Non-decreasing threshold variances in mixed generalized ordered response models: A negative correlations approach to variance reduction

Abstract

Mixed Generalized Ordered Response (MGOR) models, that allow random heterogeneity in thresholds, are widely used to model ordered outcomes such as accident injury severity. This study highlights a potential limitation of these models, as applied in most empirical research, that the variances of the random thresholds are implicitly assumed to be in a non-decreasing order. This restriction is unnecessary and can lead to difficulty in estimation of random parameters in higher order thresholds. In this study, we investigate the use of negative correlations between random parameters as a variance reduction technique to relax the property of non-decreasing variances of thresholds in MGOR models. To this end, a simulation-based approach was used (where multiple datasets were simulated assuming a known negative correlation structure between the true parameters), and two models were estimated on each dataset – one allowing correlations between random parameters and the other not allowing such correlations. Allowing negative correlations helped relax the non-decreasing variance property of MGOR models. However, maximum simulated likelihood estimation of parameters on data with correlations occasionally encountered model convergence and parameter identification issues. Comparison of the models that did converge suggests that ignoring correlations leads to an estimation of fewer random parameters in the higher order thresholds and results in bias and/or loss of precision for a few parameter estimates. However, ignoring correlations leads to an adjustment of other parameter estimates such that overall likelihood values, predicted percentage shares, and the marginal effects are similar to those from the models with correlations.