# A Latent Trait Model for Multivariate Longitudinal Data With Two Sources of Measurement Error

## TL;DR

This paper develops a latent trait model for multivariate longitudinal Likert-scale data, accounting for transient and measurement errors, and introduces two estimation methods applied to personality assessment data.

## Contribution

It proposes a novel latent trait model with two measurement error sources and compares correlation reconstruction and stochastic EM estimation methods.

## Key findings

- Both estimation methods successfully applied to real data
- Model effectively captures latent traits with measurement errors
- Provides a framework for analyzing longitudinal Likert-scale data

## Abstract

Personality traits are latent variables, and as such, are impossible to measure without the use of an assessment. Responses on the assessments can be influenced by both transient (state-related) error and measurement error, obscuring the true trait levels. Typically, these assessments utilize Likert scales, which yield only discrete data. The loss of information due to the discrete nature of the data represents an additional challenge in assessing the ability of these instruments to measure the latent trait of interest. This paper is concerned with parameter estimation in a model relating a latent variable, as well transient error and measurement error components when data are longitudinal and measured using a Likert scale. Two methods for parameter estimation are detailed: correlation reconstruction, a method that uses polychoric correlations, and maximum likelihood implemented using a Stochastic EM algorithm. These methods are applied to a motivating dataset of 440 college students taking the Big Five inventory twice in a two month period.