Operationalizing compensation over time in neurodegenerative disease

Population data were simulated according to Figure 1 of the main text. Figure 1 depicts the situation in which the Y, C, and X variables have different relationships for three phases. A regression spline approach was used for the simulation in which Y was a function of C and X, with regression coefficients varying by phase and error being allowed for the predictor variables. Suppose that í µí±Œ í µí±–í µí±— is the performance outcome for the í µí±– í µí±¡ℎ subject (í µí±– = 1, … , í µí±) at the í µí±— í µí±¡ℎ time point (í µí±— = 1, … , í µí±‡ í µí±–), with í µí±‡ í µí±– = í µí±‡, so there was no missing data for the simulation. The underlying model expresses Y as a function of C and X, so the latter two were first generated consistent with their pattern in Figure 1 allowing for errors in the variables. A linear mixed model (LMM) was used with a random intercept term and random error. X is a linear function of time, and the LMM for the simulation was í µí±‹ í µí±–í µí±— = í µí»¼ 0 + í µí±Ž 0í µí±– + í µí»¼ 1 í µí±¡ í µí±–í µí±— + í µí±’ í µí±–í µí±—. We assume í µí±Ž 0í µí±– ~í µí²©(0, í µí¼Ž í µí±Ž 0í µí±– 2) and í µí±’ í µí±–í µí±— ~í µí²©(0, í µí¼Ž í µí±’ 2). The variables were standardized to have the same initial means, so we set í µí»¼ 0 = 0. The other parameters were set to í µí»¼ 1 = −1, í µí¼Ž í µí±Ž 0í µí±– 2 = 0.05, and í µí¼Ž í µí±’ 2 = 0.01. Time values ranged from 0 to 2 incremented by a tenth, but the values were not intended to represent a specific metric (no values or units were used in the main text). A random number generator was used to obtain realized values of the random effect for each hypothetical subject, and the random error for each subject at each time point. Figure 1 indicates that C has a nonlinear trend. Therefore, C was a spline function of time, í µí° ¶ í µí±–í µí±— = í µí¼“ 0 + í µí± 0í µí±– + í µí¼“ 1 (í µí±¡ í µí±–í µí±— − í µí±‡ 1) í µí±¡ í µí±–í µí±—≥í …