On the development of the adjusted multinomial logistic Liu estimator

Abstract

The multinomial logistic regression (MNLR) model is a widely used statistical tool for predicting categorical response variables with more than two outcomes, based on multiple predictor variables. It finds applications across various fields, including healthcare, social sciences, and marketing. The maximum likelihood estimator (MLE) is the standard method for estimating parameters in MNLR models. However, when predictor variables exhibit multicollinearity, the MLE becomes inefficient, resulting in inflated variances and unstable coefficient estimates. To address this issue, a limited number of biased estimators have been proposed in the literature. This study aimed to develop an efficient estimator that reduces the impact of multicollinearity in MNLR models by introducing the adjusted multinomial logistic Liu estimator (AMLLE), which improves estimation accuracy and stability. A Monte Carlo simulation study was conducted to evaluate the performance of the MLE, multinomial logistic ridge estimator (MLRE), multinomial logistic Liu estimator (MLLE), almost unbiased multinomial logistic Liu estimator (AUMLLE), and AMLLE under moderate to high multicollinearity. The simulations considered a wide range of sample sizes and varying levels of the response variable. The findings indicated that the proposed estimator, AMLLE, outperformed the existing estimators in all scenarios considered. The relative efficiency of AMLLE compared to MLE, based on SMSE, showed substantial improvement across different correlation values. For correlations of 0.5, 0.7, 0.9, and 0.99, AMLLE achieves efficiencies of 38.25, 46.62, 68.73, and 96.79% for n=50; 21.56, 27.03, 46.25, and 90.80% for n=100; and 2.85, 3.80, 9.04, and 41.21% for n=1000, with three response levels. The corresponding efficiencies with five response levels are 43.47, 52.16, 61.86, and 97.95%; 27.66, 33.39, 52.84, and 93.95%; and 3.76, 5.23, 12.41, and 47.05%, respectively. Moreover, increasing the sample size further enhanced the performance of the proposed estimator, while higher correlation and additional response levels tend to reduce its effectiveness. In conclusion, the adjusted multinomial logistic Liu estimator provides a reliable and computationally efficient alternative for parameter estimation in multinomial logistic regression models affected by multicollinearity. It shows strong potential for practical applications, and future research could explore its extension to high-dimensional predictor settings.