  Multi-group data are commonly seen in practice. Such data structure consists
of data from multiple groups and can be challenging to analyze due to data
heterogeneity. We propose a novel Joint and Individual Component Regression
(JICO) model to analyze multi-group data. In particular, our proposed model
decomposes the response into shared and group-specific components, which are
driven by low-rank approximations of joint and individual structures from the
predictors respectively. The joint structure has the same regression
coefficients across multiple groups, whereas individual structures have
group-specific regression coefficients. Moreover, the choice of global and
individual ranks allows our model to cover global and group-specific models as
special cases. For model estimation, we formulate this framework under the
representation of latent components and propose an iterative algorithm to solve
for the joint and individual scores under the new representation. To construct
the latent scores, we utilize the Continuum Regression (CR), which provides a
unified framework that covers the Ordinary Least Squares (OLS), the Partial
Least Squares (PLS), and the Principal Component Regression (PCR) as its
special cases. We show that JICO attains a good balance between global and
group-specific models and remains flexible due to the usage of CR. Finally, we
conduct simulation studies and analysis of an Alzheimer's disease dataset to
further demonstrate the effectiveness of JICO. R implementation of JICO is
available online at https://github.com/peiyaow/JICO.
