Three-parameter log-logistic regression model: Parameter estimation and hypothesis testing

This study provides a Three-Parameter Log-Logistic Regression (LL3R) model, which is based on the Three- Parameter Log-Logistic Distribution (LL3D). This distribution is suitable for managing continuous data with strict positivity and a right-skewed distribution. LL3D is seen as a versatile alternative to the Weibull and Log-Normal distributions because of its flexibility of probability function. The LL3R model, thus formed, can be applied to a wide range of problems across various fields, including survival analysis. The primary objective of this study is to obtain parameter estimates and conduct hypothesis testing for the LL3R model. Parameter estimation for the LL3R model is accomplished using the Maximum Likelihood Estimation (MLE) technique. Because there is no closed-form expression available for the first derivative of the log-likelihood function in the LL3R model. As a result, the parameter estimation for LL3R is conducted through numerical iterations using the BFGS (Broyden-Fletcher-Goldfarb-Shanno) method. The BFGS algorithm is known for its robustness and efficiency, particularly in cases where the log-likelihood function is not easily optimized by simpler methods. The parameter estimator obtained is an asymptotic unbiased estimator for very large. Meanwhile, the covariance estimator is determined from the diagonal elements of the Hessian matrix of the parameter vector, then the standard error is determined through the square root of the covariance estimate. Hypothesis testing is carried out using the Maximum Likelihood Ratio Test (MLRT), enabling the assessment of whether multiple predictor variables in the regression model have a significant simultaneous impact on the response variable. Simultaneous test statistic is likelihood ratio which is approximated by the Chi-squared distribution with degrees of freedom k (the number of slope coefficients in LL3R). Meanwhile, the individual test statistic is derived through the central limit theorem and is approximated using a standard normal distribution.