Fig. 5

Bivariate autoregressive latent trajectory model with structured residuals. This figure illustrates a bivariate autoregressive latent trajectory model with structured residuals (ALT-SR). The manifest variables (observed in the study) are denoted as rectangles, and the latent variables (not observed) are denoted as circles, double headed arrows are the variance or covariance of variables, single headed arrows are either factor loadings (from latent variables to manifest variable) or regressions. “mval” is the abbreviation for methylation value and “rf” is the abbreviation for risk factor. So mval.IPT3 to mval.IPT9 represents the observed level of DNA methylation at different time points for one specific CpG site, and rf.IPT3 to rf.IPT9 represents the observed level of one specific cardiometabolic trait at different time points. ALT-SR comprised two parts, the autoregressive model (AR) and the latent growth curve (LGM), where LGM is initially modeled, followed by AR analysis. In LGM, the expected trajectory is established by two latent variables, the intercept (denoted as mval.i and rf.i) measuring the individual baseline levels of DNA methylation and cardiometabolic traits, respectively, and the slope (denoted as mval.s and rf.s) measuring the changing rate over time for every person. Since we fit a linear growth model for both DNA methylation and cardiometabolic traits, the factor loadings from latent intercept to manifest variables at different time points are all equal to 1, and the factor loadings from latent slope to manifest variables are set to 0, 2, 3, 5 and 6 at IPT3, IPT5, IPT6, IPT8, and IPT9, respectively. The AR part has two sets of regression paths based on the latent residuals of DNA methylation and the specific cardiometabolic trait, respectively: the autoregressive path (e.g., e.mval.IPT3 → e.mval.IPT5 → e.mval.IPT6 → e.mval.IPT8 → e.mval.IPT9) indicating the within-person changes of each of the two constructs over time, and the cross-lagged path (e.g., e.mval.IPT3 → e.rf.IPT5, e.rf.IPT3 → e.mval.IPT5) indicating whether DNA methylation at one time point predicts within-person changes of cardiometabolic traits at an adjacent later time point, and/or vice versa. Therefore, a3–a8 represent the autoregressive parameters of DNA methylation, c3–c8 represent the autoregressive parameters of cardiometabolic traits, b3–b8 represent the parameters of cross-lagged effect of cardiometabolic traits on DNA methylation, and d3–d8 represent the parameters of cross-lagged effect from DNA methylation to cardiometabolic traits. The variance and covariance of the latent residuals of DNA methylation and cardiometabolic traits are constrained to be equal across time except for the first occasion. We include time-independent covariates that influence the growth curve estimates; here sex and baseline age are common covariates when fitting the ALT-SR for different CpGs and cardiometabolic traits and include statin use as another time-independent variable when fitting the model for lipids. The relatedness of twins in the model is also adjusted for. Variance of mval.i, mval.s, rf.i, rf.s, e.mval.IPT3-e.mval.IPT9, e.rf.IPT3-e.rf.IPT9 is not displayed in the figure