Valid Likelihood Research Articles

Validation likelihood estimation method for a zero-inflated Bernoulli regression model with missing covariates

A zero-inflated Bernoulli (ZIBer) regression model is an alternative model in order to improve fitting the binary outcome data that have many more zeros than expected under a regular logistic regression model. Because some covariates often have missing values, we propose the validation likelihood (VL) method to estimate the parameters of the ZIBer regression model with covariates missing at random. We consider the true selection probability (SP) model, a parametric SP model and a nonparametric SP model to compare the efficiency of the true SP, parametric SP, and semiparametric SP VL estimators. The asymptotic results of these three estimators are studied under some regularity conditions. Both the theoretical and simulation results reveal that the parametric and semiparametric SP VL estimation methods outperform the true SP VL estimation method if the parametric SP model is correct. The simulation results also indicate that the semiparametric SP VL estimation method is preferred to the inverse probability weighting and multiple imputation estimation methods. The practical application of the proposed methodology is adorned with a survey data set of violations of traffic rules of motorcyclist respondents in Taiwan.

Assessing age‐at‐onset risk factors with incomplete covariate current status data under proportional odds models

Motivated by an epidemiological survey of fracture in elderly women, we develop a semiparametric regression analysis of current status data with incompletely observed covariate under the proportional odds model. To accommodate both the interval-censored nature of current status failure time data and the incompletely observed covariate data, we propose an analysis based on the validation likelihood (VL), which is derived from likelihood pertaining to the validation sample, namely the subset of the sample where the data are completely observed. The missing data mechanism is assumed to be missing at random and is explicitly modeled and estimated in the VL approach. We propose implementing the VL method by integrating self-consistency and Newton-Raphson algorithms. Asymptotic normality and standard error estimation for the proposed estimator of the regression parameter are guaranteed. Simulation results reveal good performance of the VL estimator. The VL method has some gain in efficiency compared with the naive complete case method. But the VL method leads to unbiased estimators, whereas the complete case method does not when missing covariates are not missing completely at random. Application of the VL approach to the fracture data confirms that osteoporosis (low bone density) is a strong risk factor for the age at onset of fracture in elderly women.