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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 non-zero values directly
from the regression coefficients which provide a direct interpretation of
covariate effects on the overall marginal mean (entire population of users and
non-users). Originally MTP models were introducted for lognormal or log skew
normal distributions and were developed for flexible generalized gamma (GG)
family of distributions by Voronca.  The GG
family can represent various types of distributions with non-negative support which
can cover different shapes and has the ability to model varied data sets with different
degrees of asymmetries and skewness. The GG family includes Weibull,  gamma, and lognormal. All of the mentioned distributions
can be used for the intensity part of MTP and TP model and they can lead to
different inferences. 

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