[Paper Review] Weighted scores method for longitudinal ordinal data

Aristidis K. Nikoloulopoulos|UEA Digital Repository (University of East Anglia)|Oct 26, 2015

Advanced Statistical Methods and Models39 references3 citations

TL;DR

This paper introduces the weighted scores method as a robust, efficient alternative to generalized estimating equations (GEE) for longitudinal ordinal data, avoiding the need to convert ordinal responses into binary indicators. By leveraging a discretized multivariate normal working model, it reduces computational burden and avoids complex correlation matrices, enabling faster and more stable estimation in high-dimensional settings with many categories.

ABSTRACT

Extending generalized estimating equations (GEE) to ordinal response data requires a conversion of the ordinal response to a vector of binary category indicators. That leads to a rather complicated association structure, and the introduction of large matrices when the number of categories and dimension of the cluster are large. To allow a richer specification of working correlation assumptions, we adopt the weighted scores method which is essentially an extension of the GEE approach, since it can also be applied to families that are not in the GLM class. The weighted scores method stems from the lack of a theoretically sound methodology for analyzing multivariate discrete data based only on moments up to second order and it is robust to dependence and nearly as efficient as maximum likelihood. There is no need to convert the ordinal response to binary indicators, thus the weight matrices have smaller dimensions and it is not necessary to guess the correlations of indicator variables for different categories. We focus on important issues that would interest the data analyst, such as choice of the structure of the correlation matrix and of explanatory variables, comparison of results obtained from our methods versus GEE, and insights provided by our method that would be missed with the GEE method. Our modelling framework is implemented in the package weightedScores within the open source statistical environment R.

Motivation & Objective

• To address the computational inefficiency and complexity of extending GEE to ordinal responses with many categories.
• To eliminate the need to convert ordinal outcomes into binary indicators, which inflates the dimension of working correlation matrices.
• To provide a robust, second-order moment-based method that is nearly as efficient as maximum likelihood but more scalable than GEE for large K or d.
• To enable reliable inference in longitudinal ordinal data with high-dimensional clusters or many categories, such as clinical scoring systems.
• To offer a practical, computationally feasible alternative to existing GEE approaches that suffer from convergence issues and slow matrix operations.

Proposed method

• The method uses a discretized multivariate normal distribution as a working model to define weights for univariate score functions.
• It constructs estimating equations by weighting score functions based on marginal distributions and pairwise associations, avoiding binary indicator conversion.
• The weight matrices are derived from second-order moments of bivariate ordinal responses under a latent variable framework.
• The method estimates regression and association parameters via solving weighted estimating equations, with variance-covariance matrices computed from empirical sandwich estimators.
• It supports flexible correlation structure selection and handles large numbers of categories (K) and cluster sizes (d) efficiently.
• The approach is implemented in the R package 'weightedScores' for practical application in biostatistical research.

Experimental results

Research questions

• RQ1 How can longitudinal ordinal data with many categories be modeled efficiently without converting them into binary indicators?
• RQ2 What are the computational and statistical advantages of the weighted scores method over traditional GEE for ordinal responses?
• RQ3 How does the method perform in terms of estimation efficiency and convergence compared to existing GEE approaches?
• RQ4 Can the weighted scores method provide insights missed by GEE, especially in high-dimensional settings?
• RQ5 What is the impact of different correlation structure assumptions on parameter estimates and inference in longitudinal ordinal models?

Key findings

• The weighted scores method avoids the need to convert ordinal responses into K−1 binary indicators, significantly reducing the dimension of working correlation matrices.
• The method demonstrates improved computational speed and stability, especially when the number of categories K or cluster size d is large.
• It provides consistent and nearly efficient estimators under second-order moment assumptions, even when the true distribution is misspecified.
• The method is robust to dependence and maintains good performance across various correlation structures, including exchangeable and autoregressive.
• Empirical comparisons show that the weighted scores method produces similar estimates to GEE but with faster convergence and reduced computational burden.
• The R package 'weightedScores' enables practical implementation, supporting variable selection, AIC/BIC-based model comparison, and correlation structure selection.