论文标题
矩阵分解观点，用于精确评估项目响应理论
Matrix Decomposition Perspective for Accuracy Assessment of Item Response Theory
论文作者
论文摘要
项目响应理论通过使用由0/1值元素组成的观察到的项目响应矩阵获得了考生能力和问题困难的估计及其置信区间。许多论文讨论了估计的表现。但是，本文没有。使用最大似然估计，我们可以重建估计的项目响应矩阵。然后，我们可以从基质分解的角度评估该重建矩阵对观察到的响应矩阵的准确性。也就是说，本文着重于重建响应矩阵的性能。为了将项目响应理论的性能与他人进行比较，我们通过近似观察到的响应矩阵提供了两种低等级响应矩阵。一个是当响应矩阵是一个完整矩阵时，通过单数值分解方法的矩阵，而另一个是通过矩阵分解方法的矩阵，当响应矩阵是不完整的矩阵时。首先，我们发现，当响应矩阵是完整的矩阵时，单数值分解方法和矩阵分解方法的性能几乎相同。在这里，性能是通过使用均方根误差和精度之间的两个矩阵之间的接近度来衡量的。次要，我们已经看到，从项目响应理论对观察到的矩阵获得的重建矩阵的接近度位于两个近似的低等级响应矩阵之间，从k = 2的矩阵分解方法获得的两个近似级别的响应矩阵与观察到的矩阵中的k = 2和k = 2获得，其中k表示k列在分数矩阵中使用了第一个k列。
The item response theory obtains the estimates and their confidence intervals for parameters of abilities of examinees and difficulties of problems by using the observed item response matrix consisting of 0/1 value elements. Many papers discuss the performance of the estimates. However, this paper does not. Using the maximum likelihood estimates, we can reconstruct the estimated item response matrix. Then we can assess the accuracy of this reconstructed matrix to the observed response matrix from the matrix decomposition perspective. That is, this paper focuses on the performance of the reconstructed response matrix. To compare the performance of the item response theory with others, we provided the two kinds of low rank response matrix by approximating the observed response matrix; one is the matrix via the singular value decomposition method when the response matrix is a complete matrix, and the other is the matrix via the matrix decomposition method when the response matrix is an incomplete matrix. We have, firstly, found that the performance of the singular value decomposition method and the matrix decomposition method is almost the same when the response matrix is a complete matrix. Here, the performance is measured by the closeness between the two matrices using the root mean squared errors and the accuracy. Secondary, we have seen that the closeness of the reconstructed matrix obtained from the item response theory to the observed matrix is located between the two approximated low rank response matrices obtained from the matrix decomposition method of k= and k=2 to the observed matrix, where k indicates the first k columns use in the decomposed matrices.
