In healthcare research, medical expenditure data for the elderly are typically semicontinuous and right-skewed, which involve a point mass at zero and may exhibit heteroscedasticity. The problem of a substantial proportion of zero values prevents traditional regression techniques based on the Gaussian, gamma, or inverse Gaussian distribution, which may lead to understanding the standard errors of the parameters and overestimating their significance. A common way to counter the problem is using zero-adjusted models. However, due to the right-skewness in the nonzeros' response, conventional zero-adjusted models such as zero-adjusted gamma, zero-adjusted Inverse Gaussian, and classic Tobit may not perform well. Here, we firstly generalize those three types of the conventional zero-adjusted model to solve the problem of right-skewness in health care. The generalized zero-adjusted models are very flexible and include the zero-adjusted Weibull, zero-adjusted gamma, zero-adjusted inverse Gaussian, and classic Tobit models as their special cases. Using the Chinese Longitudinal Healthy Longevity Survey, we find that, according to the AIC, SBC, and deviance criteria, the zero-adjusted generalized gamma model is the best one of these generalized models to predict the odds of zero cost accurately. In order to depict the predictors affecting the amount expenditure, we further discuss the situations where the mean, dispersion of a nonzero amount expenditure and model the probability of a zero amount of ZAGG in terms of predictor variables using suitable link functions, respectively. Our analysis shows that age, health, chronic diseases, household income, and residence are the main factors influencing the medical expenditure for the elderly, but the insurance is not significant. To the best of our knowledge, little study focused on these situations, and this is the first time.