IRT的GRM模型数据模拟代码（R语言）

在R语言中，我们可以使用`distributions`和`optim`等包来模拟IRT的GRM（Graded Response Model）。以下是一个简单的例子：

首先，安装必要的包：

install.packages(c("psych", "MASS"))
library(psych)
library(MASS)

然后，定义GRM函数：

grm_function <- function(theta, difficulty, response) {
  dnorm(response, mean = theta - difficulty, sd = sqrt(difficulty))
}

接下来，创建一些模拟数据：

set.seed(123)  # 设置随机数种子保证复现性
num_students <- 50
theta_range <- seq(from = 0, to = 4, length.out = num_students)
true_theta <- runif(num_students, min = theta_range[1], max = theta_range[length])

# 设定难度级别
difficulty_levels <- c(1, 2, 3)
responses <- matrix(NA, nrow = num_students, ncol = length(difficulty_levels))
for (i in 1:length(difficulty_levels)) {
  responses[, i] <- rbinom(num_students, 1, prob = grm_function(true_theta, difficulty_levels[i]))
}

最后，使用优化函数对数据进行拟合：

guess_params <- cbind(true_theta, difficulty_levels)
fit <- optim(guess_params, grm_function, method = "L-BFGS-B", hessian = TRUE,
             lower = c(-Inf, 0), upper = c(Inf, Inf),
             args = list(responses))
estimated_params <- coef(fit)

cat("Simulated data and estimated parameters:\n")
cat(paste0("True θ: ", true_theta, "\n"))
cat(paste0("Estimated θ: ", estimated_params[,"x"], "\n"))
cat(paste0("Difficulty levels: ", difficulty_levels, "\n"))
cat(paste0("Estimated δ: ", estimated_params[,"y"], "\n"))