Urnings: A new method for tracking dynamically changing parameters in paired comparison systems

We introduce a new rating system for tracking the development of parameters based on a stream of observations that can be viewed as paired comparisons. Rating systems are applied in competitive games, adaptive learning systems and platforms for product and service reviews. We model each observation as an outcome of a game of chance that depends on the parameters of interest (e.g. the outcome of a chess game depends on the abilities of the two players). Determining the probabilities of the different game outcomes is conceptualized as an urn problem, where a rating is represented by a probability (i.e. proportion of balls in the urn). This setup allows for evaluating the standard errors of the ratings and performing statistical inferences about the development of, and relations between, parameters. Theoretical properties of the system in terms of the invariant distributions of the ratings and their convergence are derived. The properties of the rating system are illustrated with simulated examples and its potential for answering research questions is illustrated using data from competitive chess, a movie review system, and an adaptive learning system for math

Urnings: A new method for tracking dynamically changing parameters in paired comparison systems

We introduce a new rating system for tracking the development of parameters based on a stream of observations that can be viewed as paired comparisons. Rating systems are applied in competitive games, adaptive learning systems and platforms for product and service reviews. We model each observation as an outcome of a game of chance that depends on the parameters of interest (e.g. the outcome of a chess game depends on the abilities of the two players). Determining the probabilities of the different game outcomes is conceptualized as an urn problem, where a rating is represented by a probability (i.e. proportion of balls in the urn). This setup allows for evaluating the standard errors of the ratings and performing statistical inferences about the development of, and relations between, parameters. Theoretical properties of the system in terms of the invariant distributions of the ratings and their convergence are derived. The properties of the rating system are illustrated with simulated examples and its potential for answering research questions is illustrated using data from competitive chess, a movie review system, and an adaptive learning system for math

Dynamic estimation in the extended marginal Rasch model with an application to mathematical computer-adaptive practice

We introduce a general response model that allows for several simple restrictions, resulting in other models such as the extended Rasch model. For the extended Rasch model, a dynamic Bayesian estimation procedure is provided, which is able to deal with data sets that change over time, and possibly include many missing values. To ensure comparability over time, a data augmentation method is used, which provides an augmented person-by-item data matrix and reproduces the sufficient statistics of the complete data matrix. Hence, longitudinal comparisons can be easily made based on simple summaries, such as proportion correct, sum score, etc. As an illustration of the method, an example is provided using data from a computer-adaptive practice mathematical environment

Tracking with (Un)certainty

One of the highest ambitions in educational technology is the move towards personalized learning. To this end, computerized adaptive learning (CAL) systems are developed. A popular method to track the development of student ability and item difficulty, in CAL systems, is the Elo Rating System (ERS). The ERS allows for dynamic model parameters by updating key parameters after every response. However, drawbacks of the ERS are that it does not provide standard errors and that it results in rating variance inflation. We identify three statistical issues responsible for both of these drawbacks. To solve these issues we introduce a new tracking system based on urns, where every person and item is represented by an urn filled with a combination of green and red marbles. Urns are updated, by an exchange of marbles after each response, such that the proportions of green marbles represent estimates of person ability or item difficulty. A main advantage of this approach is that the standard errors are known, hence the method allows for statistical inference, such as testing for learning effects. We highlight features of the Urnings algorithm and compare it to the popular ERS in a simulation study and in an empirical data example from a large-scale CAL application

Urnings: A new method for tracking dynamically changing parameters in paired comparison systems

We introduce a new rating system for tracking the development of parameters based on a stream of observations that can be viewed as paired comparisons. Rating systems are applied in competitive games, adaptive learning systems and platforms for product and service reviews. We model each observation as an outcome of a game of chance that depends on the parameters of interest (e.g. the outcome of a chess game depends on the abilities of the two players). Determining the probabilities of the different game outcomes is conceptualized as an urn problem, where a rating is represented by a probability (i.e. proportion of balls in the urn). This setup allows for evaluating the standard errors of the ratings and performing statistical inferences about the development of, and relations between, parameters. Theoretical properties of the system in terms of the invariant distributions of the ratings and their convergence are derived. The properties of the rating system are illustrated with simulated examples and its potential for answering research questions is illustrated using data from competitive chess, a movie review system, and an adaptive learning system for math

Urnings: A new method for tracking dynamically changing parameters in paired comparison systems

We introduce a new rating system for tracking the development of parameters based on a stream of observations that can be viewed as paired comparisons. Rating systems are applied in competitive games, adaptive learning systems and platforms for product and service reviews. We model each observation as an outcome of a game of chance that depends on the parameters of interest (e.g. the outcome of a chess game depends on the abilities of the two players). Determining the probabilities of the different game outcomes is conceptualized as an urn problem, where a rating is represented by a probability (i.e. proportion of balls in the urn). This setup allows for evaluating the standard errors of the ratings and performing statistical inferences about the development of, and relations between, parameters. Theoretical properties of the system in terms of the invariant distributions of the ratings and their convergence are derived. The properties of the rating system are illustrated with simulated examples and its potential for answering research questions is illustrated using data from competitive chess, a movie review system, and an adaptive learning system for math