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.