Longitudinal trand

we plan to extend GWAS to other patterns harbored in the longitudinal trajectory. A notable example is the dynamic process (upward or downward) of complex traits after specific medical treatments or surgical procedures. To characterize this process, we add a gene-environment interaction term to the linear mixed model

\[y_{ij} = X_{ij}^T \beta + G_i \beta_g + (G_i \circ T_{ij}) \beta_{g\times t} + Z_{ij}^T \gamma_i + \varepsilon_{ij}\]

where $T$ is a time-varying variable (e.g., age or medication) that changes with time. We primarily focus on testing $\beta_{g\times t}=0$.