用层级相合性指标探测反应数据中噪音大小

不同的认知诊断模型（CDM）对反应数据中噪音的抗干扰能力不同，在评估CDM性能的模拟实验中，反应数据中所含噪音的大小是十分重要的实验条件。由于噪音的内隐性，为应用中选用相应的CDM带来困难。本文在认知模型已知条件下，欲使用MHCI和NHCI指标评估认知诊断测验的反应数据（0, 1评分）中噪音的大小。模拟实验表明，两指标与噪音存在明显的统计规律。尤其是以NHCI为主要自变量对噪音进行预测的回归方程中，回归模型解释率均接近90%；以此实现对噪音的有效预测，从而为选择CDM提供一个参考。

Using hierarchical consistency indexes to evaluate the size of the noise in the 0-1 response data of cognitive diagnostic test

Assessing the performance of cognitive diagnosis model (CDM) in the simulation experiments, the size of the noise in the response data is very important experimental condition. Due to the noise being hidden, it brings difficulties to choose corresponding CDM in applications. In this article, the modified hierarchical consistency index (MHCI) and new hierarchical consistency index (NHCI) are used to evaluate the size of the noise in the 0-1 response data of cognitive diagnostic test. In order to predict the size of the noise in the response data, Monte Carlo simulation experiment was carried out to find quantitative regularity between the indexes (MHCI, NHCI) and the noise.Suppose that the reduced Q matrix (Qr) is with K-row and m-column and the test Q matrix Qt is pile of L-matrix Qr, that is Qt=(Qr,...,Qr), where L=1, 2, 3, 4, respectively. And L influences the test length essentially. Experiment investigated the changes of the mean values of MHCI(NHCI) regulation with different attribute structures(linear, convergent, divergent, independent model) under the condition of different number of attributes(5, 6, 7)×different number of Qr (L=1, 2, 3, 4)×different size of the noise(slippage belongs in {0.3 0.25 0.2 0.15 0.1 0.05}). For independent model simulation experiment was carried out with only 5 attributes and for the other K=6, 7 the amount of computing is too heavy to implement. This means the number of attributes in independent model is constant.First, in order to get quantitative regularity between the indexes（MHCI, NHCI）and the noise, through stepwise regression to build the regression equations with mean value of the indexes as the dependent variable, slippage ratio, the number of Qr and the number of attributes as the independent variable. Experimental results show that the slip ratio and the number of Qr will significantly affect the mean value of the indexes, slippage ratio is the main influence factor; the greater the slippage ratio or the larger times of tests, the smaller the mean value of indexes. Number of attributes in most cases can enter the regression equations, but small values of the effect generally. No matter what kind of attribute structures, the two factor regression models have good explanation rate, especially for NHCI index, its rate is above 90%. There is a stable and significant quantitative regularity between slip ratio and index, which provides a way to predict slippage ratio.Then, in order to predict the size of the noise, through stepwise regression to build the regression equations with slippage ratio as the dependent variable, mean value of the indexes, the number of Qr and the number of attributes as the independent variable. As well as the measurable independent variables of mean value of indexes, the number of Qr and etc. to predict implicit slip ratio, the model of regression equation were significantly. The experimental results show that the linear model is similar to the convergent model, mean value of MHCI(NHCI), the number of Qr in the regression models have bigger effect. These two kinds of models of the first factor variance explained rate has more than 65%, two factors explain the rate of close to 89%. The results for divergence and independent are similar, mean value of NHCI, the number of Qr in the regression models have bigger effect. These two kinds of models of the first factor variance explained rate has close to 83%, two factors explain the rate of more than 92%. In conclusion, such as the experimental conditions of similar cases, using the above two factor regression models to estimate the slippage ratio can achieve good effect.