dich_response_sim: Simulate Dichotomous Response Model

In mlthom/CogIRT: Cognitive Testing Using Item Response Theory

Simulate Dichotomous Response Model

Description

This function calculates the matrix of first partial derivatives, the matrix of second partial derivatives, and the information matrix for the posterior distribution with respect to theta (ability) based on theslope-intercept form of the item response theory model.

Usage

dich_response_sim(
  I = NULL,
  J = NULL,
  K = NULL,
  M = NULL,
  N = NULL,
  omega = NULL,
  omega_mu = NULL,
  omega_sigma2 = NULL,
  gamma = NULL,
  lambda = NULL,
  lambda_mu = NULL,
  lambda_sigma2 = NULL,
  nu = NULL,
  nu_mu = NULL,
  nu_sigma2 = NULL,
  zeta = NULL,
  zeta_mu = NULL,
  zeta_sigma2 = NULL,
  kappa = NULL,
  key = NULL,
  link = "probit"
)

Arguments

I	Number of items per condition.
J	Number of conditions.
K	Number of examinees
M	Number of ability (or trait) dimensions.
N	Number of contrasts (should include intercept).
omega	Examinee-level effects of the experimental manipulation (K by MN).
omega_mu	Vector of means for the examinee-level effects of the experimental manipulation (1 by MN).
omega_sigma2	Covariance matrix for the examinee-level effects of the experimental manipulation (MN by MN).
gamma	Matrix of experimental structure parameters (JM by MN).
lambda	Matrix of item structure parameters (IJ by JM).
lambda_mu	Vector of means for the item structure parameters (1 by JM)
lambda_sigma2	Covariance matrix for the item structure parameters (JM by JM)
nu	Matrix of item intercept parameters (K by IJ).
nu_mu	Mean of the item intercept parameters (scalar).
nu_sigma2	Variance of the item intercept parameters (scalar).
zeta	Condition-level effects of the experimental manipulation (K by JM).
zeta_mu	Vector of means for the condition-level effects nested within examinees (1 by JM).
zeta_sigma2	Covariance matrix for the condition-level effects nested within examinees (JM by JM).
kappa	Matrix of item guessing parameters (K by IJ). If kappa is not provided, parameter values are set to 0.
key	Option key where 1 indicates target and 2 indicates distractor.
link	Choose between logit or probit link functions.

Value

y = simulated response matrix; yhatstar = simulated latent response probability matrix; [simulation_parameters]

References

Skrondal, A., & Rabe-Hesketh, S. (2004). Generalized latent variable modeling: Multilevel, longitudinal, and structural equation models. Boca Raton: Chapman & Hall/CRC.

Thomas, M. L., Brown, G. G., Gur, R. C., Moore, T. M., Patt, V. M., Risbrough, V. B., & Baker, D. G. (2018). A signal detection-item response theory model for evaluating neuropsychological measures. Journal of Clinical and Experimental Neuropsychology, 40(8), 745-760.

Examples

# Example 1

I <- 100
J <- 1
K <- 250
M <- 1
N <- 1
omega_mu <- matrix(data = 0, nrow = 1, ncol = M * N)
omega_sigma2 <- diag(x = 1, nrow = M * N)
gamma <- diag(x = 1, nrow = J * M, ncol = M * N)
lambda_mu <- matrix(data = 1, nrow = 1, ncol = M)
lambda_sigma2 <- diag(x = 0.25, nrow = M)
zeta_mu <- matrix(data = rep(x = 0, times = M * J), nrow = 1, ncol = J * M)
zeta_sigma2 <- diag(x = 0, nrow = J * M, ncol = J * M)
nu_mu <- matrix(data = 0, nrow = 1, ncol = 1)
nu_sigma2 <- matrix(data = 1, nrow = 1, ncol = 1)
set.seed(624)
ex1 <- dich_response_sim(I = I, J = J, K = K, M = M, N = N,
                         omega_mu = omega_mu, omega_sigma2 = omega_sigma2,
                         gamma = gamma, lambda_mu = lambda_mu,
                         lambda_sigma2 = lambda_sigma2, nu_mu = nu_mu,
                         nu_sigma2 = nu_sigma2, zeta_mu = zeta_mu,
                         zeta_sigma2 = zeta_sigma2)

# Example 2

I <- 100
J <- 1
K <- 50
M <- 2
N <- 1
omega_mu <- matrix(data = c(3.50, 1.00), nrow = 1, ncol = M * N)
omega_sigma2 <- diag(x = c(0.90, 0.30), nrow = M * N)
gamma <- diag(x = 1, nrow = J * M, ncol = M * N)
key <- rbinom(n = I * J, size = 1, prob = .7) + 1
measure_weights <-
  matrix(data = c(0.5, -1.0, 0.5, 1.0), nrow = 2, ncol = M, byrow = TRUE)
lambda <- matrix(data = 0, nrow = I * J, ncol = J * M)
for(j in 1:J) {
  lambda[(1 + (j - 1) * I):(j * I), (1 + (j - 1) * M):(j * M)] <-
    measure_weights[key, ][(1 + (j - 1) * I):(j * I), ]
}
zeta_mu <- matrix(data = rep(x = 0, times = M * J), nrow = 1, ncol = J * M)
zeta_sigma2 <- diag(x = 0, nrow = J * M, ncol = J * M)
nu_mu <- matrix(data = 0, nrow = 1, ncol = 1)
nu_sigma2 <- matrix(data = .2, nrow = 1, ncol = 1)
set.seed(624)
ex2 <- dich_response_sim(I = I, J = J, K = K, M = M, N = N,
                         omega_mu = omega_mu, omega_sigma2 = omega_sigma2,
                         gamma = gamma, lambda = lambda, nu_mu = nu_mu,
                         nu_sigma2 = nu_sigma2, zeta_mu = zeta_mu,
                         zeta_sigma2 = zeta_sigma2, key = key)

# Example 3

I <- 20
J <- 10
K <- 50
M <- 2
N <- 2
omega_mu <- matrix(data = c(2.50, -2.00, 0.50, 0.00), nrow = 1, ncol = M * N)
omega_sigma2 <- diag(x = c(0.90, 0.70, 0.30, 0.10), nrow = M * N)
contrast_codes <- cbind(1, contr.poly(n = J))[, 1:N]
gamma <- matrix(data = 0, nrow = J * M, ncol = M * N)
for(j in 1:J) {
  for(m in 1:M) {
    gamma[(m + M * (j - 1)), (((m - 1) * N + 1):((m - 1) * N + N))] <-
      contrast_codes[j, ]
  }
}
key <- rbinom(n = I * J, size = 1, prob = .7) + 1
measure_weights <-
  matrix(data = c(0.5, -1.0, 0.5, 1.0), nrow = 2, ncol = M, byrow = TRUE)
lambda <- matrix(data = 0, nrow = I * J, ncol = J * M)
for(j in 1:J) {
  lambda[(1 + (j - 1) * I):(j * I), (1 + (j - 1) * M):(j * M)] <-
    measure_weights[key, ][(1 + (j - 1) * I):(j * I), ]
}
zeta_mu <- matrix(data = rep(x = 0, times = M * J), nrow = 1, ncol = J * M)
zeta_sigma2 <- diag(x = 0.2, nrow = J * M, ncol = J * M)
nu_mu <- matrix(data = c(0.00), nrow = 1, ncol = 1)
nu_sigma2 <- matrix(data = c(0.20), nrow = 1, ncol = 1)
set.seed(624)
ex3 <- dich_response_sim(I = I, J = J, K = K, M = M, N = N,
                         omega_mu = omega_mu, omega_sigma2 = omega_sigma2,
                         gamma = gamma, lambda = lambda, nu_mu = nu_mu,
                         nu_sigma2 = nu_sigma2, zeta_mu = zeta_mu,
                         zeta_sigma2 = zeta_sigma2, key = key)

# Example 4

I <- 30
J <- 2
K <- 500
M <- 1
N <- 2
omega_mu <- matrix(data = c(0, -1), nrow = 1, ncol = M * N)
omega_sigma2 <- diag(x = c(1.00, 0.25), nrow = M * N)
contrast_codes <- cbind(1, contr.treatment(n = J))[, 1:N]
gamma <- matrix(data = 0, nrow = J * M, ncol = M * N)
for(j in 1:J) {
  for(m in 1:M) {
    gamma[(m + M * (j - 1)), (((m - 1) * N + 1):((m - 1) * N + N))] <-
    contrast_codes[j, ]
  }
}
lambda <- matrix(data = 0, nrow = I * J, ncol = J * M)
lam_vals <- rnorm(I, 1.5, .23)
for (j in 1:J) {
  lambda[(1 + (j - 1) * I):(j * I), (1 + (j - 1) * M):(j * M)] <- lam_vals
}
zeta_mu <- matrix(data = rep(x = 0, times = M * J), nrow = 1, ncol = J * M)
zeta_sigma2 <- diag(x = 0.2, nrow = J * M, ncol = J * M)
nu <- matrix(data = rnorm(n = I, mean = 0, sd = .25), nrow = I * J, ncol = 1)
set.seed(624)
ex4 <- dich_response_sim(I = I, J = J, K = K, M = M, N = N,
                         omega_mu = omega_mu, omega_sigma2 = omega_sigma2,
                         gamma = gamma, lambda = lambda, nu = nu,
                         zeta_mu = zeta_mu, zeta_sigma2 = zeta_sigma2)

# Example 5

I <- 20
J <- 10
K <- 1
M <- 2
N <- 2
omega_mu <- matrix(data = c(2.50, -2.00, 0.50, 0.00), nrow = 1, ncol = M * N)
omega_sigma2 <- diag(x = c(0.90, 0.70, 0.30, 0.10), nrow = M * N)
contrast_codes <- cbind(1, contr.poly(n = J))[, 1:N]
gamma <- matrix(data = 0, nrow = J * M, ncol = M * N)
for(j in 1:J) {
  for(m in 1:M) {
    gamma[(m + M * (j - 1)), (((m - 1) * N + 1):((m - 1) * N + N))] <-
      contrast_codes[j, ]
  }
}
key <- rbinom(n = I * J, size = 1, prob = .7) + 1
measure_weights <-
  matrix(data = c(0.5, -1.0, 0.5, 1.0), nrow = 2, ncol = M, byrow = TRUE)
lambda <- matrix(data = 0, nrow = I * J, ncol = J * M)
for(j in 1:J) {
  lambda[(1 + (j - 1) * I):(j * I), (1 + (j - 1) * M):(j * M)] <-
    measure_weights[key, ][(1 + (j - 1) * I):(j * I), ]
}
zeta_mu <- matrix(data = rep(x = 0, times = M * J), nrow = 1, ncol = J * M)
zeta_sigma2 <- diag(x = 0.2, nrow = J * M, ncol = J * M)
nu_mu <- matrix(data = c(0.00), nrow = 1, ncol = 1)
nu_sigma2 <- matrix(data = c(0.20), nrow = 1, ncol = 1)
set.seed(624)
ex5 <- dich_response_sim(I = I, J = J, K = K, M = M, N = N,
                         omega_mu = omega_mu, omega_sigma2 = omega_sigma2,
                         gamma = gamma, lambda = lambda, nu_mu = nu_mu,
                         nu_sigma2 = nu_sigma2, zeta_mu = zeta_mu,
                         zeta_sigma2 = zeta_sigma2, key = key)

mlthom/CogIRT documentation built on Sept. 5, 2024, 8:18 a.m.