# Simulate Dichotomous Response Model 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. ```r 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`: Contrast effects matrix (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`: Contrast codes matrix (JM by MN). - `lambda`: Item slope matrix (IJ by JM). - `lambda_mu`: Vector of means for the item slope parameters (1 by JM) - `lambda_sigma2`: Covariance matrix for the item slope parameters (JM by JM) - `nu`: Item intercept matrix (K by IJ). - `nu_mu`: Mean of the item intercept parameters (scalar). - `nu_sigma2`: Variance of the item intercept parameters (scalar). - `zeta`: Specific effects matrix (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`: kappa Item guessing matrix (IJ by 1). 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. ## Returns y = simulated response matrix; yhatstar = simulated latent response probability matrix; [simulation_parameters] ## Examples ```r # 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 <- 25 J <- 2 K <- 200 M <- 1 N <- 2 omega_mu <- matrix(data = c(1, -2), 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 = 2), 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) ``` ## 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.

dich_response_sim function