Multivariate Multinomial Logistic Regression Models as Item Response Theory Models with Covariates

对测试和调查的反应通常被视为潜在变量的指标，如能力，成就或态度。 该项目开发了一类灵活的模型，可用于分析响应数据，这些数据还包含有关受访者的其他信息，响应选项，问题内容和其他附带信息。 观测数据的模型是对数乘法的，来自统计图形模型，给出了模型本身的示意图，以及每个项目的（多项式）逻辑模型的条件说明。 研究的两个主要焦点是估计和进一步的模型开发方法。 建模方面是以前的研究所进行的调查，它被发现，一般模型包括许多标准的项目反应理论的延伸。 鉴于这一发现，在当前的研究中，标准项目反应理论方法和对数乘法关联模型之间的关系将从理论和实证两个角度进行更充分的研究。 模型的进一步发展将包括多个相关的潜在变量和观察到的协变量。 对数乘性关联模型广泛应用的主要瓶颈是估计问题。 目前的估计方法仅限于相对简单的模型的相对较小的数据集。 本研究探讨两种可能的解决方案：伪似然估计和迭代条件估计方法。 这两种模型都适用于中等到大的数据集，并可以纳入协变量。 将编写和测试实现这些方法的计算机程序，以便对估计方法进行实证研究。 作为这项研究的一部分编写的程序将在一个专门用于该项目的网站上提供。模型的灵活性与估计的发展相结合，将为行为，社会，教育，和其他领域的统计工具，以分析对测试和问卷的响应，这些测试和问卷超出了现有能力，因为基本变量结构的复杂性以及伴随的信息. 与传统的项目反应理论估计方法不同，本项目的研究框架推导出了一个不需要数值积分进行估计的观测数据模型。 估计的这一特征允许对具有大量相关潜变量的模型进行估计。 目前研究的主要应用是针对教育测量中使用的模型，然而，模型的应用范围非常广泛。 例如，该方法可用于根据对问题的编码口头或书面答复制定一个量表或衡量标准。 该方法的图形方面使得非心理测量研究人员更容易使用该方法。 代数模型可能非常复杂，但图形表示极大地促进了模型应用以及统计人员和非统计人员之间的交流。

Responses to tests and surveys are typically viewed as indicators of underlying variables such as ability, achievement, or attitude. This project develops a flexible class of models that can be used to analyze response data that also incorporates additional information about the respondent, response options, question content, and other collateral information. The models for observed data, which are log-multiplicative, are derived from statistical graphical models that give schematic representations of the models themselves, as well as from conditional specification of a (multinomial) logistic model for each item. The two major foci of the research are estimation and further model development approach. The modeling aspect is an extension of previous research undertaken by the investigator where it was discovered that the general model encompasses much of standard item response theory. Given this discovery, in the current research the relationship between standard item response theory methods and the log-multiplicative association models will be more fully studied both from theoretical and empirical perspectives. Further development of the models will include multiple correlated latent variables and observed covariates. The major bottleneck for wide-spread application of log-multiplicative association models is estimation. The current estimation methods are limited to relatively small data sets for relatively simple models. Two potential solutions are explored in this research: pseudo-likelihood estimation and an iterative conditional estimation method. Both of these models are feasible for moderate to large data sets and can incorporate covariates. Computer programs implementing these methods will be written and tested so that the estimation methods can be empirically studied. Programs written as part of this research will be made available on a web-site devoted to this project.The flexibility of the models combined with developments in estimation will provide researchers in behavioral, social, educational, and other areas statistical tools to analyze responses to tests and questionnaires that go beyond current capabilities in terms of the complexity of the underlying variable structure as well as the inclusion of concomitant information. Unlike traditional item response theory approach to estimation, the research framework in this project derives a model for observed data that does not require numerical integration for estimation. This feature of the estimation permits models to be estimated that have a large number of correlated latent variables. The primary application of the current research is aimed at models for use in educational measurement; however, the range of applications of the models is extremely broad. For example the methodology could be used to develop a scale or measure based on coded verbal or written responses to questions. The graphical aspect of the approach makes the methodology more accessible to non-psychometric researchers. The algebraic models can be very complex, but the graphical representations greatly facilitate model applications and communication among statisticians and non-statisticians.