**2. Materials and Methods**

## *2.1. Data and Model Settings*

Consider a longitudinal study with *n* subjects and *ki* observations measured repeatedly across time for the *i*th subject (1 *i n*). Without loss of generality, we set *ki* = *k*. *Yij* denotes the response for the *i*th subject at time *j* (1 *j k*). *Xij* = (*Xij*1, ..., *Xijp*) is the *p*-dimensional vector of lipid factors. In our study, *Eij* = (*Eij*1, ..., *Eijq*) denotes the *q*-dimensional treatment factor. Suppose that the lipid factors, treatment factors, and their interactions are associated with the longitudinal phenotype through the following model:

$$\mathcal{Y}\_{\vec{i}\vec{j}} = \mathcal{B}\_0 + E\_{\vec{i}\vec{j}}^{\top}\mathcal{B}\_1 + X\_{\vec{i}\vec{j}}^{\top}\mathcal{B}\_2 + \left(X\_{\vec{i}\vec{j}} \otimes E\_{\vec{i}\vec{j}}\right)^{\top}\mathcal{B}\_3 + \epsilon\_{\vec{i}\vec{j}} = Z\_{\vec{i}\vec{j}}^{\top}\mathcal{B} + \epsilon\_{\vec{i}\vec{j}}\tag{1}$$

where *β* = (*β*0, *β* <sup>1</sup> , *β* <sup>2</sup> , *β* <sup>3</sup> ) and *Zij* = *c*(1, *E ij* , *X ij* ,(*Xij* ⊗ *Eij*)) are (1 + *q* + *p* + *pq*)-dimensional vectors that represent all the main and interaction effects. The interactions between lipids and treatment factors are modeled through *Xij* ⊗ *Eij*, the Kronecker product of the *p*-dimensional vector *Xij*, and the *q*-dimensional vector *Eij* within the following form:

$$X\_{i\bar{j}} \circledcirc E\_{i\bar{j}} = [X\_{i\bar{j}1}E\_{i\bar{j}1}, X\_{i\bar{j}1}E\_{i\bar{j}2}, \dots, X\_{i\bar{j}1}E\_{i\bar{j}q}, X\_{i\bar{j}2}E\_{i\bar{j}1}, \dots, X\_{i\bar{j}p}E\_{i\bar{j}q}]^\top$$

which is a *pq*-dimensional vector. *β*<sup>0</sup> is the intercept. *β*1, *β*2, and *β*<sup>3</sup> are unknown coefficient vectors of dimensions *q*, *p*, and *pq*, respectively. We assume that the observations are dependent within the same subject, and independent if they are from different subjects. *<sup>i</sup>* = (*i*1, ..., *iki* )*<sup>T</sup>* follows a multivariate normal distribution *Nk*(0, Σ*i*), with Σ*<sup>i</sup>* as the covariance matrix for the repeated measure of the *i*th subject across the *k* time points.
