  With a rapid increase in volume and complexity of data sets, there is a need
for methods that can extract useful information, for example the relationship
between two data sets measured for the same persons. The Partial Least Squares
(PLS) method can be used for this dimension reduction task. Within life
sciences, results across studies are compared and combined. Therefore,
parameters need to be identifiable, which is not the case for PLS. In addition,
PLS is an algorithm, while epidemiological study designs are often
outcome-dependent and methods to analyze such data require a probabilistic
formulation. Moreover, a probabilistic model provides a statistical framework
for inference. To address these issues, we develop Probabilistic PLS (PPLS). We
derive maximum likelihood estimators that satisfy the identifiability
conditions by using an EM algorithm with a constrained optimization in the M
step. We show that the PPLS parameters are identifiable up to sign. A
simulation study is conducted to study the performance of PPLS compared to
existing methods. The PPLS estimates performed well in various scenarios, even
in high dimensions. Most notably, the estimates seem to be robust against
departures from normality. To illustrate our method, we applied it to IgG
glycan data from two cohorts. Our PPLS model provided insight as well as
interpretable results across the two cohorts.
