Semi-continuous a general linear model (GLM), 2) to

Semi-continuous data with a point mass at zero is very common in
health services research. Such as medical costs, disability levels and
substance use. This data have characteristics that complicate their analysis.
The data distribution is highly skewed with positive support. In the case of
medical expenditures, the excessive zero shows a population of ‘non-users’ who
do not receive medical care in a specific time interval and therefore  do not have medical costs; also the continuous
part represents the level of  spending among
health services consumer. Common approaches to
modeling this type of data are: 1) using a simple
parametric distribution, such as Gaussian, and fit a general linear model (GLM), 2)
to remove the zero values and modeling the positive values through a GLM, 3) using a logarithmic transformation
after adding a small constant to zero values and fit a GLM or 4) fit a two-part
model (TP).

However, simple parametric distributions are not suitable to
describe such semi-continuous data (large number of zeroes an the skewness of
the data) and will lead to biased inferences. For deal to this problem multi-part
models were first introduced by Duan in the health economics studies. specifically
Two-part models are proposed to manage the semi-continuous (zero-inflated)
features of cost data. Flexibility of these models will allow two
separate models, one for zero values and another for positive values. In
general, the first part uses  a probit or
logit link function for modeling the probability of being a positive value and
the second part uses  a link function
with positive support to model the non-zero values. Despite wide use of
conventional TP models, they are limited to conditional (on positive values)
interpretation of regression coefficients from the second part; therefore,
generalization of the results from the second part is only applicable to
consumer population (e.g. positive values).

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For deal to this problem marginalized two part (MTP) mode were first
introduced by Smith and were developed by Voronca and Gebregziabher. MTP models
parameterizes the marginal mean among the all zero and nonzero values directly
from the regression coefficients which leads to a direct interpretation of
covariate effects on the marginal mean (overall population of users and
non-users).