能否提供一段用于IRT( Item Response Theory)的GRM( Graded Response Model)的数据模拟和实现代码示例?
时间: 2024-10-29 18:09:42 浏览: 7
IRT(Item Response Theory),也称为项目反应理论,是一种统计学模型,常用于测量评估中的个体能力水平。GRM(Graded Response Model)是IRT的一种简化版本,通常用于二分或多等级选择题。下面是一个简单的Python代码示例,使用numpy和scipy库来模拟数据并实现GRM:
```python
import numpy as np
from scipy.optimize import curve_fit
# 定义GRM函数,这里假设等级分布是正态分布,中心倾向依赖于能力θ,难度δ
def grm(theta, delta, response):
return (response * (1 / (np.sqrt(2 * np.pi) * delta)) *
np.exp(-(theta - delta)**2 / (2 * delta**2)))
# 假设我们有50个学生的能力分布范围(θ)从0到4,并随机生成他们的能力值
num_students = 50
theta_range = np.linspace(0, 4, num_students)
true_theta = np.random.uniform(low=theta_range.min(), high=theta_range.max(), size=num_students)
# 创建题目难度(δ)
difficulty_levels = [1, 2, 3] # 题目难易程度设定
# 模拟每个学生的答题结果,基于他们的能力和题目难度
responses = np.zeros((num_students, len(difficulty_levels)))
for i, delta in enumerate(difficulty_levels):
responses[:, i] = np.random.binomial(n=1, p=grm(true_theta, delta, 1))
# 使用曲线拟合估计每个学生的θ和δ
params_guess = np.zeros((num_students, 2))
params_guess[:, 0] = true_theta
params_guess[:, 1] = difficulty_levels
# 曲线拟合GRM函数
params, _ = curve_fit(grm, params_guess[:, 0], responses.sum(axis=1), p0=params_guess)
print("Simulated data and estimated parameters:")
print(f"True θ: {true_theta}")
print(f"Estimated θ: {params[:, 0]}")
print(f"Difficulty levels: {difficulty_levels}")
print(f"Estimated δ: {params[:, 1]}")
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