of factor analysis, \(\Phi((-\gamma + \lambda \eta) / \sqrt{\psi})\);

prob_tirt_pair()

the choice of item \(i\) over item \(k\), \(\Phi((-\gamma + \lambda_i \eta - \lambda_k \eta) / \sqrt{\psi_i + \psi_k})\);

prob_tirt_intercept()

the same comparison in intercept and slope form, \(\Phi(\alpha + (\beta_i - \beta_k) \eta)\).

Usage

prob_tirt(eta, gamma = 0, lambda = 0.5, psi = 0.5)

prob_tirt_pair(
  eta,
  gamma = 0,
  lambda_i = 0.5,
  lambda_k = 0,
  psi_i = 0.5,
  psi_k = 0.5
)

prob_tirt_intercept(eta, alpha = 0, beta_i = 0.5, beta_k = 0)

Arguments

eta

Numeric vector of trait values.

gamma

Threshold.

lambda, lambda_i, lambda_k

Factor loadings.

psi, psi_i, psi_k

Error (unique) variances.

alpha

Intercept.

beta_i, beta_k

Slopes.

Value

A numeric vector of probabilities, one for each value of eta.

Examples

eta <- c(-2, 0, 2)
prob_tirt(eta)
#> [1] 0.0786496 0.5000000 0.9213504
# the pair and intercept forms agree when alpha = -gamma / sqrt(psi_i + psi_k)
# and each beta = lambda / sqrt(psi_i + psi_k)
prob_tirt_pair(eta, gamma = 0.5, lambda_i = 1, lambda_k = 0.2)
#> [1] 0.01786442 0.30853754 0.86433394
prob_tirt_intercept(eta, alpha = -0.5, beta_i = 1, beta_k = 0.2)
#> [1] 0.01786442 0.30853754 0.86433394