A Spectral Approach to Item Response Theory

The Rasch model is one of the most fundamental models in \emph{item response theory} and has wide-ranging applications from education testing to recommendation systems. In a universe with $n$ users and $m$ items, the Rasch model assumes that the binary response $X_{li} \in \{0,1\}$ of a user $l$ with parameter $\theta^*_l$ to an item $i$ with parameter $\beta^*_i$ (e.g., a user likes a movie, a student correctly solves a problem) is distributed as $\Pr(X_{li}=1) = 1/(1 + \exp{-(\theta^*_l - \beta^*_i)})$. In this paper, we propose a \emph{new item estimation} algorithm for this celebrated model (i.e., to estimate $\beta^*$). The core of our algorithm is the computation of the stationary distribution of a Markov chain defined on an item-item graph. We complement our algorithmic contributions with finite-sample error guarantees, the first of their kind in the literature, showing that our algorithm is consistent and enjoys favorable optimality properties. We discuss practical modifications to accelerate and robustify the algorithm that practitioners can adopt. Experiments on synthetic and real-life datasets, ranging from small education testing datasets to large recommendation systems datasets show that our algorithm is scalable, accurate, and competitive with the most commonly used methods in the literature

Exploring the Use of Rasch Models to Construct Measures of Firms’ Profitability with Multiple Discretization Ratio-type Data

Ratio-type data plays an important role in real-world data analysis. Mass ratios have been created for different purposes, depending on time and people’s needs. Then, it is necessary to create a comprehensive score to extract information from those mass ratios when they measure the same concept from different perspectives. Therefore, this study adopts the same logic of psychometrics to systematically conduct scale development on ratio-type data under the Rasch model. However, it is first necessary to discretize the ratio-type data for use in the Rasch model. Therefore, this study also explores the effect of different data discretization methods on scale development by using financial profitability ratios as a demonstration. Results show that retaining more ratio categories can benefit Rasch modeling because it can better inform the model. The dynamic clustering algorithm, k-median is a better method for extracting characteristic patterns of the ratio-type data and preparing the data for the Rasch model. This study illustrates that there is no one-way good discretization method for ratio-type data under the Rasch model. It is more reasonable to use the traditional algorithm if each ratio has a target benchmark, whereas the k-median clustering algorithm achieves good modeling results when benchmark information is lacking