The average rate of change for continuous time models

The average rate of change (ARC) is a concept that has been misunderstood in the applied longitudinal data analysis literature, where the slope from the straight-line change model is often thought of as though it were the ARC. The present article clarifies the concept of ARC and shows unequivocally the mathematical definition and meaning of ARC when measurement is continuous across time. It is shown that the slope from the straight-line change model generally is not equal to the ARC. General equations are presented for two measures of discrepancy when the slope from the straight-line change model is used to estimate the ARC in the case of continuous time for any model linear in its parameters, and for three useful models nonlinear in their parameters.